LEARN ▸ 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.
1 Scientific Notation, Exponents, and Irrational Numbers
2 Rigid Motions and Congruent Figures
3 Dilations and Similar Figures
4 Linear Equations in One and Two Variables
5 Systems of Linear Equations
6 Functions and Bivariate Statistics
Student Edition: Grade 8, Module 1, Contents
Scientific Notation, Exponents, and Irrational Numbers
Topic A
Introduction to Scientific Notation
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
Topic B
Properties and Definitions of Exponents
5
Topic C
Applications of the Properties and Definitions of Exponents
Lesson 11
Small Positive Numbers in Scientific Notation
159
161
23
41
59
75
Lesson 5 77
Products of Exponential Expressions with Whole-Number Exponents
Lesson 6 93
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)
133
143
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
Topic D
Perfect Squares, Perfect Cubes, and the Pythagorean Theorem
Lesson 16 225
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
Topic E
Irrational Numbers
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
Resources Mixed Practice 1
Fluency Resources
Lesson 21
Sprint: Apply Properties of Positive Exponents
Sprint: Apply Properties of Positive and Negative Exponents
Sprint: Numerical Expressions with Exponents
Sprint: Scientific Notation and Negative Exponents
Sprint: Scientific Notation and Positive Exponents
Sprint: Squares
Sprint: Write Expressions with Exponents
Introduction to Scientific Notation
Student Edition: Grade 8, Module 1, Topic A
Million, Billion—What’s the Difference, Really?
Thousand
Seconds
About 17 minutes
Feet
The rough length of a New York block
Dollars
About $0.03 every day of your life
People
A mid-sized high school
About 12 days
The rough distance from New York to Boston
The rough distance from New York to the moon
About $35 every day of your life
San Jose, CA (approximately)
About $35,000 every day of your life
The rough population of North and South America combined
Million and billion sound awfully similar. Just one letter is different. Written out as powers of 10, they look similar too: it’s 10 6 vs. 10 9 . How different can they really be?
Extremely different, it turns out.
If a million seconds is a long vacation, then a billion seconds is longer than your life so far. If a million feet is a short flight in an airplane, then a billion feet is a journey to the moon. If a million dollars over your lifetime is a nice daily allowance, then a billion dollars over your lifetime is enough to buy a new car every day. If a million people is the population of a big city, then a billion people is the population of the whole Western Hemisphere.
Exponents let us write huge numbers by using just a few symbols. But don’t let that fool you into forgetting how huge the numbers might be and how different they are from one another.
Student Edition: Grade 8, Module 1, Topic A, Lesson 1
Name Date
Large and Small Positive Numbers
Standard Form of a Number
1. Write each number in standard form.
Writing Very Large and Very Small Positive Numbers
2. Complete the table. The table shows the approximate measurements of objects seen in the demonstration.
Approximate Measurement (meters)
Single Digit Times a Power of 10 (expanded form)
Single Digit Times a Power of 10 (exponential form)
Eiffel Tower (height)
3. Complete the table. The table shows the approximate measurements of objects seen in the demonstration. Approximate Measurement
Approximating Very Large and Very Small Positive Quantities
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.
b. Write your answer from part (a) as a single digit times a power of 10 in exponential form.
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.
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.
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.
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.
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.
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.
b. Write your answer from part (a) as a single digit times a power of 10 in exponential form.
Student Edition: Grade 8, Module 1, Topic A, Lesson 1
Student Edition: Grade 8, Module 1, Topic A, Lesson 1
Name Date
1. Consider the number 7,123,456
a. Approximate the number by rounding to the nearest million.
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.
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.
Student Edition: Grade 8, Module 1, Topic A, Lesson 1
Name Date
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)
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 .
2. Complete the table. The table shows the average speed in miles per hour of a starfish.
Each number in the Starfish row is an equivalent form of 0.01 A unit fraction has a numerator of 1
3. 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 × 10 6 square feet
Writing equivalent forms of 4,000,000 may help when writing the number as a single digit times a power of 10 in exponential form.
4,000,000 = 4 million = 4 × 1,000,000 = 4 × 10 6
Student Edition: Grade 8, Module 1, Topic A, Lesson 1
Name Date
For problems 1–5, write the number in standard form.
6. Complete the table. The table shows the approximate number of stacked pennies needed to reach the height of a given object.
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)
7. Complete the table. The table shows the average speed of a given animal.
Average Speed (miles per hour)
Standard Form Unit Form Fraction
Single Digit Times a Unit Fraction (expanded form)
Single Digit Times a Unit Fraction (exponential form)
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.
b. Write your answer from part (a) as a single digit times a power of 10 in exponential form.
9. The smallest insect on the planet is a type of parasitic wasp and measures 0.000 139 meters.
a. 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.
Tortoise
Sloth
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.
b. Write your answer from part (a) as a single digit times a power of 10 in exponential form.
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.
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.
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.
b. Write your answer from part (a) as a single digit times a power of 10 in exponential form.
Remember
For problems 13–16, add or subtract.
For problems 17–19, use the distributive property to write an equivalent expression.
