TEACH ▸ Scientific Notation, Exponents, and Irrational Numbers
What does this painting have to do with math?
Abstract expressionist Al Held was an American painter best known for his “hard edge” geometric paintings. His bright palettes and bold forms create a three-dimensional space that appears to have infinite depth. Held, who sometimes found inspiration in architecture, would often play with the viewer’s sense of visual perception. While most of Held’s artworks are paintings, he also worked in mosaic and stained glass.
Students apply their grade 5 knowledge of powers of 10 to write and operate with numbers in scientific notation. They discover the properties and definitions of exponents in the first module of grade 8 by relating the exponential form of a number, or expression, to its expanded form, such as relating 104 to 10 × 10 × 10 × 10.
Grade 7 Module 2
In this module, students extend their understanding of rational numbers from grade 7 to define irrational numbers and real numbers.
Overview
Scientific Notation, Exponents, and Irrational Numbers
Topic A
Introduction to Scientific Notation
Students relate large and small numbers to real-world measurements and contexts. They write large positive numbers as a single digit times a power of 10 before formally defining scientific notation. Students operate with large positive numbers by writing repeated factors of 10.
Topic B
Properties and Definitions of Exponents
Students extend their work with powers of 10 to expressions with integer exponents. By examining repeated factors, they make conjectures and learn the properties and definitions of exponents. Then they use these properties and definitions to simplify exponential expressions.
Topic C
Applications of the Properties and Definitions of Exponents
Students use the definition of negative exponents to write small positive numbers in scientific notation. They revisit operating with numbers written in scientific notation in topic A and apply the properties of exponents to more efficiently find products and quotients of large and small positive numbers through unique contexts.
Topic D
Perfect Squares, Perfect Cubes, and the Pythagorean Theorem
Students apply their understanding of squares and cubes to reason intuitively about square roots and cube roots. They explore and apply the Pythagorean theorem to right triangles to find the length of the hypotenuse c. Students solve problems involving values of c 2 that are not perfect squares and estimate a range of possible values for c before transitioning to using square root notation.
Topic E Irrational Numbers
Students approximate values of square roots by using number sense reasoning and by considering which whole-number interval includes the value. Then students refine the interval to consecutive tenths, hundredths, and thousandths to increase the precision of their approximation. Once students approximate numbers that are not rational, they can define irrational numbers. Students make more accurate approximations of irrational numbers, which helps them find more precise solutions to equations.
After This Module
Grade 8 Module 2
Students prove the Pythagorean theorem in module 2. They also work with right triangles to find unknown leg lengths or hypotenuse lengths.
Algebra I Module 4
Students’ knowledge of the properties and definitions of exponents and root notation supports their ability to write √ab as
or √a · √b in Algebra I. This understanding also leads students to write irrational numbers such as √12 as 2√3 in Algebra I.
Scientific Notation, Exponents, and Irrational Numbers
6
More Properties of Exponents
Achievement Descriptors:
A
Introduction to Scientific Notation
Lesson 1
Large and Small Positive Numbers
• Write very large and very small numbers in a form that uses exponents to prepare students for scientific notation.
• Approximate very large and very small quantities.
Lesson 2
Comparing Large Numbers
• Write numbers as a single digit times a power of 10 in exponential form to approximate quantities.
• Compare large and small positive numbers by using times as much as language.
Lesson 3
Time to Be More Precise—Scientific Notation
• Write numbers given in standard form in scientific notation.
Lesson 4
Adding and Subtracting Numbers Written in Scientific Notation (Optional)
• Add and subtract numbers written in scientific notation.
• Rewrite sums and differences in scientific notation.
Properties and Definitions of Exponents
Lesson 5
Products of Exponential Expressions with Whole-Number Exponents
• Apply understanding of exponential notation to write equivalent expressions for x m · x n .
• Encounter and apply properties of exponents, including raising powers to powers, raising products to powers, and raising quotients to powers.
Lesson 7
Making Sense of the Exponent of 0
• Define x 0 by confirming that the definition upholds the properties of exponents.
• Evaluate powers with an exponent of 0.
Lesson 8
Making Sense of Integer Exponents
• Explore and develop an understanding of negative exponents.
• Write equivalent expressions given an expression of the form x m x n
Lesson 9
Writing Equivalent Expressions
• Write equivalent expressions by using all the properties and definitions of exponents.
Lesson 10
Evaluating Numerical Expressions by Using Properties of Exponents (Optional)
• Simplify and evaluate exponential expressions by using the properties and definitions of exponents
Topic
C
Applications of the Properties and Definitions of Exponents
Lesson 11
Small Positive Numbers in Scientific Notation
• Write small positive numbers in scientific notation.
• Order numbers written in scientific notation.
Topic B
Lesson
Lesson 12
Operations with Numbers in Scientific Notation
• Interpret numbers in scientific notation displayed on digital devices.
• Operate with numbers written in scientific notation.
Lesson 13
Applications with Numbers in Scientific Notation
• Operate with numbers written in standard form and scientific notation.
Lesson 14
Choosing Units of Measurement
• Choose appropriate units of measurement and convert units of measurement.
Lesson 15
Get to the Point
• Model a situation by operating with numbers in scientific notation. Topic D
Perfect Squares, Perfect Cubes, and the Pythagorean Theorem
Lesson 16
Perfect Squares and Perfect Cubes
• Recognize perfect squares from 1 to 225 and perfect cubes from 1 to 125.
• Determine all numbers that square or cube to a given number.
Lesson 17
Solving Equations with Squares and Cubes
• Solve equations of the forms x 2 = p and x 3 = p, where p is a rational number and the solutions are rational numbers.
Lesson 18
The Pythagorean Theorem
• Describe the Pythagorean theorem and the conditions required to use it.
Lesson 19 .
Using the Pythagorean Theorem
• Apply the Pythagorean theorem to find the unknown length of the hypotenuse of a right triangle.
• Find two consecutive whole numbers which the length of the hypotenuse is between when the length is not rational.
• Use square root notation to express lengths that are not rational.
Lesson 20
Square Roots
• Place square roots on a number line.
Topic E
Irrational Numbers
Lesson 21
Approximating Values of Roots and π 2
• Approximate values of square roots, cube roots, and π 2 .
Lesson 22 . . .
Familiar and Not So Familiar Numbers
• Identify numbers as rational, irrational, and real by their decimal form.
• Compare the characteristics of rational and irrational numbers.
Lesson 23
Ordering Irrational Numbers
• Order irrational numbers.
• Approximate the value of expressions with irrational numbers.
Lesson 24
Revisiting Equations with Squares and Cubes
• Solve equations of the forms x 2 = p and x 3 = p, where p is a rational number and the solutions are real numbers.
Resources
Standards
Achievement Descriptors: Proficiency Indicators
Math
Materials
Fluency
Mixed Practice Solutions
Works Cited
Credits
Acknowledgments
Why
Scientific Notation, Exponents, and Irrational Numbers
Why are operations with numbers written in scientific notation in topics A and C?
Topic A sparks students’ interest through engaging contexts that utilize their exponent and place value understanding. Having students write the many factors of 10 foreshadows and drives the need for the properties and definitions of exponents. In topic C, students become fluent in applying the properties and definitions of exponents by operating with numbers written in scientific notation.
Properties of Exponents
Why are exponential expressions written as 10 5+2 and 3 5 · 4?
In lesson 5, the expression 10 5 · 10 2 is intentionally shown as the equivalent expression 10 5+2 to emphasize the use of the product of powers with like bases property. Writing 105+2 has more instructional value than writing 107. When students are ready, ask them to express the sum of the exponents as a single integer.
In lesson 6, the expression (3 4) 5 is intentionally shown as the equivalent expression 3 5 · 4 instead of 3 20 to emphasize the use of the power of a power property. Students learn that (3 4)5 is 5 factors of 34, or 5 factors of 4 factors of 3, which is 5 · 4 factors of 3. Students may wish to write expressions of the form (x m) n as x m · n rather than as x n · m. In the Properties and Definitions of Exponents graphic organizer and in future lessons, (x m) n will be written as x m n for readability. When students are ready, ask them to express the product of the exponents as a single integer.
of Exponents
Why are ( x y )n = x n y n and x m x n = x m−n not included in the Properties and Definitions of Exponents graphic organizer?
The power of a quotient, ( x y )n = x n y n , can be viewed as an extension of the definition of a power. So the power of a quotient is not included in the Properties and Definitions of Exponents graphic organizer.
The quotient of powers with like bases, x m ___ x n = x m n, is an application of the definition of a base with a negative exponent and the product of powers with like bases property.
So the quotient of powers is not included in the Properties and Definitions of Exponents graphic organizer.
Why is lesson 4 optional?
Lesson 4 provides an opportunity for students to build upon their knowledge of the distributive property and combining like terms to add and subtract numbers written in scientific notation. Students learn to rewrite numbers in scientific notation to have terms with the same powers of 10 so they can add or subtract them. Students are not expected to master this learning. Consider using it as additional learning material.
Why is lesson 10 optional?
The focus of lesson 10 is to build number sense and fluency by using the properties and definitions of exponents. Students write and simplify equivalent expressions by using the fewest number of prime bases, such as writing 45 −4 ⋅ 15 8 as 5 4, which is a highly proficient indicator. Consider using this lesson to extend students’ learning.
Why does the Pythagorean theorem appear in two modules?
Module 1 is an introduction to the Pythagorean theorem, and students solve for an unknown hypotenuse length c when given the leg lengths of a right triangle. Students solve problems involving values of c 2 that are not perfect squares and estimate a range of possible values for c before transitioning to using square root notation. In module 2, students apply rigid motions to prove the Pythagorean theorem and its converse. They also find unknown leg and hypotenuse lengths of right triangles in context.
Scientific Notation, Exponents, and Irrational Numbers
Achievement Descriptors (ADs) are standards-aligned descriptions that detail what students should know and be able to do based on the instruction. ADs are written by using portions of various standards to form a clear, concise description of the work covered in each module.
Each module has its own set of ADs, and the number of ADs varies by module. Taken together, the sets of module-level ADs describe what students should accomplish by the end of the year.
ADs and their proficiency indicators support teachers with interpreting student work on
• informal classroom observations,
• data from other lesson-embedded formative assessments,
• Exit Tickets,
• Topic Quizzes, and
• Module Assessments.
This module contains the 14 ADs listed.
8.Mod1.AD1 Determine whether numbers are rational or irrational. NY-8.NS.1, NY-8.EE.2
8.Mod1.AD2 Use rational approximations of irrational numbers to compare the size of irrational numbers. NY-8.NS.2
8.Mod1.AD3 Locate irrational numbers approximately on a number line. NY-8.NS.2
8.Mod1.AD4 Approximate the values of irrational expressions. NY-8.NS.2
8.Mod1.AD5 Apply the properties of integer exponents to generate equivalent numerical expressions. NY-8.EE.1
8.Mod1.AD6 Solve equations of the form x2 = p as √p and √p and equations of the form x3 = p as 3 √p , where p is a rational number. NY-8.EE.2
8.Mod1.AD7 Evaluate square roots of small perfect squares and cube roots of small perfect cubes. NY-8.EE.2
8.Mod1.AD8 Approximate and write very large and very small numbers in scientific notation. NY-8.EE.3
8.Mod1.AD9 Express how many times as much one number is as another when both numbers are written in scientific notation.
NY-8.EE.3
8.Mod1.AD11 Multiply or divide numbers written in standard form and scientific notation, and evaluate exponential expressions containing numbers written in standard form and scientific notation.
NY-8.EE.4
8.Mod1.AD12 Operate with numbers written in scientific notation to solve real-world problems.
NY-8.EE.4
8.Mod1.AD13 Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities.
NY-8.EE.4
8.Mod1.AD14 Interpret scientific notation that has been generated by technology.
NY-8.EE.4
8.Mod1.AD15 Apply the Pythagorean theorem to determine the unknown length of a hypotenuse in a right triangle in mathematical problems.
NY-8.G.7
Note: To ensure alignment to the New York State Next Generation Mathematics Learning Standards, some ADs have been removed.
The first page of each lesson identifies the ADs aligned with that lesson. Each AD may have up to three indicators, each aligned to a proficiency category (i.e., Partially Proficient, Proficient, Highly Proficient). While every AD has an indicator to describe Proficient performance, only select ADs have an indicator for Partially Proficient and/or Highly Proficient performance.
An example of one of these ADs, along with its proficiency indicators, is shown here for reference. The complete set of this module’s ADs with proficiency indicators can be found in the Achievement Descriptors: Proficiency Indicators resource.
ADs have the following parts:
• AD Code: The code indicates the grade level and the module number and then lists the ADs in no particular order. For example, the first AD for grade 8 module 1 is coded as 8.Mod1.AD1.
• AD Language: The language is crafted from standards and concisely describes what will be assessed.
• AD Indicators: The indicators describe the precise expectations of the AD for the given proficiency category.
• Related Standard: This identifies the standard or parts of standards from the New York State Next Generation Mathematics Learning Standards (NGMLS) that the AD addresses.
8.Mod1.AD5 Apply the properties of integer exponents to generate equivalent numerical expressions.
RELATED NGMLS
NY-8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions.
Partially Proficient Proficient Highly Proficient
Apply the properties of integer exponents to identify equivalent numerical expressions.
Which expressions are equivalent to 7 6? Choose all that apply.
Apply the properties of integer exponents to generate equivalent numerical expressions.
Use the properties of exponents to write an equivalent expression for 3 5 · 3 −9 by using only positive exponents.
Apply the properties of integer exponents to generate equivalent numerical expressions with prime factor bases.
Write an equivalent expression for 2 −10 · 16 5 by using the fewest number of prime factor bases.
8.Mod1.AD6 Solve equations of the form x 2 = p as √ p and √ p and equations of the form x 3 = p as 3 √ , where p is a rational number.
RELATED NGMLS
NY-8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Know square roots of perfect squares up to 225 and cube roots of perfect cubes up to 125. Know that the square root of a non-perfect square is irrational. Partially
Solve equations of the form x 2 = p as √p and √p and equations of the form x 3 = p as 3 √p , where √p , √p , or 3 √p is an integer.
Solve each equation.
x 2 = 16
x 3 = 64
Solve equations of the form x 2 = p as √p and √p and equations of the form x 3 = p as 3 √p , where √p , √p , or 3 √p is a rational or irrational number.
Solve each equation.
x 2 = 0.81
x 2 = 23
x 3 = 8 27
Solve two-step equations that after one step simplify to the form x 2 = p or x 3 = p where √p , √p , or 3 √p is a rational or irrational number.
Solve the equation. x 2 − 4 = 12
Related Standard
AD Indicators
Topic A Introduction to Scientific Notation
Topic A begins the year with thought-provoking and motivating contexts that spark students’ interest and create a need to further their understanding of exponents. Students use their place value understanding and knowledge of powers of 10 learned in grade 5 to write and operate with numbers in scientific notation.
Students begin the topic exploring large and small positive numbers by relating them to real-world objects and by writing the numbers in equivalent forms. They write large positive numbers as a single digit times a power of 10 as an introduction to scientific notation, which is formally defined in lesson 3. Students write small positive numbers as a single digit times a unit fraction with the denominator written as a power of 10 in exponential form. They learn to write these small positive numbers with negative exponents in topic B. After lesson 1, students work only with large positive numbers for the remainder of topic A.
By applying their understanding of place value and powers of 10, students solve how many times as much as problems. They use a more efficient method to solve these problems by approximating large positive numbers as a single digit times a power of 10.
Then students learn to write numbers in scientific notation, a × 10 n. The number a, called the first factor, is a number with an absolute value that is at least 1 but less than 10. Students use a place value chart to recall that each time a number is multiplied by 10, the digits shift one place value to the left. Therefore, the digits, not the decimal point, shift one place value to the left.
In an optional lesson, students build upon their knowledge of the distributive property to add and subtract numbers written in scientific notation. Students write out the powers of 10, such as writing 10 5 as 10 × 10 × 10 × 10 × 10, to operate with numbers written in scientific notation.
The lessons in topic A create the need for the properties and definitions of exponents. Topic A is truly meant to be an introduction to scientific notation. Therefore, do not expect mastery of operating with numbers written in scientific notation at the end of this topic.
In topic B, students learn and apply the properties and definitions of exponents. In topic C, they return to operations with numbers written in scientific notation, but this time involving integer exponents. Students use their knowledge of the properties and definitions of exponents to operate with numbers written in scientific notation instead of writing out the factors of 10.
Progression of Lessons
Lesson 1 Large and Small Positive Numbers
Lesson 2 Comparing Large Numbers
Lesson 3 Time to Be More Precise—Scientific Notation
Lesson 4 Adding and Subtracting Numbers Written in Scientific Notation (Optional)
Large and Small Positive Numbers
Write very large and very small numbers in a form that uses exponents to prepare students for scientific notation.
Approximate very large and very small quantities.
Lesson at a Glance
1. Consider the number 7,123,456
a. Approximate the number by rounding to the nearest million. 7,123,456 ≈ 7,000,000
b. Write your answer from part (a) as a single digit times a power of 10 in exponential form.
2. Consider the number 0.000 0285
a. Approximate the number by rounding to the nearest hundred thousandth. 0.000 0285 ≈ 0.00003
b. Write your answer from part (a) as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form. 0 00003 = 3
In this lesson, students use different forms of numbers to write very large and very small positive numbers. Students build an understanding of just how large or how small numbers really are by relating the sizes of known objects to powers of 10 through a digital teacher demonstration. Students lay the foundation for scientific notation by applying their understanding of the unit form of a number and powers of 10. Students write numbers as a single digit times a power of 10 in exponential form or as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form. Students write numbers in these forms after approximating very large and very small positive quantities. This lesson formally defines the term approximate
Key Questions
• Why do we write some very large positive numbers as a single digit times a power of 10 in exponential form?
• How does writing a small number in a different form help us write the number as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form?
Achievement Descriptor
8.Mod1.AD8 Approximate and write very large and very small numbers in scientific notation. (NY-8.EE.3)
Agenda
Fluency
Launch 10 min
Learn 25 min
• Standard Form of a Number
• Writing Very Large and Very Small Positive Numbers
• Approximating Very Large and Very Small Positive Quantities
Land 10 min
Materials
Teacher
• Computer or device*
• Projection device*
• Teach book* Students
• Dry-erase marker*
• Pencil*
• Personal whiteboard*
• Personal whiteboard eraser*
• Learn book*
Lesson Preparation
• None
*These materials are only listed in lesson 1. Ready these materials for every lesson in this module.
Fluency
Write Numbers by Using Powers of 10
Students write numbers as a power of 10 in exponential form to prepare for writing large positive numbers as a single digit times a power of 10 in exponential form.
Directions: Write each number as a power of 10 in exponential form.
Teacher Note
Fluency activities are short sets of sequenced practice problems that students work on in the first 3–5 minutes of class. Administer a fluency activity as a bell ringer or adapt the activity as a teacher-led Whiteboard Exchange or choral response. Directions for administration can be found in the Fluency resource.
Students relate very large and very small positive numbers to the size of real-world objects.
Open and display the Powers of 10 teacher interactive for the class. Progress through each power of 10 in the demonstration by starting with an input of 10 0 and moving upward to 10 27. The table shows a subset of the objects found in the demonstration and the corresponding inputs. Each input represents the height from which the visible object can be seen, not the size of the visible object.
UDL: Engagement
The teacher demonstration promotes relevance by connecting to possible student interests and familiar contexts. Share additional information from the Facilitation column for select objects that spark student interest.
10 2 El Castillo
10 3 Knock Nevis
El Castillo is the most recognizable ancient Mayan step pyramid in Chichén Itzá on Mexico’s Yucatán peninsula.
This is the largest ship ever built but is not in use anymore.
10 4 Mount Everest
10 7 Earth’s moon
The peak of Mount Everest is the highest point on Earth.
Our moon is bigger than Pluto. Pluto was considered a planet in our solar system until 2006, when it was downgraded to a dwarf planet.
Teacher Note
Information provided for the objects from the demonstration is based on data collected in 2020.
10 11 Polaris
10 16 Light-year
This is the name of the North Star. Earth’s geographic North Pole points right at Polaris.
10 21 Milky Way galaxy
10 27 Observable universe
This is the distance that light travels through space in one Earth year.
Our solar system is within this galaxy. However, our solar system is still very far from the center of this big disk.
The universe is a very mysterious place. We continue to make new discoveries as our technology advances.
Debrief the activity to this point by asking students what they notice and wonder. Then create a need for students to want a more efficient way to write very large positive numbers.
There are 1 billion trillion stars in the known universe. That is a 1 followed by 21 zeros. How long will it take us to write out that number ?
Write the number 1,000,000,000,000,000,000,000 for the class to see and ask students to keep track of the time.
There has to be a faster way to write out this number. But first, we will examine some very small objects.
In the teacher interactive, jump back to 10 0 in the demonstration. From there, progress down through inputs to 1 10 12 .
1 10 3 T. namibiensis This is the largest bacteria discovered.
1 10 4 Newly hatched tardigrade
Also called a water bear, this is the smallest animal discovered.
1 ___ 10 5 Spider silk Spider silk is very thin, but it is strong for its size.
1 ___ 10 6 Megavirus
This is the largest virus discovered. Its width is still only about 1 1000 of the length of the largest bacteria.
1 10 8 Semiconductor This is used in modern smartphones. 1 10 9 Water molecule
Water is made of two parts hydrogen and one part oxygen. This molecule is very simple, but it is important to life as we know it. 1 ___ 10 11 X-ray
This is where the shortest wavelength for x-rays can be seen. X-ray wavelengths measure from 0.01 to 10 nanometers.
1 10 12 Uranium-235 nucleus This is the type of uranium that is used in nuclear power plants.
Lead a class discussion by asking students what they notice and wonder. Then create a need for students to want a more efficient way to write very small positive numbers.
A proton’s diameter is 833 quintillionths of a meter. That is a decimal number written as 833 preceded by 15 zeros. How long does it take us to write out that number?
Write the number 0.000 000 000 000 000 833 for the class to see and ask students to keep track of the time.
Today, we will learn to represent both very large and very small positive numbers in a concise and efficient way.
Learn
Standard Form of a Number
Students write numbers in standard form.
Display problem 1. Work as a class to write the diameters of the planets Mercury and Venus in standard form. If needed, remind students that standard form is the common representation of a number by using digits.
Then invite students to work with a partner to complete problem 1. Circulate to listen for different strategies that students use to write the diameter of Mars, 6.8 million meters, in standard form.
• Students might write 6.8 million in unit form as 6 million 8 hundred thousand. The standard form of 6 million 8 hundred thousand is 6,800,000.
• Students might represent 6.8 million as 6.8 × 1,000,000, which is 6,800,000.
• Students might explain that in the number 6.8, the 8 is one place value to the right of the 6. In standard form, the 8 is still one place value to the right of the 6, which is 6,800,000.
Teacher Note
The dialogue shown provides suggested questions and sample responses. To maximize every student’s participation, facilitate discussion by using tools and strategies that encourage student-to-student discourse. For example, make flexible use of the Talking Tool, turn and talk, think–pair–share, and Always Sometimes Never.
Language Support
Consider using strategic, flexible grouping throughout the module.
• Pair students who have different levels of mathematical proficiency.
• Pair students who have different levels of English language proficiency.
• Join pairs to form small groups of four.
As applicable, complement any of these groupings by pairing students who speak the same native language.
1. Write each number in standard form.
Call the class back together and confirm answers. Facilitate a class discussion by asking several students to share their strategies for writing 6.8 million as 6,800,000 in standard form.
Differentiation: Support
The Place Value Chart that follows this lesson in the student book provides additional support for writing the standard form of a number. Show students how to use the chart to support their work, as needed.
• What is the
•
Writing Very Large and Very Small Positive Numbers
Students write very large positive numbers as a single digit times a power of 10 in exponential form. Students write very small positive numbers as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form.
Display problem 2, and guide the class through the Eiffel Tower row. Use the following prompts to guide students’ thinking. After students respond to each prompt, complete that part of the table and ask students to do the same.
The Eiffel Tower is one of the objects from the demonstration and is approximately 300 meters high. First, we write 300 in unit form. What kind of units do we use to write a number in unit form?
We use place value units to write a number in unit form.
How can we write 300 in unit form?
3 hundreds
The number 3 hundreds can be written as a single digit times what number?
How can we write 100 as a power of 10 in exponential form? 10 2
How can we write 300 as a single digit times a power of 10 expressed in exponential form?
3 × 10 2
Teacher Note
Students will learn the formal definition of scientific notation in lesson 3.
Have students complete problem 2 individually for Mount Everest and Venus.
2. Complete the table. The table shows the approximate measurements of objects seen in the demonstration.
Approximate Measurement (meters)
Differentiation: Support
The Place Value Chart could provide additional support for writing different forms of each number. Show students how to use the chart to support their work, as needed.
• Write 300 in the Place Value Chart.
• What is the place value of the first digit?
• What power of 10 in exponential form is associated with that digit?
When most students have completed the table, ask for volunteers to come to the board to complete the displayed table. Use the following prompt to help students recognize that each row shows the same number written in a different form.
What do you notice about the values we wrote in the Mount Everest row?
I notice that each value we wrote in the Mount Everest row represents the same number, but each column shows a different form of the number. Some forms include words, products, or exponents.
All the numbers in the Mount Everest row are equivalent to 9000, which is the approximate height in meters.
Next, students learn to write decimals as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form. Display and walk the class through the Grape row from problem 3. Use the following prompts to guide students’ thinking. After they respond to each prompt, complete that part of the table and ask students to do the same.
Now, we will look at different representations of small positive numbers. What is the unit form for the approximate width in meters of a grape?
3 hundredths
How do we write 3 hundredths as a fraction?
3 ___ 100
How do we write 3 ___ 100 as a single digit times a unit fraction? Recall that unit fractions have a numerator of 1.
3 × 1 ___ 100
How do we write the unit fraction with a denominator written as a power of 10 in exponential form?
1 10 2
How do we write 0.03 as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form?
3 × 1 10 2
Have students complete problem 3 with a partner for the approximate length of a grain of rice and width of a human hair.
Teacher Note
In topic B, students learn the meaning of a negative exponent and will then be expected to express small positive numbers by using a power of 10 in exponential form.
3. Complete the table. The table shows the approximate measurements of objects seen in the demonstration.
When most students have completed the table, ask for volunteers to come to the board to complete the displayed table. Use the following prompts to lead a class discussion.
Do you find it helpful to first write the number in unit form? Why?
Yes, writing the number in unit form helps us know which power of 10 to use for the denominator when writing the number as a fraction.
Do you find it helpful to write the number as a fraction? Why?
Yes, writing the number as a fraction helps us know to use the value of the numerator for the single digit when writing the number as a single digit times a unit fraction.
What is similar about how we write the approximate measurements of the Eiffel Tower and a grape in the last column of the tables?
They are similar because the approximate measurements of the grape and the Eiffel Tower both have a 10 2 .
Promoting the Standards for Mathematical Practice
Students reason quantitatively and abstractly (MP2) as they contextualize large and small positive numbers by relating them to the size of real-world objects.
Ask the following questions to promote MP2:
• Which real-world objects have sizes modeled by large positive numbers?
• Which real-world objects have sizes modeled by small positive numbers?
• What does writing these measurements with powers of 10 tell you about the relative sizes of these objects?
What is different about how we write the approximate measurements of the Eiffel Tower and a grape in the last column of the tables?
They are different because 10 2 is in the denominator for the approximate measurement of the grape, so 3 is multiplied by 1 10 2 . For the approximate measurement of the Eiffel tower, 3 is multiplied by 10 2 .
Because the unit forms of the numbers are written in hundreds and hundredths, there is a 102 in both numbers. A positive single digit times a power of 10 in exponential form, such as 3 × 10 2, represents a large positive number.
A positive single digit times a unit fraction with a denominator written as a power of 10 in exponential form, such as 3 × 1 ___ 102 , represents a small positive number.
Approximating Very Large and Very Small Positive Quantities
Students approximate very large and very small positive quantities. Students write these approximations as a single digit times a power of 10 in exponential form or as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form.
Direct students to problem 4. Read the problem aloud as students follow along. Use the following prompts to define approximate.
What does part (a) ask you to do?
Part (a) asks me to approximate the length of Rhode Island to the nearest ten thousand meters.
Where else in today’s lesson have you seen the word approximate?
I have seen the word approximate in the tables for problems 1–3.
What do you think the word approximate means?
I think approximate means about or close to a given value.
Yes, when we approximate, we represent the quantity by using a similar number that is found by rounding to a place value. What is another scenario where we might use approximate values?
Language Support
This is the first time the word approximate is used as a verb in the curriculum. Consider highlighting the use of the word approximate in different parts of speech. Lead a class discussion to clarify the different forms of approximate in these three examples. Emphasize the differences in pronunciation and the suffix changes, as applicable.
Part of Speech Example
Adjective
Verb
Noun
The approximate diameter of Earth is 13 million meters.
Approximate the diameter of Earth in meters.
An approximation of Earth’s diameter is 13 million meters.
Accept all reasonable answers, such as world population, the distance between Earth and other planets, or the diameter of Earth’s moon.
Then have students work on problem 4 independently. Use the following prompts to guide students’ thinking if needed.
• Is 77,249 closer to 70,000 or 80,000?
• Since we want to write the approximation as a single digit times a power of 10 in exponential form, which single digit will we use?
• How do you write 80,000 as a single digit times a power of 10 in exponential form? Explain your strategy.
4. The length of Rhode Island, from the northernmost point to the southernmost point, is 77,249 meters.
a. Approximate the length of Rhode Island by rounding to the nearest ten thousand meters.
80,000 meters
b. Write your answer from part (a) as a single digit times a power of 10 in exponential form.
80,000 = 8 × 10,000 = 8 × 10 4
8 × 10 4 meters
Confirm answers for problem 4. Then display problem 5 to students. Read the problem aloud as students follow along. Have students work independently. Use the following prompts to guide students’ thinking if needed.
• Is 0.0710 closer to 0.07 or 0.08?
• Which single digit should we use to write the approximation as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form?
• How do you write 0.07 as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form? Explain your strategy.
Differentiation: Support
The Place Value Chart could provide additional support for rounding and powers of 10. Show students how to use the chart to support their work, as needed.
• Write 77,249 in the Place Value Chart.
• What is the place value of the first digit?
• Is the number closer to 7 ten thousands or 8 ten thousands?
5. The width of a smartphone is 0.0710 meters.
a. Approximate the width of the smartphone by rounding to the nearest hundredth of a meter.
0.07 meters
b. Write your answer from part (a) as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form.
Confirm answers for problem 5. Then have students work with partners to complete problems 6–8. Circulate as students work and provide support. If students need support writing numbers in the approximate form, have them refer to the tables used in problems 2 and 3.
6. The length of California, from the northernmost point to the southernmost point, is 1,253,679 meters.
a. Approximate the length of California by rounding to the nearest million meters.
1,000,000 meters
b. Write your answer from part (a) as a single digit times a power of 10 in exponential form.
7. The diameter of a water molecule is 0.000 000 000 28 meters.
a. Approximate the diameter of a water molecule by rounding to the nearest ten billionth of a meter.
0.000 000 0003 meters
b. Write your answer from part (a) as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form.
Teacher Note
Spaces are included in very small positive numbers, such as 0.000 0084, to assist students in finding the place value of each digit.
3 × 1 10 10 meters
8. There are 3.2 ten thousand ounces in 1 ton.
a. Approximate the number of ounces in 1 ton by rounding to the nearest ten thousand ounces.
3 ten thousand ounces
b. Write your answer from part (a) as a single digit times a power of 10 in exponential form.
When most students are finished, confirm responses for problems 6–8.
Teacher Note
Some students may write 3.2 ten thousands as 32,000 and round to the nearest ten thousand, which is 30,000 or 3 ten thousands or 30 thousands. Celebrate the use of either strategy.
Land
Debrief 5 min
Objectives: Write very large and very small numbers in a form that uses exponents to prepare students for scientific notation.
Approximate very large and very small quantities.
Use the following prompts to debrief the lesson.
Why do we write some very large positive numbers as a single digit times a power of 10 in exponential form?
We write some very large positive numbers as a single digit times a power of 10 in exponential form so we can write the number faster than writing a single digit followed by all the zeros.
How does writing a number such as 0.0007 in a different form help us write the number as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form?
The unit form of the number helps us know which power of 10 will be in the denominator of the unit fraction.
The fraction form of the number helps us know which unit fraction to multiply the single digit by before writing the denominator as a power of 10 in exponential form.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
Complete the table. The table shows the average speed in miles per hour of a starfish.
Large and Small Positive Numbers
In this lesson, we
• explored large and small positive numbers by relating them to the size of real-world objects.
• analyzed equivalent forms of large and small positive numbers.
• approximated very large and very small positive numbers.
Examples
1. Complete the table. The table shows the approximate number of stacked pennies needed to reach the height of the Eiffel Tower.
Approximate Number of Stacked Pennies Standard Form Unit Form Single Digit Times a Power of 10 (expanded form) Single Digit Times a Power of 10 (exponential form) Eiffel Tower
Use place value units when writing numbers in unit form. The expanded form of one hundred thousand is 100,000 The exponential form of one hundred thousand is 10 5
Each number in the Starfish row is an equivalent form of 0.01
There are 4,356,000 square feet in 100 acres.
a. Approximate the number of square feet in 100 acres by rounding to the nearest million square feet.
4,000,000 square feet b. Write your answer from part (a) as a single digit times a power of 10 in exponential form.
= 4 million = 4 × 1,000,000 = 4 × 10 6
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
7. Complete the table. The table shows the average speed of a given animal.
6. Complete the table. The table shows the approximate number of stacked pennies needed to reach the height of a given object.
8. The deepest part of the ocean, called Challenger Deep, is 36,200 feet below sea level. a. Approximate the depth of Challenger Deep by rounding to the nearest ten thousand feet.
Approximate the length of the insect by rounding to the nearest ten thousandth of a meter.
b. Write your answer from part (a) as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form. 1 × 1 10 4 meter
EUREKA
10. There are 907,200,000 milligrams in 1 ton.
a. Approximate the number of milligrams in 1 ton by rounding to the nearest hundred million milligrams.
900,000,000 milligrams
b. Write your answer from part (a) as a single digit times a power of 10 in exponential form.
9 × 10 8 milligrams
11. There are 0.000 000 001 102 293 tons in 1 milligram.
a. Approximate the number of tons in 1 milligram by rounding to the nearest billionth of a ton.
0.000 000 001 ton
b. Write your answer from part (a) as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form.
1 × 1 10 9 ton
12. The world population is expected to reach 9.7 billion people in the year 2050.
a. Approximate the expected world population in 2050 by rounding to the nearest billion people.
10 billion people
b. Write your answer from part (a) as a single digit times a power of 10 in exponential form.
1 × 10 10 people
Remember For problems 13–16, add or subtract.
Comparing Large Numbers
Write numbers as a single digit times a power of 10 in exponential form to approximate quantities.
Compare large and small positive numbers by using times as much as language.
Lesson at a Glance
Bacterial life appeared on Earth about 4 billion years ago. Insects appeared about 400,000,000 years ago.
a. Write when bacterial life appeared on Earth as a single digit times a power of 10 in exponential form.
4 billion = 4,000,000,000 = 4 × 10 9
Bacterial life appeared on Earth about 4 × 10 9 years ago.
b. The number of years since bacterial life appeared on Earth is about how many times as much as the number of years since insects appeared?
= 4 × 10 8
Students work individually and with a partner to write unknown factor equations to answer how many times as much as questions. Students discuss why division is used to find an unknown factor in these equations and use number sense or factors of 10 to find the quotients. Students work together to determine when writing large numbers as a single digit times a power of 10 is an efficient method. All numbers in this lesson are either given as a single digit times a power of 10 in exponential form or presented for students to write quantities in that way.
Key Questions
• How can we use equations to solve how many times as much as problems?
• What forms of numbers are useful when solving how many times as much as problems with large numbers?
Achievement Descriptors
8.Mod1.AD9 Express how many times as much one number is as another when both numbers are written in scientific notation. (NY-8.EE.3)
8.Mod1.AD11 Multiply or divide numbers written in standard form and scientific notation, and evaluate exponential expressions containing numbers written in standard form and scientific notation. (NY-8.EE.4)
8.Mod1.AD12 Operate with numbers written in scientific notation to solve real-world problems. (NY-8.EE.4)
Agenda
Fluency
Launch 5 min
Learn 30 min
• Unknown Factor
• Times As Much As
Land 10 min
Materials
Teacher
• None
Students
• None
Lesson Preparation
• None
Fluency
Write Numbers in Standard Form
Students write each number in standard form to prepare for writing large numbers.
Directions: Write each number in standard form.
Students calculate an unknown factor.
Direct students’ attention to problem 1. Read the problem together and ask them to work with a partner to answer the question. As students work, look for pairs who are using different strategies to solve.
1. Analysts estimate that there were about 9 billion devices worldwide that used wireless routers to wirelessly connect to the internet in 2017 and 2018. A wireless router supports about 300 devices.
The number of devices worldwide is about how many times as much as the number of devices one router can support?
9,000,000,000 = · 300
9,000,000,000 __________ 300 = 30,000,000
The number of devices worldwide is about 30,000,000 times as much as the number of devices one router can support.
When most pairs have finished, use the following prompts to lead a class discussion about the strategies used to solve the problem.
Many of you used division to find the answer. Why?
I know the total number of devices worldwide and the number of devices one router can support, and the question has the phrase how many times as much as. So I know I have to divide to find which number I can multiply by 300 to get 9,000,000,000.
How did you divide? What method did you use?
I used the division algorithm to divide.
I divided 9 by 3 and then thought about their place values.
Do you have any concerns with using that method to divide really large numbers? Explain.
There are a lot of zeros in the numbers.
If the 9 and 3 did not divide evenly, the method could have been hard to do accurately.
Suppose we remove the context of devices and routers in problem 1. What question is being asked if you just have the numbers 9 billion and 300 in the problem?
How many 300’s are in 9 billion?
300 times what number is 9 billion?
9 billion is how many times as much as 300?
Differentiation: Support
Consider offering additional times as much as questions and equations with smaller numbers to prepare for solving with large numbers. The following table is useful to organize student reasoning.
is how many times as much as 3?
Record these questions and have students write them below problem 1 in their book. Offer the question “9 billion is how many times as much as 300?” if not already mentioned by a student.
Today, we will answer problems that ask how many times as much as to compare very large numbers.
Learn
Unknown Factor
Students determine how many times as much one number is as another.
Guide students through the work of problem 2. Consider a think-aloud style to clearly demonstrate writing the unknown factor equation and pairing the many factors of 10.
For problems 2–4, write each number as a single digit times a power of 10 in exponential form and write an unknown factor equation. Write your answer as a single digit or as a single digit times a power of 10 in exponential form.
2. 9 billion is how many times as much as 3000?
Language Support
Students who are unfamiliar with the term unknown factor equation may benefit from an explanation of the parts of the equation. product = unknown factor · known
Teacher Note
Writing the factors of 10 helps make the division simpler by allowing students to pair factors of 10 in the numerator and denominator to create quotients of 1.
9 billion is 3 × 10 6 times as much as
Pause for questions, and then have students work with a partner to solve problems 3 and 4 by using the same method. Encourage students to be flexible in their solution strategies after practicing this method in these problems.
3. 9 million is how many times as much as 3,000,000?
Differentiation: Support
The Place Value Chart that follows this lesson in the student book provides additional support. If students have difficulty writing a single digit times a power of 10, encourage them to find the place value of the first digit on the Place Value Chart. Then discuss the place value’s connection to the power of 10.
9 million is 3 times as much as 3,000,000
4. What number is 3000 times as much as
?
Confirm answers, and lead a class discussion by using the following prompts.
What is similar about the equations we wrote in our work in problems 2–4?
We wrote all the equations in the form: product = factor · factor.
How did you use the given information to solve the problems?
If we are given the product and one of the factors, we divide to find the other factor. If we are given both factors, we multiply to find the product.
Does writing a number in the form of a single digit times a power of 10 in exponential form make the calculations simpler? How?
Yes, finding pairs of 10 that divide to make 1 helps me visualize the size of the answer.
Yes, I didn’t have to use the division algorithm to find the answer.
No, when there are a lot of 10’s to write, this way doesn’t seem very efficient.
No, if I can do the calculations in my head, this method is much harder.
No, when the highest place values are the same, like in problem 7, you just divide the 9 by the 3, so writing all of the 10’s is not useful.
When would you use division with numbers in the form of single digits times a power of 10 in exponential form to calculate a quotient?
I would use this method when the numbers have a lot of zeros and when I cannot do the division in my head.
Times As Much As
Students write and solve equations with unknown factors to compare two quantities.
Direct students to the Times As Much As problems. Read the directions aloud. Have students work with a partner to approximate the quantities and write the unknown factor equations. They will solve the problems later.
Promoting the Standards for Mathematical Practice
When students write the single digit times a power of 10 using exponential notation so they can divide 10 by 10, they are breaking a mathematical object into parts and making use of structure (MP7).
Ask the following questions to promote MP7:
• Can you break 9 × 10 9 3 × 10 3 into easier problems?
• How are 9 billion 3 thousand and 9 × 10 9 3 × 10 3 related? How could that help you find the quotient?
• How are 9 million 3 million and 9 × 10 6 3 × 10 6 related? How could that help you find the quotient?
Provide students with ample time to write the unknown factor equations. Students who finish early can choose one problem to begin solving. Circulate and use the following prompts to focus students’ thinking:
• Why does the number round to ?
• Explain how you determined the unknown factor equation for this problem.
Review the approximations and unknown factor equations for each problem. Then have students choose at least two problems to solve with their partner. Circulate as students work and identify a few pairs who use a variety of solution strategies. These students can share their strategies with the class during the closing discussion.
For problems 5–8, approximate each quantity as a single digit times a power of 10 in exponential form and write the unknown factor equation. Then solve for the unknown factor by using a strategy of your choice.
5. In the fall of 2019, approximately 50,800,000 students attended school in the United States in prekindergarten through grade 12. About 11,683,000 of those students attended middle school.
The overall number of students in prekindergarten through grade 12 was about how many times as much as the number of students in middle school? 50,800,000 ≈ 50,000,000 = 5 × 10 7
Teacher Note
Encourage students to think flexibly about numbers during the partner activity, which may yield a variety of solution paths. For example, most of the sample student work shows division with powers of 10, but problem 5 uses place value thinking.
In the fall of 2019, the overall number of students in prekindergarten through grade 12 was about 5 times as much as the number of students in middle school.
UDL: Engagement
Including a variety of real-world problem contexts with varying degrees of difficulty provides an opportunity for student choice. Allowing students to select problems they find interesting and challenging puts students in charge of their learning and promotes relevance.
6. In 2019, the total outstanding consumer debt in the United States, including mortgages, auto loans, credit cards, and student loans, was about $3.9 trillion. That same year, the published national debt of the United States was $22,460,468,000,000.
The published United States national debt was about how many times as much as the total United States consumer debt?
In 2019, the published United States national debt was about 5 times as much as the total United States consumer debt.
7. As of July 1, 2018, the US Census Bureau estimated the population of New York City at 8,398,748 people and the population of New York State at 19,542,209 people.
The population of New York State was about how many times as much as the population of New York City?
2.5
In 2018, the population of New York State was about 2.5 times as much as the population of New York City.
8. The total global carbon dioxide emissions for 2018 was about 33.1 billion tons. That same year, the carbon dioxide emissions by natural gas in the United States was about 1.629 billion tons.
Approximately what fraction of the total global carbon dioxide emissions was from the use of natural gas in the United States? 33.1 billion = 33,100,000,000
In 2018, about 1 __ 15 of the total global carbon dioxide emissions was from the use of natural gas in the United States.
Invite pairs of students, particularly those who used a unique solution strategy, to present their work to the class. Have the rest of the class record the work as they follow along. Consider discussing that these are approximations, meaning that finding an approximation for each value and then calculating with that value contributes to some errors in the final answer. However, generating approximations is still a reasonable method to use when needing to calculate with very large numbers.
Land
Debrief
5 min
Objectives: Write numbers as a single digit times a power of 10 in exponential form to approximate quantities.
Compare large and small positive numbers by using times as much as language.
Facilitate a brief discussion by using some of the following questions.
How can we use equations to solve how many times as much as problems?
We can use the equation product = factor · factor to solve these problems. We could be given the product and one of the factors. To solve, we would need to divide the product by the factor to find the other factor. We could be given both factors. To solve, we would need to multiply both factors to find the product.
What forms of numbers are useful when solving how many times as much as problems with large numbers?
Numbers written as a single digit times a power of 10 in exponential form are useful because we can multiply or divide by grouping the single digits and grouping the 10’s. We can multiply or divide with the standard form of the numbers.
How does writing an unknown factor equation help to determine how many times as much one number is as another number?
The unknown factor equation organizes the information from the problem. This helps us know what operation to use to solve the problem.
Teacher Note
These questions may be more accessible by referring to problems 3 and 4 from the lesson.
Do the exponents in the powers of 10 in a quotient allow you to predict the exponent in the power of 10 in your answer? How?
Yes, I can visualize the factors of 10 in pairs, which create 1 when they are divided. The rest of the 10’s remain in the answer, so I can predict the exponent of the power of 10 in the answer.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
Comparing Large Numbers
In this lesson, we
• wrote unknown factor equations to answer how many times as much as questions.
• approximated large numbers by writing them as a single digit times a power of 10
• wrote out factors of 10 to help evaluate quotients and find unknown factors.
Examples
For problems 1 and 2, write each quantity as a single digit times a power of 10 in exponential form, and write an unknown factor equation. Then find the answer to the question.
Writing out the factors of 10 makes the division simpler because 10 10 is 1
Ask, “Which quantity is being multiplied?” to decide how to write the unknown factor equation.
The land area of Canada is about 20 times as large as the land area of France. The question can be reworded as 4000 times what number is 90,000?
Approximate the value of each land area. Then write the approximation as a single digit times a power of 10 in exponential form.
EUREKA MATH2 New York Next Gen
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
For problems 4–6, write an unknown factor equation to represent the question. Then find the answer to the question.
4.
8. The land area of the United States is about how many times as large as the land area of Brazil?
The land area of the United States is about 1 1 3 times as large as the land area of Brazil.
9. The land area of Belize is about how many times as large as the land area of Brazil?
The land area of Belize is about 3 1000 times as large as the land area of Brazil.
12. The Atlantic Ocean contains about 310,410,900 cubic kilometers of water. Lake Superior, which is the largest lake in the United States, contains about 12,000 cubic kilometers of water.
Approximately how many Lake Superiors would it take to fill the Atlantic Ocean? It would take about 30,000 Lake Superiors to fill the Atlantic Ocean.
For problems 13–16, add or subtract.
11. Which country’s land area is about 1 500 as large as the land area of the United States?
The land area of Slovenia is about 1 500 as large as the land area of the United States.
17. Approximate 8,538,206 by rounding to the nearest million. Write your approximation as a single digit times a power of 10 in exponential form.
9,000,000 = 9 × 10 6
18. Which of the following are equivalent to 5.206? Choose all that apply.
A. 5 ones 2 tenths 6 thousandths
B. 5 + 0.2 + 0.06
C. (5 × 1) + (2 × 1 10 ) + (6 × 1 100 )
D. (5 × 1) + (2 × 0.1) + (6 × 0.001)
E. 5206 thousands
F. 5206 1000
Time to Be More Precise— Scientific Notation
Write numbers given in standard form in scientific notation.
Lesson at a Glance
1. In 2018, the population of Italy was approximately 60,630,000 people. Write this number in scientific notation.
6.063 × 10 7
2. The average human body contains about 3.4 × 10 10 cells. Write
The lesson opens with a historical context that addresses the need for using scientific notation to write numbers. Students learn the definition of scientific notation and leverage their prior knowledge of place value and powers of 10 to write numbers in scientific notation and in standard form. This lesson formally defines the terms scientific notation and order of magnitude.
Key Questions
• What are the characteristics of numbers written in scientific notation?
• When is it useful to write a number in scientific notation?
Achievement Descriptor
8.Mod1.AD8 Approximate and write very large and very small numbers in scientific notation. (NY-8.EE.3)
Edition: Grade 8, Module 1, Topic A, Lesson 3
Agenda
Fluency
Launch 10 min
Learn 25 min
• Another Way to Represent Numbers
• Interpreting Scientific Notation
• Using Scientific Notation
Land 10 min
Materials
Teacher
• None
Students
• None
Lesson Preparation
• Review the Math Past resource to support the delivery of Launch.
Fluency
Expand Powers of 10
Students expand each power of 10 to prepare for writing a number in standard form and in scientific notation.
Directions: Expand each power of 10.
Launch
Students are introduced to different representations of large numbers through the history of Archimedes.
Direct students to the historical information on Archimedes. Give students 2 to 3 minutes to read silently and complete problems 1 and 2.
Archimedes was a Greek mathematician whose ideas were ahead of his time. He lived in Sicily during the third century BCE and was the first to develop fundamental concepts in geometry, calculus, and physics.
Fascinated by very large numbers and living on the coast of the Ionian Sea, Archimedes set out to determine how many grains of sand are needed to fill the known universe.
Mathematicians used Ionic Greek notation in the third century BCE. So instead of numbers, they used symbols and letters from the Greek alphabet. 10
Teacher Note
In addition to this activity, consider incorporating the Math Past material provided in the Math Past resource.
Language Support
To offer additional support, consider the following strategies:
• Read the passage aloud to a small group of students.
• Pair students together with a partner to take turns reading aloud.
• Provide highlighters for students to visually emphasize key ideas.
• Summarize the text as a class before assigning problems 1 and 2.
1. In this system, χνδ represents the number 654. What does σπε represent?
2. What does ψοζ represent?
Review the answers to problems 1 and 2 before assigning problem 3. Give students 2 to 3 minutes of silent work time before they check their work with a partner.
3. Read the story on the death of Archimedes. Then write the Greek math symbols as numbers in the blanks that follow each number. Because the exact details of his death are not confirmed, this story is one of many theories about how Archimedes died during the Second Punic War.
Archimedes’s native city of Syracuse, Italy, was captured by Roman forces in the year σιβ BCE ( 212 BCE). While the city was under siege, Archimedes was drawing a diagram of circles in the sand.
As he contemplated his work, the οε-year-old ( 75 -year-old) mathematician was approached by a Roman soldier demanding that Archimedes meet immediately with the general of the Roman Army. Archimedes refused, insisting that he would not go until he finished his math problem. This infuriated the Roman soldier, and he pulled out his sword and killed Archimedes on the spot.
Archimedes’s last words are thought to be “Do not disturb my circles.” His work would go unknown until the year φλ CE, ψμβ years ( 530 CE, 742 years) after his death.
Archimedes’s Twin Circles
Review answers by asking students to share their results and strategies. Then discuss the following questions as a class.
What is challenging about using the ancient Greek notation?
The ancient Greeks used many more characters to represent numbers. We just use 10 digits, 0 through 9. We can’t express very large numbers by using ancient Greek notation.
Consider a very large number written as a single digit times a power of 10.
Display the value 3 × 10 6 .
How can we write this number in standard form?
3,000,000
Can we write this number in Ionic Greek notation?
Have students think–pair–share to try writing the number in Ionic Greek notation. This number is impossible to write in that notation given the limited scope of that number system. Give students time to engage in a productive struggle.
Were you and your partner able to write the number 3,000,000 in Ionic Greek notation? Explain.
We were not able to write the number in Ionic Greek notation because the system cannot represent such a large number.
What is the largest number we can write by using the Greek symbols in the table?
The largest number we can write by using the Greek symbols is 999.
Archimedes wanted to know how many grains of sand are needed to fill the known universe. Is the number 999 useful to estimate the number of grains of sand needed to fill the universe? Explain.
No. It takes far more than 999 grains of sand to fill the universe.
Consider engaging students in a discussion by using the following questions:
• Would finding the number of grains of sand needed to fill the known universe be an interesting quest in Archimedes’s day?
• How many grains of sand do you think are needed to fill the universe?
• What is a number that is too high? Too low?
Archimedes had to develop a new way to write very large numbers. He called this new notation a myriad, represented by the capital letter M. He used the symbol M for the number 10,000.
With his new notation, Archimedes was able to use myriads to estimate the number of grains of sand needed to fill the known universe. He arrived at the value of 1 followed by 63 zeros.
How would we represent that value as a single digit times a power of 10?
1 × 10 63
Which is more efficient to write out: 1 followed by 63 zeros, or 1 × 10 63 ? Explain.
It is more efficient to write out 1 × 10 63 because it would take much longer to write out 1 followed by 63 zeros.
Archimedes developed the myriad when he recognized the limitations of Ionic Greek notation. Similarly, scientists developed scientific notation, which is a condensed way of writing large numbers in our current number system.
Today, we will learn how to write numbers in scientific notation.
Learn
Another Way to Represent Numbers
Students learn the definition of scientific notation and identify examples and nonexamples from a list.
Introduce students to the term scientific notation by displaying the following number with the general expression written directly underneath:
6.02 × 10 3 a × 10 n
Examine the structure of the two expressions. What do you notice?
I notice the structure is very similar. Both expressions have a power of 10. The power of 10 is multiplied by a number in the first expression and a variable in the second expression.
A number is written in scientific notation when it is represented as a number a, multiplied by a power of 10. The number a, which we call the first factor, is a number with an absolute value of at least 1 but less than 10.
The order of magnitude n is the exponent on the power of 10 for a number written in scientific notation.
What is the first factor and the order of magnitude in the number 6.02 × 10 3 ?
The first factor is 6.02, and the order of magnitude is 3.
Have students work together in pairs to capture these definitions in problems 4 and 5.
Language Support
Consider placing new terms alongside the expression a × 10 n to illustrate the significance of the new definitions.
To solidify the concept, provide examples of numbers written in scientific notation and ask students to identify the first factor and the order of magnitude of each. a × 10 n
First Factor Order of Magnitude
4. Fill in the blanks to complete the statement.
A number is written in scientific notation when it is represented as a number a multiplied by a power of 10 .
The general expression that represents a number written in scientific notation is a × 10 n .
The absolute value of a must be at least 1 but less than 10 .
5. Identify the first factor and the order of magnitude of the expression 8.86 × 10 6 .
The first factor is 8.86. The order of magnitude is 6.
Begin a short discussion to prepare students for problem 6. Display the number −7.1 × 10 3 and ask the class the following questions.
What is the first factor of the expression?
The first factor of the expression is −7.1.
What is the absolute value of the first factor?
The absolute value of −7.1 is 7.1.
Is the number −7.1 × 10 3 written in scientific notation? How do you know?
The number −7.1 × 10 +3 is written in scientific notation. It is written in the form a × 10 n , and the absolute value of the first factor is at least 1 but less than 10.
What is the order of magnitude?
The order of magnitude is 3.
As we have seen from the number −7.1 × 10 3, numbers written in scientific notation do not always need to be positive. A negative number can also be written in scientific notation, which is the form a × 10 n where a represents the first factor with an absolute value of at least 1 but less than 10 and n represents the order of magnitude.
Instruct students to complete problem 6 individually. Encourage students who finish early to share their choices with a partner.
UDL: Representation
Encourage students to identify the values of a and n for each number in problem 6. Highlight the corresponding values by using different colors to show students the parallel structure between the examples and the general form a × 10 n .
6. Use the definition of scientific notation to indicate whether each number is an example or a nonexample.
3 × 10 2 X
× 10 2 X
Review students’ reasoning in a follow-up discussion by using the following prompts.
Identify one number from the list that is written in scientific notation. Explain how you know.
The number 4.8 × 10 3 is written in scientific notation because the 4.8 represents the first factor, which is multiplied by a power of 10. The absolute value of 4.8 is at least 1 but less than 10.
Identify one number from the list that is not written in scientific notation. Explain how you know.
The number 19.04 × 10 7 is not written in scientific notation because the first factor has an absolute value of 19.04, which is greater than 10.
Language Support
Have students use the Agree or Disagree section of the Talking Tool to support discussion about why they select Example or Nonexample in problem 6.
Numbers that are correctly written in scientific notation have a first factor multiplied by a power of 10. The first factor must be a number with an absolute value of at least 1 but less than 10.
Interpreting Scientific Notation
Students write numbers presented in scientific notation in standard form.
Frame the Interpreting Scientific Notation section by posing the following rhetorical questions to the class:
Why is it useful to write numbers in scientific notation? Why does the absolute value of the first factor have to be at least 1 but less than 10? We’ll get some answers in our next activity.
Have students work in pairs on problems 7–9 without providing explicit instructions about how to write the numbers in standard form. If necessary, remind students that standard form is the common representation of a number by using digits. Circulate to note the various approaches students use.
For problems 7–9, write the number in standard form.
Differentiation: Support
For students who need to review the concept, use the Place Value Chart to illustrate a simple example like problem
Invite students to share their strategies with the class.
We can solve these problems by expanding the power of 10, or by using a place value chart to visualize this product.
Display problem 10. Ask students to turn and talk about their answers. Do not reveal the answer to problem 10 with students yet.
10. Use the Place Value Chart to write the number 9.1 × 10 3 in standard form.
Teacher Note
Students are familiar with place value charts from prior grades. Additionally, students practiced writing numbers presented in scientific notation in standard form in the previous lesson. However, these numbers were represented by a single digit times a power of 10.
Now, students work with first factors that are not single digits. A place value chart is an important tool to highlight that the first factor is what shifts left along a place value chart when multiplied by factors of 10. A common misconception is that the decimal is what moves when writing numbers presented in scientific notation in standard form.
9100
When preparing to work through problem 10 as a class, start by pointing out that the ones place is represented by 1. The other columns are written as powers of 10. Use the prompts to guide students through using a place value chart.
We can show 9.1 × 10 3 by using a place value chart. What happens to the value of a number, such as 9.1, when we multiply it by 10 ?
The value gains one place value. For example, 9.1 becomes 91.
How do we see that on a place value chart?
The digits move one place to the left when we multiply by 10.
What happens when you multiply a number by 10 3 on a place value chart?
We shift the digits to the left 3 times because 10 3 represents the product 10 × 10 × 10
Illustrate each step by showing the product 9.1 × 10 × 10 × 10 on the chart.
Have students work independently on problem 11.
11. Use the Place Value Chart to write 2.05 × 10 6 in standard form.
2,050,000
Display the answer on the chart and invite students to share their thinking.
Introduce students to problem 12 and have them complete it individually. Then bring students together in pairs to compare answers and make revisions. While in pairs, instruct students to use a place value chart to confirm their answers.
12. Logan writes the number 6.7 × 103 in standard form. He writes 67,000 because 10 3 represents thousands. Do you agree with Logan? Explain.
I disagree with Logan because 6.7 × 10 3 represents 6.7 thousands, not 67 thousands. So 6.7 × 10 3 should be written as 6700 in standard form.
Invite students to share their answers with the class. Then engage in a summarizing discussion by using the following prompts.
Promoting the Standards for Mathematical Practice
When students analyze Logan’s method for writing 6.7 × 10 3 in standard form, they construct viable arguments and critique the reasoning of others (MP3).
Ask the following questions to promote MP3:
• Which parts of Logan’s answer do you question? Why?
• What questions can you ask your partner to make sure you understand their argument?
We know we can write numbers in standard form, so why do we need scientific notation?
It is simpler to read and write very large numbers when we use scientific notation.
How does the standard form of 2.05 × 10 6 relate to its power of 10?
106 represents millions. So we know that the standard form is a number in the millions.
How does the standard form of 9.1 × 10 3 relate to its power of 10?
103 represents thousands. So we know that the standard form is a number in the thousands.
How does the standard form of 3 × 10 2 relate to its power of 10?
102 represents hundreds. So we know that the standard form is a number in the hundreds.
We can use the power of 10 to determine the highest place value of a number in standard form, which is a reason why we write numbers in scientific notation.
Using Scientific Notation
Students write numbers presented in standard form in scientific notation.
Direct students to problem 13. Display the problem and model writing a number in scientific notation by using the following questions to guide the class.
How do we write 200,000 as a single digit times a power of 10 in exponential form like we did in lesson 1?
2 × 10 5
In what form is the number 2 × 10 5 written?
2 × 10 5 is written in scientific notation.
What is the first factor of 2 × 10 5? What is the order of magnitude?
The first factor is 2. The order of magnitude is 5.
13. Write 200,000 in scientific notation.
2 × 10 5
We used a place value chart to help us write numbers in standard form. Now we are using a place value chart to write numbers in scientific notation.
Direct students to problem 14. Display the problem and the Place Value Chart and work through it together as a class.
14. Use the Place Value Chart to write 350 in scientific notation.
3.5 × 10 2
Drive the discussion with the following prompts:
When writing a number in scientific notation, what must be true about the first factor?
The absolute value of the first factor must be at least 1 but less than 10.
What number has an absolute value of at least 1 but less than 10 and can be multiplied by a power of 10 to get 350?
3.5
When writing 350 in scientific notation, our first factor must be 3.5. Now, back to our example. Write 350 in your Place Value Chart. Then write 3.5 in the next line of your chart.
How many factors of 10 do we need to multiply 3.5 by to maintain the value of 350?
We need two factors of 10.
Confirm that 3.5 × 10 × 10 equals 350 on your Place Value Chart. Then write 350 in scientific notation.
Confirm the answer with the class.
Have students complete problems 15–18 individually or with partners. Circulate to check answers and encourage students who may be struggling to use the Place Value Chart.
For problems 15–18, write the number in scientific notation. Use the Place Value Chart, if needed.
Invite students to display their work and strategies with the class. Students could use the power of 10 to identify the highest place value of the number in standard form, or they could use the Place Value Chart.
Differentiation: Support
If students are unsure what the first factor of a number should be, find the digit in the highest place value, write a decimal point after it, and then write all the other digits after the decimal point until you write the last nonzero digit. For example, when we write 5,045,000,000 in scientific notation, the first factor is 5.045.
Differentiation: Support
Students may use the Place Value Chart removable if needed in problems 15–18. Eventually, students should become comfortable without the additional support of a place value chart.
Land
Debrief 5 min
Objective: Write numbers given in standard form in scientific notation.
Review the definition of scientific notation in problem 4 before engaging in a class discussion.
What are the characteristics of numbers written in scientific notation?
Numbers written in scientific notation are in the form a × 10 n. We call a the first factor.
The absolute value of the first factor must be at least 1 but less than 10. The exponent n represents the number’s order of magnitude.
When is it useful to write a number in scientific notation?
When a very large number has many zeros, it is useful to write the number in scientific notation.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
A number is represented on a place value chart. Write the number in scientific notation.
order of magnitude is 6
In this lesson, we
• learned the definition of scientific notation.
• identified examples and nonexamples of numbers written in scientific notation.
• wrote numbers in scientific notation and in standard form.
Examples
1. Circle all the values written in scientific notation. 9.82 × 10 15 million 0.6 × 10 11 −6 × 10 5 4.2 × 10 3 4,200,000
Terminology
• A number is written in scientific notation when it is represented as a number a multiplied by a power of 10. The number a which we call the first factor, is a number with an absolute value of at least 1 but less than 10
• The order of magnitude n is the exponent on the power of 10 for a number written in scientific notation.
Numbers written in scientific notation must be in the form a × 10n The absolute value of a, the first factor, must be at least 1 but less than 10
2. Write the number 3.08 × 10 4 in standard form. 3.08 × 10 4 = 30,800
The order of magnitude, 4 shows that the highest place value of the number written in standard form is the ten thousands place.
Consider using a place value chart.
4. The earliest known mammal on Earth is a tiny mouse-like creature called the morganucodontid. It lived about 210,000,000 years ago. Write this number in scientific notation.
= 2.1 × 10 8
factor is 7.5 The first factor must have an absolute value of at least 1 but less than 10 so the first factor is 2.1 The highest place value is hundred millions, so the order of magnitude is 8
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
problems 2–7, write the number in standard form.
8. In 2017, about 8.3 × 10 12 text messages were sent and received worldwide. Write this number in standard form.
8,300,000,000,000
9. In 2014, the United States discarded a total of 5.08 × 10 9 pounds of trash. Write this number in standard form.
5,080,000,000
17. In 2019, the man considered to be the richest person in the world had a net worth totaling about $111 billion. Write this number in scientific notation.
1.11 × 10 11
18. Match each number written in standard form with its corresponding number written in scientific notation.
For problems 20–23, add or subtract.
19. The table represents the box office sales of the highest grossing movie in 2019 in various markets. Write each number in standard form and in scientific notation.
24. Consider the number 0.000 236
a. Approximate the number by rounding to the nearest ten thousandth. 0.000 236 ≈ 0.0002
b. Write your answer from part (a) as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form. 0.0002 = 2 × 1 10 4 For problems 25–30, add or subtract.
Adding and Subtracting Numbers Written in Scientific Notation (Optional)
Add and subtract numbers written in scientific notation. Rewrite sums and differences in scientific notation.
Lesson at a Glance
In this optional lesson, students extend previous understanding of combining like terms and the distributive property to add and subtract large numbers written in scientific notation. Students rewrite the product of a number greater than 10 and a power of 10 in scientific notation and explore the importance of doing so. Students also rewrite numbers in scientific notation to have terms with the same powers of 10 in order to add or subtract them.
Key Questions
• How are adding and subtracting numbers in scientific notation and combining like terms similar? How are they different?
• Why do we rewrite a number a × 10 n, where a ≥ 10, in scientific notation?
Achievement Descriptor
8.Mod1.AD12 Operate with numbers written in scientific notation to solve real-world problems. (NY-8.EE.4)
Agenda
Fluency
Launch 5 min
Learn 30 min
• Adding and Subtracting: What Is the Same?
• Rewriting Sums and Differences
• Adding and Subtracting: What Is Different?
Land 10 min
Materials
Teacher
• None
Students
• None
Lesson Preparation
• None
Fluency
Write Numbers in Scientific Notation
Students write numbers in scientific notation to prepare for calculating sums and differences of numbers written in scientific notation.
Directions: Write each number in scientific notation.
Teacher Note
Instead of this lesson’s Fluency, consider administering the Scientific Notation and Positive Exponents Sprint. Directions for administration can be found in the Fluency resource.
6. 8,104,000,000
Launch
Students activate prior knowledge by adding like terms.
Direct students to complete problems 1–6 independently.
For problems 1–6, find the sum.
3 thousands + 2 thousands + 4 thousands 9
900 When students complete the problems, use the following prompts to guide a brief discussion.
How are the expressions in problems 1 through 6 alike?
The expressions are alike because they each have three terms with the coefficients 3, 2, and 4.
How are the expressions in problems 1 through 6 different?
The expressions are different because they each have different types of addends, such as place value units, fractions, variables, and powers of 10.
Teacher Note
Some students may apply the distributive property and write the sums in problems 5 and 6 as 9 × 100 and 9 × 10 2, respectively, which is a central idea in this lesson. Consider inviting those students to be among the first to share during the upcoming discussion about problem 7.
Each expression has three addends with something in common. What is common among the addends for each expression?
Problem 1: thousands
Problem 4: m 2
Problem 2: fifths
Problem 5: 100
Problem 3: m
Problem 6: 100 or 10 2
What strategy did you use to find the sum in problem 5? In problem 6?
Expect responses related to order of operations or the distributive property. Many students will likely evaluate individual terms and then add, though some students may recognize that they can apply what they already know about combining like terms.
In each expression, we can find 3 + 2 + 4 to get 9 and write the answer with whatever the addends had in common, because the addends are like terms. Today, we will extend the concept of adding and subtracting like terms to adding and subtracting numbers written in scientific notation.
Learn
Adding and Subtracting: What Is the Same?
Students add and subtract numbers written in scientific notation with the same power of 10, where the result is also in scientific notation.
Direct students’ attention to problem 7. Read the prompt aloud, including parts (a) and (b). Give students 1 minute of individual think time. Then have students discuss by using the following prompt:
Take about 1 minute to turn and talk with a partner. Compare Pedro’s and Ava’s solution methods and discuss who is correct. Be sure to record your thoughts for parts (a) and (b) in preparation for a class discussion.
7. Pedro and Ava each find the sum 3 × 10 5 + 2 × 10 5 + 4 × 10 5. Pedro uses the order of operations, and Ava uses the distributive property.
Pedro’s work:
+ 200,000 + 400,000 = 900,000
Ava’s work:
a. Who is correct? Explain.
Both Pedro and Ava are correct. The number 900,000 is equivalent to 9 × 10 5 .
b. Compare the two methods.
Sample: Both methods arrive at answers with the same value, but Ava’s method is more efficient because it uses fewer steps and requires less calculation.
After 1 minute, invite a few pairs of students to share their thoughts about whether Pedro or Ava is correct and why. Explanations will likely focus on 900,000 and 9 × 10 5 as simply two ways to write the same number. Then ask a few other pairs of students to compare the two methods. If not addressed by students, discuss possible disadvantages of Pedro’s method as the powers of 10 become larger.
Use the following prompts to conclude discussion on problem 7.
Look at Ava’s work again. Why is (3 + 2 + 4) × 10 5 equivalent to 3 × 10 5 + 2 × 10 5 + 4 × 10 5?
If we use the distributive property with (3 + 2 + 4) × 10 5, the 10 5 can be multiplied by each term in (3 + 2 + 4) to get 3 × 10 5 + 2 × 10 5 + 4 × 10 5 .
In what form is Ava’s answer 9 × 10 5?
Ava’s answer is in scientific notation.
Have students apply Ava’s method to complete problem 8 with their partner.
For problems 8–10, add or subtract. Write the answer in scientific notation.
Teacher Note
Students may need support to recall skills from grades 6 and 7. Consider using problem 3 from Launch to demonstrate how combining like terms that have variables is an application of the distributive property.
Debrief problem 8 with the class by using the following prompts to solidify the process for adding and subtracting numbers in scientific notation.
Can we apply the distributive property to the terms in problem 8? Why?
Yes, we can apply the distributive property because the terms all have 10 12 as the power of 10.
Describe the process you used to find the sum.
I found 3 + 2 + 4 and kept 10 12 because 1012 is the same for all three terms.
When numbers in scientific notation have the same power of 10, we can add or subtract them the same way we combine like terms. We can apply the distributive property to add or subtract the first factors.
Have students complete problems 9–11 with their partner.
11. The table shows the number of views for three popular online videos.
a. How many total views do the three videos receive? Write the answer in scientific notation. 5.5 × 10 7 + 1.1 × 10 7 + 2.3 × 10 7 = (5.5 + 1.1 + 2.3) × 10 7 = 8.9 × 10 7
The three videos receive a total of 8.9 × 10 7 views.
b. How many more views does the video of the cat singing receive than the video of the baby dancing? Write the answer in scientific notation.
× 10 7 1.1 × 10 7 = (5.5 1.1) × 10 7 = 4.4 × 10 7
The video of the cat singing receives 4.4 × 10 7 more views than the video of the baby dancing.
Call the class back together and debrief problem 11 with the following prompts.
What did you need to do in problem 11 before you could add or subtract?
I needed to rewrite the numbers so that they were all in the same form.
What do the three numbers have in common?
The numbers all have the 10 7 as the power of 10.
Rewriting Sums and Differences
Students add and subtract numbers written in scientific notation with the same power of 10, where the result must be rewritten in scientific notation.
Have students complete problem 12 with their partner.
Students may leave their answers as 21 × 10 3 because they do not notice that 21 × 10 3 is not written in scientific notation. Use the following prompts to discuss the difference between 21 × 10 3 and previous answers and to develop the need for writing the answer in scientific notation.
Compare the sum 21 × 10 3 from problem 12 to the sum 9 × 10 12 from problem 8. How are the sums alike?
The sums are alike because they are both written as a factor times a power of 10.
How are the sums different?
The sums are different because the powers of 10 are different. Also, the first factor in 21 × 10 3 is greater than 10, while the first factor in 9 × 10 12 is less than 10.
Is the number 21 × 10 3 written in scientific notation? How do you know?
No. The absolute value of the first factor is greater than 10, so it does not fit the definition of scientific notation.
Promoting the Standards for Mathematical Practice
When students place requirements on the value of the first factor in scientific notation to ensure that the power of 10 represents the highest place value of the number they write, they attend to precision (MP6).
Ask the following questions to promote MP6:
• When comparing two numbers, such as 8 × 10 4 and 80 × 10 3, which steps must we follow precisely? Why?
• Is it always correct to say that 10 n represents the highest place value of a number written as a × 10 n for any value of a? What could we add to or change in that statement to make it more precise?
Let’s talk about why the first factor needs to be at least 1 but less than 10. We’ve seen that the power of 10 for a large number written in scientific notation tells us the highest place value of the number. What would we expect the highest place value to be for 21 × 10 3?
We would expect the highest place value of the number to be the thousands place because 10 3 is 1000.
Is the thousands place the highest place value for 21 × 1000?
No, the highest place value of 21 × 1000 is the ten thousands place because 21 × 1000 is 21,000.
Ask students to share their thoughts on why a discrepancy between the perceived size of a number and the actual size of a number could be problematic.
When a large number is written in scientific notation, where the first factor is a number with an absolute value of at least 1 but less than 10, the power of 10 tells us the highest place value of the number. If the absolute value of the first factor is not in that range, then the power of 10 does not tell us the highest place value, so it becomes more difficult to develop a sense of how large the number really is.
Direct students back to problem 12. Give students a minute to try to rewrite 21 × 10 3 in scientific notation with their partner. Then, as a class, work through rewriting 21 × 10 3 in scientific notation, inviting students to share their thinking. Consider color-coding or highlighting the parts of the problem to help students make the necessary connections. Ask students to record these remaining steps for problem 12 in their work.
UDL: Representation
Then have students complete problems 13–15 with their partner.
Color-coding can serve as a visual cue when rewriting the sum in scientific notation. This visual cue helps students distinguish between the first factor and the original factors of 10. It also helps them see how each step relates to the previous step.
For problems 13 and 14, add or subtract. Write the answer in scientific notation.
If students struggle to rewrite the first factor so that it is greater than 1 but less than 10, consider using the Place Value Chart that follows this lesson in the student book to illustrate the concept. For example, show that 19.05 is equivalent to 1.905 × 10.
15. In 2008, a bakery in Indonesia set a record for creating the world’s tallest cake, which was 108.27 feet tall. Bakers used about 62 thousand ounces of powdered sugar and about 5.7 × 10 4 ounces of margarine in the cake.
How many total ounces of powdered sugar and margarine were in the cake? Write the answer in scientific notation.
There were about 1.19 × 105 total ounces of powdered sugar and margarine in the cake.
Students may start to notice a pattern when writing numbers in scientific notation, which can be useful for efficiency as the powers of 10 get larger. Use the following prompts to spark an early awareness of this pattern.
The answers in this section all have initial results with a first factor greater than 10. In each case, what happens to the power of 10 between the initial result and the final result written in scientific notation?
The power of 10 increases.
By how much does the power of 10 increase?
The power of 10 increases by 1.
By how much does the power of 10 increase if you rewrite the number 400 × 10 5 in scientific notation? Explain.
The power of 10 increases by 2 because 400 = 4 × 10 × 10.
Differentiation: Support
Some students may be unsure how to extend their understanding of rewriting numbers in scientific notation to consider first factors in the hundreds or greater. Consider using the Place Value Chart to show that the power of 10 increases by the number of places the first digit of the first factor shifts on the chart.
By how much does the power of 10 increase if you rewrite the number 4000 × 10 5 in scientific notation? Explain.
The power of 10 increases by 3 because 4000 = 4 × 10 × 10 × 10.
Adding and Subtracting: What Is Different?
Students add and subtract numbers written in scientific notation with different powers of 10.
Direct students to look at problem 16. Compare problem 16 to any of problems 7 through 15. How are they alike?
The problems are alike because they involve addition or subtraction of numbers written in scientific notation.
How are they different?
They are different because the numbers in problem 16 do not have the same power of 10.
What must be true to add or subtract numbers in scientific notation?
The numbers need to have the same power of 10.
As a class, work through problem 16. Again, consider using color-coding or highlighting.
Call special attention to specific points in the solution that require applying the new skills learned throughout the lesson.
Teacher Note
Because students have not yet learned the properties of exponents, they may continue to write out all the factors of 10. By this point in the lesson, students may see this as tedious and question whether there is a more efficient way, providing a need for the properties of exponents. Students will explore the properties of exponents and delve deeper into operations with numbers written in scientific notation in topics B and C.
Have students complete problem 17 with their partner.
For problems 16 and 17, add or subtract. Write the answer in scientific notation.
Consider having students who are ready for more variation or complexity try problem 17 again by using 10 5 as the common power of 10. Students can rewrite 4 × 10 4 as (0.4 × 10) × 10 × 10 × 10 × 10.
Confirm the answer to problem 17. If students need extra support and time permits, consider going through the solution steps for problem 17 with the class, using color-coding or highlighting for clarity.
Land
Debrief 5 min
Objectives: Add and subtract numbers written in scientific notation.
Rewrite sums and differences in scientific notation.
Use the following prompts to facilitate a discussion. Encourage students to add to their classmates’ responses.
How is a strategy we use to find 2 × 10 5 + 3 × 10 5 similar to a strategy we use to find 2 x 5 + 3 x 5?
The strategies are similar because each expression contains like terms and we can apply the distributive property to find the sum.
How is a strategy we use to find 2 × 10 5 + 3 × 10 4 different from a strategy we use to find 2 x 5 + 3 x 4?
The strategies are different because neither expression has like terms to start, but we can rewrite one of the numbers in 2 × 10 5 + 3 × 10 4 so that the powers of 10 are the same. We can find the sum 2 × 10 5 + 3 × 10 4, but we cannot combine 2 x 5 + 3 x 4 any further.
Why do we rewrite numbers such as 45 × 10 6 in scientific notation?
We rewrite numbers such as 45 × 10 6 in scientific notation to provide an accurate representation for the size of the number. Since 45 is greater than 10, 10 6 does not correctly indicate the highest place value of the number 45 × 10 6 .
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
Adding and Subtracting Numbers Written in Scientific Notation (Optional)
In this lesson, we
• added and subtracted numbers written in scientific notation.
• rewrote sums and differences in scientific notation.
Examples For problems 1–3, add or subtract. Write the answer in scientific notation.
Apply the distributive property because all three terms have the same power of 10
The first factor, 17, is greater than 1 but not less than 10 Rewrite 17 as 1.7 × 10
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
For problems 3 and 4, add or subtract. Write the answer in scientific notation.
5. The table shows the estimated number of US households that had a pet in 2019.
a. What is the estimated total number of households that had a bird or a saltwater fish? Write the answer in scientific notation.
About 8 × 10 6 households had a bird or a saltwater fish.
b. About how many more households had a reptile than had a saltwater fish? Write the answer in scientific notation.
About 3 × 10 6 more households had a reptile than had a saltwater fish.
11. The table shows an estimated number of smartphone users in the three most populous countries in the world in 2018.
Country Estimated Number of Smartphone Users
× 10 8
a. What is the estimated total number of smartphone users for the three countries in 2018? Write the answer in scientific notation.
The estimated total number of smartphone users in 2018 is 1.41 × 10 9
b. In 2018, how many more smartphone users were estimated to be in China than in the United States and India combined? Write the answer in scientific notation.
In 2018, there were about 1.56 × 10 8 more smartphone users in China than in the United States and India combined.
12. Evaluate 5 × 10 5 + 3 × 10 4. Write the answer in scientific notation.
5.3 × 10 5
13. Nora creates a website password with six characters.
• There are about 3.089 × 10 8 possible six-character passwords that use only lowercase letters.
• There are about 2.177 × 10 9 possible six-character passwords that use any combination of only lowercase letters, only numbers, or both lowercase letters and numbers.
How many passwords can Nora create that use at least one number?
Nora can create about 1.8681 × 10 9 passwords that use at least one number. Remember
For problems 14–17, add or subtract.
Topic B Properties and Definitions of Exponents
Previously in topic A, students worked exclusively with bases of 10 with positive exponents. They build upon this understanding in topic B by working with properties involving positive exponents and progress to learning definitions of an exponent of 0 and negative exponents. Students build proficiency in simplifying exponential expressions and apply these properties and definitions to more complex exponential expressions.
The topic begins with students examining the products of powers of 10 to make a conjecture about the relationship between the exponents of the factors and the exponent of the product. Students then extend their conjecture for powers of 10 to include like bases and positive whole-number exponents, arriving at the product of powers with like bases property of exponents: x m · x n = x m+n. This property of exponents is fundamental, as it is used throughout the topic to develop additional properties involving powers of powers and powers of products.
Students’ previous understanding of scientific notation and place value is expanded when they learn about bases with an exponent of 0 and negative exponents. Topic B concludes with an optional lesson in which students learn to simplify expressions by rewriting certain powers’ bases in terms of other bases in the expression.
Throughout the topic, students record properties and definitions into a graphic organizer that they can use as a reference. Students may reference the organizer in topic C when they apply these properties to operate with numbers written in scientific notation.
By the end of this topic, students should be able to simplify exponential expressions by using multiple properties and definitions of exponents. These skills, which are developed in topic B, are foundational to the upcoming lessons in topic C in which students use their new knowledge of negative exponents to write small positive numbers in scientific notation and to operate with numbers written in scientific notation.
of Exponents
Progression of Lessons
Lesson 5 Products of Exponential Expressions with Whole-Number Exponents
Lesson 6 More Properties of Exponents
Lesson 7 Making Sense of the Exponent of 0
Lesson 8 Making Sense of Integer Exponents
Lesson 9 Writing Equivalent Expressions
Lesson 10 Evaluating Numerical Expressions by Using Properties of Exponents (Optional)
Products of Exponential Expressions with Whole-Number Exponents
Apply understanding of exponential notation to write equivalent expressions for x m · x n .
Lesson at a Glance
This lesson introduces students to the exponent property associated with products of powers. Students use their knowledge of powers to explore expressions of the form x m · x n for any number x and positive whole numbers m and n. Through practice with concrete examples, they make a conjecture about an equivalent expression to x m · x n and use repeated multiplication to show that x m · x n = x m+n . With this new property associated with the product of powers with like bases, students write equivalent expressions that include coefficients, multiple bases, and scientific notation.
Key Questions
• What conditions are necessary to apply the product of powers with like bases property?
• Why does x m · x n = x m+n?
Achievement Descriptor
8.Mod1.AD5 Apply the properties of integer exponents to generate equivalent numerical expressions. (NY-8.EE.1)
Agenda
Fluency
Launch 5 min
Learn 30 min
• Multiplying Powers with Like Bases
• Applying the Property of Exponents
• Multiplying Powers with Unlike Bases
Land 10 min
Materials
Teacher
• Chart paper for Properties and Definitions of Exponents anchor chart
• Markers (3)
Students
• Properties and Definitions of Exponents
Lesson Preparation
• Use the chart paper to create an anchor chart with the same tables from the Properties and Definitions of Exponents graphic organizer to display to the class. The anchor chart will be filled in throughout lessons 5–8.
Fluency
Write Expressions by Using Exponents
Students write products in exponential notation to prepare for writing equivalent expressions of the form x m · x n .
Directions: Write each product as a single base raised to an exponent.
Teacher Note
Instead of this lesson’s Fluency, consider administering the Write Expressions with Exponents Sprint. Directions for administration can be found in the Fluency resource.
Launch
Students multiply two powers with like bases.
Direct students to complete problem 1. Students should not have access to calculators. Circulate as students work and observe whether they try to write out all the 10’s by hand. The goal of this problem is to create the need for applying the more efficient property of exponents explored in the lesson.
1. Multiply. Write the product as a power of 10 in exponential form.
After about one minute of productive struggle time, engage in a brief discussion by using the following questions.
Is this problem challenging? Why?
Students who try to write out 10’s by hand and then multiply may express frustration about the problem, while a few students may recognize that the expression represents using 10 as a factor 70 times. Allow for varying opinions to be shared with the class.
Today, we will find a more efficient way to write the product of powers with like bases.
Learn
Multiplying Powers with Like Bases
Students look for patterns when multiplying powers with like bases.
Direct students to problem 2.
How is problem 2 similar to problem 1?
Problems 1 and 2 are similar because they are both products of powers of 10, and they both have exponents involving 5 and 2.
How are they different?
They are different because the exponents in problem 1 are 50 and 20, and the exponents in problem 2 are 5 and 2.
Have students complete problem 2 independently. Students who choose to write out all the factors of 10 here will quickly arrive at 7 factors of 10.
2. Multiply. Write the product as a power of 10 in exponential form.
Engage in a discussion by posing the following questions.
Which problem seemed easier to you? Why?
Allow for a variety of responses. Students who tried to write out all the 10’s in problem 1 will likely indicate that the smaller exponents in problem 2 made it much easier, while students who found a shortcut in problem 1 may not see a difference in complexity.
Often, we want to find ways to make calculations more efficient. Can we always write out all the factors of a power individually? Explain.
I suppose we could try, but it will not necessarily be simple or efficient. It would take a long time in problem 1 to write out 50 tens and 20 tens.
How do the exponents in the factors in problem 2 relate to the exponent in the product?
The exponent in the product, 7, is the sum of the exponents of the factors, 5 and 2.
Let’s explore why this works.
Revisit problem 2 and display it to the class. Consider using color coding to help students relate the equivalent expressions in each step.
How do we expand the power 10 5 as a product of 10’s?
How do we expand the power 102 as a product of 10’s?
2 = 10 · 10
Now let’s write 10 5 · 10 2 as a product of 10’s.
How many factors of 10 do we have? We have 5 + 2 factors of 10
How do we write the product of 10 5 · 10 2 as a power with base 10?
We write it as 10 5+2 .
Is it important that both powers in a product have the same base? Why?
Yes, it is important that both powers in the product have the same base. Otherwise, we could not simply add the exponents to get 5 + 2 factors of 10. For example, in a
UDL: Representation
Color coding can serve as a visual cue to help students distinguish between the factors of 10 that result from expanding each of the powers, 105 and 102.
Teacher Note
The equivalent expression 10 5 · 10 2 is intentionally shown as 10 5+2 to emphasize the use of the property. Writing 10 5+2 has more instructional value than 10 7. When students are ready, ask them to express the sum of the exponents as a single integer.
product like 10 5 · 4 2, we would have 5 factors of 10 and 2 factors of 4, which cannot be combined any further.
So the powers in a product need to have the same base for this property to work.
When we write 10 5 and 10 2 as a product of 10’s, we can see that we have 5 + 2 factors of 10. So we can write 10 5 · 10 2 as 10 5+2 .
We can use variables to make a general property for any base.
Display the expression on the board. Use color coding to help students relate the equivalent expressions in each step.
How many factors of x are represented by x m?
There are m factors of x.
How many factors of x are represented by x n?
There are n factors of x.
Label the next steps by using brackets. Express the expansion of factors of x by using ellipses:
m times n times
When we multiply x m by x n, how many factors of x do we have?
We have m + n factors of x.
Display this work to the class.
(m + n) times
Review with students the general property that results from this work.
If x is any number and m and n are positive whole numbers, then x m · x n = x m+n . For us to add the exponents, the product must have like bases.
Differentiation: Support
If students have difficulty with a variable base and variable exponents, first display x 4 · x 7
Promoting the Standards for Mathematical Practice
When students expand products of powers such as 10 m · 10 n and 10 m+n and notice the pattern x m x n = x m+n, they are looking for and expressing regularity in repeated reasoning (MP8).
Ask the following questions to promote MP8:
• What patterns did you notice when you expanded the powers?
• Will this pattern always work?
Instruct students to add this property to the first blank row in their Properties and Definitions of Exponents graphic organizer. Do the same on chart paper for the class Properties and Definitions of Exponents anchor chart. Consider using color coding for the Property and Example(s) columns.
Properties of Exponents
Language Support
Students will continue to use the Properties and Definitions of Exponents graphic organizer to formalize properties and definitions involving exponents in lessons 5–8. Display the table as an anchor chart that is gradually completed throughout the topic. Students will refer to this information throughout the topic as they identify new properties and definitions.
Encourage students to refer to their graphic organizers or the class anchor chart regularly throughout the topic. They can use it for problems within the lesson as well as Practice problems.
Definitions of Exponents
Ask students to complete problem 3 independently and to show their work by using the new property. Circulate to confirm answers.
3. Multiply. Write the product as a power of 10 in exponential form.
Teacher Note
The property names are provided to help students distinguish each property, but it is not critical that students know the property name. It is most important that students know how to correctly apply the property.
Applying the Property of Exponents
Students apply the product of powers with like bases property to write equivalent expressions with like bases.
Direct students to preview problems 4–7 and ask them to share what they notice about the problems. Have students work through problems 4–7 in pairs.
For problems 4–14, apply the property of exponents to write an equivalent expression.
Confirm the answers. Have students complete problems 8–15 independently. Then have them check their work with a partner.
Teacher Note
Students may overlook the power of 1 for problems 10 and 11. For these students, ask the following questions.
• For 72 10 · 72, by how many additional factors do you multiply 72 10?
• How are the additional factors accounted for in your answer?
Confirm answers as a class.
Introduce the Which One Doesn’t Belong? routine. Present four expressions and invite students to study them.
Give students 1–2 minutes to find a category in which three of the expressions belong, but a fourth expression does not. When time is up, invite students to explain their chosen categories and to justify why one expression does not fit. Celebrate all valid reasonings.
Which expression does not belong? Why?
The expression 10 4 · 10 4 does not belong because it is 10 4+4, which is not equivalent to the other three expressions. The expression 10 16 does not belong because it is not written as a product.
Multiplying Powers with Unlike Bases
Students apply the product of powers with like bases property to write equivalent expressions with unlike bases.
Continue the Which One Doesn’t Belong? routine. Present four expressions and invite students to study them.
Give students 1–2 minutes to find a category in which three of the expressions belong, but a fourth expression does not. When time is up, invite students to explain their chosen categories and to justify why one expression does not fit.
Which expression does not belong?
The expression 6 3 · 5 3 · 6 4 · 5 2 does not belong because it is the only expression not equivalent to 6 12 · 5 6 .
The expression 6 12 · 5 6 does not belong because it is a product of only two factors.
The expression 6 · 5 · 6 5 · 5 5 · 6 · 6 5 does not belong because it is the only expression that has factors raised to the first power.
Highlight responses that emphasize the product of powers with like bases property. Ask the following questions that invite students to use precise language, make connections, and ask questions of their own.
How are the expressions in this activity different from the expressions in problems 8–15?
For the expressions in this activity, there are two different bases. For expressions in problems 8–15, there is only one base.
How does this activity relate to the product of powers with like bases property?
The property x m · x n = x m+n applies only to products of powers with like bases. When the expression has two bases, we apply the property to each base separately.
Instruct students to complete problem 16 independently.
For problems 16–22, apply the property of exponents to write an equivalent expression.
16. 9 3 · 9 4 · 4 2 · 4 9 3+4 · 4 2+1
Problem 16 provides a rich opportunity to draw out misconceptions about multiplying powers, such as adding the bases and adding the exponents, or multiplying unlike bases. Use the following prompts to address these misconceptions.
Display the incorrect answer 13 3+4+2+1 .
Suppose Shawn applies the property to the product in problem 16 and gets the answer 13 3+4+2+1. Is Shawn correct? Why?
Shawn is incorrect. He added the bases 9 and 4 and added all the exponents. The property only applies to powers with the same base, and we only add the exponents, not the bases. So Shawn should have left his answer as 9 3+4 · 4 2+1 .
Display the incorrect answer 36 3+4+2+1 .
Suppose Lily applies the property to the product in problem 16 and gets the answer 36 3+4+2+1. Is Lily correct? Why?
Lily is incorrect. She multiplied the bases 9 and 4 and added all the exponents. The property only applies to powers with the same base, and we keep that base in the product. So Lily also should have left her answer as 9 3+4 · 4 2+1 .
Confirm the correct answer and emphasize to students that they can only apply the property to powers with like bases.
Assign students problems 17–20 to complete in pairs. When pairs have finished, have them check their work with a neighboring pair.
As a class, confirm answers and review problem 20 with the following prompts.
How can you write your answer from problem 20, 4 · 10 4+5, with the sum of the exponents?
4 · 10 9
What does this form 4 · 10 9 remind you of? Why?
It reminds me of scientific notation because there is a number multiplied by a power of 10, and that number has an absolute value of at least 1 but less than 10.
How would we express this number in standard form?
4,000,000,000
Have students work in pairs on problem 21. Circulate to look for students who may add the coefficients instead of multiplying them. When most have finished, confirm answers as a class. Ask students to point out the similarities between problem 21 and problem 22 before having them complete problem 22 in pairs.
21. 2 x 6 · 3 x 5 2 x 6 · 3 x 5 = 2 · x 6 · 3 · x 5
What do you notice about the answers to problems 21 and 22?
The structure of the answers is similar. In both problems, we multiplied 2 and 3 to get 6 in our final answer. Both the x and the 10 have the same exponent.
Land
Debrief 5 min
Objective: Apply understanding of exponential notation to write equivalent expressions for x m · x n .
Lead a class discussion about the newly learned property of exponents to debrief the lesson.
What conditions are necessary to apply the property of exponents we learned today?
Powers must be multiplied and have the same base to apply the property of exponents we learned today.
Why does 4 5 · 4 3 = 4 5+3 ?
Because 4 5 has 5 factors of 4 and 4 3 has 3 factors of 4, the product 4 5 · 4 3 has 5 + 3, or 8, factors of 4. Therefore, 4 5 · 4 3 can be written as 4 5+3 .
Exit Ticket 5
min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
In this problem, there are two different bases: a base of 7 and a base of 8
Products
of
Exponential Expressions with Whole-Number Exponents
In this lesson, we
• discovered a pattern when multiplying powers with like bases.
• learned the product of powers property.
• applied the property of exponents to write equivalent expressions.
Product of Powers with Like Bases Property
x is any number m and n are positive whole numbers when x m x n = x m+n
Examples For problems 1–3, apply the property of exponents to write an equivalent expression.
4 times 2 times Negative bases and fractional bases have parentheses to avoid confusion.
First, group the powers with like bases together. Then, apply the property of exponents to each base separately.
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
15. Sara states that when you multiply two powers with the same base, the exponents are multiplied together. She uses an example to support this claim:
Fill in the boxes to create an equation that shows that Sara’s claim is incorrect.
Find the area of the rectangle. Write the area as a single base raised to an exponent.
The product (−1) 3 ⋅ (−1) n is negative. Which of the following values of n are possible? Choose all that apply.
Fill in the boxes with digits
9.261
More Properties of Exponents
Encounter and apply properties of exponents, including raising powers to powers, raising products to powers, and raising quotients to powers.
Lesson at a Glance
Students encounter and apply properties of exponents, making use of a “mustard spot” context. Students engage with raising powers to powers by considering the repetition of a base. They examine raising products to powers by determining whether two expressions are equivalent. Finally, they make a conjecture about an equivalent expression for a quotient raised to a power. This lesson introduces the term verify.
Key Question
• How can we write equivalent expressions for (x m) n , (xy) n, and (x y )n ?
Achievement Descriptor
8.Mod1.AD5 Apply the properties of integer exponents to generate equivalent numerical expressions. (NY-8.EE.1)
Agenda Materials
Fluency
Launch 5 min
Learn 30 min
• Raising Powers to Powers
• Raising Products to Powers
• Raising Quotients to Powers
Land 10 min
Teacher
• Markers (3)
Students
• Properties and Definitions of Exponents (in the student book from lesson 5)
Lesson Preparation
• Properties and Definitions of Exponents anchor chart (from lesson 5)
Fluency
Expand Powers as a Product of the Bases
Students expand powers as products of the bases to prepare for generating expressions of the form (x m) n , (xy) n, and ( x __ y )n .