20. Match each expression to the correct power of 10 in exponential form.
17. 3(6 x + 7)
18. 5(4 + 6 a)
19. 2(4 x + 5 y)
Student Edition: Grade 8, Module 1, Topic A, Lesson 2
Name Date LESSON
Comparing Large Numbers
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?
Unknown Factor
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?
3. 9 million is how many times as much as 3,000,000?
4. What number is 3000 times as much as 30?
Times As Much As
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?
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?
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?
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?
Student Edition: Grade 8, Module 1, Topic A, Lesson 2
Student Edition: Grade 8, Module 1, Topic A, Lesson 2
Name Date
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.
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?
Student Edition: Grade 8, Module 1, Topic A, Lesson 2
Name Date
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.
1. 6,000,000 is how many times as much as 3000?
6,000,000 = 6 × 10 6 3000 = 3 × 10 3 6 ×
6 = ⋅ 3 × 10 3
Writing out the factors of 10 makes the division simpler because 10 10 is 1.
The question can be reworded as 4000 times what number is 90,000?
90,000 is 22 1 2 times as much as 4000
3. The land area of France is 212,954 square miles. The land area of Canada is 3,854,083 square miles. The land area of Canada is about how many times as large as the land area of France?
212,954 ≈ 200,000 = 2 × 10 5
3,854,083 ≈ 4,000,000 =
Ask, “Which quantity is being multiplied?” to decide how to write the unknown factor equation.
Approximate the value of each land area. Then write the approximation as a single digit times a power of 10 in exponential form.
= 2 × 10 = 20
The land area of Canada is about 20 times as large as the land area of France.
Student Edition: Grade 8, Module 1, Topic A, Lesson 2
Name Date
For problems 1–3, compute mentally.
1. 600 is how many times as much as 200?
2. 600 is how many times as much as 20?
3. 600 is how many times as much as 2?
2
For problems 4–6, write an unknown factor equation to represent the question. Then find the answer to the question.
4. 800,000 is how many times as much as 2000?
5. 60,000,000 is how many times as much as 30,000?
6. 6 × 10 5 is 2 × 10 3 times as much as what number?
For problems 7–11, use the values in the table to answer the questions.
Approximate
7. Based on land area, about how many islands the size of Jamaica does it take to equal the size of Russia?
8. The land area of the United States is about how many 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?
10. Which country’s land area is about 1000 times as large as the land area of Jamaica?
11. Which country’s land area is about 1 500 as large as the land area of the United States?
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?
Remember For problems 13–16, add or subtract.
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.
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
Student Edition: Grade 8, Module 1, Topic A, Lesson 3
Name Date
Time to Be More Precise—Scientific Notation
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.
In this system, χ νδ represents the number 654. What does σπε represent?
2. What does ψοζ represent?
1.
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 ( 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 ( -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 ( CE, years) after his death.
Archimedes’s Twin Circles
Another Way to Represent Numbers
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 of .
The general expression that represents a number written in scientific notation is × .
The absolute value of a must be at but than .
5. Identify the first factor and the order of magnitude of the expression 8.86 × 10 6 .
6. Use the definition of scientific notation to indicate whether each number is an example or a nonexample.
× 10 2
× 10 6
× 10 3
× 10 7
× 10 6
Interpreting Scientific Notation
For problems 7–9, write the number in standard form.
3 × 10 2
10. Use the Place Value Chart to write the number 9.1 × 10 3 in standard form.
7.
8. 4.8 × 10 3
9. 6.75 × 10 6
11. Use the Place Value Chart to write 2.05 × 10 6 in standard form.
12. Logan writes the number 6.7 × 10 3 in standard form. He writes 67,000 because 10 3 represents thousands. Do you agree with Logan? Explain.
Using Scientific Notation
13. Write 200,000 in scientific notation.
14. Use the Place Value Chart to write 350 in scientific notation.
For problems 15–18, write the number in scientific notation. Use the Place Value Chart, if needed.
15. 47,500,000
16. 750,000
17. 6,040,000
18. Thirty-six thousand, three hundred sixty
Student Edition: Grade 8, Module 1, Topic A, Lesson 3
Student Edition: Grade 8, Module 1, Topic A, Lesson 3
Name Date
1. In 2018, the population of Italy was approximately 60,630,000 people. Write this number in scientific notation.
2. The average human body contains about 3.4 × 10 10 cells. Write this number in standard form.
Student Edition: Grade 8, Module 1, Topic A, Lesson 3
Name Date
Time to Be More Precise—Scientific Notation
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.
• 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 × 10 n .
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.
The order of magnitude is 6
3. A number is represented on a place value chart. Write the number in scientific notation. 10 6 10 5 10 4 10 3 10 2 10 1 1 7 5 0 0 0 0 0
7,500,000 = 7.5 × 10 6
The first factor is 7.5.
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.
210,000,000 = 2.1 × 10 8
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
Student Edition: Grade 8, Module 1, Topic A, Lesson 3
1. Circle all the numbers written in scientific notation.
× 10 7
For problems 2–7, write the number in standard form.
5.505 × 10 3
6.789 × 10 2
8. In 2017, about 8.3 × 10 12 text messages were sent and received worldwide. Write this number in standard form.
9. In 2014, the United States discarded a total of 5.08 × 10 9 pounds of trash. Write this number in standard form.
10. Sara believes that the number 3 × 10 is not written in scientific notation. Do you agree or disagree? Explain.
11. Write the number shown on the place value chart in scientific notation.
For problems 12–16, write the number in scientific notation.
12. 600,000 13. −1,000,000,000
14. 57,500
15. −8,095,000
16. 892
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.
18. Match each number written in standard form with its corresponding number written in scientific notation.
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.
Remember
For problems 20–23, add or subtract.
20. 1 10 + 1 5
1 4 + 1 8
1 3 1 9
24. Consider the number 0.000 236.
a. Approximate the number by rounding to the nearest ten thousandth.
1 5 1 15
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.
For problems 25–30, add or subtract.
7 − 4
−7 + (−4)
−7 + 4
−7 − 4 29. −7 − (−4)
30. 7 − (−4)
25.
28.
Student Edition: Grade 8, Module 1, Topic A, Lesson 4
Name Date
Adding and Subtracting Numbers Written in Scientific Notation
For problems 1–6, find the sum.
Adding and Subtracting: What Is the Same?
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:
Ava’s work:
a. Who is correct? Explain.
b. Compare the two methods.
For problems 8–10, add or subtract. Write the answer in scientific notation.
11. The table shows the number of views for three popular online videos.
× 10
a. How many total views do the three videos receive? Write the answer in scientific notation.
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.
Rewriting Sums and Differences
12. Find the sum 6 × 10 3 + 7 × 10 3 + 8 × 10 3 .
For problems 13 and 14, add or subtract. Write the answer in scientific notation.
9.25 × 10 5 + 9.8 × 10 5
14. 5 × 10 6 + 9 × 10 6 − 2 × 10 6 + 8 × 10 6
13.
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.
Adding and Subtracting: What Is Different?
For problems 16 and 17, add or subtract. Write the answer in scientific notation.
16. 4 × 10 5 + 3 × 10 6
7.2 × 10 5 − 4 × 10 4
Student Edition: Grade 8, Module 1, Topic A, Lesson 4
Student Edition: Grade 8, Module 1, Topic A, Lesson 4
Name Date
The table shows the estimated number of stars in four different galaxies.
Galaxy
Milky Way
Whirlpool
Sunflower
Antennae
Estimated Number of Stars
2.5 × 10 11
1.0 × 10 11
4.0 × 10 11
3.0 × 10 11
What is the estimated total number of stars in the four galaxies? Write the answer in scientific notation.
Student Edition: Grade 8, Module 1, Topic A, Lesson 4
Name Date
Adding and Subtracting Numbers Written in Scientific Notation
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
Student Edition: Grade 8, Module 1, Topic A, Lesson 4
Name Date
For problems 1 and 2, find the sum.
1. 4 thousands + 3 thousands
× 1000 + 3 × 1000
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.
b. About how many more households had a reptile than had a saltwater fish? Write the answer in scientific notation.
For problems 6–10, add or subtract. Write the answer in scientific notation.
6. 9 × 10 8 − 2.7 × 10 8
9.6
11. The table shows an estimated number of smartphone users in the three most populous countries in the world in 2018.
Country
China
Estimated Number of Smartphone Users
783,000,000
India 375 million
United States
2.52 × 10 8
a. What is the estimated total number of smartphone users for the three countries in 2018? Write the answer in scientific notation.
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.
12. Evaluate 5 × 10 5 + 3 × 10 4 . Write the answer in scientific notation.
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?
Remember
For problems 14–17, add or subtract.
18. Write 9003 in scientific notation.
19. Which expressions are equivalent to 2 3 ( 4)? Choose all that apply.
Student Edition: Grade 8, Module 1, Topic B
Who Would Win?
The Eighth Power of 10
A Whole Squadron of Seventh Powers of 10
It’s easy to forget how different 10 7 and 10 8 really are from one another. They look like they’re just one apart. And, in a sense, they are: one factor of 10.
That means that 10 8 is a full 10 times as big as 10 7 . In the picture above, neither would win; they’re equivalent!
In this topic, we will explore different ways to combine powers. What happens if you multiply 10 8 and 10 7 ? What if you divide 10 8 and 10 7 ? These patterns will lead us to whole new frontiers where we will try to make sense of expressions like 10 0 and 10 −4
If you get lost, remember this: 10 8 is 10 times the size of 10 7 !
For problems 1–12, apply the property of exponents to write an equivalent expression.
13. Which of the following is equivalent to 5 3 ⋅ 5 5 ? Choose all that apply. A. 5 3+5 B. 5 3 · 5
14. Which of the following is equivalent to 9 12 ? Choose all that apply.
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.
16. Find the area of the rectangle. Write the area as a single base raised to an exponent.
105 ft
102 ft
For problems 17–20, indicate whether each result is a positive or a negative number.