Directions: Expand each power as a product of the bases. For example, expand 34 as 3
Launch
Students write equivalent forms of exponential expressions.
Have students complete problems 1–3 with a partner. Circulate as students work, and advance their thinking by using the following questions.
• Why does this property work?
• What happens when we multiply by another 34?
• What pattens do you notice?
1. Write an expression equivalent to 34 ⋅ 34. 34+4
2. Write an expression equivalent to 34 ⋅ 34 ⋅ 34. 34+4+4
3. Write
When most students are finished, have them turn and talk about the following question.
What is an easier way to write the given expression and your equivalent expression in problem 3?
Anticipate that not all students will think of a more efficient way to write the expressions yet. As pairs share, listen for students who may already recognize (34)9 as another way to write the given expression and recognize 39 4 as another way to write their equivalent expression.
Today, we will learn a more efficient way to write equivalent expressions for these types of expressions.
Promoting the Standards for Mathematical Practice
When students try to write equivalent expressions for repeated multiplication of a power, they are looking for and expressing regularity in repeated reasoning (MP8).
Ask the following questions to promote MP8:
• When you write these products, is anything repeating?
• How could that repeating pattern help you write the expressions more efficiently?
• Will this pattern always work?
Learn
Raising Powers to Powers
Students consider expressions of the form (x m)n .
Play the Powerful Mustard video. Then facilitate a class discussion by using the following questions.
What happened in this video?
As the character was eating lunch and working on his math homework, his friend accidently spilled mustard on the homework.
What do you think the character, Vic, is trying to figure out?
I think Vic is trying to figure out what is written on the paper. It looks like he thinks there is a 3 and a 4 in the expression, but he is not sure which operation is used.
Have students complete problems 4 and 5 with a partner. Circulate as students work and advance their thinking by using the following questions.
• Choose one possible expression Vic thinks might be under the mustard spot. What is an expression that is equivalent to your chosen expression?
• Do you think that the expression under the mustard spot will change the relationship between the base and the exponent?
4. Without knowing what is under the mustard spot, what do we know about the expression?
)5
Whatever the spot is covering, we need 5 factors of it.
UDL: Representation
Presenting the “powerful mustard” situation in video format supports students in understanding the problem context by removing barriers associated with written and spoken language.
5. Draw mustard spots to create an expression on the right side of the equal sign to make a true statement.
Sample: ( )5 = ( ) ( ) ( ) ( ) ( )
Ask students to share their thinking for problem 5. Then use the following statement to transition to problem 6.
Vic is not sure what is under the mustard spot, but he is pretty sure he remembers seeing a 3 and a 4.
Have students complete problem 6 with a partner. Encourage students to refer back to the mustard spots from problem 5 to help them complete problem 6.
6. Assume 34 is under the mustard spot.
a. Write factors of 34 to create an expression on the right side of the equal sign to make a true statement.
(34)5 = (34)(34)(34)(34)(34)
b. How is (34)5 similar to the mustard spot expression in problem 5?
These expressions are similar because they both have an exponent of 5.
c. Explain the meaning of (34)5 .
The expression (34)5 means that we multiply 5 factors of 34.
d. Apply the properties of exponents to write an equivalent expression for (34)5. Explain your reasoning.
I can write (34)5 as 35 4. I can think of the power inside the parentheses as 4 factors of 3. The exponent outside the parentheses tells me I have 5 factors of 34. This means
I have 5 factors of 4 factors of 3, which is 5 ⋅ 4, or 20, factors of 3.
Display the equation (3 4) 5 = (3 4)(3 4)(3 4)(3 4)(3 4) and ask students to share their thinking for problem 6(a).
Then display the equation (x m ) n = x m n .
If x is any number and m and n are positive whole numbers, then (xm)n = xm ⋅ n .
Have students turn and talk about this property by asking the following questions:
• Does this property accurately describe what you noticed about the expression (3 4) 5? How?
• Do you think this property is useful? How?
Then show the example (5 2) 6 = 5 2 6 .
Instruct students to add this property to the next blank row in their Properties and Definitions of Exponents graphic organizer from lesson 5. Add the new property to the Properties and Definitions of Exponents anchor chart for this topic.
Properties of Exponents
Description
Definitions of Exponents
Description Definition(s) Example(s)
Teacher Note
At this stage, have students leave the exponent in the form of a product to help make the use of the property more visible.
Students may wish to write expressions of the form (x m)n as x m n rather than as x n m
Consider highlighting the fact that both responses are correct. In the Properties and Definitions of Exponents graphic organizer and future lessons, these expressions will be written as x m n for readability.
Raising Products to Powers
Students consider expressions of the form (xy)n .
Have students work with a partner on problems 7 and 8.
7. Assume 4 ⋅ 3 is under the mustard spot. In this case, what is repeated? How many times?
If 4 ⋅ 3 is under the mustard spot, then 4 ⋅ 3 is repeated 5 times.
8. Vic notices that two problems he already completed are now also covered in mustard. What do you think the original expressions are?
Once most students are finished, lead a class discussion about their responses by asking the following questions.
Do the expressions you wrote in problem 8 evaluate to the same number? Why?
Yes. Both expressions evaluate to the same number because both expressions have seven factors of 4 and seven factors of 3
What does the fact that both expressions evaluate to the same number tell you about the original expressions?
Because the expressions evaluate to the same number, we know the original expressions are equivalent.
So (4 ⋅ 3)7 and 47 ⋅ 37 are equivalent expressions.
Direct students to problem 9. Have students turn and talk about which expression they think is correct. Then have students record their thinking for problem 9.
9. Students are discussing possible answers to a problem. What do you think?
(4x)3 = (4x) (4x) (4x) (4x)3 = 43x 3
It can’t be both, but which one is right?
Sample: Both expressions, (4x)(4x)(4x) and 43 x3, have the same number of factors of 4 and of x. Therefore, the expressions are equivalent.
Display the equation (4x)3 = (4x)(4x)(4x). Then ask the following question.
How can we make the final expression look like 43 x3?
Show the following steps to write the expression (4 x)(4 x)(4 x) as 4 3 x 3 .
(4x)3 = (4 x)(4 x)(4 x)
43 x3
Have students think–pair–share about the following prompt.
Explain how this work shows that (4x)3 is equivalent to 43 x3 .
This power has a base of 4x and the exponent is 3, so we can write the equivalent expression (4x)(4x)(4x). We can remove the parentheses because we know everything is multiplied. Next we can use the commutative property to group the factors of 4 together and the factors of x together. Then we can use exponential notation to write 43x3 .
Use the following prompts to introduce the term verify.
When we think something is correct, but we need to make sure it is right, we verify it.
Have we verified that (4x)3 = 43 x3? How?
Yes. We started with (4 x) 3, and the work we analyzed led to 43x3. So the expressions are equivalent.
Display (xy) n = x n y n .
If x and y are any numbers and n is a positive whole number, then (xy)n = xnyn.
Have students turn and talk about this property by asking the following questions:
• Does this property accurately describe what you noticed about the expression (4x)3? How?
• Do you think this property is useful? How?
Then show the example (7 · 6) 2 = 7 2 · 6 2 .
Language Support
This is the first use of the term verify in Eureka Math2. Consider previewing the meaning of the term before students see it in print. Facilitate a class discussion focusing on students’ prior experience with checking their work or making sure something is true.
Teacher Note
Some students may interpret this property as distributing the exponent. Make clear this is not what is happening. The distributive property involves multiplication over addition or subtraction. When we raise products to powers, we are finding an easier way to write an expression involving repeated multiplication.
Instruct students to add this property to the next blank row in their Properties and Definitions of Exponents graphic organizer from lesson 5. Add the new property to the Properties and Definitions of Exponents anchor chart for this topic.
Properties of Exponents
Description
Description
Definitions of Exponents
Definition(s) Example(s)
Raising Quotients to Powers
Students guess and verify equivalent expressions of the form ( x y )n .
Have students work individually on problem 10. Allow time for productive struggle.
10. Consider the expression ( x y ) 4, where y ≠ 0. Use what you know about exponents and their properties to write an equivalent expression.
When the struggle is no longer productive, have students turn and talk about how they can verify that the expression they wrote for problem 10 is equivalent to (x _ y )4 .
Display (x y )1. Show how the number of factors increases as the exponent increases up to 4 so students’ thinking about problem 10 is confirmed.
Facilitate a class discussion by using the following questions.
What do you notice about the expression as we include more factors of ( x _ y)?
I notice that each time the exponent increases by 1, one more factor of (x y) appears in the expression. This results in one more factor of x in the numerator and one more factor of y in the denominator each time.
If the base ( x _ y) has an exponent of 7, what equivalent expression do you expect?
I expect the equivalent expression to be x 7 y 7 .
If the base ( x _ y) has an exponent of 8, what equivalent expression do you expect?
I expect the equivalent expression to be x 8 y 8 .
If the base ( x _ y) has an exponent of 9, what equivalent expression do you expect?
I expect the equivalent expression to be x 9 y 9 .
If the base ( x _ y) has an exponent of n, what equivalent expression do you expect?
I expect the equivalent expression to be x n y n .
Display (x y )n = x n y n .
Why can y not be equal to zero in this equation?
The variable y cannot be equal to zero because division by 0 is undefined.
Show the example (3 _ 4 )
Have students complete problem 11 with a partner. Encourage students to use a method of their choice to write an equivalent expression.
11. Assume 3 4 is under the mustard spot. What equivalent expression should Vic write?
Confirm the answer.
Debrief 5 min
Objective: Encounter and apply properties of exponents, including raising powers to powers, raising products to powers, and raising quotients to powers.
Facilitate a class discussion by asking the following questions. Encourage students to restate or build on one another’s responses.
How can we write an equivalent expression for (x2)4?
We can write (x 2) 4 as x 4 · 2 because we can think of the power inside the parentheses as 2 factors of x, and the exponent outside the parentheses tells us we have 4 factors of x 2 . This means we have 4 factors of 2 factors of x, which is 4 ⋅ 2, or 8, factors of x.
Teacher Note
The power of a quotient, (
can be viewed as an extension of the definition of a power. So it is not included in the Properties and Definitions of Exponents graphic organizer.
How can we write an equivalent expression for (5x)4?
We can write (5 x) 4 as 5 4x 4. Because (5 x) 4 has 4 factors of 5x, we can rearrange the expression so all the 5’s are together and all the x’s are together. Then we have 4 factors of 5 multiplied by 4 factors of x, or 5 4 x 4 .
How can we write an equivalent expression for ( x _ 5 )4?
The expression ( x _ 5)4 represents 4 factors of x _ 5 . We can also write this as x 4 __ 54 , which is the same as a fraction with 4 factors of x in the numerator and 4 factors of 5 in the denominator. So we can write ( x _ 5)4 as x 4 __ 54 .
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
More Properties of Exponents
In this lesson, we
• established two new properties of exponents.
• used the properties of exponents to write equivalent expressions.
Product of Powers with Like Bases Property
x is any number m and n are positive whole numbers when
Power of a Power Property
x is any number m and n are positive whole numbers when (x m)n = x m n
Power of a Product Property
x and y are any numbers n is a positive whole number when (
Both the numerator, 9 2 , and the denominator, 10 are raised to the sixth power.
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
Do you agree? Explain your reasoning. I disagree with Jonas because the expression 5n 2 is being raised to the third power. That means there will be a product of 3 factors of
For problems 1–10, apply the properties of exponents to write an equivalent expression. Assume m is nonzero.
Which expressions are equal to 8 24? Choose all that apply.
17. The edge of a cube measures 4 n 2 inches. What is the volume of the cube? 4
18. The side length of a square measures 3 x
y 5 meters. What is the area of the square? Assume y is nonzero.
Which expression is equivalent to −11 − 7?
Making Sense of the Exponent of 0
Define x 0 by confirming that the definition upholds the properties of exponents.
Evaluate powers with an exponent of 0.
Lesson at a Glance
In this lesson, students begin by making an educated guess about the value of a power with an exponent of 0. Students build from a previous lesson to verify that their prediction preserves the property x m · x n = x m+n. Students determine the definition of x 0 and use this definition to rewrite expressions containing a power with an exponent of 0. This lesson formally defines the term exponent of 0 .
Key Questions
• What value does x 0 have to be to uphold the properties of exponents?
• How does the definition of x 0 uphold the property x m · x n = x m+n?
Achievement Descriptors
8.Mod1.AD5 Apply the properties of integer exponents to generate equivalent numerical expressions. (NY-8.EE.1)
8.Mod1.AD8 Approximate and write very large and very small numbers in scientific notation. (NY-8.EE.3)
Agenda
Fluency
Launch 10 min
Learn 25 min
• Defining the Exponent of 0
• Using the Exponent of 0
Land 10 min
Materials
Teacher
• Construction paper (4)
• Markers (3)
• Tape Students
• Properties and Definitions of Exponents (in the student book from lesson 5)
Lesson Preparation
• Properties and Definitions of Exponents anchor chart (from lesson 5)
Fluency
Apply Properties of Exponents
Students apply properties of exponents to prepare for evaluating expressions containing an exponent of 0.
Directions: Apply the properties of exponents to write an equivalent expression.
Teacher Note
If students express the exponent as an addition expression, encourage them to express the exponent as a sum.
Teacher Note
Instead of this lesson’s Fluency, consider administering the Apply Properties of Positive Exponents Sprint. Directions for administration can be found in the Fluency resource.
Launch
10
Students make a prediction about the value of a power with an exponent of 0.
Draw students’ attention to the Properties and Definitions of Exponents anchor chart.
Our last two lessons focused on these three properties of exponents. We’ve defined these properties to work with positive integer exponents. What are some numbers we haven’t yet worked with as exponents?
We haven’t worked with exponents that are negative, zero, or fractions.
Today, we are going to learn about the exponent of 0. Think for a moment about what the value of 10 0 might be.
Give students the opportunity to think–pair–share about a possible value.
Call on students to share their predictions with the class. Write each prediction on a separate piece of construction paper. The papers will be used as the signs for the Take a Stand routine. Once all predictions have been recorded, post the signs around the room.
Introduce the Take a Stand routine to the class. Draw students’ attention to the signs labeled with their predictions posted in the classroom.
Present the question “What is the value of 10 0 ?” Then invite students to stand beside the sign that best describes their thinking.
When all students are standing near a sign, allow groups to discuss the reasons why they chose that sign. Then call on each group to share reasons for their selection. Invite students who change their minds during the discussion to join a different group.
Have students return to their seats. Do not reveal that 10 0 is equal to 1. Students continue their exploration in the next segment.
Until this point, we have been working with only positive integer exponents. In today’s lesson, we will define x 0 and use it to write equivalent expressions.
Teacher Note
Remember that students have not yet learned the significance of the exponent of 0, so there will likely be a range of answers within the class. This range of answers will provide a rich discussion to introduce the concept.
Language Support
During the Take a Stand routine, direct students to use the Agree or Disagree section of the Talking Tool as a support for discussing the reasons why they chose a specific sign.
Learn
Defining the Exponent of 0
Students test their predictions about the value of an expression with an exponent of 0.
Once students return to their seats, engage them in a discussion about the exponent of 0 by using the following prompts.
Is the expression 10 0 challenging to think about when compared with an expression like 10 3 ? Why?
We know 10 3 is 10 · 10 · 10, but we can’t picture 10 0 in the same way.
When we determine the definition of a power with an exponent of 0, we want to make sure it works with our existing properties. That way, we don’t need to learn a new set of properties.
At the beginning of class, you were asked to make predictions about the value of 10 0 . Let’s test some predictions to see whether they uphold the properties of exponents.
Direct students to problem 1. Model reasoning that shows that to uphold the product of powers with like bases property, 10 0 ≠ 10.
1. Could 10 0 = 10? Use the product 10 0 · 10 3 to show whether it upholds the property x m · x n = x m+n .
By the property, we would expect this relationship: 10 0 · 10 3 = 10 0+3 = 10 3 If 10 0 = 10, then: 10 0 · 10 3 = 10 · 10 3 = 10 · 10 · 10 · 10 = 10 4 The property does not hold if 10 0 = 10 because 10 3 does not equal 10 4. So 100 ≠ 10.
Students may immediately recognize that 10 0 ≠ 10 because they know that 10 1 = 10. It is still important to model the approach given in problem 1 so students can replicate it in the upcoming problems.
We’ve determined that 10 0 does not equal 10 because the product of powers property is not upheld when we define 10 0 as 10. Now we’ll explore the other values from our opening activity.
Ask students to complete problems 2 and 3 in pairs. Circulate to check student problem solving and look for a parallel process to problem 1. If time allows, ask students to address any other predictions given at the beginning of class for 10 0 .
2. Could 100 = 0? Use the product 10 0 · 10 3 to show whether it upholds the property
x m · x n = x m+n .
By the property, we would expect this relationship:
If 10 0 = 0, then:
The property does not hold if 10 0 = 0 because
.
3. Could 10 0 = 1? Use the product 10 0 · 10 3 to show whether it upholds the property
x m · x n = x m+n .
By the property, we would expect this relationship:
If 10 0 = 1, then:
Promoting the Standards for Mathematical Practice
When students apply the properties of exponents to numerical expressions with powers of 10 to show that 10 0 = 1, they are constructing a viable argument, which they later generalize to establish the definition x 0 = 1 (MP3).
Ask the following questions to promote MP3:
• Is 10 0 = 1 (or 10 0 = 10, 10 0 = 0) a guess or do you know for sure? How do you know for sure?
• Can you find a situation where x 0 = 1 is not true?
The property holds if 10
Invite students to display and share their work.
We just found that one of our properties is upheld when we define 10 0 as 1. We can see how that definition works when we look at numbers in expanded form.
Have students work on problems 4–7 individually. If students need assistance with writing the expanded form of a number by using exponential notation for problems 6 and 7, refer them to the form of the numbers given in problems 4 and 5. Allow students to compare answers with a partner.
In problems 4 and 5, the expanded form of a number written with exponential notation is given. Write the number in standard form.
4. 4 × 10 2 + 3 × 10 1 + 6 × 10 0 4 × 10 2 + 3 ×
In problems 6 and 7, the standard form of a number is given. Write the number in expanded form by using exponential notation.
Now that we have established and applied the definition 10 0 is equal to 1, I wonder whether this definition works for any base.
Repeat the reasoning used to define 10 0 = 1 by using the expression x 0 · x n. Have students record the steps in problem 8.
8. Apply the properties of exponents to write an equivalent expression for x 0 · x n. Then show that the definition x 0 = 1 upholds the property x m · x n = x m+n for any nonzero x.
The property of exponents states that:
If x 0 = 1, then the property is upheld:
The definition of a power with an exponent of 0 is x 0 = 1 for any nonzero x.
Instruct students to add this definition to the next blank row in their Properties and Definitions of Exponents graphic organizer from lesson 5. Add the new definition to the Properties and Definitions of Exponents anchor chart for this topic.
Properties of Exponents
Description Property
Product of powers with like bases
Teacher Note
The definition x 0 = 1 applies for any nonzero x. Students may wonder what happens when x is equal to 0. The number 0 0 is considered indeterminate, which means its value cannot be determined. To help students see why, show them the following patterns and ask them what they think the value of 0 0 should be based on the patterns:
Definitions of Exponents
Description
Base with an exponent of 0 x 0 = 1 x is nonzero (10) 0 = 1
Using the Exponent of 0
Students apply the definition x 0 = 1 in exponential expressions.
Direct students to complete problems 9–12 in pairs.
Students will likely answer that the value should be 1 based on the first table and 0 based on the second table. This shows students that two valid patterns can result in two different possible values for 0 0, which is why we cannot determine its value.
Determine the value of n in problems 9–12.
The value of n is 4.
The value of n is 0.
The value of n is 1.
UDL: Engagement
Offer feedback that emphasizes the importance of finding and understanding errors. For example, discuss specific actionable steps that students might take the next time they encounter a similar problem to help avoid making errors. Consider sharing questions that students may ask themselves as they prepare to solve a similar problem.
• How have I solved problems like this before when I needed to apply the properties of exponents?
• How have I solved problems like this before when there is an exponent of 0?
Review the problems and address any misconceptions as a class. Students may say n is 0 in problem 9 if they apply the property incorrectly and multiply the exponents. Students also may answer n is 0 in problem 12 with the misconception that 5 0 = 0.
We’ve shown that this definition x 0 = 1 upholds the power of products with like bases property, but it actually works for all properties of exponents. Now we can use the properties of exponents for positive whole-number exponents and exponents of 0.
Reinforce this new definition with a Whiteboard Exchange. Present the following problems and ask students to rewrite each expression by using the exponent properties and definition of the exponent of 0. Students independently solve on the white side of their whiteboards. Students turn their whiteboards over, red side up, when finished (“Red when Ready”). Students show their work, and the teacher gives quick, individual feedback.
Teacher Note
After students have determined that 10 0 = 1 upholds the property
, consider showing students that this definition supports the other properties in the following ways:
After the Whiteboard Exchange, display the expression (10 0) 3. Ask students to record on their whiteboards how they thought about rewriting this expression, showing as many steps to illustrate their thinking as needed. Then have students show their whiteboards.
Teacher Note
Consider adding examples that include exponents of 0 to the Properties of Exponents table in the graphic organizer.
Take note of the various strategies used. Invite students to share their thinking with the class.
Expect that some students will apply the definition x 0 = 1 first, leading to 1 3, whereas others will apply a property first, leading to 10 3 · 0. Validate all acceptable strategies presented.
Display the expression ((−4) 5) 0 and ask students to record their thinking on their whiteboards. Invite students who use various strategies to share their thinking with the class. Emphasize that in this case, it makes more sense to apply properties of exponents and the definition of the exponent of 0 as a first approach rather than using the order of operations. Examples of each approach are given.
• Properties of exponents and definition of the exponent of 0: ((−4) 5) 0 = (−4) 5 · 0 = (−4) 0 = 1
• Order of operations: ((−4) 5) 0 = ((−4)(−4)(−4)(−4)(−4)) 0 = (1024) 0 = 1
Continue the Whiteboard Exchange with the following problems:
Debrief solutions to any problems in the Exchange for which multiple student answers were incorrect.
Land
Debrief 5 min
Objectives: Define x 0 by confirming that the definition upholds the properties of exponents.
Evaluate powers with an exponent of 0.
Pair students for a turn and talk. Ask one partner to explain why 10 0 does not equal 10. Have the other partner explain why 10 0 does not equal 0. Pose the following questions to the class and encourage students to rephrase or build upon their partner’s thinking.
Is 100 equal to 10? Why?
If 10 0 is equal to 10, the property x m · x n = x m+n does not work. For example, 10 0 · 10 3 should equal 10 3 by the property, but if 10 0 is 10, 10 0 · 10 3 would equal 10 4, not 10 3 .
Is 100 equal to 0? Why?
If 10 0 = 0, the property x m · x n = x m+n does not work. For example, 10 0 · 10 3 should equal 10 3 by the property, but if 10 0 is 0, 10 0 · 10 3 would equal 0, not 10 3 .
What value does x 0 have to be to uphold the properties of exponents?
The value of x 0 for any nonzero x has to be 1 to uphold the properties of exponents.
How does the definition of x 0 uphold the property x m · x n = x m+n?
The definition of x 0 upholds the property x m · x n = x m+n, because x 0 · x n should be equal to x 0+n, or x n. This is true only when x 0 is 1.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
Power of a Product Property
Making Sense of the Exponent of 0
In this lesson, we
• made predictions about the value of a power with an exponent of 0
• used properties of exponents to test which prediction was true about the value of a power with an exponent of 0
• learned the definition of the exponent of 0.
Terminology
A power with an
• applied the definition of the exponent of 0 to write equivalent expressions.
Product of Powers with Like Bases Property
is any number
are whole numbers when
Power of a Power Property
Definition of the Exponent of 0
Examples
For problems 1–4, apply the definition of the exponent of 0 to write an equivalent expression. Assume all variables are nonzero.
EUREKA
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
For problems 1–6, apply the definition of the exponent of 0 to write an equivalent expression. Assume all variables are nonzero.
For problems 7–18, apply the properties of exponents and the definition of the exponent of 0 to write an equivalent expression. Assume all variables are nonzero.
25. Choose the expression with a value of 9804
A. 9 × 10 3 + 8 × 10 2 + 4 × 10 1
B. 9 × 10 4 + 8 × 10 3 + 4 × 10 1
C. 9 × 10 4 + 8 × 10 3 + 4 × 10 0
D. 9 × 10 3 + 8 × 10 2 + 4 × 10 0
26. Find the value of
632 B. 6032 C. 6320 D. 60,302 E. 60,300
27. So-hee states that 4 0 = 4. Do you agree? Explain your reasoning by using the properties of exponents.
I disagree because 4 0 = 4 does not support the property x m x n = x m + n . If 4 0 = 4, then
4 0 ⋅ 4 2 would be 4 ⋅ 16, or 64 This does not support the property 4 0 ⋅ 4 2 = 4 0+2 = 4 2 because 4 2 is 16, not 64
For problems 28–31, add or subtract.
For problems 32–35, apply the properties of exponents to write an equivalent expression.
Which expression is equivalent to 75? Choose all that apply.
Making Sense of Integer Exponents
Explore and develop an understanding of negative exponents. Write equivalent expressions given an expression of the form x m x n
Lesson at a Glance
1. Use the definition of negative exponents to write
5
2. Use the definition of negative exponents to write 1 g−3 with a positive exponent. Assume g is nonzero. g
3. Kabir and Yu Yan each write an equivalent expression for
The table shows their work.
In this lesson, students define the meaning of negative exponents. They make conjectures about negative exponents that uphold the properties of exponents and relate negative exponents to multiplicative inverses. Then students analyze different strategies to write equivalent expressions of the form x m x n for any nonzero x. This lesson formalizes the term negative exponent.
Key Questions
• What does x n represent?
• How can we use the properties and definitions of exponents to write an equivalent expression for x m x n ?
Achievement Descriptor
8.Mod1.AD5 Apply the properties of integer exponents to generate equivalent numerical expressions. (NY-8.EE.1)
Edition: Grade 8, Module 1, Topic B, Lesson
Agenda
Fluency
Launch 5 min
Learn 30 min
• Integer Exponents
• Quotients of Powers
Land 10 min
Materials
Teacher
• Markers (3)
Students
• Properties and Definitions of Exponents (in the student book from lesson 5)
Lesson Preparation
• Properties and Definitions of Exponents anchor chart (from lesson 5)
Fluency
Find the Multiplicative Inverse
Students find the multiplicative inverse to prepare for defining negative exponents.
Directions: Write the number that completes the equation.
Launch
Students explore negative exponents.
Probe student thinking about negative exponents by using the following prompt.
In the last lesson, we used the properties of exponents to give meaning to the exponent of 0. We know the properties of exponents work for exponents that are positive whole numbers and for 0. What other types of numbers do you think can be used as an exponent?
Students may offer responses such as negative numbers, fractions, or decimals. If students mention fractions or decimals, let them know that they will discover the meaning of a fractional exponent in Algebra I. If students do not mention negative numbers, suggest that possibility to the class.
We want the properties of exponents to remain true when using negative exponents. Work with your partner and put your skills together to determine the meaning of a negative exponent. Assume that the properties of exponents remain true for negative exponents.
Direct students to problem 1. As students work in pairs, allow time for exploration, discussion, and productive struggle. Circulate and ask the following questions:
• How can you use the properties of exponents to write an equivalent expression for 10 4 · 10 −4 ?
• How can you use a known definition to write an equivalent expression for 10 0 ?
• If the product of two numbers is 1, what does that tell you about the two numbers?
• Write 10 4 in standard form. What is the multiplicative inverse for 10,000?
• How can we write the multiplicative inverse for 10 4 ?
1. Use the properties of exponents to find the value of 10 4 · 10 −4 .
When most students are finished or the struggle is no longer productive, ask a few pairs to share their answer for problem 1. Encourage students to defend their conclusion and push them to refine or correct their answer. Then use the following prompt to transition to the next segment.
Today, we will explore how certain types of exponents are related to multiplicative inverses. We will also apply the properties of exponents we have learned so far to find quotients involving exponents.
Learn
Integer Exponents
Students define negative exponents.
Engage the class in a discussion about negative exponents by using the following prompts.
If we want the properties of exponents to remain true for negative exponents, then what should be the value of 10 4 · 10 −4 ? Why?
The value of 10 4 · 10 −4 should be 1. By using the properties of exponents, 10 4 · 10 −4 should be 10 4+(−4), which is 10 0, or 1.
What is the value of 10 4 · 1 ___ 10 4 ?
The value of 104 · 1 10 4 is 1.
What do we know about two factors when their product is 1?
Two factors must be multiplicative inverses, or reciprocals, of one another.
If 10 4 · 1 ___ 10 4 = 1, then 10 4 and 1 ___ 10 4 have what relationship?
The numbers 10 4 and 1 ___ 10 4 are multiplicative inverses of one another.
If 10 4 · 10 −4 = 1, then 10 4 and 10 −4 have what relationship?
The numbers 10 4 and 10 −4 are multiplicative inverses of one another.
So if 1 ___ 10 4 and 10 −4 are both multiplicative inverses of 10 4, what does that tell us about 10 −4 and 1 ___ 10 4 ? Why?
It tells us that we can write 10 −4 as 1 10 4 because both numbers are multiplicative inverses of 10 4 .
If we want the properties of exponents to be true for negative exponents, then 10 −4 should be equivalent to 1 ___ 10 4 .
Ask students to complete problems 2 and 3 individually.
For problems 2 and 3, suppose the properties of exponents remain true for negative exponents.
2. Are 10 4 and 10 −4 multiplicative inverses? Why?
Yes, because we can use the properties of exponents to show that the product of 10 4 and 10 −4 is 1. 10 4 · 10 −4 = 10 4+(−4) = 10 0 = 1
3. Write an equivalent expression for 10 −4 .
When most students are finished, display the expression x 4 · x −4. Use the following prompts to help students think–pair–share about negative exponents.
Let’s make a general rule for negative exponents with bases other than 10. Consider a base of x. If we want the properties of exponents to work for negative exponents, then what should be the value of x 4 · x −4? Why?
The value of x 4 · x −4 should be 1. By using the properties of exponents, x 4 · x −4 should be x 4+(−4), which is x 0, or 1.
Because the product is 1, what is the relationship between x 4 and x −4 ?
The expressions are multiplicative inverses.
Teacher Note
Ask students who have the misconception that 10 −4 is equal to −10 4 to evaluate 104 · (−10 4). They find that 10 4 · (−10 4) is equal to −10 8, which is not 1.
How can you write x −4 with a positive exponent? Explain your reasoning. I can write x −4 with a positive exponent as 1 x 4 because both x −4 and 1 x 4 are multiplicative inverses of x 4 .
So we can use a positive exponent to write x −4 as 1 __ x 4 because both expressions are multiplicative inverses of x 4 .
Display x n · x n. As a class, use the prompts to help formulate the definition for a negative exponent.
What is the relationship between x n and x n?
The expressions are multiplicative inverses of one another.
How can you write x n with a positive exponent?
1 x n
We can define a negative exponent for any nonzero x and any integer n as x n = 1 __ x n . Therefore, x n is equivalent to 1 __ x n , the multiplicative inverse, or reciprocal, of x n .
Why must x be nonzero?
The value of x cannot be 0 because x is in the denominator and we cannot divide by 0.
Have students add this definition to the next blank row in their Properties and Definitions of Exponents graphic organizer from lesson 5. Add the new definition to the Properties and Definitions of Exponents anchor chart for this topic.
Teacher Note
Consider reminding students that the properties of exponents exclude special cases such as 0 0 and 0 n. Lesson 7 discusses the expression 00.
Teacher Note
After students have determined that 10 4 = 1 10 4 upholds the property x
+
, consider showing students that this definition supports the other properties in the following ways.
• Power of a power
Consider adding examples that include negative exponents to the Properties of Exponents table in the graphic organizer.
Properties of Exponents
Product of powers with like bases
Definitions of Exponents
Now, our first three properties work for exponents that are positive whole numbers, zero, and negative whole numbers. In other words, they work for all integer exponents.
Have students complete problems 4–9 with a partner.
For problems 4–9, use the definition of negative exponents to write an equivalent expression.
Assume that x is nonzero.
Confirm answers as a class. Direct students to problem 10 and use the following prompts to guide the class through the problem.
What do you notice about problems 10 and 11?
I notice there are negative exponents in the denominators of fractions.
Fractions are numbers, but a fraction can also be interpreted as what operation?
Fractions can also be interpreted as division.
So I can say problem 10 as 1 divided by 10 −2 .
Display 1 ÷ 10 −2. Show the equivalent expressions as students answer.
How can we rewrite 10 −2 as an expression with positive exponents?
evaluate
which is
. So 1 ____ 10 −2 can be written as the reciprocal of 1 ___ 10 2 , which is 10 2 .
Have students complete problem 11 with a partner. Students may show all the steps used in problem 10, or they may recognize that 1 x n = x n .
Promoting the Standards for Mathematical Practice
Students are attending to precision when they write an equivalent expression for a fraction that has a negative exponent in the denominator. They write the fraction as a division expression and then use the definition of negative exponents to write an equivalent expression (MP6).
Ask the following questions to promote MP6:
• How can we write the division expression 1 ÷ 10−2 with a positive exponent?
• How are you using the definition of negative exponents when writing an equivalent expression for 1 10 2 ?
For problems 10 and 11, use the definition of negative exponents to write an equivalent expression with positive exponents.
Teacher Note
Some students may confuse negative exponents with negative bases. If this happens, have students highlight the base in one color and the exponent in another color in each expression. Remind students that the negative of the base represents the opposite of the base. The negative exponent represents the multiplicative inverse.
Confirm the answer for problem 11. Ask students to add the definition 1 x n = x n to their Properties and Definitions of Exponents graphic organizer. Write this additional definition for a base with a negative exponent on the Properties and Definitions of Exponents anchor chart.
Definitions of Exponents
Quotients of Powers
Students evaluate quotients of powers by using the properties and definitions of exponents.
Direct students to Quotients of Powers. Have them study So-hee’s work before engaging in a class discussion.
So-hee applies the properties of exponents to write an equivalent expression for 10 3 ___ 10 7 . So-hee’s work is shown.
Use the Five Framing Questions routine to invite students to analyze So-hee’s strategy. Encourage students to add to their classmates’ responses.
Notice and Wonder
What do you notice about this work?
I notice that So-hee wrote 10 7 as 10 3 · 10 4 .
I notice that she made a simpler problem by getting a 10 3 in the numerator and denominator.
I notice that 10 3 10 3 is 1.
I notice that the difference between the exponents of the denominator and the numerator is the exponent of the answer.
From your observations, what do you wonder?
I wonder why she wrote 10 7 as 10 3 · 10 4 .
I wonder why she skipped the step that shows 10 3 ___ 10 3 is 1.
I wonder whether she could have used a negative exponent and wrote 10 −7 because 10 7 is in the denominator.
Organize
What steps did So-hee take? How do you know?
So-hee noticed a 10 3 in the numerator and decided to write the denominator as a product where one factor is 103. She did this because 10 3 ___ 10 3 is 1. When she wrote the denominator by using 10 3, she had to include the factor of 10 4 because 10 3 · 10
is
3+4 , or 10 7. She got the answer of 1 10 4 because 1 · 1
So-hee could have written the denominator as 10 5 · 10 2. Why didn’t she?
So-hee didn’t write the denominator as 10 5 · 10 2 because she wanted to make a fraction that is equal to 1 by using the factor of 10 3 in the numerator. So she wrote the denominator as 10 3 · 10 4 .
Have students complete problem 12 individually or in pairs. Circulate as students work. If students struggle, suggest that they write the fraction as a whole number times a unit fraction. Students may also reference problem 4 for support. Identify students to share their work with the class.
12. Apply the properties of exponents and the definition of negative exponents to verify So-hee’s answer.
Advance the discussion to focus on the definition of negative exponents by encouraging student thinking that makes connections between multiplicative inverses and negative exponents. Invite identified students to share their work with the class.
Reveal
Let’s focus on the definition of negative exponents. Where do you see that in your work?
My works shows the definition of negative exponents when I wrote 1 10 7 as 10 −7. Then I added the exponents of 3 and −7 to get −4. So my final answer is 10 −4, or 1 ___ 10 4 .
In So-hee’s work, I noticed the exponent of the answer was the difference between the exponents of the denominator and the numerator. Since 3 minus 7 is −4, the answer is 10 −4, or 1 10 4 .
Distill
Does using the definition of negative exponents make a difference in this work?
The work looks different, but we still get the same answer. We use properties of exponents in this work and in So-hee’s work.
When using negative exponents, I can write 10 3 10 7 as 10 3 ⋅ 10 −7. I did it this way because I used the definition of negative exponents rather than writing the same power of 10 in the numerator and denominator.
I can subtract the exponents of the numerator and the denominator and use the definition of a negative exponent to write the final answer of 10 −4 with a positive exponent as 1 10 4 .
Know
How is using the properties of exponents and the definition of negative exponents helpful for writing an equivalent expression when compared with writing out all the factors of 10 as we did in topic A?
We can write equivalent expressions by using the properties and definitions of exponents much faster than when we had to write out all the factors of 10 in topic A.
Have students complete problems 13–15 with a partner. While you circulate, ask students the following questions to advance their thinking.
• In a quotient of powers with like bases, if the exponent is greater in the numerator, will the answer have a positive exponent in the numerator? Why?
UDL: Action & Expression
Provide sentence starters to support students in sharing their thoughts and ideas about how they use negative exponents in their work.
• I know …
• My work shows …
• I did it this way because …
• I think _____ because …
Teacher Note
The quotient of powers with like bases, x m ___ x n = x m n , is an application of the definition of a negative exponent and the product of powers with like bases property.
So it is not included in the Properties and Definitions of Exponents graphic organizer.
• In a quotient of powers with like bases, if the exponent is greater in the denominator, will the answer have a positive exponent in the denominator? Why?
• Can you subtract the exponents in a quotient of powers with like bases? Why?
Identify students who use different strategies, including So-hee’s strategy and negative exponents, to share their work with the class.
For problems 13–15, apply the properties and definitions of exponents to write an equivalent expression with positive exponents. Assume all variables are nonzero.
Differentiation: Support
A common student error is to apply the properties of exponents to the coefficient of the expressions. For example, students may rewrite 8 x 5
as 4 x 2 because they apply the exponent property
to the coefficients: 8 + (−4) = 4. For extra support, consider having students complete the following problems before completing problems 13–15.
Confirm answers and invite students to share their solution strategies.
Land
Debrief 5 min
Objectives: Explore and develop an understanding of negative exponents.
Write equivalent expressions given an expression of the form x m x n .
Use the following prompts to facilitate a class discussion. Encourage students to restate or build on one another’s responses.
What was our goal today? How did we accomplish it?
We extended the properties of exponents from whole-number exponents to integer exponents. We created a definition for negative exponents where the properties of exponents still work. We used the properties of exponents to write 10 4 ⋅ 10 −4 as 10 0 , which is 1. Knowing that the product of 10 4 and 10 −4 is 1 helped us see that 10 −4 is the reciprocal of 10 4, which is 1 10 4 .
What does x n represent?
The power x n represents the multiplicative inverse of x n, which is 1 x n .
How can we use the properties and definitions of exponents to write an equivalent expression for x 3 __ x 8 ?
We can use the properties and definitions of exponents to write x 3 x 8 as x 3 ⋅ x −8, which is x −5, or 1 x 5 .
We can use the properties of exponents to write x 3 __ x 8 as x 3 _____ x 3
which
Exit
Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
Power of a Product Property
Making Sense of Integer Exponents
In this lesson, we
• related negative exponents to multiplicative inverses.
• learned the definition of a negative exponent.
• applied the definition of a negative exponent to write equivalent expressions.
Product of Powers with Like Bases Property
Definition of the Exponent of 0
Definition of Negative Exponents
of a Power Property
For problems 2–4, use the definition of negative exponents to write an equivalent expression. Assume p is nonzero.
A fraction represents a division statement.
5. Apply the properties and definitions of exponents to write an equivalent expression with positive exponents. Assume r is nonzero.
Solve by using either strategy.
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
For problems 1–9, use the definition of negative exponents to write an equivalent expression. Assume all variables are nonzero.
For problems 10–12, use the definition of negative exponents to write an equivalent expression.
Maya says
is the same as
5. Do you agree? Explain your reasoning. I disagree with Maya. The value of
Which one does not belong? Circle your answer and explain your reasoning.
For problems 13–19, apply the properties and definitions of exponents to write an equivalent expression with positive exponents. Assume all variables are nonzero.
Order the values from least to greatest.
EUREKA MATH2 New York Next Gen
For problems 23–26, add or subtract.
27. The approximate land area of Jamaica is 4 × 10 3 square miles. The approximate land area of Brazil is 3 × 10 6 square miles. The land area of Brazil is about how many times as large as the land area of Jamaica?
The land area of Brazil is about 750 times as large as the land area of Jamaica.
For problems 28 and 29, find the sum or difference.
28. 4.073 + 8.607 + 2.46
Writing Equivalent Expressions
Write equivalent expressions by using all the properties and definitions of exponents.
Lesson at a Glance
In this culminating lesson on exponent properties and definitions, students solve problems by applying multiple properties and definitions to write equivalent expressions. Students engage with the content through a relay race-like activity. Then students examine more complicated expressions and develop more efficient ways to simplify them. This lesson formally defines the phrase simplify an exponential expression
Key Question
• What are the characteristics of a simplified expression involving exponents?
Achievement Descriptor
8.Mod1.AD5 Apply the properties of integer exponents to generate equivalent numerical expressions. (NY-8.EE.1)
Agenda
Fluency
Launch 5 min
Learn 30 min
• Making Unlike Bases Alike
• Power Rounds
• Finding a More Efficient Way
Land 10 min
Materials
Teacher
• Construction paper (6)
• Markers (3)
• Tape Students
• Power Rounds cards
• Be Efficient cards
Lesson Preparation
• Prepare six signs labeled Door A, Door B, Door C, 2 6 dollars, 4 3 dollars, and 8 2 dollars. Post the three door signs around the room. Then post each dollar amount sign below its corresponding door sign with the writing facing the wall.
• Copy and cut out the Power Rounds cards (in Teach book). Fold each card along the dotted line with the print on the outside. Prepare enough sets of Rounds 1–4 for 1 per student group.
• Copy and cut out the Be Efficient cards (in Teach book). Prepare enough sets for 1 per student group.
Fluency
Write Expressions by Using a Negative Exponent
Students write each expression as a power with a negative exponent to prepare for using all the properties of exponents.
Directions: Write each expression as a power with a negative exponent. Assume all variables are nonzero.
Launch
Students examine equal powers with different bases.
Introduce students to the Mystery Door activity by drawing their attention to the posted signs in the classroom labeled Door A, Door B, and Door C. Each door leads to a theoretical cash prize. Tell students that once the dollar amounts are revealed, they will have 10 seconds to make a choice on which cash prize they prefer. Ask one student to stand next to each door sign. On your signal, have all three students reveal the dollar amount for each door by turning over the dollar amount sign.
After 10 seconds, ask students to get up and stand beside the sign that represents the cash prize they would choose. When all students are standing near a sign, call on an individual from each group to share reasons for their choice. Then give a student from each group a marker.
How many dollars did you win? Write the value under the exponential value on your sign.
Give groups 30 seconds to work together to determine the value of their prize and write the value on their sign.
Ask students to return to their seats and study the value on each sign.
What do you notice? What do you wonder?
I notice that all the values are 64 dollars. I wonder why they are all equal even though the bases are different.
Today, we will write equivalent expressions by using different bases and by applying properties of exponents.
Teacher Note
The goal of the activity is to discuss possible assumptions that students might make about the value of the expressions without having the opportunity to evaluate each expression. Some students may expect that the largest base is the largest number.
Learn Making Unlike Bases Alike
Students write numbers as powers with different bases to simplify expressions.
Direct students to problems 1 and 2 to answer independently. When most students have finished, ask them to compare answers with a partner.
1. Write each expression as a single power with a base of 2.
2. Write each expression as a single power with a base of 5.
Review answers as a class by inviting students to share their solutions and discuss their approaches.
Direct students to problems 3–6. Read the directions aloud, and ask the class to briefly analyze the problems.
What do you notice about problems 3–6?
I notice the problems are all products.
I notice the bases in each expression are different.
I notice the directions ask me to write each expression as a single power.
What do you wonder?
I wonder how I would write each expression as a single power when the given expression has different bases.
I wonder how to write 4 · 2 9 as a single power.
How can you use your answer for problem 1(a) to help you write 4 · 2 9 as a single power?
I can write 4 as 2 2 and use the properties of exponents to write 2 2 · 2 9 as 2 11 .
Have students record their answer for problem 3. Then ask them to work with a partner on problems 4–6.
For problems 3–6, use the properties and definitions of exponents to write each expression as a single power.
Review answers as a class. Consider asking students to share which problems were the most challenging and why.
Power Rounds
Students apply all the properties and definitions of exponents to write equivalent expressions.
Arrange students in groups of three and assign each group member a letter A through C. Distribute a set of Power Rounds cards to each group and instruct students to keep the round numbers faceup.
Explain the following instructions for the Power Rounds.
• Find the Round 1 card for your assigned problem. Keep it facedown until I say “Go.”
• Complete your assigned problem on your card.
• As a group, make sure all answers on the cards are correct.
• Raise your hand to signal that your team is ready to have your answers checked.
• When all groups are ready, we will check the answers as a class.
• If any answer is incorrect, work as a group for 1 minute to correct the errors.
For Round 1, display the following directions: Use the properties of exponents to write each expression as a single power.
UDL: Action & Expression
Post written directions of the Power Rounds activity as a reference for the class. Consider creating a sample set of simpler problems for the purpose of modeling the process. Before beginning the activity, have one group use the sample set to model a round for the class.
Differentiation: Challenge
As an extra scaffold for students who have mastered the concept, consider offering these additional problems:
• Write the answer to problem 4 by using a different base.
3 14
• Write 9 · 25 · 27 · 125 as a product of powers with prime bases.
3 5 · 5 5
• Write 7 11 · 121 6 · 11 3 · 1 49 2 as a product of powers with prime bases.
7 7 · 11 15
Use Round 1 as an example, and repeat the same steps for Rounds 2–4. Allow groups no more than 3 minutes of work time for each round.
For Rounds 2–4, display the following directions: Apply the properties and definitions of exponents to write an equivalent expression with positive exponents and the fewest number of bases. Assume all variables are nonzero.
Round 1
Round 3
To close the Power Rounds activity, define the word simplify for exponential expressions.
In Rounds 2 through 4, we applied the properties and definitions of exponents to write equivalent expressions with positive exponents and the fewest number of bases. In other words, we simplified.
To simplify an exponential expression, apply as many properties and definitions of exponents as needed to write an equivalent expression containing only positive exponents and the fewest number of bases.
Finding a More Efficient Way
Students apply the properties and definitions of exponents to efficiently simplify expressions that appear complicated.
Keep students in groups of three, and distribute the Be Efficient cards to each group. Each student takes two cards. Explain the structure of the activity to students by using the following directions:
• Each group member takes 1 minute to work the problem on the card independently.
• On my signal, members pass their card to another group member.
• You have 1 minute to complete the problem your group member started.
• Once a problem is completed, you can start on the next blank card.
• Repeat this process until all six problems are completed.
Answer any questions students may have about the activity, and then indicate when they can begin.