(−3) 2
(−4) 3
21. The product (−1) 3 ⋅ (−1) n is negative. Which of the following values of n are possible? Choose all that apply.
A. 2
B. 5
C. 7
D. 8
E. 10
17.
18.
19. (−5) odd number
20. (−6) even number
For problems 22–27, solve for b.
3 b = 3 11 ⋅ 3 3
28. Fill in the boxes with digits 1–6 to make each equation true. Each digit can be used only once.
Remember
For problems 29–32, add or subtract.
33. Scientists search for galaxies like ours by estimating the number of stars a galaxy includes. The estimate for the number of stars in the Milky Way galaxy is 2.5 × 10 11 , and the estimate for the number of stars in the Cartwheel galaxy is 4.8 × 10 11
How many more stars are in the Cartwheel galaxy than in the Milky Way galaxy? Write your answer in scientific notation.
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 .
2. Could 10 0 = 0? Use the product 10 0 ⋅ 10 3 to show whether it upholds the property
x m ⋅ x n = x m + n
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 .
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
5. 7 × 10 3 + 5 × 10 2 + 1 × 10 0
In problems 6 and 7, the standard form of a number is given. Write the number in expanded form by using exponential notation.
6. 982
7. 10,735
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.
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.
For problems 19–23, apply the properties of exponents to determine the value of t.
19. 8 0 ⋅ 8 2 = 8 t 20. (−9) 3 (−9) t = (−9) 3
21. (3 8 )0 (3 8 )t = (3 8 )11
23. (−0.25) 0 (−0.25) t = 1
24. Fill in the boxes with digits 0–5 to make each equation true. Each digit can be used only once.
a. (−2) 0 (−2) = (−2) 5 b. (1 4 )(1 4 )0 = 1
(6 0 ) 2 =
d. (m n ) 0 (m n ) = 1
c.
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 6 × 10 4 + 3 × 10 2 + 2 × 10 0 .
A. 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.
Remember
For problems 28–31, add or subtract.
28. 1 6 + 1 4
29. 2 9 + 7 6
30. 5 6 1 8 31. 11 10 1 25
For problems 32–35, apply the properties of exponents to write an equivalent expression.
36. Which expression is equivalent to 75? Choose all that apply.
1. Use the properties of exponents to find the value of 10 4 ⋅ 10 −4 .
Integer Exponents
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?
3. Write an equivalent expression for 10 −4 .
For problems 4–9, use the definition of negative exponents to write an equivalent expression. Assume that x is nonzero.
For problems 10 and 11, use the definition of negative exponents to write an equivalent expression with positive exponents.
10. 1 10 2 11. 1 4 6
Quotients of Powers
So-hee applies the properties of exponents to write an equivalent expression for 103 10 7 . So-hee’s work is shown.
12. Apply the properties of exponents and the definition of negative exponents to verify So-hee’s answer.
For problems 13–15, apply the properties and definitions of exponents to write an equivalent expression with positive exponents. Assume all variables are nonzero.
• 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
x is any number m and n are integers when
Power of a Power Property
x is any number m and n are integers when (x m)n = x m n
Power of a Product Property
x and y are any numbers n is an integer when (xy)n = x ny n
Definition of the Exponent of 0 x is nonzero x 0 = 1
Definition of Negative Exponents x is nonzero n is an integer when x –n = 1 x n
Examples
1. How can we show 10 −4 = 1 10 4 ?
By using the properties of exponents, we know 10 4 ⋅ 10 −4 = 10 4+(−4) = 10 0 = 1
We also know 10 4 ⋅ 1 10 4 = 1
This means 10 −4 and 1 10 4 are both multiplicative inverses of 10 4 . So 10 −4 = 1 10 4 . Two factors that have a product of 1 are multiplicative inverses.
For problems 2–4, use the definition of negative exponents to write an equivalent expression. Assume p is nonzero.
2. 17 −4 1 17 4 3. 1 p 6 p −6
The multiplicative inverse of p 6 can be expressed as 1 p 6 or p −6 . 4. 1 5 3
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. 2 r 5 16 r 11
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.
For problems 13–19, apply the properties and definitions of exponents to write an equivalent expression with positive exponents. Assume all variables are nonzero.
20. Maya says 10 −5 is the same as (−10) 5 . Do you agree? Explain your reasoning.
21. Which one does not belong? Circle your answer and explain your reasoning.
22. Order the values from least to greatest.
Remember
For problems 23–26, add or subtract.
23. 5 6 + 3 4 24. 1 9 + 5 18
25. 3 4 7 10 26. 7 15 4 9
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?
For problems 28 and 29, find the sum or difference.
• 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.
Write
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.