Language Support
Students are familiar with the term simplify when working with fractions, but it may be new for them to consider simplifying expressions. Tell students they have been simplifying exponential expressions throughout this topic. Consider presenting the following examples to illustrate this point:
• In lesson 5, x 4 · x 5 simplifies to x 9 .
• In lesson 6, (10 4) 2 simplifies to 10 8 .
• In lesson 7, 6 0 simplifies to 1.
• In lesson 8, y −5 simplifies to 1 y 5 .
For additional clarification, present the following nonexamples:
• The expression 1 3 6 is not simplified because it can be written with a positive exponent.
• The expression x 5 · y · x 2 is not simplified because it can be written with fewer bases.
The term simplify is defined now because all the properties and definitions of exponents have been utilized in this lesson.
Simplify.
(6 x 2 y) 2 (6 x 2 y) 3 (6 x 2 y) −2
( 6 x 2 y ) 2 ( 6 x 2 y ) 3 ( 6 x 2 y ) −2 = ( 6 x 2 y ) 2+3+( −2 ) = ( 6 x 2 y ) 3 = 6 3 x 2·3y 3 = 6 3 x 6 y 3
Simplify. (4 m 3 n −2 ) −1 (4 m 3 n −2 )(3 m 4 n3 ) 2 ( 4 m 3 n −2 )−1 ( 4 m 3 n −2 )( 3m 4 n 3 ) 2 = ( 4 m 3 n −2 ) −1+1 ( 3 m 4 n 3 ) 2 = ( 4 m 3 n −2 ) 0 ( 3 2 m 4·2n 3·2 ) = 3 2 m 8n 6
Simplify. 8 a 5 b 3 · 2 a b −1 ( 4 a 3 b) 2 8 a 5 b 3 · 2 ab−1 ( 4 a 3 b ) 2 = 8 · 2 a 5+1 b 3+(−1) 4 2 a 3·2b 2 = 16 a 6 b 2 16 a 6 b 2 = 1
Simplify. ((3 h −2 ) 2 (9 h −4) ) 3 ((3 h −2) 2 9 h −4 ) 3 = (3 2 h −2 · 2 9 h −4 ) 3 = (9 h −4 9 h −4 ) 3 = ( 1 ) 3 = 1
Simplify. xy 2 ( xy 2 ) −1 ________ 5 x y 2 x y 2 ( x y 2 ) −1 5 x y 2 = ( x y 2 ) 1+(−1) 5 x y 2 = ( x y 2 ) 0 5 x y 2 = 1 5 x y 2
After students complete all six problems, display the answers. Ask the following questions to facilitate a discussion that concludes the activity:
• Which properties or definitions of exponents did you use to simplify the problems?
• Did you follow a certain order when simplifying the problems?
• Is there a different order that makes the simplifying process shorter?
• What patterns in the structure of the problems allowed you to more efficiently simplify the problems?
Debrief 5 min
Objective: Write equivalent expressions by using all the properties and definitions of exponents.
Facilitate a class discussion to conclude the lesson. Invite multiple students to expand on or refine their classmates’ answers. Consider capturing student responses by displaying them to the class.
What are the characteristics of a simplified expression involving exponents?
I know an expression is simplified when I can’t apply any additional properties or definitions of exponents.
An expression is simplified when it is written with the fewest number of bases. A simplified expression must have positive exponents.
Display the expression 3 3 · 27.
What are some different equivalent expressions for 33 · 27? Explain your strategies to find each equivalent expression.
I can write 27 as 3 3. Then I can apply the properties of exponents to write 3 3 · 3 3 as 3 6 .
I can write 27 as 3 3 and then apply the properties of exponents to write 3 3 · 3 3 as (3 3) 2, or 27 2 .
I can write 3 3 as 27 and then apply the properties of exponents to write 27 · 27 as 27 2 .
I can write 3 3 as 27. Then I can multiply 27 by 27 to get 729.
Promoting the Standards for Mathematical Practice
When students apply their understanding of the properties and definitions of exponents to write equivalent expressions for 3 3 · 27, they are making use of structure (MP7).
Ask the following questions to promote MP7:
• How can you use what you know about exponents to help you write an equivalent expression?
• What’s another way you could apply the properties of exponents that would help you write a different equivalent expression?
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time permits. Refer students to the Recap for support.
Writing Equivalent Expressions
In this lesson, we
• wrote numerical expressions with unlike bases as a single power.
• applied multiple properties and definitions of exponents to write equivalent expressions.
Examples
For problems 1 and 2, apply the properties and definitions of exponents to write the expression as a single power.
Terminology
To simplify an exponential expression, apply as many properties and definitions of exponents as needed to write an equivalent expression containing only positive exponents and the fewest number of bases.
For problems 3–7, simplify. Assume all variables are nonzero.
Simplify the coefficients by writing them as a separate factor.
Use the commutative property to rearrange factors.
MATH2 New York Next
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
For problems 1–6, use the properties and definitions of exponents to write the expression as a single power.
For problems 17–22, match each expression with an equivalent expression. Each answer can be used more than once.
For problems 7–16, simplify. Assume all variables are nonzero.
Which of the following is equal to 12 g h ? Assume g and h are nonzero. Choose all that apply.
Evaluating Numerical Expressions by Using Properties of Exponents (Optional)
Simplify and evaluate exponential expressions by using the properties and definitions of exponents.
Lesson at a Glance
In this optional lesson, students continue to build fluency by using the properties of exponents when writing numerical expressions with different bases. As a class, students use strategies for making powers of 10 to evaluate and write expressions with the fewest prime bases to simplify. Student pairs share their work so that the class becomes mindful of different solution strategies.
Key Questions
• What is the difference between evaluating and simplifying a numerical expression?
• Is there an advantage to writing exponential expressions with only prime bases? Explain.
Achievement Descriptor
8.Mod1.AD5 Apply the properties of integer exponents to generate equivalent numerical expressions. (NY-8.EE.1)
Agenda
Fluency
Launch 5 min
Learn 30 min
• Make a Power of 10
• Write with Fewest Bases
• Compare and Connect
Land 10 min
Materials
Teacher
• None
Students
• None
Lesson Preparation
• None
Fluency
Evaluate Each Expression
Students evaluate each expression to prepare for writing equivalent expressions by using the properties of exponents.
Directions: Evaluate.
Teacher Note
Instead of this lesson’s Fluency, consider administering the Numerical Expressions with Exponents Sprint. Directions for administration can be found in the Fluency resource.
Launch
5
Students write numerical expressions by using properties of exponents and powers of 10.
Have students find the value for problem 1 individually. Allow students to work on problem 1 until everyone has finished or the struggle is no longer productive. Look for students who are creating 10’s with the factors of 2 and 5 as you circulate.
1. Find the value of 2
Call the class together and display the make a power of 10 strategy work for finding the value of 2 4 · 15 3. The work shown applies the properties of exponents and uses powers of 10 in a manner that makes evaluation of the expression a mental math exercise.
Language Support
As students complete work in this lesson, English learners may benefit from completing a graphic organizer as a support for familiar terminology.
Consider including the following terms:
• Base
• Exponent
• Positive exponents
• Prime base
• Properties of exponents
• Simplify
What do you notice?
I notice that the answer here is the same answer that I got. I notice that the work shows the properties of exponents. I notice that the work shows 10 as a base to get the answer.
What do you wonder?
I wonder why the work does not show all of the factors written out. I wonder why the 2’s and 5’s are grouped together. I wonder why the problem was done this way.
If students do not wonder why the person grouped the 2’s and 5’s together, nor notice that the person used 10 as a base to get the answer, then guide the discussion in that direction. If you observed a student creating 10’s by using multiple factors of 2 and 5, ask the student to explain why they chose that strategy to evaluate.
Keep the make a power of 10 strategy work displayed and debrief the problem by using the following prompts.
The work shown uses a make a power of 10 strategy. How does the make a power of 10 strategy compare to the work you did when you evaluated the expression?
I had to multiply 16 and 3375 so this make a power of 10 strategy was simpler.
I factored each base and then multiplied the 2’s and 5’s to create 10’s. So making a power of 10 is similar to what I did, but I did not use the properties of exponents.
We evaluated this numerical expression. Is there a difference between evaluating a numerical expression and simplifying a numerical expression? Explain.
There is a difference. Evaluate means “to find the value,” so we want the answer in standard form. Simplify means “to write the expression with the fewest number of bases and only positive exponents.”
To evaluate means to find a value, or one number. To simplify an exponential expression means to write the expression with the fewest number of bases and use positive exponents.
Today, we will add to the strategies we have to evaluate and simplify numerical expressions with exponents.
Learn
Make a Power of 10
Students evaluate numerical expressions by using properties of exponents and powers of 10.
Assign students Make a Power of 10 problems 2 and 3 to complete with a partner. Encourage students to discuss their anticipated solution pathways with a partner before they begin evaluating.
For problems 2 and 3, make a power of 10 to evaluate each expression. Write your answer in standard form. 2.
Differentiation: Support
As students evaluate the expressions, they need to write the answers in standard form. If they get to 27 1000 but have difficulty writing it as 0.027, consider using a place value chart to support their thinking.
• How do you say the fraction in unit form?
• Find the thousandths place on the chart.
• In what place value do you write the 2? The 7?
Moving from fraction form to decimal form is a helpful skill that is applied in the next lesson.
Confirm answers and invite one or two students to share their solution pathways. Then debrief by using the following prompts.
What is similar about the way you evaluated each expression?
I had to factor in both problems. Both problems used the properties of exponents to make a power of 10. Both problems included a power of 3.
What is different about the way you evaluated each expression?
Since there are negative exponents in problem 3, I used the definition of a negative exponent to write the problem as a fraction. Problem 2 evaluated to a large number and problem 3 evaluated to a small positive number.
Write with Fewest Bases
Students factor and use the properties of exponents to simplify numerical expressions.
Use the following prompts to prepare students to write exponential expressions by using the fewest number of prime bases. If necessary, remind students that they simplified expressions in the last lesson by using the fewest number of bases and only positive exponents. Now, the bases must be prime.
In the last lesson, we simplified exponential expressions. Now, we will simplify further by using only prime bases. In problem 2, is 5 2 · 6 2 written with the fewest number of prime bases?
No, 6 is not a prime number because it can be factored as 2 · 3.
Prime means that a number’s only factors are 1 and itself, so 6 is not prime. Review your solution to problem 2. Look for any line where each base is prime or is written as a product of primes.
In 5 2 · (2 · 3) 2, each base is prime or is written as a product of primes.
Can this expression be further simplified by using the properties of exponents? How?
It can be simplified further as 5 2 · 2 2 · 3 2. Now, it is simplified by using the fewest number of prime bases.
When simplifying in this lesson, we will apply as many properties of exponents as needed so the final answer has only prime bases and only positive exponents. The bases 2, 3, and 5 are all prime and all of their exponents are positive, so we know the expression is fully simplified.
Assign problems 4 and 5. Consider working as a class to complete problem 4. Then have students complete problem 5 individually or with a partner.
For problems 4 and 5, simplify the expression by using the fewest number of prime bases.
UDL: Representation
Direct instruction on squares and cubes is in module 1 topic D. However, students have seen these numbers in grades 6 and 7. To activate prior knowledge, create and display a chart with common squares and cubes.
Confirm answers and invite students to discuss how they simplified the expressions. Then discuss simplification more thoroughly.
Describe what it means to simplify an expression by using the fewest number of prime bases.
First, factor each base into prime bases. Next, use the properties of exponents to make all the exponents positive. Then, apply properties of exponents as needed until there are no further properties to use.
Compare and Connect
Students simplify and evaluate numeric expressions and compare their solutions with other students.
Direct students to the Compare and Connect problems. Assign each student pair one problem from the set. Student pairs will present their work to the class and compare their solution strategies with the strategies of their classmates.
Circulate to ensure students are following the given directions to evaluate or simplify. As students work, advance their thinking by asking any of the following questions.
• Why did you factor that number to __________?
• How did you use the properties of exponents for this line in your work?
• How do you know there are no more ways to simplify your final answer? How do you know you evaluated the expression?
• Is there another way to do this work and still get the same answer?
For problems 6–8, simplify by using the fewest number of prime bases.
Teacher Note
The problems in Compare and Connect are ordered from simple to complex. When assigning problems, consider the fluencies of your students. Two of the simpler problems may need to be assigned to take the same time as one of the more difficult problems.
Teacher Note
For problem 6, some students may not recognize that 8 can be written as 23. Assist these students in writing 8 with the factors of 2 and 4.
For problems 9–11, evaluate each expression. Write your answer in standard form.
Differentiation: Challenge
Consider giving these problems as an additional challenge to students who finish early.
Choose one problem from the Compare and Connect problems. Tell all student pairs who were not assigned that problem to begin simplifying or evaluating the expression. Invite all pairs who completed that problem to come together and select one pair of students to present their solution. When most students have finished working, allow the selected pair to present their solution. Then have all students compare their solution with the one presented. Use any of the following questions to guide the discussion.
• Who can restate and ’s reasoning in a different way?
• Does anyone want to add on to and ’s strategy?
• Do you agree or disagree? Why?
After two student pairs have presented, the following questions can guide student thinking.
• Why does this approach include and this one does not?
• Where is factoring done in each approach?
• Where are the properties of exponents in each approach?
Land
Debrief 5 min
Objective: Simplify and evaluate exponential expressions by using the properties and definitions of exponents.
Facilitate a brief discussion by using the following questions. Encourage students to restate or build upon one another’s responses.
What is the difference between evaluating and simplifying a numerical expression?
Evaluate means finding the final value, or one number in standard form. Usually, simplify means writing expressions with the fewest number of bases and only positive exponents. But today, we simplified by writing expressions with the fewest number of prime bases.
Promoting the Standards for Mathematical Practice
When students listen to and analyze their peers’ methods for simplifying or evaluating numerical expressions, they are constructing viable arguments and critiquing the reasoning of others (MP3).
Ask the following questions to promote MP3:
• What parts of your strategy are effective? Convince your classmates.
• What parts of the other students’ work do you question? Why?
• What questions can you ask your classmates to make sure you understand their solution method?
Is there an advantage to writing exponential expressions with only prime bases? Explain.
Yes, you can simplify an expression in new ways when writing exponential expressions with only prime bases. The bases may seem to be different, such as 2 4 · 8 2, but the bases may actually be the same when writing an equivalent expression by using the properties of exponents such as 2 4 · 8 2 = 2 4 · (2 3) 2 .
Exit Ticket
5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
Edition: Grade 8, Module 1, Topic B, Lesson 10
Evaluating Numerical Expressions by Using Properties of Exponents (Optional)
In this lesson, we
• wrote equivalent numerical expressions by using prime factors of the given bases.
• evaluated numerical expressions mentally by making bases of 10
• simplified numerical expressions by using prime bases.
Examples
1. Make a power of 10 to evaluate the expression 32 50 Hint: 32 can be written as 2 5 and 50 can be written as 2 25
make a base of 10 the 2’s and the 5’s need to have the same exponents. Use the properties of exponents
2. Simplify the expression by using the fewest number of prime bases.
Apply the definition of negative exponents to
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
8. Noor writes 2 3 4 3 8 3 = 64 9, and Mr. Adams marks it wrong.
a. Explain Noor’s error(s).
Noor multiplied all of the bases and added the exponents. These powers do not have like bases, so she cannot add the exponents.
b. Simplify 2 3 4 3 8 3 to show Noor how to correctly simplify the expression.
9. Use the properties of exponents to show the steps to simplify 6 3
10. Consider the expression
a. What are some ways can you rewrite the expression by using exponents?
Samples:
b. Which way do you prefer to rewrite the expression and why?
Sample: I prefer to rewrite the expression as 7 6 ⋅ 2 6 because it is simplified by using prime bases.
Remember For problems 11–14, multiply.
Write an equivalent expression for 1
with a negative exponent.
Topic C Applications of the Properties and Definitions of Exponents
In topic C, students order and operate with numbers written in scientific notation by combining the properties and definitions of exponents from topic B with scientific notation from topic A. By applying the properties of exponents, students can now operate with large and small positive numbers without needing to write out the many factors of 10.
Students begin the topic by writing small positive numbers in scientific notation. They recall from lesson 1 how to write a small positive number as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form. Students now use the definition of negative exponents to write the number in scientific notation. Now that students can write large and small numbers in scientific notation, they can order them by using each number’s order of magnitude and first factor. Students use previous learning from topic A about unknown factors to broaden their ability to operate with all numbers written in scientific notation.
By the end of the topic, students can multiply and divide numbers written in scientific notation, which includes raising to a power. They learn that measurements can be written in scientific notation and that the units of those measurements are important. Choosing an appropriate unit of measurement, such as using years instead of seconds for measuring an adult's lifespan, ensures the relevance of the quantity to the reader. Using appropriate units of measurement also makes it easier to interpret comparisons of quantities. Students end this topic by exploring pieces of art and painting techniques in an open-ended activity that applies their understanding of scientific notation and properties of exponents.
In topic D, students continue to work with exponents, namely squares and cubes. The ability to recognize numbers written in equivalent forms assists students when they encounter square root notation and cube root notation in topics D and E.
Progression of Lessons
Lesson 11 Small Positive Numbers in Scientific Notation
Lesson 12 Operations with Numbers in Scientific Notation
Lesson 13 Applications with Numbers in Scientific Notation
Lesson 14 Choosing Units of Measurement
Lesson 15 Get to the Point
3.
Small Positive Numbers in Scientific Notation
Write small positive numbers in scientific notation.
Order numbers written in scientific notation.
Lesson at a Glance
Students use the definition of negative exponents to write small positive numbers in scientific notation. They learn a new strategy that focuses on place value to write numbers in both scientific notation and standard form. Through an activity in which students move to designated locations in the room, pairs of students are challenged to determine whether a given number written in scientific notation is less than or greater than a given number. The set of given numbers continues to grow, challenging students to refine their number’s place in the ordered set. During this activity, students briefly experience negative numbers written in scientific notation and their place in the order of a set of numbers.
Key Questions
• What features of a number written in scientific notation indicate it has a small positive value?
• How do you order positive numbers written in scientific notation from least to greatest?
Achievement Descriptor
8.Mod1.AD8 Approximate and write very large and very small numbers in scientific notation. (NY-8.EE.3)
Agenda
Fluency
Launch 5 min
Learn 30 min
• Small Positive Numbers
• Ordering Numbers in Scientific Notation
Land 10 min
Materials
Teacher
• Construction paper (2)
• Marker
• Tape Students
• Scientific Notation Ordering cards (1 card per student pair)
Lesson Preparation
• Prepare two signs labeled Less Than and Greater Than. Post the Less Than sign at the front left corner of the room and post the Greater Than sign at the front right corner of the room.
• Copy and cut out one set of Scientific Notation Ordering cards (in Teach book). Separate the four cards that are shaded gray to use for display.
Fluency
Order Numbers
Students order numbers from least to greatest to prepare for ordering numbers written in scientific notation.
Directions: Place the following numbers in order from least to greatest. 1.
Launch
Students write small positive numbers as a number times a unit fraction with a denominator written as a power of 10.
Open and display the Powers of 10 teacher interactive for the class. Progress through each power of 10 in the interactive. Start with an input of 100 and move through to 1 1012
Briefly discuss what students notice and wonder. Then, guide students through the Grain of Rice row of the table. Next, direct students to complete the Spider Silk row independently or in pairs for additional support.
1. Complete the table as directed during the lesson. The table shows an approximate measurement of various objects.
Approximate Measurement (meters)
000 0003
Blood Cell (diameter)
Call the class back together to discuss their work. The remainder of the table is filled in during Learn.
Today, we will use the definitions of exponents to learn how to write and compare small positive numbers by using scientific notation.
Learn
Small Positive Numbers
Students recognize and write small positive numbers in scientific notation.
Begin this segment by asking students the following questions.
In the table, label the last column header Scientific Notation. What is the definition of scientific notation?
A number is written in scientific notation when it is represented as a number a, which has an absolute value of at least 1 but less than 10, multiplied by a power of 10.
Is 1 ___ 10 3 a power of 10? Why?
No, there is a power of 10 in the denominator, but the whole expression is not a power of 10.
No, because I can write 1 10 3 as ( 1 10 )3, which is a power of 1 10 , not a power of 10.
What is another way to write 1 ___ 10 3 so it is a power of 10? Explain.
I can write 1 10 3 as 10 −3 by using the definition of negative exponents.
Use the definition of negative exponents to write the first two measurements from the table in scientific notation.
What are the first two measurements from the table in scientific notation?
6 × 10 −3 and 4 × 10 −6
Look more closely at the number 6 × 10 −3. What do you notice about the power of 10 in scientific notation and the unit of the number in unit form?
I notice that the power of 10 is 10 −3 and the unit of the number is thousandths.
How do we know 10 −3 represents thousandths?
Since 10 3 = 1 10 3 = 1 1000 , we know 10 −3 represents thousandths.
What is the order of magnitude? What does it tell us about the number 6 × 10 −3 ?
The order of magnitude is −3, and it tells us that the 6 is in the thousandths place when written in standard form.
Teacher Note
For the work of this segment, students use the definition of scientific notation that is formalized in lesson 3. They also apply the definition of negative exponents, which is taught in lesson 8.
We can use the power of 10 to determine the highest place value of the number written in standard form. This is one reason we write numbers in scientific notation.
What does the exponent on the power of 10 tell us about the number 4 × 10 −6?
The number has an order of magnitude of −6, which means the 4 will be in the millionths place.
Direct students to the last two rows of the table. Ask them to write the water molecule diameter in scientific notation and the white blood cell diameter in standard form. Guide students to use any part of the table to help them write the numbers. Note that students do not have to fill in all the highlighted cells in the four columns.
Approximate Measurement (meters)
Call the class back together and confirm answers. If needed, guide students through using the highlighted cells to help them write each number in its new form. Use the following prompts to engage students in a discussion about writing numbers in standard form or scientific notation.
Were you able to write each number immediately, or did a certain part of the table help you? Explain.
I was able to write each number immediately by using place values.
I chose to write a number times a unit fraction to help me write each number in scientific notation.
Can you use the order of magnitude to write any number that is in scientific notation in standard form? How?
Yes, the order of magnitude helps us find the highest place value of the number written in standard form for any number because the order of magnitude is the exponent on the power of 10 in scientific notation.
A number in scientific notation is written as the product of a number, called the first factor, and a power of 10. Scientific notation requires that the first factor be a number with an absolute value of at least 1 but less than 10. As a result, the power of 10 always indicates the highest place value of a number written in standard form.
Have students work with a partner on problems 2–7.
For problems 2–7, write the number in scientific notation or standard form as indicated in the table.
Display the completed table and allow students to check their answers. Then use the following prompts to facilitate a discussion about number representations.
In problem 5, why is the order of magnitude −6 if the number written in standard form has 7 digits after the decimal point?
The order of magnitude of −6 tells us that the first digit of our number is 2 and is in the 10−6 place value in standard form. It does not tell us the place value of the last digit of the number in standard form.
In problem 5, the order of magnitude of −6 indicates that the first nonzero digit of the number is found 6 places after the decimal point. So the 2 in 0.000 0023 is in the sixth place after the decimal point.
Looking only at the numbers in the table written in scientific notation, how can we tell these numbers are small positive values?
The power of 10 has a negative exponent. We know we can rewrite any of these numbers as a fraction with a denominator written as a power of 10 with a positive exponent. In that fraction, the denominator is a large number that is greater than the numerator. This means the whole fraction is a small number between 0 and 1.
All the numbers in problems 2–7 have a negative order of magnitude. Why are none of the numbers negative when written in standard form?
All the numbers have a positive first factor, so the numbers are positive. The order of magnitude just helps us find the highest place value of the number written in standard form.
How would we write negative numbers in scientific notation? For example, if we changed the standard form number in problem 2 to −0.0007, how would the scientific notation change?
The first factor would be negative, but the order of magnitude would stay the same. We would write −0.0007 as −7 × 10 −4 in scientific notation.
The power of 10 in scientific notation tells us the highest place value of a number written in standard form. If the first factor is positive, the number is positive. If the first factor is negative, the number is negative. However, scientific notation is generally used to represent measurements, so most of the scientific notation we encounter will represent positive numbers instead of negative numbers.
Ordering Numbers in Scientific Notation
Students order numbers given in scientific notation from least to greatest.
Distribute one white Scientific Notation Ordering card to each pair of students. The gray cards are used for display.
Look at the number on your card. Discuss with your partner whether the number is small or large.
While partners discuss, tape the 3 × 10 2 card at the front of the room, halfway between the Less Than and Greater Than signs.
Figure out whether the number on your card is less than or greater than the number on the card I just displayed at the front of the room. Once you know, stand by the appropriate sign.
In your group, discuss why your number is less than or greater than the number I have displayed. If you find that you are in the wrong group, simply move to the other group and explain why your number belongs in that location.
As students explain, listen for the terms first factor and order of magnitude. If you do not hear these terms, encourage students to use the them as a way to explain their number’s location.
When students have finished explaining, display the cards with the numbers 7 × 10 −2 and 6 × 10 4. Discuss how to order these two numbers with 3 × 10 2 by using their orders of magnitude. Tape these numbers appropriately between the Less Than sign and 3 × 10 2 and between 3 × 10 2 and the Greater Than sign.
Ask student pairs to reorganize themselves in relation to 7 × 10 −2 , 3 × 10 2, and 6 × 10 4. Since there are now four groups, students may need more support understanding where to stand. If needed, point out the four groups to students as less than 7 × 10 −2, between 7 × 10 −2 and 3 × 10 2, between 3 × 10 2 and 6 × 10 4, and greater than 6 × 10 4. When pairs are settled in their new groups, direct them to discuss why their number is in that location. Again, if any students realize they belong in another group, have them move to and participate with that group.
Promoting the Standards for Mathematical Practice
When students explain why a number written in scientific notation is less than or greater than a second number written in scientific notation, they are constructing viable arguments (MP3).
Ask the following questions to promote MP3:
• Is your sign choice of Less Than (or Greater Than) a guess, or do you know for sure?
• What questions can you ask another pair of students to make sure you understand their reasoning?
UDL: Representation
Consider using a horizontal number line to model the values of the numbers, smaller numbers to the left and larger numbers to the right. Including the numbers 0 and 1 on the number line can further support learning about small positive numbers. Alternatively, use a vertical number line with larger numbers near the top.
When students finish discussing, ask them to organize themselves in their current group from least to greatest. Invite each student pair to tape their number in the correct location starting at one of the given numbers, 7 × 10 −2 , 3 × 10 2, or 6 × 10 4. Then have them explain to the class why their number is in that location. Again, encourage students to use the terms first factor and order of magnitude to explain their number’s location.
Have students return to their seats. Direct them to problem 8, which includes some of the numbers used in the activity. Have students complete the problem with their partner.
8. Order the given numbers from least to greatest in the table. Use the order of magnitude to explain why each number is greater than the number before it.
Numbers Ordered from Least to Greatest
Number
3.6
3 × 10 2
3.2 × 10 2
6 × 10 4
7.02 × 10 4
Explanation
3.6 has an order of magnitude of 0, which is greater than the order of magnitude of −2 in 7 × 10 −2 .
3 × 10 2 has an order of magnitude of 2, which is greater than the order of magnitude of 0 in 3.6.
3.2 × 10 2 and 3 × 10 2 both have the same order of magnitude, but 3.2 is greater than 3.
6 × 10 4 has an order of magnitude of 4, which is greater than the order of magnitude of 2 in 3.2 × 10 2 .
7.02 × 10 4 and 6 × 10 4 both have the same order of magnitude, but 7.02 is greater than 6.
When students have finished writing their explanations in the table, display the last gray card with the number −2 × 10 3. Use the following prompts to engage in a class discussion.
Is −2 × 10 3 written in scientific notation? Explain how you know.
Yes, the absolute value of −2 is at least 1 but less than 10, and it is multiplied by a power of 10.
What does the first factor of −2 tell us?
The −2 tells us that the number written in standard form is negative.
Where would −2 × 10 3 be placed in our ordering activity? Explain.
It would be less than 4.6 × 10 −2 since negative numbers are less than positive numbers.
Differentiation: Challenge
To further challenge students, create another card with −2 × 10 −3. Have students use their knowledge of order of magnitude to explain why −2 × 10 3 is less than −2 × 10 −3 .
If the numbers in the table represent measured distances, would we still include −2 × 103 in the table? Why?
No, we would not include −2 × 10 3 in the table. Distances are always positive. Since the first factor is negative, the number is negative and cannot represent a distance.
To debrief this segment, facilitate the discussion in Land.
Land
Debrief 5 min
Objectives: Write small positive numbers in scientific notation.
Order numbers written in scientific notation.
Facilitate a discussion by using the following prompts.
What features of a number written in scientific notation indicate it has a small positive value?
Small positive values written in scientific notation feature a negative exponent and a positive first factor.
Can a number written in scientific notation have a negative value? If so, how is that number written?
Yes, a number written in scientific notation can have a negative value if the first factor is negative.
How do you order positive numbers written in scientific notation from least to greatest?
The power of 10 indicates the highest place value of the number written in standard form. Larger orders of magnitude mean larger powers of 10, larger place values, and therefore, larger positive numbers. If the orders of magnitude are the same, the value of the number is dependent upon the first factor; the number with the larger first factor is the larger number.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Students may benefit from a list of numbers to explain their process. Consider displaying the numbers 7.2 × 10 15 , 7.2 × 10 −12 , 3.1 × 10 −12, and 7.25 × 10 15 .
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
For problems 9–13, compare the numbers by using the < or > symbol. Explain your answer by using the order of magnitude.
14. Before 2006, Pluto was considered to be one of the planets in our solar system. Many people thought it should be excluded from the list of planets because of its size. The table lists the planets, including Pluto, and their approximate diameters in meters.
Name the planets, including Pluto, in order from the smallest diameter to largest diameter. Pluto, Mercury, Mars, Venus, Earth, Neptune, Uranus, Saturn, Jupiter
15. Ava writes 0.00056 in scientific notation as 5.6 × 10 −4 and shows the following work. Explain what Ava is doing in her work.
Ava is using the place value of each digit to determine the place value of the 5. Because 5 is in the ten thousandths place, 10 −4 will be the power of 10 when she writes the number in scientific notation.
16. Nora is comparing the diameter of a grain of salt with the length of a flea.
• The average diameter of a grain of salt is 3 × 10 −4 meters.
• The average length of a flea is 1.5 × 10 −3 meters.
Nora believes that the diameter of a grain of salt is greater than the length of a flea because 3 is greater than 1.5. Explain why you agree or disagree with Nora.
I disagree with Nora. Nora only compared the first factors of each number, but she needed to first compare the orders of magnitude. The order of magnitude for 1.5 × 10 −3 is −3, which is greater than the order of magnitude for 3 × 10 −4, which is −4. Because 1.5 × 10 −3 is greater than 3 × 10 −4, the length of a flea is greater than the diameter of a grain of salt.
17. The mass of a neutron is about 1.67493 × 10 −27 kg. The mass of a proton is about 1.67262 × 10 −27 kg. Explain which is heavier.
Because the numbers have the same order of magnitude, the first factor determines which is heavier. A neutron is heavier because 1.67493 is greater than 1.67262
For problems 18–21, multiply.
22. The table represents box office sales of the highest-grossing movies in the years 2017 to 2019. Write each number in scientific notation.
23. The area of the base of a right rectangular prism is 16 square inches. The height of the prism is 4 inches. What is the volume of the prism?
Interpret numbers in scientific notation displayed on digital devices.
Operate with numbers written in scientific notation.
Lesson at a Glance
Students explore a number in scientific notation displayed on a digital device. They learn how to input numbers written in scientific notation into devices as a means to check their answer after calculations are made. Students use the properties of exponents to efficiently operate with numbers written in scientific notation, including numbers with an order of magnitude that is negative.
Key Questions
• How is a number in scientific notation displayed on a digital device?
2. Which number is greater, 93 million or (239,900) 2? Show calculations by using scientific notation, and explain your answer.
• How are the properties of exponents useful when operating with numbers written in scientific notation?
Achievement Descriptors
8.Mod1.AD11 Multiply or divide numbers written in standard form and scientific notation, and evaluate exponential expressions containing numbers written in standard form and scientific notation. (NY-8.EE.4)
8.Mod1.AD14 Interpret scientific notation that has been generated by technology. (NY-8.EE.4)
Agenda
Fluency
Launch 5 min
Learn 30 min
• Power to a Power
• How Many Times
Land 10 min
Materials
Teacher
• Scientific calculator
Students
• Scientific calculator
Lesson Preparation
• Prepare a display of the scientific calculator so keystrokes and answers can be shown to the class.
Write in Scientific Notation
Students write each number in scientific notation to prepare for operating with numbers written in scientific notation.
Directions: Write each number in scientific notation.
Teacher Note
Instead of this lesson’s Fluency, consider administering the Scientific Notation and Negative Exponents Sprint. Directions for administration can be found in the Fluency resource.
0.00056
Launch
Students become familiar with entering numbers written in scientific notation into digital devices and reading numbers in scientific notation displayed on digital devices.
Distribute a calculator to each student. Display a calculator so the class can see the keys pushed to enter data. Direct students to problem 1 and read the problem aloud. Have students enter the problem into their calculators.
What do you notice? What do you wonder?
I notice there is an “e” on Liam’s screen and a “+10”. We don’t generally see a “+” on the answer screen.
I wonder what the “e” stands for. I wonder why the calculator displays the answer like this.
Let’s calculate this answer by hand and see whether that clears up any of our questions. Have students calculate the product by writing the numbers in scientific notation.
1. Liam enters 200,000 × 450,000 into his calculator, and the screen shows the following display. 9e+ 10
Calculate 200,000 × 450,000 by writing the numbers in scientific notation.
Teacher Note
Digital devices display numbers in scientific notation in many ways. If possible, have students use the calculator or digital device they will use throughout the year, including on assessments, to become familiar with its input and output characteristics.
Teacher Note
Depending on the type of calculator, students may notice that their screen displays large numbers differently from Liam’s screen.
Confirm answers and lead the class in a discussion by using the following prompts. Note that students may interpret the +10 in the display to be the base of 10 instead of the order of magnitude. This misinterpretation can lead to a rich class debate. Use the properties of exponents shown in the work for problem 1 to convince students that the +10 represents the order of magnitude.
What do you think the “e” represents?
The “e” tells us that the answer is in scientific notation.
What about the “+10”?
The “+10” says that the order of magnitude is positive 10.
The calculator displays numbers in scientific notation with an “e” and the order of magnitude. Why do you think the calculator displays the number in this way?
Just like us, calculators use scientific notation to show numbers with a lot of digits.
Why not just display the answer in standard form?
Calculators have a limited space to display, or show, an answer.
If the number is very large or very small with many digits when written in standard form, the calculator must display the answer in a different way.
Try to input Liam’s calculations into your calculator by using scientific notation.
Use the displayed calculator to guide students to input (2 × 10 5) (4.5 × 10 5).
Today, we will practice interpreting scientific notation on our calculator. We will learn to operate with numbers in scientific notation by using the properties of exponents.
Teacher Note
If students are not convinced that the +10 represents the order of magnitude, consider asking them to calculate 2,000,000 · 4,500,000 on their devices to determine the interpretation of the “e” and the +12.
Promoting the Standards for Mathematical Practice
Students are using appropriate tools strategically (MP5) to explore and deepen their understanding of concepts when they use calculators for part of a calculation or to check calculations.
Ask the following questions to promote MP5:
• Is it appropriate to use a calculator to help evaluate this expression? Why?
• Is it appropriate to use a calculator to multiply 2 and 4.5? Why?
• Your calculator says the answer is _____. Does that seem about right?
Learn
Power to a Power
Students use properties of exponents to calculate squares and cubes of numbers written in scientific notation.
Have students work with a partner on problems 2 and 3, but do not have them check their answers yet. Tell students you will let them know when to check their answers. Circulate as they work to see whether their answers are written in scientific notation.
For problems 2–8, use the properties of exponents to evaluate the expression. Write the answer in scientific notation. Check the answer with a calculator.
2. (3 × 105)2
(3 × 105)2 = 32 × (105)2 = 9 × 1010
3. (5 × 103)2
(5 × 103)2 = 52 × (103)2 = 25 × 106 = 2.5 × 107
If students leave their answer to problem 3 as 25 × 106, facilitate a class discussion about writing the answer in scientific notation.
Is the number 25 × 106 written in scientific notation? How do you know?
No. The absolute value of the first factor is greater than 10, so it does not fit the definition of scientific notation.
Differentiation: Support
If students answer 3 × 1010 for problem 2, they may have incorrectly applied the power of a product property. Consider having them write (3 × 105)2 as (3 × 105) (3 × 105).
Let’s talk about why the first factor needs to be at least 1 but less than 10. We’ve seen that the power of 10 for a large number written in scientific notation tells us the highest place value of the number. What would we expect the highest place value of 25 × 106 to be?
We would expect the highest place value of the number to be the millions place because 106 is 1,000,000.
Is the millions place the highest place value of 25 × 1,000,000? Explain.
No. The highest place value of 25 × 1,000,000 is the ten millions place because 25 × 1,000,000 is 25,000,000.
Ask students to share their thoughts about why a discrepancy between the perceived size of a number and the actual size of a number could be problematic.
When a large number is written in scientific notation, where the first factor is a number with an absolute value of at least 1 but less than 10, the power of 10 tells us the highest place value of the number. If the absolute value of the first factor is not in that range, then the power of 10 does not tell us the highest place value, so it becomes more difficult to develop a sense of how large the number really is.
Direct students back to problem 3. If needed, have them revise their answer by rewriting it in scientific notation. Then, as a class, work through rewriting 25 × 106 in scientific notation, inviting students to share their thinking. Consider color-coding or highlighting the parts of the problem to help students make the necessary connections.
Once students’ answers to problems 2 and 3 are written in scientific notation, guide students through checking each answer on their calculator by inputting the original expression into the displayed class calculator.
Direct students to problems 4–8. To build students’ sense of operating with numbers written in scientific notation, have students turn and talk to make predictions about what the order of magnitude in each answer will be. Then have students complete the problems in pairs. Instruct students to use a calculator to find powers of decimals such as 3.722 and to check their final answers.
UDL: Representation
Color-coding can serve as a visual cue when writing the answer in scientific notation. It helps students distinguish between the first factor and the original power of 10. It also helps students see how the lines of work relate to one another.
Another way to represent the relationship between the numbers is by using annotations such as circling or underlining.
4. (3.72 × 10 5)2
Teacher Note
Students use a calculator throughout the lesson to operate with numbers efficiently, to practice inputting numbers, and to interpret the answer displayed in scientific notation.
5. (4 × 10 6) 2
(2 × 10 −9)
8. (2.6 × 10 −4) 3
Confirm answers. Then have pairs turn and talk about whether the predictions they made for the orders of magnitude matched the actual orders of magnitude in the answers and why. Listen for students to mention the influence of the first factor.
Facilitate a class discussion by using the following prompts.
What properties of exponents did you apply to evaluate these expressions? Explain.
I used the property (xy)n = xn y n to raise the first factor to an exponent and the power of 10 to the same exponent.
I used the property (x m)n = xm n to evaluate the power of 10 raised to an exponent.
I used the property xm ⋅ xn = xm+n to multiply the powers of 10.
What property of operations did you apply to evaluate these expressions? Explain.
I used the associative property of multiplication to group the powers of 10 as I wrote the answer in scientific notation.
When checking your work on your calculator, is it important to include the parentheses? Why?
Yes, it is important to include the parentheses. Without the parentheses, the calculator applies the outside exponent to the power of 10 only and not to the entire number.
How do you know your answer is correctly written in scientific notation?
In scientific notation, a number is written as a number a times a power of 10. The first factor a has an absolute value that is at least 1 but less than 10.
We need to always remember that a number written in scientific notation has a first factor with an absolute value that is at least 1 but less than 10. That information is important for all the operations we use in today’s lesson.
Teacher Note
For certain calculators, if students use the scientific notation button, the parentheses surrounding the number in scientific notation are not needed. However, it is important for students to understand when to use parentheses to prevent computational errors.
How Many Times
Students use properties of exponents to calculate quotients and products of numbers written in scientific notation.
Direct students to the table in the How Many Times section. Invite them to turn and talk with a partner about what they notice and wonder. Then have students read problem 9. If needed, guide students through the creation of the multiplication sentence and the use of the properties of exponents to find the unknown factor for problem 9. Instruct students to use a calculator to find the quotient of the first factors and to check their final answer. Then direct them to complete problems 10 and 11 with their partner. Circulate to ensure that all students correctly show the properties of exponents work before checking with a calculator.
For problems 9–11 use the values from the table to answer the questions. Write the answer in scientific notation. Use a calculator to check your answers. Object
Measurement
Differentiation: Support
If students have difficulty creating multiplication sentences or solving for the unknown factor, guide them to their work from lesson 2.
9. A blue whale is about how many times as long as a tube of lip balm?
A blue whale is about 4.18 × 10 2 times as long as a tube of
10. About how many grains of salt positioned side by side would it take to equal the length of a tube of lip balm?
It would take about 2.23 × 10 2 grains of salt positioned side by side to equal the length of a tube of lip balm.
11. Which object from the table is about 50 million times as large as a blue whale? ( 50 × 10 6) ( 2.8 × 10 ) = ( 50 · 2.8 ) × ( 10 6 · 10 ) = 140 × 107 = 1.40 × 10 2 × 10 7 = 1.40 × 10 9
The diameter of the sun is about 50 million times as large as a blue whale.
Facilitate a discussion about this segment by using the following questions.
Is problem 11 different from the other two in this segment? How?
Yes, problem 11 is different from the others. We know both of the factors for problem 11 instead of the product and one factor. So instead of dividing to find a factor, we need to multiply to find the product.
When you are given two factors, you multiply to find the product. When you are given a product and one factor, you divide to find the unknown factor.
Land
Debrief 5 min
Objectives: Interpret numbers in scientific notation displayed on digital devices.
Operate with numbers written in scientific notation.
Facilitate a discussion by using the following prompts. Encourage students to restate or build upon one another’s responses.
When does a digital device display answers in scientific notation?
A digital device screen has a limited number of digits. When the answer has more digits than will fit on the screen, the calculator displays the answer in scientific notation.
How is a number in scientific notation displayed on a digital device?
To display an answer in scientific notation, most digital devices display the first factor followed by an “e” and the order of magnitude.
How are the properties of exponents useful when operating with numbers written in scientific notation?
When operating with numbers written in scientific notation, the properties are useful in different ways:
For problems involving a power of a number in scientific notation, we can use the property (xy) n = x n y n to raise the first factor to an exponent and the power of 10 to the same exponent. Then we can use the property (x m) n = x m n to evaluate the power of 10 raised to an exponent.
When the result is not written in scientific notation, we can write the first factor so it has an absolute value of at least 1 and less than 10. Then we can use the property x m · x n = x m+n to multiply the powers of 10.
For division problems, we can use the definition x n = 1 _ x n to understand the quotient of the powers of 10 as a multiplication expression. Then we can multiply by using the property x m · x n = x m+n .
For multiplication problems, we can use the property x m · x n = x m+n to multiply the powers of 10.
Teacher Note
Accept all answers based on the displays of the calculators or digital devices students used in this lesson.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
For the Practice problems, students should have access to a scientific calculator or similar digital device.
Operations with Numbers in Scientific Notation
In this lesson, we
• interpreted scientific notation displayed on digital devices.
• used the properties of exponents to efficiently operate with numbers written in scientific notation.
Examples
1. A calculator displays 3.45e−4. Interpret the meaning of this number by writing it in scientific notation and in standard form.
3.45 × 10 −4 = 0.000 345
2. Use the table of approximate animal weights to answer the question.
The scorpion weighs more than the termite because the order of magnitude of −2 is greater than the order of magnitude of −6
An emperor scorpion is about 2 × 10 4 times as heavy as a termite.
Use the property of exponents to square both factors of the product within the parentheses.
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
For problems 5–7, use the table of animal weights to answer the questions. Use a calculator to check your answers.
7. A zebra is about how many times as heavy as a hummingbird?
A zebra is about 100,000 times as heavy as a hummingbird.
8. Which number is greater, (1.1 × 10 −3) 2 or 1.1 × 10 −3? Explain.
The number (1.1 × 10 −3) 2 has an order of magnitude of −6 because (1.1 × 10 −3) 2 = 1.21 × 10 −6
The number 1.1 × 10 −3 has an order of magnitude of −3. So 1.1 × 10 −3 is greater than (1.1 × 10 −3) 2 because it has a larger order of magnitude.
9. Which number is less, (3 × 10 6) 2 or (6 × 10 3) 3? Explain.
The number (3 × 10 6) 2 has an order of magnitude of 12 because (3 × 10 6) 2 = 9 × 10 12
The number (6 × 10 3) 3 has an order of magnitude of 11 because (6 × 10 3) 3 = 2.16 × 10 11
So (6 × 10 3) 3 is less than (3 × 10 6) 2 because it has a smaller order of magnitude.
10. Use the following table from lesson 11 to answer the questions.
a. Comparing only diameters, Jupiter is about how many times as large as Pluto? Round your answer to the nearest whole number.
The diameter of Jupiter is about 60 times as large as the diameter of Pluto.
b. Comparing only diameters, Jupiter is about how many times as large as Mercury? Round your answer to the nearest whole number.
The diameter of Jupiter is about 29 times as large as the diameter of Mercury.
c. Assume you are a voting member of the International Astronomical Union (IAU) and the classification of Pluto is based entirely on the length of its diameter. Would you vote to keep Pluto a planet or to reclassify it as a dwarf planet? Why?
Sample: I would vote to reclassify it as a dwarf planet. Knowing that Jupiter is 29 times larger than Mercury means Mercury is pretty small. Jupiter is about 60 times larger than Pluto, which means Pluto is even smaller than Mercury. For that reason, I’d vote that the length of the diameter of Pluto is too small compared to the other planets, even the next smallest one.
Remember For problems 11–14, multiply.
15. Simplify the expression 14
14
. Write the expression with positive exponents. 14
Plot all the points from the table in the coordinate plane.
Applications with Numbers in Scientific Notation
Operate with numbers written in standard form and scientific notation.
Lesson at a Glance
Students operate with numbers written in standard form and scientific notation by engaging with unconventional contexts. Students compare the operations and strategies they used to solve problems to recognize that there are various paths to a correct response. Scaffolds involving less complex mathematical operations prepare students to persevere with more challenging questions.
Key Question
• What tools or strategies can help you solve problems in contexts that use numbers written in scientific notation?
Achievement Descriptors
8.Mod1.AD11 Multiply or divide numbers written in standard form and scientific notation, and evaluate exponential expressions containing numbers written in standard form and scientific notation. (NY-8.EE.4)
Agenda
Fluency
Launch 5 min
Learn 30 min
• Operating with Numbers in Scientific Notation
Land 10 min
Materials
Teacher
• None
Students
• Scientific calculator
Lesson Preparation
• None
Fluency
Write in Scientific Notation
Students write each number in scientific notation to prepare for operating with numbers written in scientific notation.
Directions: Write each expression in scientific notation. 1. (3
Launch
Students make predictions about relative size by using the volume of a balloon and the volume of air in one breath.
Display the outline of an inflated balloon.
Show how the volume of the balloon increases with each breath, up to 100 breaths. Then facilitate a brief discussion by asking the following questions.
What do you notice? What do you wonder?
I notice that the balloon does not inflate much with each breath.
I wonder how many breaths it will take to reach the goal.
I wonder how long it will take to reach the goal.
What information do you need to determine how many breaths it will take to reach the goal?
I need to know the volume of air in one breath and the total volume of air the balloon can hold.
Knowing the sizes of the objects allows us to compare the objects. Today, we will answer questions about unusual contexts by operating with numbers in scientific notation. 5
UDL: Engagement
The contexts for problems in this lesson are intentionally unconventional. These contexts are meant to capture students’ imaginations while engaging students in meaningful mathematics.