5. 8 hk 0 (2h 4 k −3 ) 3 6. ( m 2 10 n −1 )3
8 hk 0 ( 2 h 4 k 3 ) 3 = 8 ⋅ h ⋅ k 0 ⋅ 2 3 ⋅ h 4 3 ⋅ k 3 3
= 2 3 ⋅ h ⋅ 1 ⋅ 2 3 ⋅ h 12 ⋅ k 9
= 23 ⋅ 23 ⋅ h ⋅ h12 ⋅ k 9
= 23+3 ⋅ h1+12 ⋅ k 9
= 2 6 h 13 k 9
Another way to simplify the numerical bases is 8 ⋅ 2 3 = 8 ⋅ 8 = 64. ( m2 10 n 1 )3 = m 2 3 103 n 1 ⋅ 3 = m6 103 n 3 = 1 103 ⋅ m6 ⋅ n3 = m 6 n3 10 3
7. p 3 r 0 s p −5 r s 4 p 3 r 0 s p 5 r s 4 = p 3 1 s p 5 ⋅ r ⋅ s 4 = p 3 ⋅ s ⋅ p 5 ⋅ r 1 ⋅ s 4 = p 3 ⋅ p 5 ⋅ r 1 ⋅ s ⋅ s 4
= p 3+5 ⋅ r 1 ⋅ s 1+( 4)
= p 8 ⋅ r 1 ⋅ s 3
= p 8 r s 3
Rewrite a power in the denominator with a negative exponent as a power in the numerator with a positive exponent.
a. What are some ways can you rewrite the expression by using exponents?
b. Which way do you prefer to rewrite the expression and why?
Remember
For problems 11–14, multiply.
15. Write an equivalent expression for 1 ____ 3854 with a negative exponent.
For problems 16 and 17, find the area of the figure.
Student Edition: Grade 8, Module 1, Topic C
How Many Words Have Ever Been Spoken?
Some questions are so dizzying that we’ll never know the answers—at least not exactly, anyway.
Take this one: “How many words have ever been spoken on Earth?” It’s impossible to know for sure. Nobody has been counting. You don’t even know how many words you’ve spoken today; how could we know the number of words for all people ever?
Well, we can’t. But we can estimate the number of words.
1011
Estimated number of humans who ever lived:
(i.e., 100 billion)
Estimated number of words spoken in an average lifetime:
108
(i.e., 100 million)
Estimated number of words ever spoken on Earth:
1019
(i.e., 10 quintillion)
These numbers are estimates. But that’s okay. With numbers this large, tiny errors don’t matter much. Even big errors may start to seem small by comparison.
Did we miss a billion people? That’s only 1% of our total, which barely changes the answer!
1. Complete the table as directed during the lesson. The table shows an approximate measurement of various objects.
Approximate Measurement (meters)
Small Positive Numbers
For problems 2–7, write the number in scientific notation or standard form as indicated in the table. Number in Standard Form
in Scientific Notation 2. 0.0007
× 10 −6
0.000 0062
Ordering Numbers in Scientific Notation
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.
For problems 9–13, compare the numbers by using the < or > symbol. Explain your answer by using the order of magnitude.
12. 3 × 10
2 × 10
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.
× 10 6
Name the planets, including Pluto, in order from the smallest diameter to largest diameter.
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.
0. 00056
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.
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.
Remember
For problems 18–21, multiply.
18(1 3 )
8(1 2 )
45(1 5 )
80( 1 10 )
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?
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.
Power to a Power
For problems 2–6, use the properties of exponents to evaluate the expression. Write the answer in scientific notation. Check the answer with a calculator.
2. (3.72 × 10 5 ) 2
3. (4 × 10 6 ) 2
4. (8.85 × 10 3 ) 3
5. (2 × 10 −9 ) 2
6. (2.6 × 10 −4 ) 3
Find the Difference
For problems 7–13, use the values from the table to answer the questions. Write the answer in scientific notation. Use a calculator to check your answers.
Object
Approximate Measurement
7. About how much longer is the state of California than the state of Rhode Island?
8. About how much longer is a tube of lip balm than the width of a human hair?
• 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 questions.
Animal
Approximate Weight (pounds)
Aphid 4.4 × 10 −7
Emperor Scorpion 6.6 × 10 −2
Gray Tree Frog 1.6 × 10 −3
Termite 3.3 × 10 −6
a. About how much heavier is a gray tree frog than an aphid? ( 1.6 × 10 3) ( 4.4 × 10 7) = ( 1.6 × 10 3) ( 4.4 × 10 4 × 10 3) = ( 1.6 × 10 3) ( 0.00044 × 10 3) = ( 1.6 0.00044 ) × 10 3
How much heavier means we have to find the difference of the weights.
= 1.59956 × 10 3
Write 10 −7 as 10 −4 × 10 −3 . Then apply the distributive property to the same powers of 10
A gray tree frog is about 1.59956 × 10 −3 pounds heavier than an aphid.
b. An emperor scorpion is about how many times as heavy as a termite? 6.6 × 10 −2 = ⋅ (3.3 × 10 −6 ) 6.6 × 10 2 3.3 × 10 6 = (6.6 3.3 ) × (10 2 10 6 ) = 2 × 10 4
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.
3. Is the value of (2.6 × 10 −5 ) 2 greater than or less than 2.6 × 10 −5 ? Explain. ( 2.6 × 10 5) 2 = 2. 6 2 × ( 10 5) 2 = 6.76 × 10 10
Use the property of exponents to square both factors of the product within the parentheses.
The value of (2.6 × 10 −5 ) 2 is 6.76 × 10 −10 , which is less than 2.6 × 10 −5 because the order of magnitude of −10 is less than the order of magnitude of −5.
3. 1.386471e9 4. 1.9321 * 10 18 (1.39 * 10 9 )2 Name Date
For problems 1–4, write the given answer for each calculator display in scientific notation and in standard form.
1. . 678e+ 51 7 2. 5 0.0000023*0.0000 1.15E-10
For problems 5–10, use the table of animal weights to answer the questions. Use a calculator to check your answers.
5. About how much more does a hummingbird weigh than a monarch butterfly?
6. About how much heavier is a wood mouse than a monarch butterfly?
7. A wood mouse is about how many times as heavy as a monarch butterfly?
8. About how many more pounds does a zebra weigh than a wood mouse?
9. Which animal or insect is about 8 times as heavy as a monarch butterfly?
10. A zebra is about how many times as heavy as a hummingbird?
11. Which number is greater, (1.1 × 10 −3 ) 2 or 1.1 × 10 −3 ? Explain.
12. Which number is less, (3 × 10 6 ) 2 or (6 × 10 3 ) 3 ? Explain.
13. 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.
b. Comparing only diameters, Jupiter is about how many times as large as Mercury? Round your answer to the nearest whole number.
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?
Remember
For problems 14–17, multiply.
14. 15(11 5 )
15. 22(13 11 )
16. 18(11 9 )
17. 44(10 11 )
18. Simplify the expression 14 11 14 6 . Write the expression with positive exponents.
19. Plot all the points from the table in the coordinate plane.
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.
b. What volume of air is in 10 breaths? Write your answer in scientific notation.
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?
3. Complete the comparison statement.
The volume of the goal is cubic meters. That is times as much as the volume of a single breath, which is 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.
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.
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.
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.
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 Tokyo Skytree Tower is 6.34 × 10 2 meters tall. A paperclip is about 3.17 × 10 2 meters long. If placed end to end, approximately how many paperclips will it take to measure the Tokyo Skytree Tower?
• 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.
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.
c. Find the difference between your place value guess from part (a) and the actual answer from part (b). Did you overestimate or underestimate?
1. Use the following information to fill in the blanks. On Earth, there are about 1 quadrillion (1,000,000,000,000,000) ants and about 8 billion people.
a. On Earth, there are about more ants than people.
b. On Earth, there are about times as many ants as people.
2. Complete parts (a)–(c) to estimate, calculate, and compare 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.
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.
c. Find the difference between your place value guess from part (a) and the actual answer from part (b).
3. Complete parts (a)–(c) to estimate, calculate, and compare 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.
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)
c. Find the positive difference between your place value guess from part (a) and the actual answer from part (b).
Remember
For problems 4–7, multiply.
4. 4(5 8 ) 5. 4( 7 24 )
6. 11( 7 33 ) 7. 10(14 80 )
8. Simplify (x −6 y 5 z 0 ) −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.
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?
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?
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?
Seconds of Life
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?
Two-Way Radio Problem
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.
Use this space to write your advertisement comparison statement.
• 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.
b. Earth’s circumference is about 4 × 10 4 kilometers. Using your answer from part (a), about how many days would it take for a monarch butterfly to travel a distance equal to the length of Earth’s circumference? (1 mile ≈ 1.6 kilometers)
240 ⋅ 1.6 = 384. A monarch butterfly travels about 384 kilometers each day.
Convert the number of miles traveled in one day to kilometers.
Divide the circumference of Earth by the distance traveled by the butterfly in one day.
It would take about 100 days for a monarch butterfly to travel a distance equal to the length of Earth’s circumference.
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.
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.
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.
Canis Majoris (Sirius)
Canis Minoris (Procyon)
Lyrae (Vega)
Eridani (Rana)
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)
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.
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.
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.
1. What assumptions did you make when you modeled the number of brush strokes in the painting La Corne d’Or. Matin?
2. Was writing numbers in scientific notation helpful in the lesson? If so, when? If not, why not?
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?
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?
b. What assumptions did you make in answering part (a)?
c. Do you think your estimation for part (a) is reasonable?
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?
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.
Remember For problems 6–9, multiply.
10. Simplify x 5 y 2 x 7 y 4 , where x and y are nonzero.
For problems 11–14, evaluate.
9 2
Student Edition: Grade 8, Module 1, Topic D
Jealous of the Perfect Squares
Hey, 16? Why does everyone call you “perfect”? Well, not to brag, but I technically am. I’m a perfect square.
Some numbers are special.
But... doesn’t anybody think that I’m perfect?
Oh,
17... you’re
a perfect... um... a perfectwhatever-you-are.
Well—scratch that. All numbers are special.
It’s just that some numbers are especially special.
Take 16. Because it’s equal to 4 · 4, it can be represented as a 4-by-4 square. That’s why we say 4 2 as four squared—because we are, in a sense, using 4 as the side length to create a new square.