Promoting the Standards for Mathematical Practice
When students examine a new context and determine missing information, they are making sense of problems (MP1).
Ask the following questions to promote MP1:
• What information or facts do you need to solve this problem?
• What are some things you can try to start solving the problem?
• What is your plan to solve the problem?
Learn
Operating with Numbers in Scientific Notation
Students operate with numbers written in standard form and scientific notation.
Have students complete problem 1 with a partner.
1. On average, an eighth grader’s lungs can hold 3.6 × 10−3 cubic meters of air per breath.
a. What volume of air is in 2 breaths? Write your answer in scientific notation.
2(3.6 × 10−3) = 2(3.6) × 10−3 = 7.2 × 10−3
There are 7.2 × 10 3 cubic meters of air in 2 breaths.
b. What volume of air is in 10 breaths? Write your answer in scientific notation.
There are 3.6 × 10−2 cubic meters of air in 10 breaths.
When most students are finished, invite a few pairs to share how they determined the answers to the questions.
Display the volume of the balloon and lead students in a discussion about how they could calculate the number of breaths needed to fill the balloon to reach the stated goal.
Do we have enough information now to determine how many breaths it takes to reach the goal? How do you know?
Yes, now that we know the volume of the goal and the volume of each breath, we can determine how many breaths it takes to reach the goal.
Direct students to complete problem 2 in pairs.
Teacher Note
Students use a calculator to operate with numbers efficiently throughout the lesson.
Circulate and consider asking the following questions to guide students’ thinking:
• What do you notice about the first factors and the powers of 10?
• Are there similarities between the two numbers that make calculations easier? How?
2. The volume of each breath is about 3.6 × 10 3 cubic meters. How many breaths does it take to reach the goal of filling the balloon with 3.6 × 100 cubic meters of air?
1000 It takes 1000 breaths to reach the goal.
Once most students are finished, invite a few pairs to share their answers with the class and confirm responses.
Direct student pairs to complete problems 3 and 4. Circulate and consider asking the following guiding questions to support students in completing the comparison statement in problem 3:
• If you had a balloon that you inflated with one breath, how would its volume compare with the larger balloon’s volume of 3 × 100 cubic meters?
• How is the number of breaths represented in the comparison statement?
For problem 4, provide various levels of challenge by empowering students to respond with different levels of precision. For example, some students may approximate their age to the nearest year, while others may calculate their age down to the minute.
3. Complete the comparison statement.
The volume of the goal is 3.6 × 100 cubic meters. That is 1000 times as much as the volume of a single breath, which is 3.6 × 10−3 cubic meters.
4. On average, an eighth grader breathes 14 times per minute. Assume you have been breathing at about the same rate your whole life.
a. About how many breaths have you taken so far in your life? Write your answer in scientific notation.
Sample:
I am 13 years old.
13(365) = 4745 = 4.745 × 103
I have been alive for 4.745 × 103 days. There are 1440 minutes in a day.
I have been alive for 6.8328 × 106 minutes. I assume I take 14 breaths per minute.
14(6.8328 × 106) = 95.6592 × 106 = 9.56592 × 107
I have taken about 9.6 × 107 breaths so far in my life.
b. If the volume of one breath is about 3.6 × 10 3 cubic meters, what is the approximate total volume of breaths you have taken so far in your life? Write your answer in scientific notation.
The total volume of breaths I have taken so far in my life is about 3.5 × 105 cubic meters.
Differentiation: Support
Some students may benefit from a brief review of equivalent units of time to aid their calculation in problem 4. Consider asking the following questions to support students’ thinking:
• How many minutes are there in an hour?
• How many hours are there in a day?
• How many days are there in a year?
Differentiation: Challenge
For students who finish early, encourage them to find a more precise number of breaths by approximating their age to a smaller unit, such as minutes.
Once most students have finished, bring the class together and model a solution to the problem by using a typical age for an eighth grader or an age from a student volunteer.
Then display the Earth, Moon, and dollar bill.
Show how the height of the stack of dollar bills changes when up to 100 dollar bills are added. Then facilitate a brief discussion by asking the following questions.
What do you notice? What do you wonder?
I notice that the stack grows more slowly than I expected as we add more dollar bills.
I wonder how many dollar bills it would take to reach the moon.
I wonder how far away the moon is.
What information do you need to determine how many dollar bills it takes to reach the moon?
I need to know the thickness of a dollar bill and the distance between the earth’s surface and the moon.
Have students work in pairs on problem 5.
Encourage students to work on the problem before offering them any guidance. If students do not notice the different units right away, consider pointing out that the thickness of a dollar bill is given in millimeters.
5. One dollar bill is 0.11 millimeters thick. About how many dollar bills will it take to make a stack 1 meter tall? Write your answer in scientific notation.
Differentiation: Support
Some students may find it helpful to explore the meaning of the prefix milli in different ways. Consider asking the following questions to support students’ thinking:
• How many millimeters are there in a meter?
• How can we represent the prefix milli as a power of 10 in exponential form?
It will take about 9.1 × 103 dollar bills to make a stack 1 meter tall.
• How can we write 0.11 millimeters in meters as a power of 10 in exponential form?
Once most students are finished, invite a few pairs to share their responses with the class. Then have students work in pairs to complete problems 6 and 7.
6. The moon is approximately 3.84 × 108 meters away from Earth. One dollar bill is 0.11 millimeters thick. About how many dollar bills will it take to make a stack from Earth to the moon? Write your answer in scientific notation.
It will take about 3.49 × 1012 dollar bills to make a stack from Earth to the moon.
7. In February 2019, the US national debt was about 22 trillion dollars, or 2.2 × 1013 dollars. Make a comparison statement between the value of the Earth-to-moon stack of dollar bills in problem 6 and the national debt.
The US national debt is equivalent to 6.3 of the Earth-to-moon stacks of dollar bills.
Once most students are finished, invite pairs to share and discuss different strategies they used to solve the problems.
Land
Debrief 5 min
Objective: Operate with numbers written in standard form and scientific notation.
Facilitate a class discussion by using the following questions. Encourage students to restate and build upon one another’s responses.
What tools or strategies can help you solve problems in contexts that use numbers written in scientific notation?
I can use properties of exponents to solve problems. I sometimes use technology to help calculate my answer.
What did you find interesting about today’s lesson?
Some things I found interesting were how many stacked dollar bills it takes to reach the moon and that our national debt is equivalent to more than 6 of these stacks.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
For the Practice problems, students should have access to a scientific calculator or similar digital device.
• performed operations with numbers written in scientific notation.
• recognized there are various paths to solve a problem by comparing solution strategies.
Example
Mount Denali, located in Alaska, is the highest mountain peak in North America. Complete parts (a)–(c) to estimate, calculate, and compare the approximate number of pink erasers you must stack to reach the height of Mount Denali.
a. Estimate by making an educated guess about the place value of the answer. For example, describe the place value as in the thousands or in the tens of millions.
In the millions
b. The thickness of a pink eraser is approximately 3.1 × 10 −2 feet. The height of Mount Denali is approximately 2.031 × 10 4 feet. Calculate the approximate number of pink erasers that must be stacked to reach the height of Mount Denali.
2.031 × 10 4 = (3.1 × 10 −2)
2.031 × 104
3.1 × 10−2 = (2.031 3.1 ) × ( 10 4 10 −2)
≈ 0.655 × 10 6
= 0.655 × 10 × 10 5
= 6.55 × 10 5
The definition of negative exponents tells us that
About 6.55 × 10 5, or 655,000, pink erasers must be stacked to reach the height of Mount Denali.
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
1. Use the following information to fill in the blank. On Earth, there are about 1 quadrillion (1,000,000,000,000,000) ants and about 8 billion people.
b. A stack of 200 sheets of paper measures approximately 5 × 10 −2 feet. The height of Mount Everest is approximately 2.9035 × 10 4 feet. Calculate the approximate number of sheets of paper you must stack to reach the height of Mount Everest.
About 1.1614 × 10 8, or 116,140,000 sheets of paper must be stacked to reach the height of Mount Everest.
On Earth, there are about times as many ants as people.
× 10
2. Complete parts (a) and (b) to estimate and calculate the approximate number of sheets of paper you must stack to reach the height of Mount Everest.
a. Estimate by making an educated guess about the place value of the answer. For example, describe the place value as in the thousands or in the tens of millions
Sample: In the billions
3. Complete parts (a) and (b) to estimate and calculate the approximate number of people it would take standing shoulder to shoulder to fill the entire United States.
a. Estimate by making an educated guess about the place value of the answer. For example, describe the place value as in the thousands or in the tens of millions
Sample: In the hundred millions
b. The United States has a land area of about 3.797 × 10 6 square miles. Each person takes up about 4 square feet of area. Calculate the number of people that could stand shoulder to shoulder to fill the United States. (1 square mile ≈ 2.788 × 10 7 square feet)
About 2.646509 × 10 13 or 26,465,090,000,000, people could stand shoulder to shoulder to fill the United States.
Remember For problems 4–7, multiply. 4.
Simplify (x −6 y 5 z
) −4. Assume all variables are nonzero.
9. The length of a line segment is 12 units. One endpoint of the line segment is (−3, 6). Find four points that could be the other endpoint of the line segment. Use the coordinate plane as needed.
Choosing Units of Measurement
Choose appropriate units of measurement and convert units of measurement.
A popular television series is added to your favorite online streaming service. It will take 1.902 × 10 5 seconds to watch the entire series.
a. Choose a more appropriate unit of measurement to describe the amount of time needed to watch the entire series. Explain why you chose that unit.
A more appropriate unit of measurement is days. It gives a better idea of the amount of time it will take to watch the entire series.
Lesson at a Glance
In this lesson, students predict which unit of measurement is appropriate for a variety of situations and share their predictions by engaging in a Take a Stand routine. They test their predictions by converting units of measurement to examine whether their choice makes the most sense for each situation. Students watch a video and are challenged to compare two advertisements for two-way radios, each providing information in different units of measurement. This activity allows students to better understand the importance of choosing an appropriate unit of measurement.
b. Convert the unit from seconds to the unit of measurement you chose in part (a). Round your answer to the nearest tenth of a unit.
There are 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute.
60 60 = 86,400 = 8.64 × 10 4
There are 8.64 × 10 4 seconds in a day.
Key Questions
• Is choosing an appropriate unit of measurement important? Why?
• Does choosing a unit of measurement influence the comparison of two quantities? How?
Achievement Descriptor
8.Mod1.AD13 Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. (NY-8.EE.4)
• Prepare three signs labeled Smallest Unit, Medium Unit, and Largest Unit. Colors for the wording should match for like-measured items: length, time, and weight.
◦ On the Smallest Unit sign, write Minutes, Ounces, and Millimeters.
◦ On the Medium Unit sign, write Hours, Pounds, and Centimeters.
◦ On the Largest Unit sign, write Days, Tons, and Meters.
• Post the signs in three different areas of the classroom.
Fluency
Convert to the Equivalent Unit of Measurement
Students convert to equivalent units of measurement to prepare for choosing appropriate units of measurement.
Directions: Convert to the equivalent unit of measurement.
Teacher Note
Consider giving students a conversion chart to highlight the relationships in this Fluency.
Launch
Students choose the appropriate unit of measurement.
Guide students to problems 1–3. Ask them to think about and record which unit of measurement they would choose for each situation. No calculations are necessary yet. Have students make predictions and write their explanations.
For problems 1–3, choose a unit of measurement to use for each problem. Then explain why you chose that unit of measurement.
1. You wonder how long you can listen to your favorite music before you hear the same song twice. You have 1000 songs, and the average song length is 4 minutes.
Would you want to display the time in minutes, hours, or days of music?
Displaying the time in minutes does not make sense to me because I know the value will be in the thousands place. That will still be a lot of hours, so I would want to display the time in days.
2. You wonder about the weight of all the oranges used for commercial production in the United States in a year. Your research indicates that each box of oranges weighs about 85 pounds, and about 92 million boxes of oranges were used for commercial production in 2017 and 2018.
Would you want to display the weight in ounces, pounds, or tons of oranges?
Displaying the weight in ounces or pounds does not make sense to me because I know the value will be in at least the billions place. So I would want to display the weight in tons.
3. The seafloor spreads at a rate of approximately 10 centimeters per year. You collect data on the spread of the seafloor each week.
Would you want to record the data in millimeters, centimeters, or meters?
The seafloor spreads 10 centimeters per year, which is less than 1 centimeter per month. Since I’m collecting data each week, it makes the most sense to record the seafloor spread in millimeters.
UDL: Action & Expression
By prompting students to stop and think before they begin the Take a Stand routine, you are supporting students in planning and strategizing.
After most students finish, introduce the Take a Stand routine to the class. Draw students’ attention to the signs hanging in the classroom: Smallest Unit, Medium Unit, Largest Unit. Read problem 1 aloud, and then invite students to stand beside the sign that best describes their choice for a unit of measurement.
Smallest Unit
Medium Unit Hours Pounds Centimeters
Largest Unit
Language Support
During the Take a Stand routine, direct students to use the Agree or Disagree section of the Talking Tool as a support for discussing the reasons why they chose a specific sign.
When all students are standing near a sign, allow 2 minutes for groups to discuss why they chose that sign. Then call on each group to share reasons for their selection. Invite students who change their minds during the discussion to join a different group.
Continue the routine for problems 2 and 3.
Have students return to their seats after choosing a unit of measurement for all three problems. Reflect on key takeaways and have students revise their explanations if necessary.
When choosing a unit of measurement, what did we have to consider?
We want a value and a unit of measurement that makes sense to us for the situation. It should be a familiar unit of measurement of a reasonable size.
Sometimes there was a frequency, like every year or every week.
Sometimes there was a number of items that would be multiplied many times by another unit of measurement. The value, or size of that product, would need to be considered when deciding which unit of measurement to use.
Sometimes, we can better interpret measurements when we convert their units of measurement. Today, we will learn how to choose an appropriate unit of measurement to make sense of very small and large quantities.
Learn
Converting Units of Measurement
Students convert units of measurement within a given situation.
In this segment, students work in pairs to verify whether their choice of unit of measurement makes sense in each situation given in problems 1–3. Distribute calculators to ease conversion from one unit of measurement to another. This will allow students to focus on the choice of unit of measurement and not on the calculations required to convert the units. Guide students’ work by using the following prompt.
Convert the units of measurement in problems 1–3 to determine whether we made the right choices.
As students work, listen for student pairs to identify the conversion fact and the operation they will use to convert the unit of measurement. Solutions for each unit of measurement are provided.
Problem 1:
There are 1000 songs at 4 minutes each. There are 1000 · 4, or 4000, minutes of music.
There are 60 minutes in an hour. There are 4000 ____ 60 , or 66 2 _ 3 , hours of music.
There are 24 hours in a day. There are 66 2 3 ÷ 24, or about 2.8, days of music.
Problem 2:
Change the number of boxes from 92 million to scientific notation, which is 9.2 × 10 7 .
Each box weighs 85 pounds. There are 85 · (9.2 × 10 7), or 7.82 × 10 9, pounds of oranges.
There are 16 ounces in a pound. There are 16 · (7.82 × 10 9), or 1.2512 × 10 11 , ounces of oranges.
There are 2000 pounds in a ton. There are 7.82 × 10 9 2 × 10 3 , or 3.91 × 10 6, tons of oranges.
Problem 3:
There are 52 weeks in a year. The seafloor spreads 10 52 , or about 0.192 307 6923, centimeters per week.
There are 10 millimeters in a centimeter. The seafloor spreads 0.192 307 6923 · 10, or 1.923 076 923, millimeters per week.
There are 0.01 meters in a centimeter. The seafloor spreads 0.192 307 6923 · 0.01, which is 0.001 923 076 923, or 1.923 076 923 × 10 −3, meters per week.
Debrief the work of problems 1–3 by using the following prompts. Encourage students to build upon one another’s responses.
Why do the measurements for problem 1 make more sense when using the units of measurement we chose?
It’s easier to relate to the values in the answer by using our choice of unit of measurement. For example, it is easier to make sense of 2.8 days than 66 2 3 hours.
In problem 2, is 3.91 × 10 6 tons easier to understand than the other choices? Why?
No, tons is still difficult to understand in this problem because the value is so large. But tons is the largest unit of weight we have, so we cannot convert to a larger unit of measurement.
In which problems, if any, did you use the properties of exponents in your calculations? Why?
I used the properties of exponents in problem 2 because it was simple to write 2000 and the number of boxes of oranges in scientific notation.
Seconds of Life
Students recognize that the unit of measurement choice is important to appreciate the size of a measurement.
Introduce problem 4 and have students work with a partner to find the answer. When most students are finished, call the class together to review the solution.
4. Mr. Jacobs just welcomed a baby into his family. Since birth, the baby has been alive for 1 million seconds. Mr. Jacobs has been alive for 1 billion seconds. How many seconds older is Mr. Jacobs than the baby? Write your answer in scientific notation.
1 million = 1,000,000
1 billion = 1,000,000,000
1,000,000,000 − 1,000,000 = 999,000,000
= 9.99 × 10 8
Mr. Jacobs is 9.99 × 108 seconds older than the baby.
Confirm answers and discuss the converting of units of measurement by using the following prompts.
How long do you think 9.99 × 10 8 seconds is?
I think 9.99 × 10 8 seconds is about 2 years.
I think 9.99 × 10 8 seconds is about 40 years.
Should we convert the units of measurement from seconds to another unit of measurement? If so, what unit of measurement should we use? Why?
Yes, we should convert seconds to years because it is hard to make sense of someone’s age being 9.99 × 10 8 seconds.
Circulate as pairs of students convert 9.99 × 10 8 seconds to years. Look for students who divide by using the properties of exponents. Offer guidance to students who have trouble determining which operation to use to convert units of measurement.
There are 365 days in a year, 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute.
365 · 24 · 60 · 60 = 31,536,000 = 3.1536 × 10 7
Teacher Note
During preparation for this lesson, consider reviewing grade 6 module 1 topic D for suggested dialogue that will help students recall how to convert units.
There are 3.1536 × 10 7 seconds in a year. 9.99 × 10 8 3.1536 × 10 7 = ( 9.99 3.1536 ) × (10 8 10 7 ) ≈ 3.1678 × 10 = 31.678
When most students complete the problem, continue the class discussion by using the following prompts.
What is the difference in the ages of Mr. Jacobs and the baby in the new unit of measurement you chose?
The difference in the ages of Mr. Jacobs and the baby is about 31.678 years.
Is years an appropriate unit to measure the difference in their ages? Why?
Yes, we usually compare ages in years, so it makes sense to measure the difference in years.
Yes, 31.678, or about 32, years makes more sense than 9.99 × 10 8 seconds.
How accurate was your prediction for the value of 9.99 × 10 8 seconds?
My prediction was not very accurate. I had no idea the actual difference would be close to 32 years.
Why is the choice of unit of measurement important?
It is hard to understand the size of a measurement when the value of the measurement is too large, too small, or unrelatable.
Just as we would not measure large things like the width of the United States in inches, we also would not measure small things like the diameter of a blood cell by using miles. We want the measurement to be relatable.
Two-Way Radio Problem
Students explore the importance of choosing an appropriate unit of measurement.
Play the Miscommunication video. Then facilitate a discussion by using the following prompts. What was the problem the teenager encountered when purchasing the two-way radios?
The teenager didn’t look at the units of measurement and only looked at the numbers on the boxes.
When purchasing an item, it is important to have all of the information. Units of measurement are an important part of any comparison.
Facilitate a class discussion about why the two-way radios did not work in the video.
Why did Henry’s Handhelds, the two-way radios that were purchased, not work?
Henry’s Handhelds did not work because the friends live 20 miles apart, and the range of the handhelds is 85,000 feet, or approximately 16 miles. Because 20 > 16, the range is not far enough to connect the friends.
We want to develop an advertisement that shows Winnie’s Walkie Talkies are a much better choice than Henry’s Handhelds. What type of comparisons can we make between the two companies?
We can say how much farther the range is for Winnie’s Walkie Talkies than the range is for Henry’s Handhelds.
We can say how many times as far the range of Henry’s Handhelds is as the range of Winnie’s Walkie Talkies.
Tell student pairs to create a persuasive advertisement for either company by using a comparison statement. Circulate and use the following prompts to guide students to the different options for comparison:
• What other units of measurement can you use to compare the ranges of the two-way radios?
• Do you think the comparison would be different if you used feet instead of miles? Miles instead of feet?
• Why did you make that final decision regarding how you would compare the two companies?
Promoting the Standards for Mathematical Practice
When students work to compare the two quantities from the advertisements, they are using a real-world context to aid in reasoning with mathematical objects or reasoning abstractly and quantitatively (MP2).
Ask the following questions to promote MP2:
• What is the problem asking you to do?
• Does your answer make sense when comparing the range of the two-way radios?
• Does the solution you found make sense mathematically?
As students work, identify which pairs will share their comparison statement with the class. Select pairs that use difference in distance statements and how many times as far as statements for both miles and feet. Ask the student pairs that finish early to write another comparison statement and choose the most persuasive advertisement.
5. Consider the given information about Henry’s Handhelds and Winnie’s Walkie Talkies. Create a persuasive advertisement for either company by using a comparison statement. Use the space provided to show your work in making the comparison statement.
Henry’s Handhelds range in miles: 85,000 5280 ≈ 16.098
Winnie’s Walkie Talkies range in feet: 30(5280) = 158,400
Difference in distance:
Winnie’s Walkie Talkies has a range that is about 14 miles farther than the range of Henry’s Handhelds because 30 − 16.098 ≈ 14.
Winnie’s Walkie Talkies has a range that is 73,400 feet farther than the range of Henry’s Handhelds because 158,400 − 85,000 = 73,400.
How many times as far as:
Winnie’s Walkie Talkies has a range that is about 2 times as far as the range of Henry’s Handhelds because 30 16.098 ≈ 1.864.
Winnie’s Walkie Talkies has a range that is about 2 times as far as the range of Henry’s
Handhelds because 158,400 85,000 ≈ 1.864.
Use this space to write your advertisement comparison statement.
Winnie’s Walkie Talkies has a range that is about 14 miles farther than the range of Henry’s Handhelds.
Winnie’s Walkie Talkies has a range that is 73,400 feet farther than the range of Henry’s Handhelds.
Winnie’s Walkie Talkies has a range that is about 2 times as far as the range of Henry’s Handhelds.
Invite selected student pairs to share their choice of advertisement statement with the class. Have them explain their reasoning by displaying their work. After the selected pairs share, have all students choose the most persuasive comparison statement shared by their classmates. Then have a class discussion about which advertisement comparison statement they chose, and ask them to justify their response.
Help students see that there is not one correct answer by providing the following examples.
• One student pair may choose to advertise the range of Winnie’s Walkie Talkies as being 14 miles farther than the range of Henry’s Handhelds because they feel customers can more readily understand that distance.
• Another pair may choose to advertise Winnie’s Walkie Talkies as having a range of 73,400 feet farther than Henry’s Handhelds because it is the largest number of the three comparison statements.
• Yet another pair may choose to advertise that the range of Winnie’s Walkie Talkies is 2 times as far as the range of Henry’s Handhelds because that comparison makes the most sense to customers.
Discuss that all of these comparisons are accurate and that good consumers look critically at advertised comparisons.
Language Support
Consider supporting students in sharing thoughts and ideas by providing students with sentence frames for the comparison statements.
• Winnie’s Walkie Talkies has a range that is farther than the range of Henry’s Handhelds.
• Winnie’s Walkie Talkies has a range that is times as far as the range of Henry’s Handhelds.
Differentiation: Challenge
Have students research products offered in your area, and then compare features of those products. Some examples include the following:
• Connectivity speeds of Wi-Fi
• Data space on flash drives
• Advertising space in newspapers
• Students should determine which product offers the best feature and create an advertisement plan that includes the comparison statement to show how much better the suggested product is than another.
To make a connection to art and design, students could design an advertisement poster for the company by using their comparison statement.
Land
Debrief 5 min
Objective: Choose appropriate units of measurement and convert units of measurement.
Lead a class discussion by using the following prompts. Encourage students to provide an example to help explain their reasoning.
Is choosing an appropriate unit of measurement important? Why?
It is important to choose an appropriate unit of measurement that makes sense for the situation and is easy for us to understand. The number portion of the measurement can help determine whether the most appropriate unit of measurement was chosen. For example, if the number portion is really large, we might switch to a larger unit of measurement so that the number is smaller and easier to interpret. For example, it is easier for us to understand the age difference between Mr. Jacobs and the baby as about 32 years instead of 999 million seconds.
Does choosing a unit of measurement influence the comparison of two quantities? How?
Yes, choosing a unit of measurement can influence the comparison of two quantities. There are different ways to compare quantities, such as by their difference or by how many times as large one quantity is as another quantity. Depending on the units of measurement used, the value in one difference comparison may look very different from the value in another difference comparison. However, the how many times as large as comparisons do not change even if we convert the unit of measurement.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
For the Practice problems, students should have access to a scientific calculator or similar digital device.
Choosing Units of Measurement
In this lesson, we
• determined the appropriate unit of measurement.
• converted one unit of measurement to a more appropriate unit of measurement.
Example
A monarch butterfly can travel about 1.552 × 10 7 inches each day.
a. If you report this data, what unit of measurement would you choose? Explain why you chose that unit.
I would choose miles to report this data.
1 foot = 12 inches
1 mile = 5280 feet
There are 12 ⋅ 5280 or 63,360 inches in 1 mile. Divide 1.552 × 10 7 inches by 63,360 inches to find how many miles the butterfly can travel each day.
I chose miles because 2.4 × 10 2 miles, or 240 miles, is easier for me to visualize than 1.552 × 10 7 inches, or 15,520,000 inches.
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
For problems 1–4, circle the unit of measurement that makes the most sense to use for the situation.
1. Total area of the continent of Africa
A. Square feet
B. Square miles
C. Square inches
3. Weight of a newborn baby
A. Ounces
B. Pounds
C. Tons
2. Age of a statue made last week
A. Days
B. Months
C. Years
4. Weight of a car
A. Milligram
B. Gram
C. Kilogram
5. Human hair grows at a rate of about 15 centimeters per year. If you measure the hair lengths for ten different people each month, which unit should you use to record your data? Explain.
Because I measure the hair lengths each month, and the hair grows at a rate of about 15 centimeters per year, I can record my data in centimeters.
I would use hours to report this data because 8.61624 × 10 4 seconds equals 23.934 hours, which is nearly a complete day. I would use hours as the unit of measurement because 23.934 hours is easier for me to understand than 1436.04 minutes or 8.61623 × 10 4 seconds. Name Date
7. A light-year is the distance that light travels in 1 year. For example, the distance from the closest star, Proxima Centauri, to Earth is 4.243 light-years because it takes light 4.243 years to reach Earth from Proxima Centauri.
Light travels at a speed of approximately 9.46 × 10 15 meters per year. Determine which stars given in the table are between 1.3 × 10 17 and 2.5 × 10 17 meters from Earth.
Star Name Distance from Earth (light years) Alpha Canis Majoris (Sirius) 8.6 Alpha Canis Minoris (Procyon) 11.41
Lyrae (Vega)
Alpha Lyrae (Vega) and Delta Pavonis are between 1.3 × 10 17 and 2.5 × 10 17 meters from Earth.
8. Shawn’s Wi-Fi signal has a range of 380 kilometers. Ava’s Wi-Fi signal has a range of 13,200 feet. The range of Ava’s Wi-Fi signal is how many times as much as the range of Shawn’s Wi-Fi signal? (1 km ≈ 3281 ft)
Ava’s Wi-Fi signal has about 0.010 587 times as much range as Shawn’s Wi-Fi signal.
6. It takes Earth 8.61624 × 10 4 seconds to complete a single rotation on its axis. If you report this data, what unit of measurement would you use? Explain why you chose that unit.
EUREKA MATH
9. Maya bought a fish tank that has a volume of 175 liters. She read a fun fact that it would take 7.43 × 10 18 fish tanks of water at that tank size to fill all the oceans in the world.
a. Use the fun fact to determine the total volume of water in the world’s oceans in liters. Write your answer in scientific notation.
The total volume of the water in the world’s oceans is 1.30025 × 10 21 liters.
b. Given that 1 liter = 1 × 10 −12 cubic kilometers, find the total volume of water in the world’s oceans in cubic kilometers. Write your answer in scientific notation.
The total volume of the water in the world’s oceans is 1.30025 × 10 9 cubic kilometers.
c. The Atlantic Ocean has a volume of about 323,600,000 cubic kilometers. You bought a fish tank that holds 75 more liters than Maya’s tank. About how many tanks like yours would it take to fill the Atlantic Ocean? Write your answer in scientific notation. It would take about 1.2944 × 10 18 tanks like mine to fill the Atlantic Ocean.
Choose all the values that are greater than 409.806.
Get to the Point
Model a situation by operating with numbers in scientific notation.
Lesson at a Glance
This lesson is an open-ended modeling exploration. Students learn about an impressionist painting technique and then examine a piece of artwork composed of small distinct dots. Operating with very large and very small positive numbers, students estimate how many brush strokes the painting has and compare their estimates with their initial guesses. Students discover the selling price of the painting and estimate the value of each brush stroke. Groups of students present their solution strategies to the class.
Key Question
• What types of problems benefit from using numbers in scientific notation?
Achievement Descriptor
8.Mod1.AD12 Operate with numbers written in scientific notation to solve real-world problems. (NY-8.EE.4)
Agenda
Fluency
Launch 5 min
Learn 30 min
• Point by Point
• How Large?
• Expensive Dots
Land 10 min
Materials
Teacher
8 of each of the following:
• Highlighters
• Markers
• Paper (lined, grid, and blank)
• Rulers, inch and metric
• Transparency film
• Scientific calculators
Students
• None
Lesson Preparation
• Arrange materials in a designated area of the classroom.
Fluency
Operate with Numbers Written in Scientific Notation
Students operate with numbers written in scientific notation to prepare for modeling a situation with numbers written in scientific notation.
Directions: Evaluate. Write your answer in scientific notation.
Launch
Students learn about artistic techniques and generate questions about a piece of art.
Direct students to complete problem 1 independently.
1. Match the painting technique to the work of art. Fill in each blank with the correct artist.
Painting Technique
Pointillism Fauvism
Works of Art
Paul
Impressionism
Vincent
Pointillism: Fauvism: Impressionism:
Paul Signac Vincent van Gogh Claude Monet
Claude Monet Weeping Willow
Signac Notre-Dame de la Garde
van Gogh Landscape from Saint-Rémy
Engage in a class discussion by posing some of the following questions:
• What similarities and differences do you notice about the painting techniques?
• What similarities and differences do you notice among the paintings?
• Did you make connections between the painting techniques and the paintings? How?
Today, we will examine the painting technique of pointillism, which involves the application of paint in carefully placed brush strokes to form dots. Because we are able to see each brush stroke, we are able to answer questions by using information we learned in this topic about exponents and scientific notation.
Learn
Point by Point
Students choose tools to estimate the number of brush strokes in a painting.
Display the painting La Corne d’Or. Matin by Paul Signac. Direct students to problem 2. Have them study the painting more closely and record their questions.
UDL: Action and Expression
Consider providing a graphic organizer with the following sections to help students organize their plan:
• Focus question
• Needed information
• Useful tools
• Plan
Record some questions that come to mind.
After most students have a few questions recorded, lead the class in a discussion by using the following prompts.
When you look at the painting, what questions come to mind?
Resist the urge to answer the questions. Instead, use this time to encourage student curiosity.
2. Study the painting La Corne d’Or. Matin by Paul Signac.
Record questions, one at a time, and display them for the class. Expect students to be curious about the colors, the painting technique, or the location depicted in the painting. When a student volunteers a question, ask the class who else has that same question. Record the number of students who have the same question by placing check marks, plus signs, or other counting marks next to each question.
If students do not question how many dots or brush strokes are in the painting, offer that as your question. Find out how many students also find that question interesting.
We won’t be able to address all these questions today. Let’s focus on estimating the number of brush strokes in Signac’s painting.
Direct students to record the focus question in problem 3.
Without counting, make an educated guess that answers the focus question. Write your guess in scientific notation.
3. Focus Question:
How many brush strokes are in Signac’s pointillism painting?
Guess:
Sample: 2 × 10 6
Now that you have a guess, let’s estimate how many brush strokes are in Signac’s pointillism painting.
Divide the class into groups of four. Distribute the following tools to each group: calculator, highlighter, marker, paper (lined, grid, and blank), ruler, and transparency. Give groups time to brainstorm a plan for how to use the provided tools to estimate the number of brush strokes in the painting. Circulate to monitor progress as groups discuss their planning. Expect students to generate plans similar to the following:
• We can use a ruler to measure a small area of the painting and count the brush strokes by using a highlighter within that area. Then we can multiply to estimate the number of brush strokes in the entire painting.
• We can count the number of brush strokes along the base and along the height of the painting. Then we can multiply those numbers to estimate the number of brush strokes in the entire painting.
Promoting the Standards for Mathematical Practice
When students apply their understanding of area to estimate the total number of brush strokes in the painting La Corne d'Or. Matin and model the painting as a rectangle and each brush stroke as a unit of area, they are modeling with mathematics (MP4).
Ask the following questions to promote MP4:
• What assumptions can you make to simplify the problem of finding the number of brush strokes in the painting?
• How can you improve your method of estimation to better determine the number of brush strokes in the painting?
• What math can you write to represent this problem?
Have students work with their groups to complete problem 4.
Gather data to make an informed estimate that answers the focus question. Record your strategy and work in the space provided in problem 4. Describe which tools you used and how you used them to gather your information and make an estimate.
Note that estimates will vary, but they should be within a range of 15,000 to 25,000 brush strokes.
4. Strategy:
Sample: We can use a ruler to measure a small area of the painting and count the brush strokes by using a highlighter within that area. Then we can multiply to estimate the number of brush strokes in the entire painting.
Tools Used:
Sample: Ruler and highlighter
Estimate:
Sample: 19,000 brush strokes
When all groups of students finish, invite them to share their solution strategies. After all groups have shared, engage students in a discussion by asking the following questions:
• Are the answers you found reasonable? Are you surprised by the results?
• What assumptions did you make? How did those assumptions affect your solution?
• How is your solution strategy similar to another group’s strategy? How is it different?
• Were your estimates close to each other?
• How could you have increased the accuracy of your estimates?
How Large?
Students compare their guess and estimate for the focus question.
Have students complete problem 5 individually. Then have students discuss their answers with their group. After the group discussions, ask groups to share their answers with the class.
5. Consider your answers from problems 3 and 4.
a. Which is larger: your guess from problem 3 or your estimate from problem 4?
Sample: My guess is larger.
b. How many times as large is it as your other answer?
Sample:
My guess is about 1.05 × 10 2 , or 105, times as large as my estimate.
Expensive Dots
Students estimate the dollar value of each brush stroke in the painting.
Display the following value: 8 × 10 6 .
This number represents the painting in some way. What do you think the number 8 × 10 6 represents?
Record the answers students share. Then share an additional piece of information with the class to launch the next focus question.
Signac’s painting appraises for about 8 million dollars. What questions do you have now?
Record the questions that students generate. Note the number of students who have the same question by placing check marks, plus signs, or other counting marks next to each question.
If students do not question the dollar value of each brush stroke, offer that as your wondering and find out how many students also find that question interesting.
We will not be able to address all of these questions. Let’s focus on estimating the dollar value of each brush stroke in the painting by using your estimate from problem 4.
Direct students to record the focus question in problem 6. Then instruct them to work with their groups to gather the needed information to answer the focus question. Allow students access to the same tools available throughout the lesson.
6. Focus Question:
What is the dollar value of each brush stoke?
Estimate:
Sample:
Teacher Note
Although the dollar value of a painting is not based on the number of brush strokes, it serves as an intriguing context for students to explore.
The value of each brush stroke is about
When groups finish, invite them to share their estimates with the class. Then use the following prompts to engage students in a class discussion:
• Did you work out the problem by using numbers written in scientific notation? Why?
• Did you all calculate similar dollar values?
• What could account for the variety of results in the class?
Land
Debrief
5 min
Objective: Model a situation by operating with numbers written in scientific notation.
Use the following questions to help students recognize where they used operations in scientific notation. Encourage students to add to their classmates’ responses.
What types of problems benefit from using numbers in scientific notation? Why?
Problems that involve very large numbers can benefit from the numbers being written in scientific notation before we operate with them. Writing numbers in scientific notation sometimes makes operations simpler because we can apply properties of exponents to solve the problem.
Select from the following questions to further debrief the lesson:
• What assumptions did your group make about the painting, and how did that affect your results?
• Where did your group struggle?
• How did your group overcome obstacles?
• Is there anything your group would do differently?
• What was most helpful?
• How could we be more precise in the estimates we made today?
Teacher Note
The Exit Ticket asks students to reflect on this lesson. Consider guiding students’ reflection with a question that was not asked during the debrief.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson.
For the Practice problems, students should have access to a scientific calculator or similar digital device.
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
1. What assumptions did you make when you modeled the number of brush strokes in the painting La Corne
Matin?
We assumed that each brush stroke was the same size.
2. Was writing numbers in scientific notation helpful in the lesson? If so, when? If not, why not?
Yes, in problem 6 when I had to find the cost of each brush stroke, writing the number of brush strokes and the appraised value of the painting in scientific notation was helpful.
3. If you had more time to explore another question related to Paul Signac’s paintings, what question would you choose? What would be your plan to determine the answer?
Student responses will vary.
For problems 4 and 5, use your estimations from the lesson.
4. Paul Signac completed about 500 pointillism paintings during his career as a painter.
a. Paul Signac painted about how many brush strokes during his entire career?
Paul Signac painted about 9.5 × 10 6, or 9,500,000, brush strokes during his entire career.
b. What assumptions did you make in answering part (a)?
Sample: I assumed all of Signac’s paintings have the same amount of brush strokes. I also assumed he never had to repaint a brush stroke.
c. Do you think your estimation for part (a) is reasonable?
Sample: No, I do not think my estimate is reasonable. Signac likely did not use the same number of brush strokes in each of his 500 paintings. Also, I think Signac worked on more than 500 paintings, but he may not have completed all those paintings.
5. You want to recreate the La Corne d’Or. Matin painting. Each brush stroke takes approximately 20 seconds to paint, which includes mixing the paint, painting the canvas, and washing your brush.
a. How long does it take you to recreate the painting if you paint at a constant rate?
Sample:
Each brush stroke takes 20 seconds. There are 1.9 × 10 4 brush strokes. It takes (2 × 10)(1.9 × 10 4), or 3.8 × 10 5, seconds to recreate the painting.
There are 60 seconds in 1 minute. It takes 3 8 × 105 60 , or approximately 6.3 × 10 3, minutes to recreate the painting.
There are 60 minutes in 1 hour. It takes 6 3 × 103 60 , or approximately 105, hours to recreate the painting.
There are 24 hours in 1 day. It takes 105 24 , or 4 3 8 , days to recreate the painting.
b. What unit of measurement did you choose to report the amount of time it will take in part (a)? Explain why you chose this unit of measurement.
Sample: I chose to report the amount of time in days because the approximate time was over 24 hours.
Remember For problems 6–9, multiply.
EUREKA MATH2 New
d’Or.
Topic D
Perfect Squares, Perfect Cubes, and the Pythagorean Theorem
Topic D introduces the Pythagorean theorem to motivate and support the need for expressing numbers that are not rational and to develop students’ early awareness of the meaning of these numbers. Working with the Pythagorean theorem inherently builds on and extends students' prior exploration in grade 7 of the conditions that determine a unique triangle.
Students begin topic D by exploring geometric representations of perfect squares and perfect cubes and relate these concepts to area and volume before expanding to consider squares and cubes of any rational number. Students rely on previous understanding of squares and cubes as products of equal factors to reason intuitively about square roots and cube roots before any notation is introduced. Students apply similar intuitive reasoning to solve equations of the forms x 2 = p and x 3 = p, where p is a perfect square or a perfect cube, respectively. Students revisit these equations where p is any rational number at the end of topic E.
Students engage with the story What's Your Angle, Pythagoras? to explore, anecdotally and experimentally, the Pythagorean theorem, and emerge with an assumption that the relationship a 2 + b 2 = c 2 holds true for any right triangle. They will formally prove the Pythagorean theorem in module 2. Students apply the Pythagorean theorem to solve for hypotenuse lengths by concentrating first on values of c 2 that are perfect squares. Students focus on finding only hypotenuse lengths in module 1 and solve for leg lengths in module 2.
Later in module 1, they encounter values of c 2 that are not perfect squares in the context of finding hypotenuse lengths, which naturally creates the need for square roots. Students first develop an understanding of square roots by finding two consecutive whole numbers that the value c must lie between. Square root notation is then formalized, and students use it to express hypotenuse lengths.
Students create their own Spiral of Theodorus and use the lengths from the spiral to observe the placement of square roots on a number line, which lays the groundwork for approximating values of irrational numbers. In topic E, students expand on their number line work to approximate values of square roots to the nearest hundredth and to explore and approximate cube roots. Topic E also formally defines irrational numbers. Students revisit the Pythagorean theorem in module 2.
Progression of Lessons
Lesson 16 Perfect Squares and Perfect Cubes
Lesson 17 Solving Equations with Squares and Cubes
Lesson 18 The Pythagorean Theorem
Lesson 19 Using the Pythagorean Theorem
Lesson 20 Square Roots
Perfect Squares and Perfect Cubes
Recognize perfect squares from 1 to 225 and perfect cubes from 1 to 125.
Determine all numbers that square or cube to a given number.
Lesson at a Glance
Students explore the concepts of a perfect square and a perfect cube through geometric representations and connections to area and volume. Then students widen their scope based on the definitions of perfect squares and perfect cubes to classify a variety of other numbers as perfect squares, perfect cubes, both, or neither. Throughout the lesson, students rely on an intuitive sense of the relationships between squares and square roots and cubes and cube roots. Students apply this understanding to find the squares and cubes of decimals and fractions. This lesson formally defines perfect square and perfect cube.
Key Questions
• How do you know whether a number is a perfect square?
• How do you know whether a number is a perfect cube?
Achievement Descriptor
8.Mod1.AD7 Evaluate square roots of small perfect squares and cube roots of small perfect cubes. (NY-8.EE.2)
Agenda
Fluency
Launch 5 min
Learn 30 min
• Area and Volume
• Which Category?
• Squares and Cubes of Fractions and Decimals
Land 10 min
Materials
Teacher
• None
Students
• Perfect Squares and Perfect Cubes cards (1 per student pair)
Lesson Preparation
• Copy and cut out the Perfect Squares and Perfect Cubes cards (in Teach book). Prepare enough sets for 1 per student pair.
Fluency
Multiply Rational Numbers
Students multiply an odd or an even amount of negative factors to prepare for determining which number when squared or cubed is a given number.
Directions: Multiply.
Launch
Students explore geometric representations of perfect squares and cubes.
Display the Perfect Squares interactive.
Here is one tile.
Ask students the following questions.
Suppose we have 49 tiles. Can we arrange all 49 tiles to form a square? How do you know?
Give students a few moments to talk with a partner. Then invite a pair to share their answer and reasoning. Use the interactive to verify the answer. Repeat the process for 27 tiles and 81 tiles. Consider having students predict another number of tiles that can form a square and another number of tiles that cannot form a square.
Display the Perfect Cubes interactive.
Here is one block.
Ask students the following questions.
Suppose we have 27 blocks. Can we arrange all 27 blocks to form a cube? How do you know?
Give students a few moments to talk with a partner. Then invite a pair to share their answer and reasoning. Use the interactive to verify the answer. Repeat the process for 49 blocks and 125 blocks. Consider having students predict another number of blocks that can form a cube and another number of blocks that cannot form a cube.
Some numbers of tiles or blocks form squares or cubes and some do not. Today, we will learn to identify these special numbers. 5
UDL: Action & Expression
Consider supporting students to express learning in flexible ways. For example, students may benefit from a tactile experience. Consider providing grid paper, square tiles, and unit cubes for students to draw, arrange, and stack as they try to form squares and cubes by hand.
Learn
Area and Volume
Students apply prior knowledge of area and volume to explore perfect squares and perfect cubes.
Direct students’ attention to the Area and Volume problems. Have students complete problems 1−4 independently.
1. Use what you know about the area of a square to complete the table.
2. Suppose a square has an area of 25 square units. What is its side length? Explain how you know.
The side length is 5 units. The side lengths of a square, which represent the base and height, are equal. The area of a square is the product of the base and height. Since the area is 25 square units, the side lengths must be 5 units because 5 · 5 = 25.
3. Use what you know about the volume of a cube to complete the table.
4. Suppose a cube has a volume of 125 cubic units. What is its edge length? Explain how you know.
The edge length is 5 units. The edge lengths of a cube, which represent the length, width, and height of the cube, are equal. The volume of a cube is the product of the length, width, and height. Since the volume is 125 cubic units, the edge length must be 5 units because 5 · 5 · 5 = 125.
Have students check answers with a partner. Then use the following prompts to facilitate a discussion about problems 1−4.
If you know the side length of a square, how do you find its area?
To find the area of a square, multiply two side lengths.
If you know the edge length of a cube, how do you find its volume?
To find the volume of a cube, multiply three edge lengths.
Problem 2 gives the area of a square. Describe how you can determine the square’s side length from its area.
I know that squares have equal side lengths. So I have to think of a product with two equal factors that multiply to 25.
Problem 4 gives the volume of a cube. Describe how you can determine the cube’s edge length from its volume.
I know that cubes have equal edge lengths. So I have to think of a product with three equal factors that multiply to 125.
When we multiply two equal factors, we say we are squaring the number. When we multiply three equal factors, we say we are cubing the number.
Which Category?
Students identify whether a number is a perfect square, a perfect cube, both, or neither.
Students work in pairs on a card sort activity. Distribute one set of Perfect Squares and Perfect Cubes cards to each pair. Ask students to review the cards, identify relationships among them, and then sort them into two categories. Tell students that they should be prepared to describe their categories to the class.
Circulate as students work to observe how they are sorting the cards. At this point, honor any categories into which students choose to sort the cards. However, identify pairs of students who sort the cards into the following two categories:
• Numbers that are perfect squares
• Numbers that are perfect cubes
Language Support
For students who may need support defining squaring and cubing a number, consider using the following statements.
• We say we are squaring a number when we multiply a number by itself.
• We say we are cubing a number when we multiply a number by itself and then by itself again.
Then display a square with a side length of 5 units. Find the area of the square by using both equations 5 · 5 = 25 and 5 2 = 25. Repeat for a cube with an edge length of 5 units.
Invite these student pairs to share with the class which cards are in each of their categories. Then have them describe what the cards in each of their categories have in common and how the two categories differ from each other.
Use the following prompts to define perfect squares and perfect cubes.
The numbers in one category of cards are all perfect squares. A perfect square is the square of an integer. What are some examples of perfect squares?
Some examples are 4, 9, and 16.
What types of numbers are integers?
Integers are all the positive and negative whole numbers and zero.
How do you know the number 25 is a perfect square?
I know 25 is a perfect square because it is the square of the integer 5.
Are there any other integers we can square to get 25?
Yes, we can square −5 to get 25.
The integers 5 and −5 can both be squared to get the perfect square 25.
Direct students to the Venn diagram in their book. Have them use the label Perfect Squares for the left section in their Venn diagram and record the examples of perfect squares from the card sort. Then continue the class discussion by using the following prompts.
The numbers in the other category of cards are all perfect cubes. A perfect cube is the cube of an integer. What are some examples of perfect cubes?
Some examples are 8, 27, and 125.
How do you know the number 8 is a perfect cube?
I know 8 is a perfect cube because it is the cube of the integer 2.
Are there any other integers we can cube to get 8?
No, there are no other integers we can cube to get 8.