Most numbers are not so lucky. Take 16’s neighbor, 17. You can’t arrange 17 little squares to form a bigger square. In fact, you can’t arrange them to form a nice rectangle of any kind! You can make a 4-by-4 or 8-by-2 rectangle with one little square left over; or you can make a 6-by-3 or 9-by-2 rectangle with one little square missing; but other than the dull 1-by-17 single row of little squares option, there’s no nice rectangle of 17 squares.
We have a name for this imperfection: prime. Perhaps 17 is special after all!
For problems 4 and 5, determine all numbers that square to the given number.
4. 81
9 and −9
5. 0.64
0.8 and −0.8
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
For problems 6 and 7, determine all numbers that cube to the given number.
6. −27
Only one integer has a cube of −27.
( 3 ) 2 = ( 3 ) ( 3 ) ( 3 ) = (9 ) ( 3 ) = 27
Look for familiar numbers. This fraction includes the perfect cubes 8 and 125, which result from cubing 2 and 5.
For problems 1 and 2, find a pattern and complete the table.
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?
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?
For problems 5–10, evaluate.
For problems 11–14, determine all the numbers that square to the given number.
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.
Remember
For problems 20−23, multiply.
24. Simplify (x 2 y −4 z −1 ) 5 . Assume all variables are nonzero.
25. Indicate whether the decimal equivalent for the rational number is a repeating decimal or a terminating decimal.
• solved equations of the form x 2 = p, where p is either a perfect square or a fraction or decimal related to a perfect square.
• solved equations of the form x 3 = 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.
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 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.
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.
Examples
For problems 1–6, solve the equation. 1. x 2 = 121 11 and −11
3. 25 49 = m 2 5 7 and 5 7
k 2 = −25
Use the perfect squares 25 and 49 to help determine the solutions.
The phrase solve the equation means to find the value(s) that, when substituted for the variable, make the equation true.
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?
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.
b. How does the information from part (a) relate to a right triangle?
Remember
For problems 3–6, divide.
28 ÷ 7
15 ÷ (−3)
7. Which number is greater, 18 million or (13,000) 2 ?
For problems 1 and 2, find the length of the hypotenuse c. If the answer is not a whole number of units, use square root notation to express the length.
3. The length of the hypotenuse c in problem 2 is between which two consecutive whole numbers? Explain.
• determined which two consecutive whole numbers the length of a hypotenuse is between.
• used square root notation to express hypotenuse lengths.
Examples
RECAP
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.
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.
For problems 1–4, find the length of the hypotenuse c
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: a 2 + b 2 = c 2
2 + 9 2 = c 2
The length of the hypotenuse is 112.5 units.
a. Describe all errors So-hee makes.
b. Describe all errors Vic makes.
Vic’s Work:
c. Find the correct length of the hypotenuse.
The length of the hypotenuse is 42 units.
For problems 6–9, determine which two consecutive whole numbers the length of the hypotenuse c is between.
For problems 10–13, determine whether the expression represents a whole number. Indicate yes and state the whole number that the expression represents, or indicate no
6. 75 = c 2 7. c 2 = 116
10 6
5 5
10. √16
11. √144
12. √125
13. √40
For problems 14 and 15, use square root notation to express the length of the hypotenuse c.
Remember For problems 16–19, multiply.
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.
21. Complete the table by drawing an example of each geometric figure.
For problems 5–10, find the length of the hypotenuse c.
11. Which point represents the approximate location of √8 on a number line?
12. Which numbers are between 3 and 4 on a number line? Choose all that apply.
A. √2
B. √4
C. √10 D. √12
E. √15
13. For each leg length, a and b, choose a digit from 5, 6, 7, 8, and 9 so that the length of the hypotenuse c is closest to 9 units. A digit can only be used once.
Remember
For problems 14–17, divide.
14. −156 ÷ 4
15. 132 ÷ 11
16. −247 ÷ (−19)
17. 170 ÷ (−5)
18. Solve the equation h 2 = 9 49 .
19. A map of a playground is shown in the coordinate plane.
a. What is the ordered pair for the location of the slides?
b. What is located at (3, 3)?
c. How far is the picnic area from the climbing area?
Irrational Numbers
Student Edition: Grade 8, Module 1, Topic E
Special Delivery
Hello, square root of 2? I have a package for you, but I’m having trouble finding the address.
Oh, I’m between 1.4 and 1.5.
I’m here. Where are you?
Okay, now go between 1.41 and 1.42.
LATER
Okay, now go between 1.41421356 and 1.41421357.
Look, can I just leave the package with one of your neighbors?
Irrational numbers are strange.
Very strange.
Irrational means “not a ratio.” In other words, an irrational number can’t be written as a nice fraction. Even writing it as a decimal is hard because the digits will continue forever, without repeating.
This makes it rather hard to find a number like √ 2 on a number line. No matter how far you zoom in, it always falls just between your nicely labeled intervals.
Student Edition: Grade 8, Module 1, Topic E, Lesson 21
Name Date
Approximating Values of Roots and π 2
Approximating Square Roots
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. Place Value
Hundredths
Approximating Cube Roots
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.
Approximating π 2
3. Complete the tables.
a. Determine the intervals that contain π and π 2
Interval That Contains ��
Interval That Contains �� 2 and and and and and and and and
b. Use your results from part (a) to approximate the value of π 2 to the nearest given place value.