The integer 2 is the only integer we can cube to get the perfect cube 8.
Have students use the label Perfect Cubes for the right section in their Venn diagram and record the examples of perfect cubes from the card sort.
Teacher Note
Square roots that result in numbers that are not rational are introduced later in this topic, and cube roots that result in numbers that are not rational are discussed in topic E.
Promoting the Standards for Mathematical Practice
When students notice that there are generally two integers you can square to get a perfect square, the exception being 0, but there is only one integer you can cube to get a perfect cube, they are looking for and expressing regularity in repeated reasoning (MP8).
Ask the following questions to promote MP8:
• Will there always be two integers you can square to get a perfect square?
• Will there always be one integer you can cube to get a perfect cube?
Next, have students use the label Both for the overlapping section of the Venn diagram and use the label Neither for the outside of the Venn diagram. Use the following prompts to transition the class into the next activity.
We saw some examples of perfect squares and perfect cubes in the card sort. Let’s use the definitions of perfect square and perfect cube to identify whether other numbers fit into the categories you labeled on your Venn diagram.
Have students complete problem 5 with their card sort partner. Tell them that several pairs of students will be called upon to share their reasoning for placements of numbers.
Differentiation: Support
To support students with understanding why −2 is not identified as an integer that when cubed results in 8, consider reviewing the rules about multiplying two numbers with the same sign and two numbers with different signs from grade 7. Help students examine and compare the products that generate perfect squares and perfect cubes by guiding students step-by-step through the products of (−5) 2 and (−2) 3 .
5. Record each number from the following list in the correct part of the Venn diagram.
UDL: Representation
The Venn diagram helps students visually organize the numbers into the correct categories and supports clear identification of numbers that are both perfect squares and perfect cubes.
After most students have finished, invite a pair of students to share with the class where they placed a number on the diagram and explain why. Repeat this process for each number. Consider having others indicate agreement by using a thumbs-up or another nonverbal signal. For students who disagree with the placement on the diagram, ask them to share their reasoning.
Depending on the responses and explanations provided by students, consider using any, or all, of the following prompts to debrief problem 5 with the class or with individual pairs.
If the explanations for the nonzero perfect squares mention squaring only a positive integer, probe further and ask the following question.
• Is there another number you can square to result in that perfect square? If so, what is it?
If students recognize that 81 100 and 0.027 result from squaring 9 10 and cubing 0.3, respectively, but wonder why those numbers are considered neither a perfect square nor a perfect cube, ask the following questions.
• What is the definition of a perfect square and a perfect cube?
• What type of number is 9 10 ? What type of number is 0.3?
If students classify 0 as neither a perfect square nor a perfect cube, ask the following questions.
• What is the definition of a perfect square and a perfect cube?
• Which number has a square of 0? Which number has a cube of 0?
• What type of number is 0?
If students classify −64 as both a perfect square and a perfect cube, ask the following questions.
• What sign does the square of a positive number have?
• What sign does the square of a negative number have?
• Which number can you square to result in −64?
After correct responses and explanations have been shared, ensure that students make any corrections to their Venn diagrams so they are accurate for future reference.
Language Support
Some students may benefit from additional support in providing their explanations verbally. Consider providing the following sentence frame examples.
The number is a perfect square because it is the square of .
The number is a perfect cube because it is the cube of .
The number is both a perfect square and a perfect cube because it is the square of and the cube of .
The number is neither a perfect square nor a perfect cube because
Teacher Note
Numbers that are squares or cubes of rational numbers, such as 81 100 and 0.027, are explored in detail in the next segment.
Conclude by asking students to look at their Venn diagrams and share what they notice about the signs of perfect squares and perfect cubes and where 0 fits. Summarize their findings.
A perfect square can be positive or 0, but never negative. A perfect cube can be positive, negative, or 0.
Squares and Cubes of Fractions and Decimals
Students apply what they know about perfect squares and perfect cubes to explore squares and cubes of fractions and decimals.
Direct students to the Squares and Cubes of Fractions and Decimals segment.
Recall that perfect squares and perfect cubes are the results of squaring and cubing integers only. However, we can square and cube any number.
Have students complete problems 6−9 individually and check their answers with a partner. Circulate to provide support as needed as students work on the calculations.
Although students can apply standard multiplication algorithms to evaluate the expressions, encourage them to use their understanding of known perfect squares and known perfect cubes, as well as properties of exponents, to make the computations simpler.
For example, (5 8 )
For decimal squares such as (−1.1) 2, encourage students to notice its relationship to (−11) 2. Because (−11) 2 = 121, the product (−1.1) 2 will also be positive and contain the digits 1, 2, and 1. Then use place value reasoning to place the decimal point.
For problems 6–9, evaluate.
Differentiation: Challenge
For students ready for an additional challenge, consider having them discuss with their partners the following questions.
• Are all perfect squares integers? Are all integers perfect squares? Explain.
• Are all perfect cubes integers? Are all integers perfect cubes? Explain.
Confirm answers to problems 6–9.
Then refer students back to their Venn diagrams. Use the following prompts to facilitate a discussion.
In which category did we decide the number 81 ___ 100 belongs? Why?
The number 81 ___ 100 is neither a perfect square nor a perfect cube because it is not the square or cube of an integer.
What do you notice about both the numerator and the denominator of 81 ___ 100 ?
Both 81 and 100 are perfect squares.
If both 81 and 100 are perfect squares, there must be a fraction, or fractions, that we can square to get 81 ___ 100
Demonstrate and display for the class how this line of thinking looks symbolically.
Recall from earlier in the lesson that the square of any nonzero integer is positive. Can this apply to fractions also? Explain.
Yes, it applies to fractions because squaring any number is multiplying two equal factors, and any positive number times a positive number is positive and any negative number times a negative number is also positive.
Is there another number we can square to get
If so, which number? If not, explain why.
Yes, we can square
Again, demonstrate and display this symbolically for the class, showing that it works for −9 in the numerator and then for −10 in the denominator.
Next, refer students back to their Venn diagrams. Use the following prompts to highlight similar thinking for cubes.
In which category did we decide the number 0.027 belongs? Why?
The number 0.027 is neither a perfect square nor a perfect cube because it is not the square or cube of an integer.
What can we identify about the number that reminds us of perfect squares or perfect cubes?
There is a 27 as part of the decimal number, and 27 is a perfect cube.
Students may discover that finding a decimal that squares or cubes to a given number feels less intuitive than finding a fraction that squares or cubes to a given number. Support students to help them visualize how the number of decimal places in the base relates to the number of decimal places in the product. Encourage students to think in terms of fractions by demonstrating the following reasoning.
Teacher Note
Some students may wonder about the case where 9 and 10 are both negative:
Help those students draw upon grade 7 learning about dividing integers. Guide students to recognize that 9 10
or −9 ÷ (−10), is equivalent to 9 __ 10 because dividing two numbers with the same sign is always positive.
Is there another number we can cube to get 0.027? If so, which number? If not, explain why.
No, there is no other number we can cube to get 0.027 because we cannot cube a negative number that results in a positive number.
Have students complete problems 10−13 with a partner. Circulate as students work to provide support, particularly for the decimal work.
12. What are all the numbers that square to 1.69?
and −1.3
13. What are all the numbers that cube to −0.125?
As students finish, confirm answers and address any remaining questions.
Land
Debrief 5 min
Objectives: Recognize perfect squares from 1 to 225 and perfect cubes from 1 to 125.
Determine all numbers that square or cube to a given number.
Use the following prompts to guide a discussion.
How do you know whether a number is a perfect square?
A number is a perfect square if it is the square of an integer.
How do you know whether a number is a perfect cube?
A number is a perfect cube if it is the cube of an integer.
How can you use what you know about perfect squares to find the number, or numbers, that square to 25 __ 64 ?
The numerator and the denominator of 25 __ 64 are perfect squares. I can look at the numerator and the denominator individually to determine the positive numbers I can square to get 25 and 64. Next, I can write those positive numbers as the numerator and the denominator of a fraction. So to get 25 64 , I square 5 8
Then, because I know that the square of any positive or negative number is positive, I can also square 5 8 to get 25 64 .
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
Perfect Squares and Perfect Cubes
In this lesson, we
• recognized perfect squares from 1 to 225 and perfect cubes from 1 to 125
• determined all numbers that square or cube to a given number. Perfect squares from 1 to
Perfect cubes from 1 to
Examples
For
1–3, evaluate.
• A perfect square is the square of an integer.
• A perfect cube is the cube of an integer.
Two numbers with the same sign result in a positive product. This means there are two numbers, one positive and one negative, that have the same positive square. 9 2 = 81 and (−9) 2 = 81
Only one integer has a cube of −27
Look for familiar numbers. This fraction includes the perfect cubes 8 and 125, which result from cubing 2 and 5
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
problems 1 and 2 find a pattern and complete
4. Describe the pattern you used to complete the table in problem 2 How did you use the pattern to find the unknown numbers in Column B? In Column A?
The pattern I noticed in the table is that the numbers in Column B are perfect cubes found by cubing the numbers in Column A. To find a number in Column B, I cubed the number in the corresponding row in Column A. To find a number in Column A, I determined the number that, when cubed, results in the number in the corresponding row in Column B.
For problems 5–10, evaluate.
3. Describe the pattern you used to complete the table in problem 1. How did you use the pattern to find the unknown numbers in Column B? In Column A?
The pattern I noticed in the table is that the numbers in Column B are perfect squares found by squaring the numbers in Column A. To find a number in Column B, I squared the number in the corresponding row of Column A. To find a number in Column A, I determined the positive number that, when squared, results in the number in the corresponding row of Column B.
For problems 11–14, determine all the numbers that square to the given number.
EUREKA MATH
For problems 15–18, determine all the numbers that cube to the given number.
19. Liam says that when you square or cube any number, the result is always a greater number. Do you agree with Liam? Explain why or why not, and provide an example to support your thinking.
No, I do not agree with Liam. When you square or cube a positive fraction less than 1 or a positive decimal less than 1, you get a number that is smaller. For example, 0.2 2 is 0.04 and 0.2 3 is 0.008
Solve equations of the forms x 2 = p and x 3 = p, where p is a rational number and the solutions are rational numbers.
Lesson at a Glance
In this lesson, students engage in a gentle introduction to solving equations of the forms x 2 = p and x 3 = p. Students solve these equations intuitively, without square root or cube root notation, by building on and applying the thinking developed in lesson 16. Students begin with equations for which p is a perfect square or a perfect cube and progress to equations for which p is rational and the solutions are rational. Students also consider cases in which p is negative and justify their reasoning about the solutions via the Take a Stand routine.
Key Questions
• How many solutions does an equation of the form x 2 = p have? Why?
• How many solutions does an equation of the form x 3 = p have? Why?
Achievement Descriptors
8.Mod1.AD6 Solve equations of the form x 2 = p as √— p and √— p and equations of the form x 3 = p as 3 √p , where p is a rational number. (NY-8.EE.2)
8.Mod1.AD7 Evaluate square roots of small perfect squares and cube roots of small perfect cubes. (NY-8.EE.2)
Agenda
Fluency
Launch 5 min
Learn 30 min
• Solving Equations of the Form x 2 = p
• Solving Equations of the Form x 3 = p
• Solving Equations with Rational Numbers
Land 10 min
Materials
Teacher
• Construction paper (4)
• Marker
• Tape
Students
• None
Lesson Preparation
• Prepare four signs labeled 2 and −2, −2, 2, and No Solution. Post one sign in each corner of the classroom.
Fluency
Evaluate Squares and Cubes
Students evaluate squares and cubes to prepare for solving equations of the forms x 2 = p and x 3 = p.
Directions: Evaluate each expression.
Launch
Students create true statements to reinforce understanding of perfect squares and perfect cubes.
Have students complete problems 1–7 independently. Circulate and identify students who demonstrate a strong grasp of perfect squares and perfect cubes.
For problems 1–4, determine the value that makes the equation true.
For problems 5–7, determine all values that make the equation true.
Use the following prompts to debrief the activity.
Describe how you determined the unknown values. What sorts of questions did you ask yourself?
Invite identified students to share their thinking as they approach these types of problems. Students may ask themselves the following questions:
• What is squared?
• What is cubed?
• What number do I square to get ?
• What number do I cube to get ?
UDL: Action & Expression
Think-alouds provide an opportunity to support students in strategizing by encouraging those who have mastered the concepts to model sound mathematical thinking for peers. This also provides an opportunity for any student to receive immediate feedback to dispel any misconceptions.
The directions for problems 5 through 7 say to determine all values that make the equation true. Does more than one value work for any of the problems? Explain.
Yes. In problem 5, both 4 and −4 work because 4 2 = 16 and (−4) 2 = 16.
Today, we will apply some of the same type of thinking used for problems 5 through 7 to solve equations involving squares and cubes.
Learn
Solving Equations of the Form
x 2 =
p
Students solve equations of the form x 2 = p, where p is a perfect square.
Direct students’ attention to problem 8 and use the following prompts to facilitate a discussion.
The directions say to solve the equation. What does it mean to solve an equation?
Solving an equation means finding a value for the variable that makes the equation true.
The number 100 is a special number. What type of number is 100?
The number 100 is a perfect square.
What value can we substitute for x to make this equation true? How do you know?
We can substitute 10 for x to make this equation true. I know this because 10 2 = 100.
So 10 is a solution to the equation. Is there another value we can substitute for x that makes the equation true? Explain how you know.
Yes, −10 also makes the equation true because (−10) 2 = 100.
Two values for x make this equation true: 10 and −10. So the solutions to the equation x 2 = 100 are 10 and −10.
Teacher Note
In topic E, students revisit solving equations of the form x 2 = p, where p is any rational number.
Teacher Note
The concept of an equation having more than one solution may surprise some students. Because students solve only linear equations prior to grade 8, some may have the misconception that an equation can have only one solution. Consider taking a moment to acknowledge this new development with students.
Ask students to complete problems 9–13 independently or with a partner. Circulate as students work and look for those who may need support for the equations presented with the variable on the right side in problems 9 and 12.
For problems 8–14, solve the equation. 8. x 2 = 100
Share answers with the class. Use the following prompts to probe for recognition that equations of the form x 2 = p have two solutions when p is positive but only one solution when p is 0
How many solutions does each equation have in problems 8 through 13? Explain why.
Each equation in problems 8 through 12 has two solutions because there is one positive value and one negative value that makes the equation true. The square of a positive number is always positive, and the square of a negative number is also always positive. The equation in problem 13 has only one solution because 0 is the only value that makes the equation true. The square of 0 is 0.
Next, have students complete problem 14 independently. Although some confusion or questions may arise, refrain from providing any immediate assistance.
14. w 2 = −4
No solution
Teacher Note
In grade 8, students work only within the real number system and conclude that there is no solution for equations such as w 2 = −4
Note that real numbers are defined in topic E of this module. In later courses, students eventually discover that such equations do have solutions, and those solutions are part of the complex number system.
After students have had a minute to work, introduce the Take a Stand routine to the class. Draw students’ attention to the four signs hanging in the classroom: 2 and −2, −2, 2, No Solution.
Invite students to stand beside the sign that best describes the solution to the equation in problem 14.
When all students are standing near a sign, allow one minute for groups to discuss the reasons why they chose that sign.
Then call on each group to share reasons for their selection. Invite students who change their minds during the discussion to join a different group.
Have students return to their seats. As a class, reflect on the reasons why the equation has no solution.
Watch for students who may insist that −2 is the solution because −2 2 = −4. If this misconception arises, review the order of operations to help students recognize the difference between −22, which is the opposite of 2 squared, and (−2) 2, which is the square of −2.
When we square any nonzero number, the result is always positive because the product of two positive numbers is positive and the product of two negative numbers is also positive. So with the numbers we know, it is impossible to get −4 from squaring a number. Because there are no numbers that make the equation w 2 = −4 true, the equation has no solution.
Solving Equations of the Form x 3 = p
Students solve equations of the form x 3 = p, where p is a perfect cube.
Direct students’ attention to problem 15 and use the following prompts to facilitate a discussion.
We can also solve equations involving cubes. The number 27 is a special number. What type of number is 27?
The number 27 is a perfect cube.
Promoting the Standards for Mathematical Practice
When students justify their own reasoning and then listen to and analyze their peers’ reasoning in the Take a Stand routine, they are constructing viable arguments and critiquing the reasoning of others (MP3).
Ask the following questions to promote MP3:
• Is your choice of sign a guess, or do you know for sure? How do you know for sure?
• What questions can you ask your peers who are standing by other signs to make sure you understand the reason for their choice?
Teacher Note
In topic E, students revisit solving equations of the form x 3 = p, where p is any rational number.
What value can we substitute for x to make this equation true? How do you know?
We can substitute 3 for x to make this equation true because 3 3 = 27.
So 3 is a solution to the equation. Is there another integer that we can substitute for x that makes the equation true? Explain how you know.
No, there is not another integer that makes the equation true because (−3) 3 = −27.
The equation in problem 15 has only one solution because 3 is the only value that makes the equation true. So the solution to the equation x 3 = 27 is 3.
Have students complete problems 16–20 independently or with a partner. Circulate as students work and listen for any misconceptions arising from problem 20. Some students may see the negative number and think that the equation has no solution because problem 14 has no solution.
For problems 15–20, solve the equation.
Share answers with the class. Use the following prompts to reinforce an understanding that these equations have only one solution. Also reflect on why equations of the form x 3 = p, where p is negative, have a solution.
How many values make each equation in problems 15 through 20 true? How do you know?
Only one value makes each equation true. When I cube a positive number, I get a positive number. But when I cube a negative number, I get a negative number.
How is the equation in problem 20 similar to the equation in problem 14? How is it different?
The equations are similar because they both have a negative number. They are different because one is a square and one is a cube. Also, the cube equation has a solution and the square equation does not.
Why is it that h 3 = −8 has a solution, but w 2 = −4 does not have a solution?
We can cube −2 to get −8, but there are no numbers we can square to get −4.
When we cube a positive number, the result is positive. When we cube a negative number, the result is negative. When we cube 0, the result is 0. So equations of the form x 3 = p have one solution for any value of p.
Solving Equations with Rational Numbers
Students solve equations of the forms x 2 = p and x 3 = p, where p is rational.
Organize students into pairs and present the Solving Equations with Rational Numbers problems.
We can apply what we learned today about solving equations with perfect squares and perfect cubes to solve equations with other rational numbers.
Have students complete problems 21–26 with a partner. Circulate as students work and provide support on the fraction and decimal concepts from lesson 15. As needed, encourage students to relate the fractions and decimals to the perfect squares and perfect cubes they recognize.
Note that this section also contains more practice with equations of the forms x 2 = p and x3 = p, where p is negative.
For problems 21–26, solve the equation.
Differentiation: Support
If students are still progressing toward proficiency in determining the number that is squared or cubed to result in a decimal, provide support by using the strategies introduced in lesson 16. Consider providing the following examples:
Invite students to share their responses and confirm the answers as a class. If time permits, consider having students share thoughts about which problems they found easiest and which they found most challenging.
Land
Debrief
5 min
Objective: Solve equations of the forms x 2 = p and x 3 = p, where p is a rational number and the solutions are rational numbers.
Initiate a class discussion by using the following prompts. Encourage students to add on to their classmates’ responses.
How many solutions does an equation of the form x 2 = p have? Why?
It depends. When p is positive, the equation x 2 = p has two solutions because there are two numbers, one negative and one positive, that we can square to get p.
When p is 0, the equation x 2 = p has one solution because only 0 2 is 0.
When p is negative, the equation x 2 = p has no solution because if we square any number we know, we can only get a positive number or 0.
How many solutions does an equation of the form x 3 = p have? Why?
The equation has only one solution. When p is positive, negative, or 0, the equation x 3 = p has one solution because there is only one number that we can cube to get p.
Teacher Note
Some students may give 1 8 and 1 8 as solutions for the equation in problem 23. Encourage students to look very carefully at the exponent on the variable.
If students initially respond with answers such as “it depends,” or if they focus on only one case (positive, negative, or zero) for p, probe further. Consider scaffolding by asking a series of questions for each case, including actual values for p, if necessary. For example, consider asking the following questions:
• When p is a positive number such as 16, how many solutions does the equation x 2 = 16 have? Why?
• When p is a positive number such as 1, how many solutions does the equation x 3 = 1 have? Why?
Continue as needed for the other cases.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
Solving Equations with Squares and Cubes
In this lesson, we
• solved equations of the form x2 = p where p is either a perfect square or a fraction or decimal related to a perfect square.
• solved equations of the form x3 = p where p is either a perfect cube or a fraction or decimal related to a perfect cube.
Solutions to x 2 = p
When p is positive: the equation has two solutions because both positive and negative numbers have positive squares.
When p is negative: the equation has no solution because no numbers have negative squares.
When p is 0 the equation has one solution because only 0 has a square of 0
Solutions to x 3 = p
When p is positive: the equation has one solution because only positive numbers have positive cubes.
When p is negative: the equation has one solution because only negative numbers have negative cubes.
When p is 0 the equation has one solution because only 0 has a cube of 0
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
For problems 1–4, determine all the values that make the equation true.
For problems 5–18, solve the equation.
Fill in the box to create an equation that has the solutions −13 and 13 x 2 = 169
Fill in the box to create an equation that has 1 4 as its only solution. x 3 = 1 64
Remember For problems 21–24, multiply.
Order the numbers from least to greatest.
The Pythagorean Theorem
Describe the Pythagorean theorem and the conditions required to use it.
1. To which type of triangle does the Pythagorean theorem apply? The Pythagorean theorem applies only to right triangles.
Lesson at a Glance
2. Identify the sides of each right triangle by labeling each side as a leg or a hypotenuse.
Students follow the plot of the book What’s Your Angle, Pythagoras? and engage in hands-on activities to explore the Pythagorean theorem. Students use string to represent Pythagoras’s knotted rope from the story as they consider what combinations of side lengths create a right triangle. Then they use the areas of squares to explore a relationship among the side lengths of a right triangle. As the story unfolds, students come to recognize this relationship as the Pythagorean theorem. This lesson formally defines the Pythagorean theorem and the terms leg and hypotenuse.
Key Questions
• What sorts of problems did Pythagoras solve? How did he solve them?
• What relationship does the Pythagorean theorem show?
Achievement Descriptor
8.Mod1.AD15 Apply the Pythagorean theorem to determine the unknown length of a hypotenuse in a right triangle in mathematical problems. (NY-8.G.7)
Agenda
Fluency
Launch 5 min
Learn 30 min
• Knotted Rope
• Squares and Right Triangles
• Pythagoras the Problem Solver
Land 10 min
Materials
Teacher
• White or light-colored string
• Marker
• Ruler
• Scissors
Students
• String loop (1 per student pair)
• Right Triangle Squares
• Colored pencils (1 red and 1 blue)
• Scissors
Lesson Preparation
• Cut 14-inch pieces of string. Starting 1 inch in from each end, use a marker to make small marks 1 inch apart on each piece of string for a total of 13 marks. Tie the ends of each piece of string together so that the first and last mark come together to form a single mark. Each tied string loop should have 12 marks total. (See image in the Knotted Rope segment.) Prepare one string loop for each student pair.
Fluency
Identify the Right Triangle
Students identify right triangles to prepare for applying the Pythagorean theorem to right triangles.
Directions: Determine whether the triangle is a right triangle. Write Yes or No.
Launch
Students encounter problems that can be solved by using the Pythagorean theorem.
Throughout the lesson, transition between reading the book What’s Your Angle, Pythagoras? and engaging in discussion and activities. Consider individual class needs and time constraints to decide whether to read the story aloud to the class, have students read to themselves, or use some combination of both.
Begin the lesson by reading pages 3–7 aloud or by having students read to themselves. Then have students think–pair–share about the following question.
What could Saltos and Pepros do to make the pillars stand straight?
Encourage students to draw from their prior experience and consider a wide variety of answers so that students can feel personally invested in the story. If students mention specific modern tools, ask whether Pythagoras had access to those tools in ancient Greece.
Today, we will discover what Pythagoras used to solve this problem.
Learn
Knotted Rope
Students explore whether a knotted rope can form a right triangle.
Read pages 8–14 aloud or have students read to themselves.
Organize students into pairs and provide each student pair with a prepared loop of string.
The marks on the string represent the knots on the knotted rope Nef gave to Pythagoras.
UDL: Engagement
Students engage with a fictional story about Pythagoras by reading the book What’s Your Angle, Pythagoras? The book provides contexts to frame the activities in this lesson.
Invite students to think–pair–share about the following question.
What do you notice about the knots?
Expect students to notice that there are 12 knots total. If students do not mention that the knots are evenly spaced, encourage them to use a ruler or to mark on a piece of paper to see if there is a pattern.
Can you find a way to make a right triangle with a knot on each corner of the triangle?
Allow a few minutes for pairs to explore. Have them use their fingers or pencils to hold the corners of their triangle in place. Consider having students use the corner of a piece of paper or an index card to help guide their string loop to form a right angle. Encourage students to sketch their triangle with the knots in the space available in their book. Expect all pairs to create the same right triangle but with different orientations.
Teacher Note
This activity serves as an intuitive preview of the converse of the Pythagorean theorem, which students formally explore in module 2. If time is limited, consider exploring this activity as a whole class by using one large, prepared rope or string.
Watch for pairs who form a right triangle without a knot in each corner. Remind those students that the goal in using this tool is to have a knot in each corner. While there are other right triangles that can be formed with this string, there is one unique triangle with a knot in each corner. Students will learn more precise language to communicate this idea in module 2 when congruent triangles are introduced.
Differentiation: Challenge
Use the following prompts to summarize students’ findings.
Let’s measure the sides of the right triangles. What do you think would be a convenient unit of measurement?
The distance between each knot represents 1 unit.
What are the side lengths of the right triangle you formed with the string?
Our right triangle has side lengths of 3 units, 4 units, and 5 units.
Ask pairs who finish quickly whether they can form another right triangle. Then ask what the right triangles they formed have in common.
Do you think any right triangle you make from this loop of string with a knot in each corner will have those same side lengths? Why?
Yes. I think any right triangle we make with this string will be a triangle with the side lengths of 3 units, 4 units, and 5 units because the number of knots on the loop does not change. So once we form a right angle at a knot, the side lengths have to be 3 units, 4 units, and 5 units to make the sides straight.
Yes. We tried to make a right triangle with one side length of 2 units and one side length of 4 units, but the last side was not straight.
Squares and Right Triangles
Students explore how the areas of squares relate when the side lengths of the squares are the lengths of the sides of a right triangle.
Read pages 15–17 aloud or have students read to themselves.
Have students remove the Right Triangle Squares page from their books. If working in pairs is preferred, each student pair needs only one page. For the image that includes a right triangle, have students shade the top left square that contains 16 smaller squares with their blue colored pencil. Have them shade the top right square that contains 9 smaller squares with their red colored pencil.
Then have students think–pair–share about the following question.
Pythagoras made one small square with red tiles. Then he made one medium square with some blue tiles that he found. If Pythagoras moved these tiles into the third square, how many tiles do you think would fit? Why?
Some students may reason that the third square can fit 25 tiles, while others may attempt to estimate with spatial reasoning. Encourage different types of responses and rationales.
Let’s try actually moving the tiles to see if our predictions are correct.
Direct students back to their Right Triangle Squares page and have them use their scissors to cut off the bottom section that contains the two squares. Instruct students to use their colored pencils to shade these squares to match the shaded squares on top of the triangle. Then have them cut the two squares from the bottom section into square tiles. Allow students a few minutes to arrange the tiles to determine how many tiles fit into the third square on the top of the page.
Invite students to share their findings with the class. Then use the following question to engage in a brief discussion.
What is the length of the third side of the triangle? How do you know?
It is 5 units because exactly 5 square tiles fit along the third side of the triangle.
The length of the third side of the triangle is 5 units. There are 25 tiles and they fit perfectly into the third square, which means it has an area of 25 square units. So the side length must be 5 units.
Read pages 18–23 aloud or have students read to themselves.
Then have students turn and talk about the following question.
Pythagoras is excited to tell Saltos and Pepros about the “the secret of the right triangle.” What do you think the secret is? Why?
Pythagoras the Problem Solver
Students apply their emerging understanding of the relationships between the sides of a right triangle.
Read page 24 aloud or have students read to themselves. Then have students think–pair–share about the following question.
How did Pythagoras apply what he learned from making tile squares on the sides of the statue base to find the length of the ladder?
As students discuss and share, have them sketch and label in their book a diagram of the situation involving the wall, the ground, and the ladder.
Read the remainder of the story aloud or have students read to themselves.
If time allows, have students discuss how the map situation on page 29 is similar to the situations with the tiles and the ladder.
Differentiation: Challenge
Encourage students to consider whether the relationship among the side lengths would still be true if the third side length were not 5 units. Consider asking the following questions:
• Suppose the third side length is not 5 units. Would all the tiles fit exactly into the third square? Explain your reasoning.
• What happens to the angle across from the third side if the side length is not 5 units? Why?
Promoting the Standards for Mathematical Practice
Students reason quantitatively and abstractly (MP2) as they consider the ladder situation and the map situations in the story.
Ask the following questions to promote MP2:
• What does a right triangle represent in each situation?
• Does Pythagoras’s answer make sense for each situation?
Land
Debrief 5 min
Objective: Describe the Pythagorean theorem and the conditions required to use it.
Use the following prompts to introduce the Pythagorean theorem.
The book What’s Your Angle, Pythagoras? illustrates the idea that the side lengths in a right triangle have a special relationship. This relationship is called the Pythagorean theorem. The Pythagorean theorem is named for Pythagoras, a philosopher and mathematician in ancient Greece.
Display the Pythagorean theorem. Point to and label the legs and the hypotenuse of the triangle when describing them.
A leg of a right triangle is a side adjacent to the right angle. The hypotenuse of a right triangle is the side opposite the right angle.
The Pythagorean theorem describes the relationship in a right triangle in which the sum of the squares of the leg lengths is equal to the square of the hypotenuse length. In a right triangle, if a and b are the two shorter sides and c is the longest side, then a 2 + b 2 = c 2 .
Emphasize that a and b must represent the legs of the triangle but that it does not matter which leg is a and which leg is b. Inform students that c must represent the hypotenuse, which they can identify as the side opposite the right angle, or the longest side.
Then lead a class discussion by asking the following questions.
What sorts of problems did Pythagoras solve? How did he solve them?
Pythagoras found the correct ladder height and the distance from Samos to Crete. He modeled each situation by using right triangles and the relationship among the sides of a right triangle.
Teacher Note
If time allows, consider displaying the painting School of Athens by Raphael. Ask students what they notice and wonder about the painting. Then share the following information:
• School of Athens is a famous painting by Raphael that features many prolific mathematicians and philosophers.
• Though there is some debate over who Raphael was portraying in this painting, it is commonly thought that the crouching man with the book in the bottom left is Pythagoras.
• In the bottom right corner, the man in a red robe stooped over a chalkboard is Archimedes, another prolific mathematician.
Encourage interested students to research this painting and learn about other mathematicians, philosophers, and scientists commonly identified in the painting, such as Plato and Aristotle in the middle.
What relationship does the Pythagorean theorem show?
The Pythagorean theorem shows a relationship among the side lengths of a right triangle. If the lengths of the legs of the triangle are a and b and the length of the hypotenuse is c, then a 2 + b 2 = c 2 .
If students answer only with an equation, push their thinking further by asking guiding questions:
• What do a, b, and c represent?
• When does the Pythagorean theorem apply?
Exit Ticket 5
min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
The Pythagorean Theorem
In this lesson, we
• explored the Pythagorean theorem through a story.
• examined situations that can be modeled with the Pythagorean theorem.
• stated the conditions required for using the Pythagorean theorem.
The Pythagorean Theorem
In a right triangle, the sum of the squares of the leg lengths is equal to the square of the hypotenuse length.
a 2 + b 2 = c 2
Terminology
The hypotenuse of a right triangle is the side opposite the right angle. It is also the longest side of a right triangle.
A leg of a right triangle is a side adjacent to the right angle. c
EUREKA MATH2
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
1. According to the story of Pythagoras’s discoveries and your own exploration during the lesson, when does the relationship a 2 + b 2 = c 2 hold true?
The relationship a 2 + b 2 = c 2 holds true for right triangles when the leg lengths are a and b and the hypotenuse length is c
2. Suppose Saltos and Pepros need a ladder for another temple building project.
a. Describe the information they can use to make sure they choose a ladder that is the correct length.
Saltos and Pepros can use the distance from the ground to the place on the temple where the ladder reaches. They can also use the distance from the base of the temple to the base of the ladder.
b. How does the information from part (a) relate to a right triangle?
Assuming the temple forms a right angle with the ground, the distance from the ground to the place on the temple where the ladder reaches represents one leg of a right triangle, and the distance from the base of the temple to the base of the ladder represents the other leg. The length of the ladder represents the hypotenuse.
For problems 3–6, divide.
Using the Pythagorean Theorem
Apply the Pythagorean theorem to find the unknown length of the hypotenuse of a right triangle.
Find two consecutive whole numbers which the length of the hypotenuse is between when the length is not rational.
Use square root notation to express lengths that are not rational.
Lesson at a Glance
Students engage in the Which One Doesn’t Belong? routine to build on the relationships in right triangles they explored in lesson 18. They apply the Pythagorean theorem to find the length of the hypotenuse for a right triangle by operating with side lengths expressed as whole numbers, decimals, or fractions. Students continue to employ the intuitive understanding of square roots developed early in topic D. They encounter cases where c 2 is not a perfect square and estimate a range of possible values for c before transitioning to using square root notation. This lesson formally defines the term square root
Key Questions
• What information do we need to find the length of the hypotenuse of a right triangle? How do we use that information to find the length of the hypotenuse?
• When do we use square root notation to represent a hypotenuse length?
Achievement Descriptor
8.Mod1.AD15 Apply the Pythagorean theorem to determine the unknown length of a hypotenuse in a right triangle in mathematical problems. (NY-8.G.7)
Agenda
Fluency
Launch 5 min
Learn 30 min
• Finding the Length of the Hypotenuse
• Reasoning About the Length of the Hypotenuse
• Using Square Root Notation
Land 10 min
Materials
Teacher
• None
Students
• None
Lesson Preparation
• None
Fluency
Solve Equations of the Form x 2 = p
Students solve equations of the form x 2 = p to prepare for applying the Pythagorean theorem to find the unknown length of the hypotenuse of a right triangle.
Directions: Solve each equation. 1.
Launch
Students identify similarities and differences in various representations related to the Pythagorean theorem.
Introduce the Which One Doesn’t Belong? routine. Direct students to the table of four items in their book and invite them to study the items.
Give students about 2 minutes to find a category in which three of the items belong, but the fourth item does not.
When time is up, invite students to explain their chosen categories and to justify why one item does not fit.
Highlight responses that emphasize reasoning about the relationships explored in lesson 18, such as connecting squares and area to right triangles.
Consider asking the following questions that invite students to use precise language, make connections, and ask questions of their own.
What do the equation 3 2 + 4 2 = 5 2 and the square figure have in common?
They both show the same number of square units. 32 is the area covered by the red squares, and 42 is the area covered by the blue squares. 5 2 is the total area covered by both the red and blue squares.
How are the areas of squares related to right triangles?
On a right triangle, I can draw a square touching each leg, and I can draw a square touching the hypotenuse. The total area of the two squares on the legs is equal to the area of the square on the hypotenuse.
9 + 16 = 25
Use the following prompt to debrief the activity.
Was it difficult to determine a category in which only three of the items belong? Why?
Ask several students to share opinions. Steer the discussion toward students recognizing that although there are ways to rule one item out, as they were asked to do in the activity, there is also one common link: All items represent the relationship among the side lengths of a right triangle with side lengths of 3 units, 4 units, and 5 units.
In the last lesson, we explored a special relationship between the squares of the side lengths of a right triangle called the Pythagorean theorem. Today, we will learn how to find the length of the hypotenuse of a right triangle by using the Pythagorean theorem.
Learn
Finding the Length of the Hypotenuse
Students find the length of the hypotenuse of a right triangle c, where c is rational, when given the lengths of the legs of the triangle.
Direct students to the unlabeled right triangle at the beginning of the Finding the Length of the Hypotenuse problems. Once labeled, the triangle can serve as a visual reference for students as they work through the rest of the lesson.
Display the triangle. Use the following prompts to guide the class in labeling the triangle. Label the sides as you speak.
In this right triangle, the leg lengths are a and b, and the hypotenuse length is c.
Have students label their triangles at the same time.
What does the Pythagorean theorem tell us about side lengths a, b, and c in this triangle?
a 2 + b 2 = c 2
Write the equation below the triangle as students say it aloud. Have students record the equation below their labeled triangle. Repeat the equation aloud again, pointing to each side as you refer to it.
Teacher Note
Students solve only for the length of the hypotenuse in module 1. They will revisit the Pythagorean theorem and solve for leg lengths in module 2.
The Pythagorean theorem:
a 2 + b 2 = c 2
Direct students to problem 1.
What length is unknown for the triangle in problem 1?
The length of the hypotenuse c is unknown.
This is a right triangle, so we can use the Pythagorean theorem to find the length of the hypotenuse.
Have students write a 2 + b 2 = c 2 below the triangle in problem 1.
What numbers can we substitute for a and b? Why?
We can substitute 8 for a and 6 for b because these values are the lengths of the legs.
Is that the only way we can substitute for a and b? Explain.
No, we can substitute 6 for a and 8 for b. The order in which we substitute the lengths of the legs does not matter as long as we substitute them for a and b.
Have students finish the remainder of problem 1 and complete problem 2, individually or in pairs.
Teacher Note
The realization that the leg lengths can be substituted into the equation in either order opens an opportunity to engage in a rich discussion with students around properties of operations and equivalence.
For problems 1–4, find the length of the hypotenuse c.
The length of the hypotenuse is 10 units
The length of the hypotenuse is 13 units.
Once most students are finished with problems 1 and 2, confirm answers by using the following prompts.
What are the solutions to the equation 100 = c 2?
The solutions are −10 and 10.
So what is the length of the hypotenuse in problem 1?
The length of the hypotenuse is 10 units.
What are the solutions to the equation 169 = c 2?
The solutions are −13 and 13.
So what is the length of the hypotenuse in problem 2?
The length of the hypotenuse is 13 units.
The equations each have two solutions. Why are we only using one of the solutions for the length of the hypotenuse?
We only use the positive solution for c because we are finding a length, which cannot be negative.
Two solutions make each of the equations in problems 1 and 2 true, but only one of the solutions makes sense for describing a side length of a right triangle. So when using the Pythagorean theorem, only the positive solution is valid.
Have students complete problems 3 and 4 in pairs. Circulate as students work to support them with solving the equations. Encourage them to use what they know about related perfect squares. For example, use facts such as 5 2 = 25 to help students understand the relationship between 0.5 and 0.25.
Promoting the Standards for Mathematical Practice
Students reason abstractly and quantitatively (MP2) as they find the length of the hypotenuse of a right triangle. Students decontextualize when they apply the Pythagorean theorem to represent the situation symbolically, and they contextualize when they consider whether the mathematical answers are valid.
Ask the following questions to promote MP2:
• Do both solutions of −10 and 10 make sense in the context of side lengths?
• What real-world situations are modeled by right triangles?
The length of the hypotenuse is 0.5 units.
The length of the hypotenuse is 13 __ 4 units.
Confirm responses. Use the following prompts to facilitate a brief discussion about the similarities and differences between problems 1–4.
How are problems 1 through 4 alike?
The problems are alike because they all have right triangles and an unknown hypotenuse length.
How are problems 1 through 4 different?
The problems are different because the side lengths in problems 1 and 2 are whole numbers, and the side lengths in problems 3 and 4 are not. Also, the triangles in problems 2 and 4 are turned so that the legs are not perfectly vertical and horizontal.
How do you know that you can use the Pythagorean theorem to find the length of the hypotenuse in problems 1 through 4?
The triangle in each problem is a right triangle.
We can use the Pythagorean theorem to find the length of the hypotenuse for any right triangle when we know the lengths of the legs, even when the leg lengths are expressed as fractions or decimals.
Reasoning About the Length of the Hypotenuse
Students find which two whole numbers the length of the hypotenuse c is between when c is not rational.
The goal of problem 5 is for students to naturally develop questions about values of c 2 that are not perfect squares. The problem also sets up the need for square root notation to express values of c, which is introduced later in the lesson.
Have students complete problem 5 individually.
5. Find the length of the hypotenuse c. c 4
Teacher Note
At this point, students only need to recognize that the value of c is between two consecutive whole numbers. Students will delve into closer approximations of square roots in topic E. To prepare for this thinking, you may consider asking students to make a guess as to whether the value for c is closer to 9 or 10
Note that students will likely look for guidance when they get to 97 = c 2, because this is their first encounter with a solution that is not rational. Use the following prompts to facilitate a discussion.
How is problem 5 different from any of the previous problems in this lesson?
I cannot determine a value for c in this problem. Is 97 a perfect square? How do you know?
No, there are no integers that can be squared to get 97.
So we don’t currently have a way to describe the number with a square of 97. Let’s use reasoning to determine a range in which the square of 97 must be located. What is the perfect square that is closest to, but less than, 97?
The number 81 is the closest perfect square that is less than 97.
What is the perfect square that is closest to, but greater than, 97?
The number 100 is the closest perfect square that is greater than 97.
We just identified that the value of c 2, which is 97, is between the perfect squares 81 and 100. So this tells us that the value of c must be between which two whole numbers? Why?
The value of c must be between 9 and 10 because length is positive and 9 2 = 81 and 10 2 = 100.
The length of the hypotenuse for this triangle is between 9 units and 10 units. The numbers 9 and 10 are consecutive whole numbers because they follow each other in order. So the length of the hypotenuse is between the consecutive whole numbers 9 and 10.
Have students complete problems 6–8 in pairs. Consider providing student pairs who finish early with an opportunity for extra practice in determining which two consecutive whole numbers c is between. Have one student make up a value for c 2 and have the other student determine which two consecutive whole numbers c is between. Then switch roles and repeat the activity.
Language Support
Though students may be familiar with the term consecutive, they may need support in understanding the term as it relates to numbers within a mathematical context. Consider displaying the words consecutive and nonconsecutive. Ask students to provide examples and nonexamples of consecutive whole numbers. Record responses next to the appropriate word.
For problems 6–8, determine which two consecutive whole numbers the length of the hypotenuse c is between.
6. c 3 3
The value of c is between 4 and 5 because length is positive and 4 2 = 16 and 5 2 = 25, which are the perfect squares closest to 18.
The length of the hypotenuse is between 4 units and 5 units.
7. 1 c 10
The value of c is between 10 and 11 because length is positive and 10 2 = 100 and 11 2 = 121, which are the perfect squares closest to 101. The length of the hypotenuse is between 10 units and 11 units.
The value of c is between 13 and 14 because length is positive and 13 2 = 169 and 14 2 = 196, which are the perfect squares closest to 193. The length of the hypotenuse is between 13 units and 14 units.
Confirm responses to problems 6–8.
Using Square Root Notation
Students use square root notation to express the length of the hypotenuse of a right triangle.
Refer students back to problems 5–8.
Problems 5 through 8 all had something in common. We could not state an exact value for the length of the hypotenuse. We could only find the two consecutive whole numbers that the length of the hypotenuse was between.
However, we can use a special symbol in these cases to denote an exact value.
Display √ .
This is called a square root symbol.
Then display √x .
A square root of a nonnegative number x is a number with a square that is x. The expression √x represents the positive square root of x when x is a positive number. When x is 0, √0 = 0.
Display the following equations. Then use the prompts to help students understand the new notation, pointing to each line of the equations as appropriate.
Teacher Note
This is the students’ initial introduction to the notation and definition of square root. Students will explore square roots as values on the number line in lesson 20 and investigate closer approximations in topic E.
The equation 97 = c 2 in problem 5 indicates that the value of c is the number that, when squared, results in 97. We do not know what that number is exactly, but we can use square root notation to communicate the value of c as √97 . So the length of the hypotenuse is √97 units.
We can practice reading this new notation. If √97 represents the positive number that we square to get 97, how would you describe each of the following numbers?
Display the following numbers: √5 , √16 , √19 , and √25 . Point to them one at a time.
The number √5 is the positive number we square to get 5.
The number √16 is the positive number we square to get 16.
The number √19 is the positive number we square to get 19.
The number √25 is the positive number we square to get 25.
Do any of these expressions represent a whole number? Explain.
Yes, √16 represents 4 because we can square 4 to get 16, and √25 represents 5 because we can square 5 to get 25.
Have students complete problem 9 individually.
9. Circle all the expressions that do not represent a whole number.
Teacher Note
Some students may wonder why √16 and √25 do not also represent −4 and −5, respectively, since (−4)2 = 16 and (−5)2 = 25. This is a common misconception. Support students by restating that, by definition, the expression √x represents the positive square root of x when x is a positive number, and when x is 0,
Invite students to turn and talk briefly with a partner about which expressions they circled and why. Then confirm responses and use the following prompt to transition to applying the notation in the context of the Pythagorean theorem.
Teacher Note
Look for students who confuse roots with division. Because the root symbol resembles a long division symbol, some students may mistakenly divide by 2 rather than take the square root. If students circle all the expressions with odd radicands, they are likely working under that misconception.
Now that we have some familiarity with square root notation, let’s use it to exactly express hypotenuse lengths.
Have students complete problems 10 and 11. Circulate and look for students who may need extra support with the correct use of the square root notation.
For problems 10 and 11, use square root notation to express the length of the hypotenuse c.
The length of the hypotenuse is √73 units.
The length of the hypotenuse is √61 units.
Discuss responses, as needed. Then use the following prompt to solidify understanding about the need for the square root notation.
Why did we use square root notation to express the length of the hypotenuse in problems 10 and 11?
We used square root notation because we could not determine exact lengths.
The square root notation allows us to exactly represent a hypotenuse length, rather than finding two consecutive whole numbers that the length falls between.
Land
Debrief 5
min
Objectives: Apply the Pythagorean theorem to find the unknown length of the hypotenuse of a right triangle.
Find two consecutive whole numbers which the length of the hypotenuse is between when the length is not rational.
Use square root notation to express lengths that are not rational.
Initiate a class discussion by using the following prompts.
What information do we need to find the length of the hypotenuse of a right triangle?
We need to know the lengths of the two legs to find the length of the hypotenuse.
How do we use that information to find the length of the hypotenuse?
We can substitute the lengths of the legs for a and b in a 2 + b 2 = c 2 and solve for c, which is the length of the hypotenuse.
When do we use square root notation to represent a hypotenuse length?
We use square root notation to represent a hypotenuse length when we cannot determine an exact value.
What does √x represent?
The expression √x represents the positive square root of x when x is a positive number.
When x is 0, √0 = 0.
When is √x a whole number?
The expression √x is only a whole number when x is a perfect square.
If students have difficulty articulating responses to the last two questions, consider providing the questions with numerical examples, either in place of, or leading up to, the general cases. For example, consider providing students with the following questions:
• What does √7 represent? What does √4 represent?
• Is √7 a whole number? Is √4 a whole number?
• How are √7 and √4 different?
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
Using the Pythagorean Theorem
In this lesson, we
• found hypotenuse lengths of right triangles.
• determined which two consecutive whole numbers the length of a hypotenuse is between.
• used square root notation to express hypotenuse lengths.
Examples
Terminology
• A square root of a nonnegative x is a number with a square that is x The expression √x represents the positive square root of x when x is a positive number. When x is 0, √0 = 0
For problems 1 and 2, find the length of the hypotenuse c. If the answer cannot be written as a rational number, use square root notation to express the length.
Substitute the values of the leg lengths for a and b in either order.
The length of the hypotenuse is 5 2 units.
5 c What number squared is 106? Use the square root symbol to represent the exact value.
The length of the hypotenuse is √106 units.
Read this expression as the square root of 106
3. Between which two consecutive whole numbers is the length of the hypotenuse c in problem 2? Explain.
The value of c is between 10 and 11 because length is positive and 10 2 = 100 and 112 = 121, which are the perfect squares closest to 106
The length of the hypotenuse is between 10 units and 11 units.
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
5. So-hee and Vic each make errors while finding the length of the hypotenuse c of the following right triangle.
So-hee’s Work:
Vic’s Work:
The length of the hypotenuse is 42 units.
The length of the hypotenuse is
a. Describe all errors So-hee makes. So-hee divides 225 by 2 instead of finding the number you square to get 225
b. Describe all errors Vic makes.
Vic multiplies 12 and 9 by 2 instead of squaring each number. He also says that the length of the hypotenuse is the value of c 2 instead of c
c. Find the correct length of the hypotenuse. The length of the hypotenuse is 15 units.
20. The distance between Los Angeles and New York City is approximately 4.928 × 10 6 yards. Choose a more appropriate unit of measurement to describe the distance between the two cities. Then convert from yards to the unit you choose. 1
The distance between Los Angeles and New York City is approximately 2800 miles.
21. Complete the table by drawing an example of each geometric figure.
20
Square Roots
Place square roots on a number line.
Lesson at a Glance
In this lesson, students apply the Pythagorean theorem to engage in a hands-on activity, recreating the Spiral of Theodorus, to interpret square roots as length measurements. Students then use the lengths created on the spiral to place square roots on a number line. This prompts an initial shift toward recognizing square roots of whole-number nonperfect squares as approximate decimal values relative to whole numbers.
Key Question
• How do we determine the location of √x on a number line, for any whole-number value of x that is not a perfect square?
Achievement Descriptors
8.Mod1.AD6 Solve equations of the form x 2 = p as √p and √p and equations of the form x 3 = p as 3 √p , where p is a rational number. (NY-8.EE.2)
8.Mod1.AD15 Apply the Pythagorean theorem to determine the unknown length of a hypotenuse in a right triangle in mathematical problems. (NY-8.G.7)
Agenda
Fluency
Launch 5 min
Learn 30 min
• The Spiral of Theodorus
• Square Roots on a Number Line
Land 10 min
Materials
Teacher
• Sticky notes (1 per student pair)
Students
• Spiral of Theodorus
• Index card
Lesson Preparation
• Prepare 16 sticky notes by writing the following labels: √1 , √2 , √3 , ... and √16 . Prepare fewer notes if the number of student pairs is less than 16.
• Create a number line from 0 to 4 (or 0 to 3 for classes with 9 student pairs or less) with a scale of 1 to display in class. Leave ample space between the whole numbers so the sticky notes can be placed on the number line.
Fluency
Reasoning About Length
Students determine which two consecutive whole numbers the length of the hypotenuse of a right triangle is between to prepare for placing square roots on a number line.
Directions: Determine which two consecutive whole numbers the length of the hypotenuse c is between.
1. c 2 = 15
2. c 2 = 38
3. 99 = c 2
4. c 2 = 27
10 units and 11 units 6. c 2 = 12
3 units and 4 units
Launch
5
Students intuitively reason about the values of square roots.
Divide students into pairs. Distribute one prepared sticky note to each student pair. Then direct their attention to the displayed number line.
Decide where on the number line your number belongs. After everyone decides, I will ask one partner from each pair to come forward and place your sticky note on the number line. Both partners should know where the number belongs and be able to explain why.
If students have difficulty getting started, ask them to describe the way they reasoned about hypotenuse lengths before they learned about square root notation.
Allow students about 1 minute of discussion time before inviting them to come forward.
Have students return to their seats once they place all the sticky notes. Then ask a few student pairs to describe how they decided where to place their sticky note. Include at least one pair of students that had a perfect square radicand.
Consider leaving the number line displayed until the end of the lesson. That way, students will have an opportunity to see how close they came to finding the correct location of their number.
Today, we will use side lengths to help us more accurately locate these numbers on a number line.
Teacher Note
Because students are reasoning with a very new understanding of square roots, errors and inaccuracies may arise in this activity. Based on their work in lesson 19, students should be able to place the square roots between the appropriate consecutive whole numbers on the number line.
By the end of this lesson, students will have an opportunity to refine those locations. They will approximate values of square roots to the nearest hundredth more strategically and formally in lesson 21.
Learn
The Spiral of Theodorus
Students construct the Spiral of Theodorus to relate square roots to lengths.
Have students remove the Spiral of Theodorus from their books.
Invite students to study the triangle, and then lead them through a class discussion by using the following prompts.
How can we classify the triangle?
The triangle is a right triangle. It is also isosceles.
How do we know that the triangle is a right, isosceles triangle?
The triangle has a right-angle mark, and the two legs have the same lengths.
How can we find the length of the hypotenuse?
We can use the Pythagorean theorem.
Use the Pythagorean theorem to find the length of the hypotenuse. Let a and b represent the length of each leg, and let c represent the length of the hypotenuse.
Once most students are finished, display the work shown ending in 2 = c 2 .
Continue the class discussion by using the given prompts.
Is there a whole number we can square to equal 2? Explain.
No, because 1 2 = 1 and 2 2 = 4, the number we square to equal 2 must be between 1 and 2, which would not be a whole number.
How can we represent the exact length of the hypotenuse c ?
We can use square root notation.
What is the value of c for this triangle?
The value of c is √2 .
Label the length of the hypotenuse √2 .
We will create a spiral made by connecting right triangles, called the Spiral of Theodorus, starting from the triangle we already have. Each triangle has one leg with a length of 1 unit and another leg with the same length as the hypotenuse of the triangle before it.
Guide students through the creation of the next triangle.
Instruct students to use their index card to measure one leg of the triangle and mark a length of 1 unit on the card. Then have students line up the corner of their index card along the hypotenuse to make a right angle.
Then, by using the index card as a straightedge, have students draw a 1-unit segment from the mark on the card to the top vertex of the triangle.
UDL: Action & Expression
Consider offering the following options for completing the Spiral of Theodorus.
• Allow students to work with a partner to complete the Spiral of Theodorus.
• Provide visual and written directions that students can reference as they create the Spiral of Theodorus.
Instruct students to mark a right angle and a length of 1 for the new segment. Finally, have students use the index card as a straightedge to connect the end of the new segment to the center point of the spiral to form the hypotenuse of the new triangle.
• Open and display the Spiral of Theodorus teacher interactive, pausing as desired so that students can calculate hypotenuse lengths. Then provide students with a copy of Completed Spiral of Theodorus (in the teacher edition). Have them use an index card to transfer lengths from the spiral to the number line.
These options may decrease barriers posed by the fine motor and executive function demands of the task. They may also make sense for classes with limited time available.
When the first drawn triangle of the spiral is complete, calculate the length of the hypotenuse c together as a class.
Students may need support in understanding why (√2 ) 2 = 2. Use the following prompts and show the steps to help students make sense of this fact by first using a perfect square example. Then proceed to reasoning with √2 .
If c 2 = 25 for a right triangle, then we know by the definition of a square root that c = √25 . We also know that c = 5, because 5 2 = 25.
Teacher Note
Consider reminding students that equations of the form x 2 = p have two solutions when p is positive. However, because they are finding the length of the hypotenuse of a triangle, only the positive solution makes sense.
Students solve the general form of x 2 = p by using square root notation in lesson 24.
We know that √25 = 5. We can use this fact to evaluate
Since we do not actually know what √2 is, we cannot write the middle step in the same way we did for √25 .
Refer students back to the displayed (√25 ) 2 = (5) 2 = 25 and cover the middle expression (5)2. Have students turn and talk to a partner about what they notice regarding the remaining two expressions. Then invite a few students to share what they notice. If prompting is needed, ask students what they notice about the number in the square root symbol and the final answer.
Because we notice that (√25 ) 2 = 25, what is the value of (√2 ) 2?
2
If students need additional support with this reasoning, work through another example with a perfect square.
Have students label the hypotenuse of their new triangle √3 . Then tell students to repeat the entire process for the next triangle and each triangle after that. If students struggle to get started on the next triangle, guide them through one more triangle before they continue on their own.
The goal is for each student to create their own Spiral of Theodorus, but they may check in with another student as they work.
Circulate as students work to assist as needed in the construction process and to offer support in squaring square roots in the calculations. Students may use the workspace provided to do the hypotenuse length calculations.
Promoting the Standards for Mathematical Practice
When students repeatedly solve for the value of c in a 2 + b 2 = c 2 for each subsequent triangle in the spiral and recognize that the radicand increases by 1 each time, they are looking for and expressing regularity in repeated reasoning (MP8).
Ask the following questions to promote MP8:
• What patterns do you notice when you solve for c from one triangle to the next?
• How can this pattern help you solve for c more efficiently?
• Will this pattern always work? Explain.
Allow students to work on their spirals for about 10–12 minutes. Students do not need to complete the spiral all the way to the triangle with the hypotenuse length of √17 units if time does not permit.
Use the following prompts to debrief the activity.
What did you notice about the hypotenuse lengths as you continued to add triangles to the spiral?
The number under the square root symbol increased by 1 each time.
Do we know a whole-number value for any of the lengths? Explain.
Yes. We know that √4 = 2, √9 = 3, and √16 = 4 because 4, 9, and 16 are perfect squares.
Teacher Note
The completed spiral shown is where Theodorus stopped. The spiral could go on forever, but the 17th triangle begins to overlap with the rest of the spiral. Although there are plausible theories, the reason why Theodorus stopped at the triangle with a hypotenuse length of √17 units is unknown.
Square Roots on a Number Line
Students use the lengths from the Spiral of Theodorus to place square roots on a number line.
Direct students back to their spiral. Make sure they have their index card accessible.
The spiral shows us that each square root has a numerical value because each square root represents the length of a hypotenuse. Let’s use our index card to measure each hypotenuse and then mark the square root lengths on the number line.
Recall that the √2 is not a whole number. Between which two whole numbers is √2 ?
How do you know?
We know that (√2 )2 = 2, so √2 is between 1 and 2 because 12 = 1 and 22 = 4.
Let’s see whether we are correct by measuring the length of √2 on the spiral and transferring the length to the given number line.
Have students line up the corner of their index card on the center point of the spiral and line up the edge on the hypotenuse labeled √2 . Then, have the students mark the length of the hypotenuse on their card with a tick mark. Next, move the index card to the number line, aligning the corner of the card at 0, and plot a point at √2 on the number line. Label the point √2 .
Recall that the value of √2 is between 1 and 2. Does your number line verify this estimate?
Yes, the number line verifies this estimate because the point for √2 is between 1 and 2.
Can we say with certainty which whole number √2 is closer to? Why?
Yes, we can say with certainty that √2 is closer to 1. We can verify this by measuring on the number line. We also know that (√2 ) 2 = 2, and 2 is between the perfect squares 1 and 4. Because 2 is closer to 1 than it is to 4, √2 must be closer to 1 than it is to 2.
What might be a good estimate for the value of √2 to the nearest tenth based on your number line? How can we check our estimate?
Student estimates may vary, but listen for estimates around 1.3 and 1.4. Students can check their estimates by squaring their value and seeing how close their estimate is to 2, which is the basis of lesson 21.
Have students use their index card to repeat the transfer process through at least √9 , or for as many lengths as completed on their spiral. If short on time, consider displaying the finished number line after students have had the opportunity to transfer at least two lengths to the number line.
Once students complete the transfer process or are viewing a finished number line, have them think–pair–share about the following question. Give students a brief think time and then have them turn to a partner to discuss. Invite several pairs to share explanations.
The number line shows that the values of √5 , √6 , √7 , and √8 all fall between 2 and 3. Explain why this makes sense.
We know that (√5 ) 2 = 5, (√6 ) 2 = 6, (√7 ) 2 = 7, and (√8 ) 2 = 8. The numbers 5, 6, 7, and 8 are between two perfect squares, 4 and 9. Because 22 = 4 and 3 2 = 9, the numbers you square to get 5, 6, 7, and 8 must be between 2 and 3.
Next, have students compare their number line with their partner’s number line. Students may discover that the points on their number line are in slightly different locations than their partner’s number line. They may also notice that the points for √4 and √9 are not exactly on 2 and 3 as expected. Such discoveries provide a basis for rich discussion. Use the following questions to engage the class in a discussion about accuracy:
• Why might your points be placed slightly differently than your partner’s points?
• In what ways might the materials or the method we used to create the spiral lead to slight inaccuracies on our number lines?
• What kind of human errors may have affected our accuracy?
Differentiation: Challenge
For students ready for more of a challenge, ask questions about other square roots similar to the questions posed to the class about √2 .
• Is closer to or to ?
• Based on your number line, what might be a good estimate for to the nearest tenth?
Consider extending even further to foreshadow lesson 21.
• How might we come up with more precise approximations of numbers such as √2 ?
Land
Debrief 5 min
Objective: Place square roots on a number line.
Have students compare the placement of the numbers on their number lines to those on the class number line from Launch. This allows students to assess how accurate their original placements were. Facilitate a class discussion by using the following prompt. Encourage students to restate or build upon one another’s responses.
How do we determine the location of √2 on the number line?
We know that (√2 ) 2 = 2, and 2 is between the two perfect squares 1 and 4. Because 12 = 1 and 22 = 4, the number you square to get 2 must be between 1 and 2.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
Square Roots
In this lesson, we
• found the squares of square roots.
• created the Spiral of Theodorus to relate square roots to length measurements.
• placed square roots on a number line. Examples
2. Find the length of the hypotenuse
(
12 ) 2 = 12, and 12 is between the two perfect squares 9 and 16 Because 3 2 = 9 and 4 2 = 16, the number squared to get 12 must be between 3 and 4
represents the approximate location of
on the number
length of the hypotenuse is √8 units.
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
Topic E
Irrational Numbers
Now that students have experienced square roots of nonperfect squares and cube roots of nonperfect cubes, the next natural step is to find a more accurate approximation for these values. In this topic, students begin by approximating the length of a hypotenuse of a right triangle, such as √2 . They approximate values of square roots by using a number line to consider which whole-number interval includes the value. Then, they continue to increase the precision of the interval to consecutive tenths, hundredths, and thousandths. After students gain more insight about decimal approximations of square roots and cube roots, they begin to classify real numbers as rational or irrational.
Students experience connections to geometry when they approximate the edge length of a cube when given its volume and the area of a square when given an irrational side length. Students approximate square roots and cube roots throughout the topic to order and place real numbers on a number line. In module 4, students use algebraic understanding to write the fraction form of a repeating decimal.
The sequence of topic E allows students to experience irrational numbers, and then name and classify them as part of the real numbers. Students use number sense to approximate irrational numbers and determine when a more precise approximation is necessary. Students then solve equations involving square roots and cube roots using square root and cube root notation. Additionally, students use the relationship between the square and the square root, and the cube and the cube root, to determine the number of solutions for equations expressed as x 2 = p and x 3 = p, where p is a rational number and the solutions are real numbers.
In module 2, students apply solving equations of the form x 2 = p to find the length of a leg or the hypotenuse of a right triangle. The equations increase in complexity as students find the length of a leg or the length of the hypotenuse when given side lengths that are irrational. In module 5, students revisit square roots to find the unknown radii of circles given their areas and cube roots to find the unknown radii of spheres given their volume.
Progression of Lessons
Lesson 21 Approximating Values of Roots and π 2
Lesson 22 Familiar and Not So Familiar Numbers
Lesson 23 Ordering Irrational Numbers
Lesson 24 Revisiting Equations with Squares and Cubes
Approximating Values of Roots and π 2
Approximate values of square roots, cube roots, and π2 .
Explain how you approximate the value of √17 by rounding to the nearest tenth.
First, I find which perfect squares 17 is between. Because 17 is between 16 and 25, the value of √17 is between 4 and 5
Then, I explore the tenths from 4 to 5 through guess and check. I find that 17 is between 4.1 2 and 4.2 2 , or 16.81 and 17.64.
Next, I explore the hundredths from 4.1 to 4.2 through guess and check. I find that 17 is between 4.12 2 and 4.13 2 , or 16.9744 and 17.0569. Both 4.12 and 4.13 round to 4.1, so √17 ≈ 4.1
Lesson at a Glance
In this lesson, students consider the values of roots and other expressions such as π 2 before placing those values in their approximate locations on the number line. Students find the approximate locations of these less familiar numbers by using rational number benchmarks such as perfect squares and cubes. This lesson introduces the term cube root.
Key Questions
• Can we write all square roots or cube roots as rational numbers? Why?
• How would we begin to approximate the square root of a whole number that is not a perfect square?
Achievement Descriptors
8.Mod1.AD3 Locate irrational numbers approximately on a number line. (NY-8.NS.2)
8.Mod1.AD4 Approximate the values of irrational expressions. (NY-8.NS.2)
Student Edition: Grade 8, Module 1, Topic E, Lesson 21
Agenda
Fluency
Launch 5 min
Learn 30 min
• Approximating Square Roots
• Approximating Cube Roots
• Approximating π 2
Land 10 min
Materials
Teacher
• None
Students
• Number Lines
• Scientific calculator
Lesson Preparation
• None
Fluency
Plot on a Number Line
Students plot square roots of perfect squares on a number line to prepare for computing the approximate value of square roots.
Directions: Plot and label the number on the number line.
Teacher Note
Students may use the Number Lines: 0 to 10 removable.
Launch
Students guess and check the value of √2 .
Display a right triangle with leg lengths of 1 unit.
Have students use their personal whiteboard as needed to calculate the answer to the following question. Some students may simply recall this answer from Lesson 20.
What is the exact length of the hypotenuse?
√2 units
Confirm the exact length and then have students think–pair–share about the following question.
What do you think is an approximate length of the hypotenuse in units?
Invite pairs to share their approximate lengths. Then facilitate a discussion by using the following prompts.
√2 represents an exact number we can square to get 2. Let’s approximate the value of √2 .
What number is too large for an approximation? How do you know?
The number 2 is too large because 22 is 4.
What number is too small for an approximation? How do you know?
The number 1 is too small because 12 is 1.
We know the value of √2 is between 1 and 2 because 12 = 1 and 22 = 4. What is a more precise approximation for the value of √2 ?
Sample: 1.5
How could we test our approximations to see if they are close approximations for the value of √2 ?
We could square the numbers and see if the results are close to 2. 5
If they have not already done so, have pairs test their approximations by squaring them.
What could improve our approximations of the value of √2 ?
We could use more decimal places.
Today, we will learn a strategy that helps us approximate the values of roots more precisely.
Learn
Approximating Square Roots
Students approximate the values of square roots.
Have students remove the Number Lines page from their book and place it into their personal whiteboard. Ensure that students have calculators, but restrict the use of the calculator to squaring numbers. Do not allow them to use the square root feature on their calculator.
We know the value of √2 is between 1 and 2. Let’s use a number line to help visualize the approximate location of √2 . We can focus on smaller intervals on the number line to find more precise approximations.
Direct students to use the number lines and their calculators to complete problem 1 with a partner.
Circulate and help students use number sense and guess and check to identify the appropriate intervals to find the approximate location for the value of √2 on the number line. Students can label each subsequent number line based on the interval they identify on the previous number line. Have students continue until the number line is measured in thousandths.
As students work, listen and look for the following reasoning and work.
We know the value of √2 is between 1 and 2 because 12 = 1 and 22 = 4.
We know the value of √2 is between 1.4 and 1.5 because 1.42 = 1.96 and 1.52 = 2.25.
Promoting the Standards for Mathematical Practice
When students repeatedly find the square of two values to determine a better approximation of √2 , they are looking for and expressing regularity in repeated reasoning (MP8).
Ask the following questions to promote MP8:
• What is the same about your reasoning as you refine your approximation of √2 ?
• Will this process of finding a more precise interval always work?
We know the value of √2 is between 1.41 and 1.42 because 1.412 = 1.9881 and 1.422 = 2.0164.
We know the value of √2 is between 1.414 and 1.415 because 1.4142 = 1.999396 and 1.4152 = 2.002225.
Ones
1. Approximate the value of √2 by rounding to the nearest given place value. Use the number lines and a calculator to complete the table.
UDL: Representation
Encourage students to highlight or circle the interval on each number line that contains the value of √2 . This can support students by helping them keep track of the interval they are zooming in on and by helping them label the first and last tick marks of the next number line.
When most students are finished, confirm the answers. Consider inviting one or two pairs to show their number lines and to describe their process for approximating the value of √2 . Then use the following prompts to facilitate a brief discussion.
How can we approximate the value of √2 by rounding to the nearest hundredth?
Because the value must be less than 1.415, we know the value of the thousandths place is less than 5. So we can approximate the value of √2 by rounding to 1.41.
Based on our work, can we approximate the value of √2 by rounding to the nearest thousandth? Why?
No, we cannot round to the thousandths place based on our work because we do not know the digit in the ten thousandths place.
How can we approximate the value of √2 to the nearest thousandth?
We need to continue the same process as before with the interval 1.414 to 1.415.
Approximating Cube Roots
Students approximate the values of cube roots.
Display the cube and its volume, 8 cubic units.
How can we find the volume of a cube?
We can multiply the length, width, and height of the cube.
How can we find the edge length of a cube if we know that the volume of the cube is 8 cubic units?
Because we know all the edges of a cube have the same length, we can find a number with a cube of 8.
What is the edge length of the cube? How do you know?
The edge length is 2 units because the volume is 8 cubic units and 23 = 8.
When we look for a number with a cube of 8, we are finding the cube root of 8. In general, the cube root of a number x is a number that when cubed is x.
Language Support
Consider comparing cube roots to square roots to access students’ existing understandings of roots. For example, highlight that both squares and cubes make use of repeated multiplication.
The goal is to find a number that results in the given number after repeated multiplication.
Display the following equations.
23 = 8
2 = 3 √ 8
Model the language used to describe the equations while pointing to them.
• 2 cubed is 8.
• 2 is the cube root of 8.
Suppose the volume of a cube is 18 cubic units. What is the edge length of the cube?
The edge length of the cube is the number of units we cube to get 18 cubic units.
The edge length of the cube is 3 √ 18 units.
The number 3 √ 18 represents a number with a cube of 18.
How is finding the value of 3 √18 different from finding the value of 3 √ 8 ?
Finding the value of 3 √18 is different because 18 is not a perfect cube.
Because 18 is not a perfect cube, let’s approximate the value of 3 √ 18 .
Have students erase the work from problem 1 on their personal whiteboards. Then direct them to complete problem 2 with their partner by using a method similar to the one they used to approximate the value of √2 . Ensure that students have calculators but restrict their use to cubing numbers. Do not allow them to use the cube root feature on their calculator.
Circulate and support students in narrowing in on appropriate intervals on the number lines. As students work, listen and look for the following reasoning and work.
We know the value of 3 √ 18 is between 2 and 3 because 23 = 8 and 33 = 27.
We know the value of 3 √ 18 is between 2.6 and 2.7 because 2.63 = 17.576 and 2.73 = 19.683.
We know the value of 3 √ 18 is between 2.62 and 2.63 because 2.623 = 17.984728 and 2.633 = 18.191447.
We know the value of 3 √ 18 is between 2.62 and 2.621 because 2.623 = 17.984728 and 2.6213 = 18.005329061.
Thousandths
2. Approximate the value of 3 √ 18 by rounding to the nearest given place value. Use the number lines and a calculator to complete the table.
Place Value Rounded Value
When most students are finished, confirm the answers. Have students turn and talk with their partner to describe how they would approximate the value of 3 √ 18 to the nearest thousandth.
Approximating π2
Students approximate the value of π 2 .
Display π = 3.1415926535… .
What is this number?
The number is pi.
What does pi represent?
Pi is the circumference of a circle divided by the diameter.
Pi is the value of the ratio of a circle’s circumference to its diameter. The value of pi is a decimal that neither terminates nor repeats.
Between what two whole numbers is the value of π on a number line?
The value of π is between 3 and 4.
What two whole numbers is the value of π 2 between? How do you know?
The value of π 2 must be between 9 and 16 because 32 = 9 and 42 = 16.
Have students complete the table in problem 3(a). Then encourage them to use their number lines as needed to narrow down the intervals that contain π to tenths, hundredths, and thousandths to complete the rest of the table. Then have students complete problem 3(b).
3. Complete the tables.
a. Determine the intervals that contain π and π 2 .
b. Use your results from part (a) to approximate the value of π 2 to the nearest given place value.
Confirm answers to problem 3. Students may have expected the value of π 2 to be closer to 9 because of how close the value of π is to 3. Consider using this opportunity to discuss the influence of rounding values before calculations rather than after calculations by asking the following question.
We can round numbers either before calculations or after calculations. How does this choice affect the results of the calculation?
Rounding after calculations will be much more accurate than rounding before.
Land
Debrief 5 min
Objective: Approximate values of square roots, cube roots, and π 2 .
Lead a class discussion by using the following prompts.
If x 2 = 12, what is the exact value of x ?
√12
Can we write the value of √12 as a rational number? Why?
No, we have to approximate the value of √12 because 12 is in between two perfect squares, so its square root is not a whole number. We also can’t write 12 as a fraction with perfect squares to find its square root like we have done in other lessons.
Can we write the value of 3 √ 12 as a rational number? Why?
No, we have to approximate the value of 3 √ 12 because 12 is in between two perfect cubes, so its cube root is not an integer. We also can’t write 12 as a fraction with perfect cubes to find its cube root like we have done in other lessons.
How can we begin to approximate the value of √12 ?
We can begin by identifying the closest perfect squares to 12, which are 9 and 16. So we know the value of √12 is between √9 and √16 , or between 3 and 4.
How can we get a better approximation?
We can approximate the value of the number by finding two numbers that are close to that value. We can identify one number that is greater than the value and one number that is less than the value we are approximating and then square the two numbers. We can make these approximations as precise as we want by choosing decimal numbers in the tenths, hundredths, and thousandths.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
For the Practice problems, students should have access to a scientific calculator or similar digital device.
Student Edition: Grade 8, Module 1, Topic E, Lesson 21
Approximating Values of Roots and π 2
In this lesson, we
• approximated values of square roots, cube roots, and π 2 .
• explored and defined cube root notation and approximated the values of cube roots.
Terminology
The cube root of a number x is a number with a cube that is x. The expression 3 √x represents the cube root of x
Examples
For problems 1 and 2, determine the two consecutive whole numbers each value is between.
1. 8 < √68 < 9 68 is between the perfect squares 64 and 81
problems 3 and 4, round each root to the nearest whole number.
2. 2 < 3 √ 12 < 3 12 is between the perfect cubes 8 and 27
3 √8 < 3 √12 < 3 √27
2 < 3 √12 < 3
5. Explain how you approximate the value of 3 √46 by rounding to the nearest tenth.
First, I find which perfect cubes 46 is between. Because 46 is between 27 and 64, the value of 3 √46 is between 3 and 4
Then, I explore the tenths from 3 to 4 through guess and check. I find that 46 is between 3.5 3 and 3.6 3, or 42.875 and 46.656
Next, I explore the hundredths from 3.5 to 3.6 through guess and check. I find that 46 is between 3.58 3 and 3.59 3, or 45.882712 and 46.268279. Both 3.58 and 3.59 round to
EUREKA MATH2 New York Next Gen
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
For problems 1–4, determine the two consecutive whole numbers each value is between.
For problems 5–8, round each root to the nearest whole number.
9. Henry states that the value of √30 is between 5.5 and 5.6 Do you agree or disagree with Henry?
Explain your reasoning.
I disagree with Henry. The value of √30 is not between 5.5 and 5.6 because 5.5 2 = 30.25 and 5.6 2 = 31.36, which are both greater than 30. The value of √30 is between 5.4 and 5.5
10. Explain how you approximate the value of √23 by rounding to the nearest tenth.
First, I find which perfect squares 23 is between. Because 23 is between 16 and 25 the value of √23 is between 4 and 5
Then, I explore the tenths from 4 to 5 through guess and check. I find that 23 is between 4.7 2 and 4.8 2 , or 22.09 and 23.04
Next, I explore the hundredths from 4.7 to 4.8 through guess and check. I find that 23 is between 4.79 2 and 4.80 2 , or 22.9441 and 23.04. Both 4.79 and 4.80 round to 4.8, so √23 ≈ 4 8
For problems 11–13, approximate each root by rounding to the nearest whole number, tenth, and hundredth.
EUREKA MATH2 New York Next
14. Find the length of the hypotenuse c. Approximate the length by rounding to the nearest tenth.
19. Use the definition of negative exponents to write 1 7 4 with positive exponents.
20. Use the coordinate plane to answer the following questions.
a. Which point is on the y-axis?
b. What is the ordered pair for the point on the y-axis? (0, 9)
c. Which point is located at (6, 1)?
d. What is the ordered pair for point T ?
6)
Familiar and Not So Familiar Numbers
Identify numbers as rational, irrational, and real by their decimal form. Compare the characteristics of rational and irrational numbers.
Lesson at a Glance
1. Compare the decimal forms of a rational number and an irrational number. The decimal form of a rational number terminates or repeats, and the decimal form of an irrational number never terminates or repeats.
2. Indicate whether the number is rational or irrational.
In this lesson, students study the decimal forms of a set of real numbers. Students classify numbers by determining whether the next decimal digit can be identified by repetition in the given digits. They use that distinction and their prior knowledge of rational numbers to develop a definition of irrational numbers. Students work on refining the definition while classifying rational and irrational numbers given in various forms. This lesson formally defines the terms irrational numbers and real numbers.
Key Questions
• How can we tell whether a number is rational or irrational?
• How are rational and irrational numbers related to real numbers?
Achievement Descriptor
8.Mod1.AD1 Determine whether numbers are rational or irrational. (NY-8.NS.1, NY-8.EE.2)
Agenda
Fluency
Launch 5 min
Learn 30 min
• Terminating and Repeating
• Is It Rational?
Land 10 min
Materials
Teacher
• Construction paper (3)
• Marker
• Tape Students
• None
Lesson Preparation
• Prepare three signs labeled Always Rational, Always Irrational, and Both. Post each sign on a different wall in the classroom.
Fluency
Write Decimals as Fractions
Students write decimals as fractions to prepare to categorize numbers.
Directions: Write the number as a fraction.
Launch
Students identify the next decimal digit for repeating and nonrepeating decimal forms of numbers.
Have students complete problems 1–7 independently.
For problems 1–7, identify the decimal digit that comes next in the decimal form of each number. If you cannot identify the next decimal digit, write cannot identify
Bring the class together and prompt discussion with the following questions:
• How did you identify which decimal digit came next in the decimal form of each number?
• Was it difficult to identify the next decimal digit in any of the decimal forms of each number? Why?
Focus the discussion on whether the decimal digits of each number eventually repeat. The decimal digits in the decimal form of the numbers in problems 1, 3, 4, and 7 eventually repeat, but the decimal digits in problems 2, 5, and 6 do not repeat.
Based on the observations you made, how would you classify the numbers in problems 1 through 7?
Some numbers have decimal digits that eventually repeat. Other numbers do not have any decimal digits that repeat.
Today, we will learn to define two new types of numbers based on how we classified the numbers in problems 1 through 7.
Learn
Terminating and Repeating
Students classify numbers in decimal form by using the descriptions of rational and irrational numbers.
Use the following prompts to engage students in a discussion where they use the definition of a rational number from grade 7 to develop a definition for the new terms irrational number and real number.
What do we call numbers that have a decimal form that eventually repeats?
We call them rational numbers.
A rational number has a decimal form that terminates or eventually repeats. Any rational number can be written as a fraction.
What could we call numbers that have a decimal form that does not terminate and does not repeat?
Allow students to brainstorm and invite several suggestions. Consider writing a list of these suggestions for the class to see and engage in a discussion around them.
We call these numbers irrational numbers. An irrational number is a number that is not rational and cannot be expressed as p __ q for integer p and nonzero q. An irrational number has a decimal form that is neither terminating nor repeating.
Allow students time to work in pairs on problem 8. If pairs finish early, invite them to compare answers with a neighboring pair.
8. Identify whether numbers from problems 1–7 are rational or irrational. Justify your reasoning.
The numbers in problems 1, 3, 4, and 7 are rational because each decimal form eventually repeats.
The numbers in problems 2, 5, and 6 are irrational because each decimal form does not terminate or repeat.
Confirm answers as a class.
Engage in a class discussion about rational, irrational, and real numbers.
If a number is written in decimal form, how can we tell if it is rational?
If the decimal form terminates or eventually repeats, the number is rational because it can be written as a fraction. If the decimal form does not terminate or repeat, the number is irrational.
The digits in the decimal form of a fraction eventually repeat or terminate. Because all fractions are rational, all repeating or terminating decimal forms of numbers are rational.
Any number that is either rational or irrational is considered a real number.
Direct students to problem 9 and have them work in pairs. Encourage pairs to use examples different from problems 1–7.
Language Support
Consider calling students’ attention to the prefix ir-, meaning not in words such as irrelevant and irregular. This will help students understand the distinction between rational and irrational.
Teacher Note
In grade 7, a terminating decimal is defined as a decimal that can be written with a finite number of nonzero digits.
A decimal is defined as repeating if, after a certain digit, all remaining digits consist of a block of one or more digits repeated indefinitely.
Consider playing the Long Division with Repeating Decimals video to reactivate prior learning.
Teacher Note
If students ask about numbers that are not real, celebrate this question and tell them that non-real numbers do exist. These numbers are called imaginary numbers and are studied in later courses.
9. Complete the table by including a definition, examples, and features for each type of number.
Rational Numbers Irrational Numbers Real Numbers
Definition A number with a decimal form that terminates or eventually repeats A number with a decimal form that is neither terminating nor repeating A number that is either rational or irrational
1.23444444… 5.678678678… 3.4642548… 0.121314151…
Features Any fraction
Any decimal form with bar notation Cannot be written as a fraction Decimal form does not terminate or repeat All the numbers we know
When most pairs have finished, ask each pair to share with the class an example from each column.
Continue student analysis and classification of different types of numbers by using the following examples and prompts.
Display the numbers 0.25 and 0.25.
Which of these numbers is rational? Why?
Both numbers are rational. The first number is rational because the decimal form terminates. The second number is rational because it has bar notation, which means it is a repeating decimal.
If the decimal form of a number terminates or can be written with bar notation, then the number is rational.
Have students include these numbers in the Examples row of the table in problem 9. Have them also include bar notation in the Features row of the table. Then engage in an open discussion about square roots.
Are all square roots irrational?
Invite a variety of opinions and then display the following examples.
2 = 1.41421356…
25 = 5
Give students time to think–pair–share about the following discussion questions:
• Are these numbers rational or irrational? How do you know?
• Are all square roots irrational? How do you know?
After pairs share their answers with the class, instruct students to add each number, √2 and √25 , to the Examples row of the table in problem 9.
Differentiation: Support
For a further review of bar notation, consider having students return to problems 1–7 to place the bar notation appropriately.
Is It Rational?
Students classify numbers by using the definitions of rational and irrational numbers.
Have students complete problem 10 individually.
10. Indicate whether the number is rational or irrational.
Once most students have finished, have them compare answers with a partner. Facilitate a discussion about the difference between a decimal form that eventually repeats and a decimal form that follows a pattern.
Did you classify 0.121314151… as rational or irrational?
Anticipate that students may categorize the number as rational because they are able to predict the next digit based on a pattern.
Rational numbers require that the decimal form repeats. The number 0.121314151… follows a pattern, but not a repeating pattern, so it is irrational.
Next, prepare students for the Take a Stand routine by showing the three posted signs titled Always Rational, Always Irrational, and Both. For each type of number displayed, direct students to stand near the sign that classifies the type of number. Ask students to share and justify their opinions, making sure to hear from people near different signs. If students hear a justification that changes their mind, allow them to move to reflect that change.
Display the following types of numbers one by one with the following prompt.
Determine whether the following types of numbers are always rational, always irrational, or could be both.
• Integers
Always rational
• Square roots
Both
• Negative numbers
Both
• Fractions
Always rational
• π
Always irrational
To close the activity, invite students to share which types of numbers were the most challenging to classify and why.
Land
Debrief
5 min
Objectives: Identify numbers as rational, irrational, and real by their decimal form.
Compare the characteristics of rational and irrational numbers.
Invite students to share their descriptions of the terms rational and irrational. Encourage students to expand their classmates’ descriptions and revise or enhance their own.
Promoting the Standards for Mathematical Practice
When students justify their own reasoning and then listen to and analyze their peers’ reasoning in the Take a Stand routine, they are constructing viable arguments and critiquing the reasoning of others (MP3).
Ask the following questions to promote MP3:
• Is your type of number choice a guess, or do you know for sure? How do you know for sure?
• What questions can you ask your peers standing by other types of numbers to make sure you understand the reason for their choice?
How can we tell whether a number is rational or irrational?
A number is rational if it can be expressed as a fraction. A number is rational if its decimal form is either terminating or repeating.
A number is irrational if it cannot be expressed as a fraction. A number is irrational if its decimal form does not terminate or repeat.
How are rational and irrational numbers related to real numbers?
All rational and irrational numbers can be classified as real numbers.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
Familiar and Not So Familiar Numbers
In this lesson, we
• identified numbers as rational, irrational, and real.
• compared the features of rational and irrational numbers.
Examples
For problems 1–7, identify whether the number is rational or irrational.
1. 4 3
An irrational number is a number that is not rational and cannot be expressed as p q for integer p and nonzero integer q An irrational number has a decimal form that is neither terminating nor repeating.
A real number is any number that is either rational or irrational.
A rational number can be written as a fraction. When the decimal form of the number neither terminates nor repeats, the number cannot be written as a fraction.
Some roots are irrational, and some roots are rational.
Student Edition: Grade 8, Module 1, Topic E, Lesson 22
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
Student Edition: Grade 8, Module 1, Topic E, Lesson 22
1. Complete the table.
a. Identify the decimal digit you think comes next in the decimal form of each number. If you cannot identify the next decimal digit, write cannot identify
b. Identify whether the number is rational or irrational.
2. Sort each value into the appropriate column on the table.
3. Abdul can predict the next decimal digit in the decimal form of the number 0.010305070… so he concludes that the number is rational. Explain the error in Abdul’s thinking.
The decimal digits in the number follow a pattern, but the pattern does not repeat. The decimal expansion of a rational number must either terminate or repeat. So this number is irrational.
4. Ethan states that √5 is an irrational number. Nora states that √5 is a real number. Who is correct? Explain.
Both Ethan and Nora are correct. Ethan is correct because √5 has a decimal form that is neither terminating nor repeating. Nora is correct because all rational and irrational numbers are real numbers.
5. Eve states that all square roots are irrational numbers. Provide three examples that show Eve is incorrect.
Not all square roots are irrational numbers. Some square roots can be written as whole numbers such as √4 = 2, √9 = 3, and √16 = 4
For problems 6–9, divide.
EUREKA MATH2 New York Next Gen
10. Two vertices of a rectangle are located at (−2, −9) and (3, −9). The area of the rectangle is 100 square units. Name the possible locations of the other two vertices. Use the coordinate plane as needed.
(−2, 11) and (3, 11) or (−2, −29) and (3, −29)
1.
Ordering Irrational Numbers
Order irrational numbers.
Approximate the value of expressions with irrational numbers.
Lesson at a Glance
A playful game of Battle Cards allows students to compare rational and irrational numbers, one pair of numbers at a time. Rather than evaluating each number, students consider which whole-number interval includes the number on the card to determine the winning card. They then consider those same cards as a larger group, order them from least to greatest and place them on a number line. Students again use wide-ranging approximation and number sense to order a set of expressions that include irrational numbers.
2. Compare the numbers by using the
Explain your reasoning.
Key Questions
• How do we order irrational and rational numbers?
• Can we use the same strategy to order expressions that include irrational numbers? Why?
Achievement Descriptors
8.Mod1.AD2 Use rational approximations of irrational numbers to compare the size of irrational numbers. (NY-8.NS.2)
8.Mod1.AD3 Locate irrational numbers approximately on a number line. (NY-8.NS.2)
8.Mod1.AD4 Approximate the values of irrational expressions. (NY-8.NS.2)
Agenda
Fluency
Launch 10 min
Learn 25 min
• Ordering Numbers
• Ordering Expressions
Land 10 min
Materials
Teacher
• None
Students
• Set of Battle Cards (1 per student pair)
Lesson Preparation
• Copy and cut out the Battle Cards (in the teacher edition). Prepare enough sets for 1 per student pair.
Fluency
Evaluate Roots
Students evaluate square roots and cube roots to prepare for approximating irrational numbers and expressions that include roots.
Directions: Evaluate the expression.
Launch
Students compare pairs of rational and irrational numbers.
Arrange students in pairs and hand out a set of Battle Cards to each pair. Instruct pairs to shuffle the cards and split them equally into two piles laid facedown, with one pile in front of each student. Have students flip over the top card in their pile. The student who turns over the card with the greater value collects both cards and sets them aside in their own winning pile. The game ends when students have gone through all their cards once.
Students should keep the cards in their winning pile to use in the next activity.
Circulate to listen for strategies that pairs of students use to compare their Battle Cards. When most pairs have finished playing, discuss the comparison strategies as a class.
When you compared two cards, how did you determine which card had the greater value?
For each card, we determined which two whole numbers the value on the card was between and then determined which whole number that value was closer to. Then we compared these whole numbers to determine which card had the greater value.
We just compared rational and irrational numbers. Today, we will place these numbers on a number line and approximate the values of expressions with irrational numbers.
Learn
Ordering Numbers
Students order rational and irrational numbers.
When all pairs finish playing, direct students to arrange the cards in their winning pile from least to greatest. After most students have their winning pile in order, tell pairs to order all the cards from least to greatest.
UDL: Action & Expression
To support students in monitoring their own progress, consider providing structure for pair sharing. Suggest that students take turns thinking aloud by explaining their reasoning and then providing feedback to one another.
Teacher Note
If time allows, students can play another round of Battle Cards at the end of the lesson.
After most pairs have their cards in order, explain that students have 2 minutes to study the card arrangement of a neighboring pair of students. Have pairs make revisions to the order of their cards if necessary.
Show the correct order, one card at a time. Note reactions from students and ask the reason for any moments of surprise, confusion, or realization. Instruct pairs to rearrange their cards as necessary. The following questions may be useful to debrief the activity:
• What differences did you notice between the card arrangements of your peers?
• Did you make any revisions to your work?
• Which cards did you find the most challenging to order?
• Which cards did you find the easiest to order?
Direct students to complete problem 1.
Plot and label each value on the given number line.
Display the correct locations of the numbers on the number line.
Differentiation: Challenge
Students who finish early may benefit from placing additional numbers from the set of Battle Cards on the number line. A number line with all 16 numbers is shown.
Ordering Expressions
Present problem 2 and use the Math Chat routine to engage students in mathematical discourse.
Give students 2 minutes of silent think time to compare the expressions. Have students give a silent signal to indicate they are finished.
Have students discuss their thinking with a partner. Circulate and listen as they talk. Ask a few students to share their thinking, and have the class make connections between shared strategies.
Then facilitate a class discussion. Invite students to share their thinking with the whole group and record their reasoning.
Continue the Math Chat routine for problems 3 and 4.
For problems 2–4, order the expressions from least to greatest. Explain your reasoning. 2. π, π 2 , 3π
Teacher Note
Some students may try to evaluate √28 5 as √23 or 2 · √17 as √34 . Address this misconception by discussing with students that we approximate the root first and then apply the other operations.
Promoting the Standards for Mathematical Practice
When students order expressions with irrational numbers from least to greatest without finding an approximate value for each expression, they are making use of structure (MP7).
I know that the value of π is the least because multiplying by 3 will make the value greater. Because the value of π is greater than 1, squaring π will also make the value greater.
I know that the value of π is greater than 3, so squaring π will be a greater value than multiplying π by 3.
Ask the following questions to promote MP7:
• How are π and π 2 related? How can that relationship help you compare the expressions in problem 2?
• How are 28 and 22 related? How can that relationship help you compare the expressions in problem 3?
• How can you use what you know about subtraction to help you compare √17 and √17 2 in problem 4?
3. √28 , √22 , √28 5, √22 + 5
√28 5, √22 , √28 , √22 + 5
I know that the value of √28 is between 5 and 6 but closer to 5. Subtracting 5 from √28 gives a value between 0 and 1 but closer to 0. So √28 is greater than √28 5.
I know that the value of √22 is between 4 and 5 but closer to 5. So √22 is less than √28 and greater than √28 5. Adding 5 to √22 gives a value between 9 and 10 but closer to 10. So √22 + 5 is greater than √28 .
4. √17 − 2, 2 · √17 , √17 + 2
√17 2, √17 + 2, 2 · √17
I know that the value of √17 is between 4 and 5 but closer to 4. Subtracting 2 from √17 gives a value between 2 and 3 but closer to 2. Multiplying √17 by 2 gives a value between 8 and 10 but closer to 8. Adding 2 to √17 gives a value between 6 and 7 but closer to 6. So √17 + 2 is less than 2 · √17 and greater than √17 2.
Differentiation: Challenge
Consider asking students to change √28 to 3 √28 . Students can approximate 3 √28 to the nearest whole number to compare √22 and 3 √28 . The rest of the comparison work will be similar to that shown for problem 3.
Land
Debrief 5 min
Objectives: Order irrational numbers.
Approximate the value of expressions with irrational numbers.
Facilitate a brief discussion by using the following questions. Encourage students to build on one another’s responses.
How do we order irrational and rational numbers?
We approximate irrational numbers before comparing them to rational numbers. Sometimes we can compare irrational numbers just by knowing which whole numbers they fall between. After approximating and comparing two numbers in this way, we can order a set of irrational and rational numbers.
Can we use the same strategy to order expressions that include irrational numbers? Why?
Yes, we can approximate the values of expressions that include irrational numbers. Because we are ordering, we just need to know which value is greater, not the exact value of each expression.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
Ordering Irrational Numbers
In this lesson, we
• compared pairs of rational and irrational numbers.
• ordered and plotted rational and irrational numbers on a number line.
• used whole-number intervals to approximate the value of expressions which include irrational numbers.
Examples
For problems 1–4, compare the numbers by using the < or > symbol. 1.
know
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
Consider the diagram in which two lines and a ray all meet at point C
18. Find the length of the hypotenuse
a. Which angle relationships will help you solve for x and y?
ACD and ∠ FCE are vertical angles. ∠ DCF and ∠ FCE are supplementary angles.
b. Find the value of y. 38
c. Find the value of x 142
Revisiting Equations with Squares and Cubes
Solve equations of the forms x2 = p and x3 = p, where p is a rational number and the solutions are real numbers.
Lesson at a Glance
In this lesson, students encounter more complex equations of the forms x 2 = p and x 3 = p than previously in the module. The complexity builds from p being a perfect square or perfect cube in lesson 17, to p now being any rational number. Students engage in the Critique a Flawed Response routine to analyze others’ work, correct errors, and clarify meaning of the solutions to an equation. Students also solve two-step equations, resulting in the forms x 2 = p and x 3 = p, in anticipation of skills required for finding the leg length of a right triangle in module 2.
Key Questions
• Given that p is a perfect square and q is not a perfect square, how are the solutions to x 2 = p and x 2 = q similar? How are the solutions different?
• Given that p is a perfect cube and q is not a perfect cube, how are the solutions to x 3 = p and x 3 = q similar? How are the solutions different?
Achievement Descriptor
8.Mod1.AD6 Solve equations of the form x 2 = p as √p and √p and equations of the form x 3 = p as 3 √p , where p is a rational number. (NY-8.EE.2)
Agenda
Fluency
Launch 5 min
Learn 30 min
• Revisiting Equations of the Form x 2 = p
• Revisiting Equations of the Form x 3 = p
• Solving Two-Step Equations
Land 10 min
Materials
Teacher
• None
Students
• None
Lesson Preparation
• None
Fluency
Rational or Irrational
Students classify square roots and cube roots as rational or irrational numbers to prepare for solving equations with rational and irrational solutions.
Directions: State whether each number is rational or irrational.
Launch
Students determine whether given values are solutions to equations involving x 2 and x 3 .
Have students work in pairs to complete problem 1.
1. Indicate whether the value of x is a solution to the given equations.
Have each pair of students turn to another pair of students and compare answers. If the answers differ, have students discuss within their group which answers are correct. After most students have finished, bring the class together for discussion. Use any, or all, of the following questions:
• Are there any equations for which both values are solutions? If so, explain.
• Are there any equations for which only one value is a solution? If so, explain.
• Are there any equations for which neither value is a solution? If so, explain.
In this lesson, we will revisit equations involving squares and cubes. This time, we will use square root and cube root notation to provide exact solutions to equations we could not solve before.
Teacher Note
Substitution may be the best approach for students who are still building proficiency with solving equations with perfect squares and perfect cubes. If students have trouble getting started, consider asking them how they can check whether a solution makes an equation true.
Differentiation: Support
If students need more specificity in the questions to engage effectively in the discussion, consider asking the following questions.
• Why are both 1 and −1 solutions to x 2 = 1, but only 1 is a solution for x 3 = 1?
• Why is −1 a solution to x 3 = −1, but neither 1 nor −1 are solutions for x 2 = −1?
Learn
Revisiting Equations of the Form x 2 = p
Students solve equations of the form x 2 = p, where p is a rational number and the solutions are real numbers.
Have students complete problems 2 and 3 either individually or in pairs.
For problems 2–7, solve the equation. Identify all solutions as rational or irrational.
2. x 2 = 25
The solutions are 5 and −5. Rational
3. 81 = m 2
The solutions are 9 and −9. Rational
Confirm answers with the class. Because students have most recently solved equations in the context of the Pythagorean theorem, they may not remember to include the negative solution to an equation of the form x 2 = p. Use the following prompts to reinforce the need to include all solutions.
What questions can we ask ourselves to solve the equations in problems 2 and 3?
For problem 2, we can ask the question, What numbers can we square to get 25? For problem 3, we can ask the question, What numbers can we square to get 81?
How many numbers can we square to get 25? To get 81? Explain.
We can square two numbers, 5 and −5, to get 25. We can square two numbers, 9 and −9, to get 81. The square of a positive number is always positive, and the square of a negative number is also always positive.
What do you notice about the two solutions to equations such as x 2 = 25?
The two solutions are opposites of each other.
UDL: Representation
Reviewing equations similar to those students have seen before helps activate prior knowledge and supports making connections to more complex equations.
When we solved equations by using the Pythagorean theorem, how many solutions did we have? Why?
We had only one solution because we were finding the hypotenuse length of a triangle, which is always positive.
Recall that when an equation is not representing quantities within a context, we need to consider all possible values of the variable. So equations of the form x 2 = p have two solutions for positive values of p, and those two solutions are opposites of each other.
Direct students to problem 4. Use the following prompts to discuss the similarities and differences between x 2 = 35 and the equations in problems 2 and 3.
How is the equation in problem 4 similar to the equations in problems 2 and 3? How is it different?
The equation in problem 4 is similar because it has a variable squared. It is different because 35 is not a perfect square, but the numbers 25 and 81 in the other equations are perfect squares.
We have solved equations before that did not involve a perfect square. With those equations, how did we write the exact value of x?
We used a square root symbol.
Recall that the square root symbol represents the positive square root. But what do we know about equations of this form when they are not representing a context?
The equation has two solutions: a positive number and its opposite.
Guide the class to express the two possible values of x as a compound statement by doing problem 4 together. Emphasize that the square root symbol represents only the positive square root and, again, highlight the need to consider both the positive and negative values of x. Also point out that √35 and √35 are opposites and irrational.
Teacher Note
In grade 8, students write solutions to x 2 = p as √p and √p . Students transition to the shorthand notation ± √p in later courses.
Have students complete problems 5–7, individually or in pairs.
4. x 2 = 35
x 2 = 35
x = √35 or x = − √35
The solutions are √35 and − √35 . Irrational
6. m 2 = 144
m 2 = 144
m = √144 m = 12 or m = √144 m = −12
5. 110 = h2 110 = h 2 √110 = h or √110 = h
The solutions are √110 and √110 . Irrational
The solutions are 12 and −12. Rational 7. t 2 = 27 t 2 = 27 t = √27 or t = −√27
The solutions are √27 and √27 . Irrational
Check responses as a class. Problem 6 provides a nice opportunity to allow for debate about whether the solutions are √144 and √144 or 12 and −12. Some students may be so focused on solving the equation that they do not notice 144 is a perfect square. Other students may recognize that 144 is a perfect square immediately. Invite students to offer opinions about which solutions are correct and why. Discussion should culminate with students recognizing that the solutions are equivalent and rational.
Teacher Note
When p is a perfect square, some students might apply the intuitive approach from lesson 17 and skip the step of writing the solutions with square root notation. For example, in problem 6, students may proceed directly to identifying 12 and −12 as the solutions. Allow for either approach.
Revisiting Equations of the Form x 3 = p
Students solve equations of the form x 3 = p, where p is a rational number and the solutions are real numbers.
Have students complete problems 8 and 9 individually or in pairs.
For problems 8–13, solve the equation. Identify all solutions as rational or irrational.
8. t 3 = 27
The solution is 3. Rational
9. −125 = k 3
The solution is −5. Rational
Confirm responses as a class. Then use the following prompts to reinforce that equations of the form x 3 = p have only one solution.
What questions can we ask ourselves to solve the equations in problems 8 and 9?
For problem 8 we can ask, What numbers can we cube to get 27? For problem 9 we can ask, What numbers can we cube to get −125?
How many numbers can we cube to get 27? To get −125? Explain.
We can cube only one number, the positive integer 3, to get 27. We can cube only one number, the negative integer −5, to get −125. The cube of a positive number is always positive, and the cube of a negative number is always negative.
Equations of the form x 3 = p have only one solution for any value of p.
Direct students’ attention to problem 10. Use the following prompts to guide a discussion.
How is the equation in problem 10 similar to the equations in problems 8 and 9? How is it different?
The equation in problem 10 is similar because it has a variable cubed. It is different because 7 is not a perfect cube, but the numbers 27 and −125 in the other equations are perfect cubes.
So far, we have only solved equations with perfect cubes. However, we can apply what we know about solving equations with squares to solving equations with cubes.
What does the equation x 3 = 7 tell us about x ?
It tells us that x is a number we can cube to get 7.
How can we write an exact value of x?
We can use a cube root symbol.
The solution to the equation x 3 = 7 is
. Is 3 √7 rational or irrational? Irrational
Have students complete problems 11–13, individually or in pairs. Look for students who need support understanding that there is only one solution to each equation.
Check responses as a class. Consider engaging students in a debate about the solution to problem 11, similar to the debate earlier in the lesson for problem 6.
Introduce the Critique a Flawed Response routine and present problem 14. Read the prompt aloud as students follow along.
Give students 1 minute to identify the error(s) in Dylan’s work. Then invite students to share what they noticed. Responses may include the following:
• The equation x 3 = −64 has only one solution.
• Dylan found a square root rather than a cube root.
Give students 1 minute to correctly solve the equation. Circulate and ensure that students are getting only −4 as the solution.
14. Dylan solved the equation x 3 = −64 on his homework, but he made some errors. His work is shown. Help Dylan correct his math homework.
3 = −64
The solutions are −8 and 8.
solution is −4.
Use the following prompts to debrief the activity.
Why does the equation x 3 = −64 have only one solution?
There is only one number that when cubed is −64. That number is −4.
How can we verify that −4 is a solution and 4 is not a solution?
We can substitute the numbers −4 and 4 into the equation and see whether they each make the equation true. Only −4 makes the equation true.
Solving Two-Step Equations
Students solve two-step equations involving x 2 or x3 .
Have students complete problems 15 and 16 with a partner. The equation in problem 15 is a scaffold for the equation in problem 16. Circulate to identify students who need additional support with problem 16. Encourage them to notice the similarities and differences between the two equations. Then ask them to describe how they solved problem 15 and tell them to begin problem 16 the same way.
For problems 15 and 16, solve the equation. Show all steps.
15. 2 x + 1 = 17
The solution is 8. 16. x 2 + 1 = 17
2 x + 1 = 17
2 x + 17 − 1 = 17 − 1 2 x = 16 1 2 (2 x) = 1 2 (16) x = 8
2 + 1 = 17
2 + 1 − 1 = 17 − 1 x 2 = 16
= √16 or x = −√16
= 4 x = −4
The solutions are 4 and −4.
When most students are finished, guide students through the solutions to the two equations. Then engage the class in a discussion by using the following prompts.
Teacher Note
The two-step equations in this lesson prepare students to use the Pythagorean theorem to solve for leg lengths of right triangles in module 2.
How are problems 15 and 16 similar?
Both equations start as being equal to 17, and there is a 1 added to both 2x and x2 .
How are they different?
The first equation has a 2x while the second equation has an x2. There is only one solution to the first equation and the second equation has two solutions. We need to use a square root to find the solutions to the second equation.
We can solve equations involving x 2 and x 3 similar to when we solve equations with x. However, once we have only x 2 or x 3 on one side of the equation, we use the definition of a square root or a cube root to find the solutions.
Direct students to complete problems 17–20 in pairs. Circulate as students work to ensure that they isolate the variable term before they take the square root or cube root. Look for students who identify the solutions in problems 19 and 20 as rational. Encourage them to look closely at the exponent on the variable so that they are clear on which root they are using.
For problems 17–20, solve the equation. Identify all solutions as rational or irrational.
17. x 2 + 2 = 11
x 2 + 2 = 11
x 2 + 2 − 2 = 11 − 2
x 2 = 9
x = √9 or x = −√9
x = 3 x = −3
The solutions are 3 and −3. Rational
The solutions are √10 and √10 . Irrational
Promoting the Standards for Mathematical Practice
When students solve equations involving x 2 and x3 with irrational solutions expressed by using square roots and cube roots, they are attending to precision (MP6).
Ask the following questions to promote MP6.
• How are you using square root symbols and cube root symbols in your work?
• Where is it easy to make mistakes when solving equations with x 2 or x 3?
19. p 3 − 8 = 41 p 3 − 8 = 41 p 3 − 8 + 8 = 41 + 8 p 3 = 49 p = 3 √49
The solution is 3 √49 . Irrational
The solutions are √125 and − √125 . Irrational
When most students have finished, invite pairs to share responses with the class. Address any questions that arise.
Differentiation: Challenge
If students are ready for more complex equations, consider having them solve the following equations involving square roots and coefficients on the variable term.
Debrief 5 min
Objective: Solve equations of the forms x 2 = p and x 3 = p, where p is a rational number and the solutions are real numbers.
Initiate a class discussion by using the following prompts. Also, encourage students to add to their classmates’ responses. If students provide a basic answer for the first question, such as the solutions both have 10 in them, acknowledge their findings, but encourage them to think further by asking about how many solutions the equations have and the types of the solutions. Support students in utilizing precise language to describe the solutions as rational or irrational.
How are the solutions to x 2 = 100 and x 2 = 10 similar? How are the solutions different?
The solutions are similar because both equations have two solutions: a positive number and its opposite. The solutions are different because the solutions to x 2 = 100, 10 and −10, are rational numbers, and the solutions to x 2 = 10, √10 and √10 , are irrational numbers.
How are the solutions to x 3 = 27 and x 3 = 3 similar? How are the solutions different?
The solutions are similar because both equations have one positive solution. The solutions are different because the solution to x 3 = 27, 3, is a rational number, and the solution to x 3 = 3, 3 √3 , is an irrational number.
How are the solutions to x 2 = 15 and x 3 = 15 similar? How are the solutions different?
The solutions are similar because they both have roots. The solutions to the equations are also all irrational numbers. The solutions are different because the equation x 2 = 15 has two solutions, √15 and √15 , which are square roots, and the equation x 3 = 15 has only one solution, 3 √15 , which is a cube root.
Exit Ticket 5 min
Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.
Language Support
As the discussion unfolds, consider providing students with sentence frames and key terminology as a scaffold. For example, post the terms positive, negative, square root, cube root, rational, and irrational, and post the following sentence frames.
The solutions are similar because .
The solutions are different because .
Teacher Note
Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.
Student Edition: Grade 8, Module 1, Topic E, Lesson 24
Revisiting Equations with Squares and Cubes
In this lesson, we
• solved equations of the forms x 2 = p and x 3 = p.
• expressed irrational solutions by using square roots and cube roots.
Examples
For problems 1–3, solve the equation. Identify all solutions as rational or irrational.
1. m 2 = 35
m 2 = 35
m = √35 or m = √35
The solutions are √35 and √35
Irrational
2. 100 = r 3 100 = r 3 3 √100 = r
The solution is 3 √100
Irrational
3. x 2 + 11 = 36 x 2 + 11 = 36 x 2 = 25 x =
The solutions are 5 and −5.
Rational
There are two numbers that have a square of 35 a positive number and its opposite. The equation has two solutions. Pay close attention to the exponent of the variable. Although 100 is a perfect square, it is not a perfect cube.
Subtract 11 from each side, which is similar to solving the equation x + 11 = 36
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
For problems 17–20, divide.
21. Approximate the value of √96 by rounding to the tenths place. Explain your answer.
First, I find which perfect squares 96 is between. Because 96 is between 81 and 100, the value of √96 is between 9 and 10
Then, I explore the tenths from 9 to 10 through guess and check. I find that 96 is between 9.7 2 and 9.8 2 or 94.09 and 96.04
Next, I explore the hundredths from 9.7 to 9.8 through guess and check. I find that 96 is between 9.79 2 and 9.80 2 , or 95.8441 and 96.04. Both 9.79 and 9.80 round to 9.8, so √96
22. A faucet drips water at a constant rate of 2 3 quarts in 15 minutes. What is the rate in quarts per hour?
The rate is 2 2 3 quarts per hour.
Teacher Edition: Grade 8, Module 1, Standards
Standards
Module Content Standards
Know that there are numbers that are not rational, and approximate them by rational numbers.
NY-8.NS.1 Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion eventually repeats. Know that other numbers that are not rational are called irrational.
NY-8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line, and estimate the value of expressions.
Work with radicals and integer exponents.
NY-8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions.
NY-8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p, where p is a positive rational number. Know square roots of perfect squares up to 225 and cube roots of perfect cubes up to 125. Know that the square root of a non-perfect square is irrational.
NY-8.EE.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.
NY-8.EE.4 Perform multiplication and division with numbers expressed in scientific notation, including problems where both standard decimal form and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. Interpret scientific notation that has been generated by technology.
Understand and apply the Pythagorean Theorem.
NY-8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
Standards for Mathematical Practice
MP1 Make sense of problems and persevere in solving them.
MP2 Reason abstractly and quantitatively.
MP3 Construct viable arguments and critique the reasoning of others.
MP4 Model with mathematics.
MP5 Use appropriate tools strategically.
MP6 Attend to precision.
MP7 Look for and make use of structure.
MP8 Look for and express regularity in repeated reasoning
Module Achievement Descriptors and Content Standards by Lesson
Pre-test standard Post-test standard
A Topic 1 Lesson
Achievement Descriptor Aligned NGMLS
8.Mod1.AD1 NY-8.NS.1 NY-8.EE.2
8.Mod1.AD2 NY-8.NS.2
8.Mod1.AD3 NY-8.NS.2
8.Mod1.AD4 NY-8.NS.2
8.Mod1.AD5 NY-8.EE.1
8.Mod1.AD6 NY-8.EE.2
8.Mod1.AD7 NY-8.EE.2
8.Mod1.AD8 NY-8.EE.3
8.Mod1.AD9 NY-8.EE.3
8.Mod1.AD11 NY-8.EE.4
8.Mod1.AD12 NY-8.EE.4
8.Mod1.AD13 NY-8.EE.4
8.Mod1.AD14 NY-8.EE.4
8.Mod1.AD15 NY-8.G.7
Why are operations with numbers written in scientific notation in this module if NY-8.EE.3 and NY.8.EE.4 are post-test standards?
The module begins with students using their knowledge of exponents and place value, which creates an accessible entry point to the module’s learning. When students write the many factors of 10, there is an opportunity for a productive struggle that leads to a need for the properties and definitions of exponents. Students operate with numbers written in scientific notation to become fluent in applying the properties and definitions of exponents.
Achievement Descriptors: Proficiency Indicators
8.Mod1.AD1 Determine whether numbers are rational or irrational.
RELATED NGMLS
NY-8.NS.1 Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion eventually repeats. Know that other numbers that are not rational are called irrational.
NY-8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p, where p is a positive rational number. Know square roots of perfect squares up to 225 and cube roots of perfect cubes up to 125. Know that the square root of a non-perfect square is irrational.
Partially Proficient Proficient Highly Proficient
Determine whether numbers are rational or irrational.
Determine whether each number is rational or irrational.
8.Mod1.AD2 Use rational approximations of irrational numbers to compare the size of irrational numbers.
RELATED NGMLS
NY-8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line, and estimate the value of expressions.
Partially Proficient Proficient
Approximate the values of irrational numbers. What is the approximate value of √23 to the nearest tenth?
Approximate the values of irrational numbers to compare their sizes.
Compare the numbers by using the <, >, or = symbol.
Highly Proficient
8.Mod1.AD3 Locate irrational numbers approximately on a number line.
RELATED NGMLS
NY-8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line, and estimate the value of expressions.
Partially Proficient
Proficient Highly Proficient
Locate irrational numbers approximately on a number line.
Consider the number √29 .
Part A
Between what two whole numbers does √29 lie?
Part B
Approximate the value of √29 to the tenths place.
Part C
Plot √29 on a number line.
8.Mod1.AD4 Approximate the values of irrational expressions.
RELATED NGMLS
NY-8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line, and estimate the value of expressions.
Partially Proficient
Proficient Highly Proficient
Approximate the values of irrational expressions. Ethan claims that the best approximation for the value of √46 to the hundredths place is 6.71. Is he correct? Explain your answer.
8.Mod1.AD5 Apply the properties of integer exponents to generate equivalent numerical expressions.
RELATED NGMLS
NY-8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions.
Apply the properties of integer exponents to identify equivalent numerical expressions.
Which expressions are equivalent to 7 6? Choose all that apply.
Apply the properties of integer exponents to generate equivalent numerical expressions.
Use the properties of exponents to write an equivalent expression for 3 5 · 3 −9 by using only positive exponents.
Apply the properties of integer exponents to generate equivalent numerical expressions with prime factor bases.
Write an equivalent expression for 2 −10 · 16 5 by using the fewest number of prime factor bases.
8.Mod1.AD6 Solve equations of the form x 2 = p as √ p and √ p and equations of the form x 3 = p as 3 √ , where p is a rational number.
RELATED NGMLS
NY-8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Know square roots of perfect squares up to 225 and cube roots of perfect cubes up to 125. Know that the square root of a non-perfect square is irrational.
Partially Proficient Proficient Highly Proficient
Solve equations of the form x 2 = p as √ p and √ p and equations of the form x 3 = p as 3 √ p , where √ p , √ p , or 3 √ p is an integer.
Solve each equation.
x 2 = 16
x 3 = 64
Solve equations of the form x 2 = p as √ p and √ p and equations of the form x 3 = p as 3 √ p , where √ p , √ p , or 3 √ p is a rational or irrational number.
Solve each equation.
x 2 = 0.81
x 2 = 23
x 3 = 8 27
Solve two-step equations that after one step simplify to the form x 2 = p or x 3 = p where √ p , √ p , or 3 √ p is a rational or irrational number.
Solve the equation.
x 2 − 4 = 12
8.Mod1.AD7 Evaluate square roots of small perfect squares and cube roots of small perfect cubes.
RELATED NGMLS
NY-8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p, where p is a positive rational number. Know square roots of perfect squares up to 225 and cube roots of perfect cubes up to 125. Know that the square root of a non-perfect square is irrational.
Evaluate square roots of small perfect squares and cube roots of small perfect cubes.
Evaluate each expression.
16 3 √64
8.Mod1.AD8 Approximate and write very large and very small numbers in scientific notation.
RELATED NGMLS
NY-8.EE.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.
Write very large or very small numbers in scientific notation.
Write each number in scientific notation.
0.00008
2,500,000
Approximate and write very large or very small numbers in scientific notation.
Round 6,190,000 to the nearest million. Then write that value in scientific notation.
Round 0.000 0489 to the nearest hundred thousandth. Then write that value in scientific notation.
Compare very large and very small numbers written in scientific notation.
Consider the numbers 4.11 × 10 −6 and 4.11 × 10 −5 . Which number is smaller? Explain how you know.
8.Mod1.AD9 Express how many times as much one number is as another when both numbers are written in scientific notation.
RELATED NGMLS
NY-8.EE.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.
Identify how many times as much one number is as another, when both numbers are written in scientific notation.
Consider the numbers 5 × 10 9 and 5 × 10 4. Which statement is true?
A. 5 × 10 9 is 10,000 times as large as 5 × 10 4 .
B. 5 × 10 9 is 100,000 times as large as 5 × 10 4 .
C. 5 × 10 9 is 1,000,000 times as large as 5 × 10 4 .
D. 5 × 10 9 is 10,000,000 times as large as 5 × 10 4 .
Express how many times as much one number is as another when both numbers are written in scientific notation.
A desktop computer costs about 5 × 10 2 dollars. A supercomputer costs about 1 × 10 8 dollars.
The cost of a supercomputer is about how many times as much as the cost of a desktop computer ?
Round numbers to express how many times as much one number is as another by using scientific notation.
During a musician’s concert tour, fans spend 13 million dollars on tickets. If 1.9 × 10 5 tickets are sold, about how much does the average ticket cost? Write the cost in scientific notation.
8.Mod1.AD11 Multiply or divide numbers written in standard form and scientific notation, and evaluate exponential expressions containing numbers written in standard form and scientific notation.
RELATED NGMLS
NY-8.EE.4 Perform multiplication and division with numbers expressed in scientific notation, including problems where both standard decimal form and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. Interpret scientific notation that has been generated by technology.
Partially Proficient
Multiply or divide numbers written in scientific notation, and evaluate exponential expressions containing numbers written in scientific notation.
Write a number in scientific notation that is equivalent to the given expression.
(2 × 10 −3) 3
(3 × 10 −2)(2 × 10 5)
Proficient Highly Proficient
Multiply or divide numbers written in standard form and scientific notation, and evaluate exponential expressions containing numbers written in standard form and scientific notation.
Write a number in scientific notation that is equivalent to the given expression.
(0.0009)(7.5 × 10 −4)
(8.4 × 10 5) ÷ 2100
8.Mod1.AD12 Operate with numbers written in scientific notation to solve real-world problems.
RELATED NGMLS
NY-8.EE.4 Perform multiplication and division with numbers expressed in scientific notation, including problems where both standard decimal form and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. Interpret scientific notation that has been generated by technology.
Partially Proficient
Proficient
Operate with numbers written in scientific notation to solve real-world problems.
A giraffe weighs approximately 2.2 × 103 pounds. A monarch butterfly weighs approximately 1.1 × 10−3 pounds. A giraffe is about how many times as heavy as a monarch butterfly? Write your answer in scientific notation.
Highly Proficient
Note: To ensure alignment to the New York State Next Generation Mathematics Learning Standards, some ADs have been removed.
8.Mod1.AD13 Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities.
RELATED NGMLS
NY-8.EE.4 Perform multiplication and division with numbers expressed in scientific notation, including problems where both standard decimal form and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. Interpret scientific notation that has been generated by technology.
Partially Proficient
Select units of appropriate size for measurements of very large or very small quantities.
The diameter of a penny is 1.905 × 10 −5 kilometers.
Which is the best unit of measurement to describe the diameter of a penny ?
A. Feet
B. Yards
C. Meters
D. Millimeters
Proficient
Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities.
Jonas’s house is next to a store. He lives about 5 × 10 −2 miles from the store.
Write the distance from Jonas’s house to the store by using a more appropriate unit of measurement.
Highly Proficient
Explain choices in units of appropriate size for measurements of very large or very small quantities. Eve is a newborn baby who is 2.16 × 10 6 seconds old.
Part A
Explain why seconds is not an appropriate unit for this measurement. Then choose a more appropriate unit of measurement.
Part B
Write Eve’s approximate age by using the unit chosen in part A.
8.Mod1.AD14 Interpret scientific notation that has been generated by technology.
RELATED NGMLS
NY-8.EE.4 Perform multiplication and division with numbers expressed in scientific notation, including problems where both standard decimal form and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. Interpret scientific notation that has been generated by technology.
Interpret scientific notation that has been generated by technology.
A calculator is used to perform several operations. The result is shown.
375e + 31 2
.
Write the result by using scientific notation.
8.Mod1.AD15 Apply the Pythagorean theorem to determine the unknown length of a hypotenuse in a right triangle in mathematical problems.
RELATED NGMLS
NY-8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. Partially Proficient
Apply the Pythagorean theorem to determine the unknown whole number length of a hypotenuse in a right triangle in mathematical problems in two dimensions.
Find the length of the hypotenuse c.
Apply the Pythagorean theorem to determine the unknown rational length of a hypotenuse in a right triangle in mathematical problems in two dimensions.
Find the length of the hypotenuse c.
Apply the Pythagorean theorem to determine the unknown irrational length of a hypotenuse in a right triangle in mathematical problems in two dimensions.
Find the length of the hypotenuse c.
Teacher Edition: Grade 8, Module 1, Terminology
Terminology
The following terms are critical to the work of grade 8 module 1. This resource groups terms into categories called New, Familiar, and Academic Verbs. The lessons in this module incorporate terminology with the expectation that students work toward applying it during discussions and in writing.
Items in the New category are discipline-specific words that are introduced to students in this module. These items include the definition, description, or illustration as it is presented to students. At times, this resource also includes italicized language for teachers that expands on the wording used with students.
Items in the Familiar category are discipline-specific words introduced in prior modules or in previous grade levels.
Items in the Academic Verbs category are high-utility terms that are used across disciplines. These terms come from a list of academic verbs that the curriculum strategically introduces at this grade level.
New cube root
The cube root of a number x is a number with a cube that is x. The expression 3 √x represents the cube root of x. (Lesson 21)
exponent of 0
A power with an exponent of 0 is x 0 = 1 for any nonzero x. (Lesson 7)
hypotenuse
The hypotenuse of a right triangle is the side opposite the right angle. It is also the longest side of a right triangle. (Lesson 18)
irrational number
An irrational number is a number that is not rational and cannot be expressed as p _ q for integer p and nonzero integer q. An irrational number has a decimal form that is neither terminating nor repeating. (Lesson 22)
leg
A leg of a right triangle is a side adjacent to the right angle. (Lesson 18)
negative exponent
For any nonzero x, and for any integer
order of magnitude
(Lesson 8)
The order of magnitude n is the exponent on the power of 10 for a number written in scientific notation. (Lesson 3)
perfect cube
A perfect cube is the cube of an integer. (Lesson 16)
perfect square
A perfect square is the square of an integer. (Lesson 16)
Pythagorean theorem
In a right triangle, the sum of the squares of the leg lengths is equal to the square of the hypotenuse length. (Lesson 18)
a 2 + b 2 = c 2
real number
A real number is any number that is either rational or irrational. (Lesson 22)
scientific notation
A number is written in scientific notation when it is represented as a number a multiplied by a power of 10. The number a, which we call the first factor, is a number with an absolute value of at least 1 but less than 10. (Lesson 3)
simplify an exponential expression
To simplify an exponential expression, apply as many properties and definitions of exponents as needed to write an equivalent expression containing only positive exponents and the fewest number of bases. (Lesson 9)
square root
A square root of a nonnegative number x is a number with a square that is x. The expression √x represents the positive square root of x when x is a positive number. When x is 0, √0 = 0. (Lesson 19)
Familiar
bar notation
base
cubed
exponent integer
multiplicative inverse power
rational number repeating decimal right triangle terminating decimal squared
Academic Verbs approximate verify
Math Past
Archimedes: The Sand Reckoner
How many grains of sand are there on the beach?
How many grains of sand would fill the world?
How many grains of sand would fill the visible universe?
Ask your students how they would go about estimating the number of grains of sand needed to fill a container. This may spark memories of “guess the number of jelly beans in the jar” or some similar contest. Students may remember trying to count the jelly beans in one layer and then trying to count the layers. That would be pretty hard to do with sand!
More than 2000 years ago, Greek mathematician and scientist Archimedes set out to determine how many grains of sand are needed to fill the ultimate container—the entire known universe!
Archimedes (287–211 BCE) lived in the Greek city-state of Syracuse on the island of Sicily. He is considered to be one of the greatest mathematicians and scientists of all time.
In his book The Sand Reckoner, Archimedes determined that no more than 1 × 10 63 grains of sand are needed to fill the universe.
The universe of Archimedes’s time was a much smaller space than the known universe of today. People assumed that Earth was at the center of the universe, and that the moon, the planets, and the sun orbited Earth. Archimedes’s universe was the sphere containing the sun’s apparent orbit. The distant stars were thought to lie on the surface of this sphere.
However, when thinking about the size of the universe, Archimedes considered a different opinion that was expressed by his colleague Aristarchus. Remarkably, Aristarchus anticipated by centuries the findings of the famous astronomer Copernicus by asserting that the sun, not Earth, was at the center of the universe. Aristarchus’s calculations implied a universe that was 10,000 times as large in diameter as the universe of Archimedes. So Archimedes, wanting to err on the high side, calculated the sand needed to fill Aristarchus’s larger universe. This led to the figure 1 × 10 63 .
Archimedes didn’t actually write 1 × 10 63 for his answer. Here, we use the modern representation of an extremely large number by writing it as a power of 10 with a coefficient greater than or equal to 1 and less than 10. We call this method of writing numbers scientific notation.
Archimedes knew that he needed very large numbers to answer the question about grains of sand. But there was no such notation for large numbers in the time of Archimedes. He had to invent it.
Ordinary Greek numbers (written by using letters) went as high as 1 myriad, or 10,000. A myriad of myriads produced 10,000 2 , or 100,000,000. That was the limit for using Greek notation. Help your students see that a myriad of myriads (i.e., 100,000,000) would be 1 × 108 in modern notation.
Since this number is so large, let’s use the symbol ℳ to stand for a myriad of myriads. Archimedes treated the number we are calling ℳ as a counting base for a new number system. Instead of place value based on powers of 10, Archimedes’s numbers had place value based on powers of ℳ. Show your students this example of how Archimedes’s base-ℳ system would represent a large number.
Have your students notice Archimedes’s “digits”—the multipliers in front of the powers of ℳ. According to Archimedes’s base-ℳ system, any number between 0 and ℳ – 1 can appear as a multiplier. This is the same idea as in our base-10 number system, where any digit from 0 to 9 (i.e., 10 − 1) can appear as a multiplier of a power of 10. But Archimedes had 100,000,000 possible multipliers!
Archimedes called the numbers from 0 up to ℳ – 1 the first order. A number of the first order multiplied by ℳ gives a number of the second order. A number of the first order multiplied by ℳ 2 gives a number of the third order, and so on. Archimedes would express the large example number this way:
247,368 units of the third order, plus 1009 units of the second order, plus 43,216 units of the first order.
Let’s get back to the grains of sand. How did Archimedes relate them to the universe? He did what your students might think of doing—he got there in stages.
Grains of sand are pretty small. Archimedes decided that it would take no more than 10,000 of them to make a cluster the size of a poppy seed. He figured that no more than 40 poppy seeds laid side
by side equal 1 fingerbreadth. Finally, Archimedes determined that no more than 10,000 fingerbreadths equal a stadium (i.e., approximately 600 feet—a standard unit of length for the Greeks). Archimedes separately determined that the diameter of his universe was no larger than 10,000,000,000 stadia in diameter, with the Aristarchus universe having a diameter 10,000 times that much.
So Archimedes put it all together one step at a time. When he finally reached the end of his calculations, he declared that the number of grains of sand that fill the (Aristarchus) universe is at most 10,000,000 units of the eighth order of numbers.1
This is 10,000,000 × ℳ 7, or equivalently, 1 × 10 63 when written in scientific notation.
Fast-forward to modern times. The universe is vastly larger than either Archimedes or Aristarchus imagined. All matter is made of particles called atoms. Today, scientists put the number of atoms in the known universe at around 1 × 1082. Suppose that all those atoms became sand.
A single grain of sand has about 1 × 10 19 atoms. Turning all the atoms in the universe into sand would therefore make approximately (1 × 10 82) ÷ (1 × 10 19) = 1 × 10 63 grains.
1
Heath, Works of Archimedes, 232.
Surprise! It’s the same as Archimedes’s number. How is it possible that the same number could emerge from two vastly different interpretations of the universe? A lively student discussion might generate an explanation for this apparent paradox. Some students may observe that the atoms in the known universe are highly dispersed, meaning that they are not concentrated as Archimedes assumed.
In any case, it is amusing to note that the number of grains of sand that Archimedes said would be needed to fill the Aristarchus
universe is actually all the grains of sand the real universe could possibly provide!
As noted, Archimedes is regarded as one of the greatest mathematicians and scientists of all time. His book The Sand Reckoner, with its novel number system and careful calculations, shows that Archimedes deserves the honor of that title.
Teacher Edition: Grade 8, Module 1, Fluency
Fluency
Fluency activities allow students to develop and practice automaticity with fundamental skills so they can devote cognitive power to solving more challenging problems. Skills are incorporated into fluency activities only after they are introduced conceptually within the module.
Each lesson in A Story of Ratios begins with a Fluency segment designed to activate students’ readiness for the day’s lesson. This daily segment provides sequenced practice problems on which students can work independently, usually in the first few minutes of a class. Students can use their personal whiteboards to complete the activity, or you may distribute a printed version, available digitally. Each fluency routine is designed to take 3–5 minutes and is not part of the 45-minute lesson structure. Administer the activity as a bell ringer or adapt the activity as a teacher-led Whiteboard Exchange or choral response.
Bell Ringer
This routine provides students with independent work time to determine the answers to a set of problems.
1. Display all the problems at once.
2. Encourage students to work independently and at their own pace.
3. Read or reveal the answers.
Whiteboard Exchange
This routine builds fluency through repeated practice and immediate feedback. A Whiteboard Exchange maximizes participation by having every student record solutions or strategies for a sequence of problems. These written recordings allow for differentiation: Based on the answers you observe, you can make responsive, in-the-moment adjustments to the sequence of problems. Each student requires a personal whiteboard and a whiteboard marker with an eraser for this routine.
1. Display one problem in the sequence.
2. Give students time to work. Wait until nearly all students are ready.
3. Signal for students to show their whiteboards. Provide immediate and specific feedback to students one at a time. If revisions are needed, briefly return to validate the work after students make corrections.
4. Advance to the next problem in the sequence and repeat the process.
Choral Response
This routine actively engages students in building familiarity with previously learned skills, strengthening the foundational knowledge essential for extending and applying math concepts. The choral response invites all students to participate while lowering the risk for students who may respond incorrectly.
1. Establish a signal for students to respond to in unison.
2. Display a problem. Ask students to raise their hands when they know the answer.
3. When nearly all hands are raised, signal for the students’ response.
4. Reveal the answer and advance to the next problem.
Count By
This routine actively engages students in committing counting sequences to memory, strengthening the foundational knowledge essential for extending and applying math concepts.
1. Establish one signal for counting up and counting down and another signal for stopping the count.
2. Tell students the unit to count by. Establish the starting and ending numbers between which they should count.
3. Begin the count by providing the signals. Be careful not to mouth the words as students count.
Sprints
Sprints are activities that develop mathematical fluency with a variety of facts and skills. A major goal of each Sprint is for students to witness their own improvement within a very short time frame. The Sprint routine is a fun, fast-paced, adrenaline-rich experience that intentionally builds energy and excitement. This rousing routine fuels students’ motivation to achieve their personal best and provides time to celebrate their successes.
Each Sprint includes two parts, A and B, that feature closely related problems. Students complete Sprint A followed by two count by routines—one fast-paced and one slow-paced—that include a stretch or other physical movement. Then students complete Sprint B, aiming to improve their score from Sprint A. Each part is scored but not graded.
Sprints can be given at any time after the content of the Sprint has been conceptually developed and practiced. The same Sprint may be administered more than once throughout a year or across grade levels. With practice, the Sprint routine takes about 10 minutes.
Directions
1. Have students read the instructions and sample problems. Frame the task by encouraging students to complete as many problems as they can—to do their personal best.
2. Time students for 1 minute on Sprint A. Do not expect them to finish. When time is up, have students underline the last problem they completed.
3. Read the answers to Sprint A quickly and energetically. Have students call out “Yes!” if they answered correctly; have them circle the answer if they answered incorrectly.
4. Have students count their correct answers and record that number at the top of the page. This is their personal goal for Sprint B.
5. Celebrate students’ effort and success on Sprint A.
6. To increase success with Sprint B, offer students additional time to complete more problems on Sprint A or ask discussion questions to analyze and discuss the patterns in Sprint A.
7. Lead students in the fast-paced and slow-paced count by routines. Include a stretch or other physical movement during the count.
8. Remind students of their personal goal from Sprint A.
9. Direct students to Sprint B.
10. Time students for 1 minute on Sprint B. When time is up, have students underline the last problem they completed.
11. Read the answers to Sprint B quickly and energetically. Have students call out “Yes!” if they answered correctly; have them circle the answer if they answered incorrectly.
12. Have students count their correct answers and record that number at the top of the page.
13. Have students calculate their improvement score by finding the difference between the number of correct answers in Sprint A and in Sprint B. Tell them to record the number at the top of the page.
14. Celebrate students’ improvement from Sprint A to Sprint B.
Sample Dialogue
Have students read the instructions and complete the sample problems. Frame the task:
• You may not finish, and that’s okay. Complete as many problems as you can—do your personal best.
• On your mark, get set, think!
Time students for 1 minute on Sprint A.
• Stop! Underline the last problem you did.
• I’m going to read the answers quickly. As I read the answers, call out “Yes!” if you got it right. If you made a mistake, circle the answer.
Read the answers to Sprint A quickly and energetically.
• Count the number of answers you got correct and record that number at the top of the page. This is your personal goal for Sprint B.
Celebrate students’ effort and success.
Provide 2 minutes to allow students to complete more problems or to analyze and discuss patterns in Sprint A using discussion questions.
Lead students in the fast-paced and slow-paced count by routines. Include a stretch or other physical movement during the count.
• Point to the number of answers you got correct on Sprint A. Remember, this is your personal goal for Sprint B.
Direct students to Sprint B.
• On your mark, get set, improve! Time students for 1 minute on Sprint B.
• Stop! Underline the last problem you did.
• I’m going to read the answers quickly. As I read the answers, call out “Yes!” if you got it right. If you made a mistake, circle the answer.
Read the answers to Sprint B quickly and energetically.
• Count the number of answers you got correct and record that number at the top of the page.
• Calculate your improvement score and record it at the top of the page.
Celebrate students’ improvement.
The table below provides implementation guidance for the Sprints recommended in this module.
Sprint Name Administration Guidelines Discussion Questions
Apply Properties of Positive Exponents
Apply Properties of Positive and Negative Exponents
Students apply the properties and definitions of exponents to write equivalent expressions with a single base.
Administer after module 1 lesson 7 or in place of the module 1 lesson 7 Fluency.
Students apply the properties and definitions of exponents to write equivalent expressions with positive exponents and a single base.
Administer after module 1 lesson 8.
Numerical Expressions with Exponents
Scientific Notation and Negative Exponents
Students evaluate expressions to generate equivalent expressions by using properties and definitions of exponents.
Administer after module 1 lesson 10 or in place of the module 1 lesson 10 Fluency.
Students write numbers in scientific notation to prepare for operating with numbers written in scientific notation.
Administer after module 1 lesson 12 or in place of the module 1 lesson 12 Fluency.
How are problems 10 and 11 different from problems 1–9?
Count By Routines
Fast-paced: Count by twenties from −80 to 0.
Slow-paced: Count by threes from 0 to 63.
How do the answers to problems 1–4 compare with the answers to problems 10–13?
What do you notice about problem 18?
How can you use problem 6 to answer problems 7–9?
What do you notice about problems 15–17?
Fast-paced: Count by twos from −30 to 10.
Slow-paced: Count down by halves from 0 to −10 by using mixed numbers.
Fast-paced: Count by fives from −50 to 10.
Slow-paced: Count by tenths from 0 to 1.
What pattern do you notice in problems 1–5?
What do you notice about problems 10–14?
Fast-paced: Count by fourths from 0 fourths to 12 fourths.
Slow-paced: Count by negative fourths from 12 fourths to 0 fourths.
Sprint Name Administration Guidelines
Scientific Notation and Positive Exponents
Squares
Write Expressions with Exponents
Students write numbers in scientific notation to prepare for operating with numbers written in scientific notation.
Administer after module 1 lesson 4 or in place of the module 1 lesson 4 Fluency.
Students evaluate squares to prepare for using the Pythagorean theorem.
Administer after module 1 lesson 15.
Students write expressions by using exponents to prepare for generating equivalent expressions of the form x m . x n .
Administer before module 1 lesson 5 or in place of the module 1 lesson 5 Fluency.
Discussion Questions
What pattern do you notice in problems 1–5?
What are the similarities and differences between problems 6–9 and 10–13?
Count By Routines
Fast-paced: Count by tens from 0 to 120.
Slow-paced: Count by sixes from 0 to 72.
How can you use problem 5 to answer problem 6?
How can you use problem 11 to answer problem 12?
What do you notice about problems 1–4 and problems 5–8?
Fast-paced: Count by thirds from 0 thirds to 12 thirds.
Slow-paced: Count by thirds from 0 thirds to 4 by using mixed numbers.
Fasted-paced: Count by hundreds from 0 to 1000.
Slow-paced: Count by fours from 0 to 72.
Apply Properties of Positive Exponents
Apply Properties of Positive Exponents
Apply the properties and definitions of exponents to write an equivalent expression with a single base. Let all variables represent nonzero numbers. Number Correct:
2 New York Next Gen
Sprint ▸ Apply Properties of Positive Exponents
Number Correct: Improvement: Apply the properties and definitions of exponents to write an equivalent expression with a single base. Let all variables represent nonzero numbers.
Apply Properties of Positive and Negative Exponents
Apply the properties and definitions of exponents to write an equivalent expression with a positive exponent and a single base. Let all variables represent nonzero numbers.
Apply Properties of Positive and Negative Exponents
Sprint
M1
Number Correct: Improvement: Apply the properties and definitions of exponents to write an equivalent expression with a positive exponent and a single base. Let all variables represent nonzero numbers.
Numerical Expressions with Exponents
Notation and Positive Exponents
Write Expressions with Exponents
Write Expressions with Exponents
AWrite the expression with an exponent. Let all variables represent nonzero numbers. Number Correct:
Write Expressions with Exponents
Write Expressions with Exponents
Number Correct: Improvement: Write the expression with an exponent. Let all variables represent nonzero numbers.
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
Student Edition: Grade 8, Module 1, Mixed Practice 1
2. Jonas and Kabir each evaluate the expression 4 + (−12)
a. Model the expression on the vertical number line.
3. Which expression is equivalent to −11 − 7?
A. −11 − (−7)
B. 11 + 7
C. −11 + (−7)
D. 11 − (−7)
4. Lily walks 1.5 miles up a trail, turns around, and then walks 1.5 miles down the same trail. This situation can be modeled by the equation 1.5 + (−1.5) = 0
a. What does the number 1.5 represent in the equation?
The 1.5 represents the distance in miles that Lily walks up the trail.
b. What does the number −1.5 represent in the equation?
The −1.5 represents the distance and direction that Lily walks after turning around. The 1.5 is the distance in miles, and it is negative because Lily walks in the opposite direction.
c. What does the number 0 represent in the equation?
The 0 represents that Lily returns to where she started on the trail.
b. Jonas says the answer is −8. Kabir says the answer is −16. Who is correct? Explain. Jonas is correct
4 + (−12) = −8
What is the width of the rectangle?
9. In the given diagram, two lines meet at a point that is also the vertex of an angle.
a. Use complete sentences to describe the angle relationships that can help you solve for x and y
∠ BAG and ∠ EAD are vertical angles. ∠ BAC and ∠ EAC are supplementary angles, so the measure of ∠ EAC is 90°
b. What is the value of y? Explain your answer.
The value of y is 15 because ∠ BAG and ∠ EAD are vertical angles.
c. Find the value of x
The value of x is 75
10. Yu Yan does not know how to answer 2 multiple-choice questions on a test. She randomly guesses on both questions. If there are 5 choices for each multiple-choice question, what is the probability that Yu Yan guesses the correct answer on both questions?
11. The letters of the word MISSISSIPPI are written on note cards and placed in a bag. One card is drawn at random from the bag.
a. What is the sample space for this chance experiment?
M, I, S, P
b. Are the outcomes of this experiment equally likely? Explain your answer. No, the outcomes are not equally likely. For example, the probability of drawing the letter I from the bag is 4 11 and the probability of drawing a P is 2 11 .
Teacher Edition: Grade 8, Module 1, Works Cited
Works Cited
CAST. Universal Design for Learning Guidelines version 2.2. Retrieved from http://udlguidelines.cast.org, 2018.
Common Core Standards Writing Team. 2022. Progressions for the Common Core State Standards for Mathematics, May 24, 2023. Tucson, AZ: Institute for Mathematics and Education, University of Arizona. https://mathematicalmusings.org/wp -content/uploads/2023/05/Progressions.pdf.
Ellis, Julie. What’s Your Angle, Pythagoras? Watertown, MA: Charlesbridge, 2004.
Heath, Thomas L., trans. The Sand Reckoner of Archimedes. London: 2008.
Heath, Thomas L., ed. The Works of Archimedes. New York: Dover Publications, 2002.
National Governors Association Center for Best Practices, Council of Chief State School Officers (NGA Center and CCSSO). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010.
Zwiers, Jeff, Jack Dieckmann, Sara Rutherford-Quach, Vinci Daro, Renae Skarin, Steven Weiss, and James Malamut.
Principles for the Design of Mathematics Curricula: Promoting Language and Content Development. Retrieved from Stanford University, UL/SCALE website: https://ul.stanford.edu/resource/principles-design -mathematics-curricula-and-mlrs, 2017.
Teacher Edition: Grade 8, Module 1, Credits
Credits
Great Minds® has made every effort to obtain permission for the reprinting of all copyrighted material. If any owner of copyrighted material is not acknowledged herein, please contact Great Minds for proper acknowledgment in all future editions and reprints of this module.
From the New York State Education Department. New York State Next Generation Mathematics Learning Standards. Available from https://www.nysed.gov/sites/default/files/programs /curriculum-instruction/nys-next-generation-mathematics-p-12 -standards.pdf; accessed 19 September, 2023.
For a complete list of credits, visit http://eurmath.link /media-credits.
(1680–1770); engraved for the Encyclopaedia Britannica (1st Edition, 1771; facsimile reprint 1971), Volume 1, Fig. 2 of Plate XL facing page 449 by Andrew Bell, public domain via Wikimedia Commons; page 459, “A simulated view of the entire observable universe” by Andrew Z. Colvin, Courtesy
Wikimedia Commons, is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported license (CC BY-SA 3.0).
Accessed November 14, 2023, https://commons.wikimedia.org /wiki/File:Observable_Universe_with_Measurements_01.png; All other images are the property of Great Minds.
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What does this painting have to do with math?
Abstract expressionist Al Held was an American painter best known for his “hard edge” geometric paintings. His bright palettes and bold forms create a three-dimensional space that appears to have infinite depth. Held, who sometimes found inspiration in architecture, would often play with the viewer’s sense of visual perception. While most of Held’s artworks are paintings, he also worked in mosaic and stained glass.