Place Value Rounded Value Ones Tenths Hundredths
Student Edition: Grade 8, Module 1, Topic E, Lesson 21
Name Date
Explain how you approximate the value of √17 by rounding to the nearest tenth.
Student Edition: Grade 8, Module 1, Topic E, Lesson 21
Name Date
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.
Examples
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.
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
2. 2 < 3 √ 12 < 3 , 12 is between the perfect cubes 8 and 27
√8 < 3 √12 < 3 √27
< 3 √12 < 3
For problems 3 and 4, round each root to the nearest whole number.
14
200 is between the perfect squares 196 and 225.
Both 14.1 and 14.2 round to 14.
Both 3.7 and 3.8 round to 4.
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 3.6, so 3 √46 ≈ 3.6.
Student Edition: Grade 8, Module 1, Topic E, Lesson 21
Name
PRACTICE
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.
10. Explain how you approximate the value of √23 by rounding to the nearest tenth.
For problems 11–13, approximate each root by rounding to the nearest whole number, tenth, and hundredth.
14. Find the length of the hypotenuse c. Approximate the length by rounding to the nearest tenth.
Remember
For problems 15–18, divide.
45 ÷ ( 1 5 )
80 ÷ ( 1 10 )
17.
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?
c. Which point is located at (6, 1)?
d. What is the ordered pair for point T ?
Student Edition: Grade 8, Module 1, Topic E, Lesson 22
Name Date
Familiar and Not So Familiar Numbers
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.
1. 1 3 = 0.333333333…
2. 3 √12 = 2.2894284…
3. 144 99 = 1.45454545…
4. 3 5 12 = 3.41666666…
5. √42 = 6.480740698…
6. π = 3.1415926…
7. 3 7 = 0.4285714285…
Terminating and Repeating
8. Identify whether numbers from problems 1–7 are rational or irrational. Justify your reasoning.
9. Complete the table by including a definition, examples, and features for each type of number.
Rational Numbers Irrational Numbers Real Numbers
Definition Examples
Features
Is It Rational?
10. Indicate whether the number is rational or irrational.
Student Edition: Grade 8, Module 1, Topic E, Lesson 22
Name
1. Compare the decimal forms of a rational number and an irrational number.
2. Indicate whether the number is rational or irrational.
Student Edition: Grade 8, Module 1, Topic E, Lesson 22
Name Date
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.
0.24680153…
Terminology
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.
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.
4. Ethan states that √5 is an irrational number. Nora states that √5 is a real number. Who is correct? Explain.
5. Eve states that all square roots are irrational numbers. Provide three examples that show Eve is incorrect.
Remember
For problems 6–9, divide.
10. Add. Write the answer in scientific notation.
(1.2 × 10 −3 ) + (2.8 × 10 −4 )
11. 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.
Student Edition: Grade 8, Module 1, Topic E, Lesson 23
Name Date
Ordering Irrational Numbers
Ordering Numbers
Plot and label each value on the given number line.
Ordering Expressions
For problems 2–4, order the expressions from least to greatest. Explain your reasoning.
Student Edition: Grade 8, Module 1, Topic E, Lesson 23
Name
1. Plot and label each value on the given number line. √24 , 4.2 3, √16 , 14 3 , 3 √125
2. Compare the numbers by using the < or > symbol. Explain your reasoning.
Student Edition: Grade 8, Module 1, Topic E, Lesson 23
Name Date RECAP
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. 14.2 > √ 196 2. 3. 94 < 3 √ 65
Because √196 is 14, 14.2 is greater than √196 . Because 3 √64 is 4, 3 √65 is greater than 4. So 3. 94 is less than 3 √65 3. √50 > 62 9 4. 3 √ 120 > √ 18
Because 63 9 is 7, 62 9 is less than 7
Because √49 is 7, √50 is greater than 7
Because 3 √64 is 4, and 3 √125 is 5, 3 √120 is between 4 and 5 but closer to 5.
5. Which value is greater, 3 √25 + 4 or 2 ⋅ 3 √25 ? Explain.
Because √16 is 4 and √25 is 5, √18 is between 4 and 5 but closer to 4
I know that the value of 3 √25 is between 2 and 3 but closer to 3. Adding 4 to 3 √25 gives a number that is between 6 and 7. Multiplying 3 √25 by 2 gives a number that is between 4 and 6. So 3 √25 + 4 is greater than 2 3 √25
Student Edition: Grade 8, Module 1, Topic E, Lesson 23
For problems 1–9, compare the numbers by using the < or > symbol.
For problems 10 and 11, plot and label each value on the given number line.
10. √27 , 3 √27 , √11 , √20 , 3 √15 , √4
11. 4.1 6,
12. Liam states that √75 is greater than 3 √100 . Do you agree or disagree with Liam? Explain your reasoning.
13. Is the value of √88 2 greater than or less than the value of √88 6? Explain your reasoning.
Remember
For problems 14–17, divide. 14. 3 5 ÷ 1 4
1 6 ÷ 7 11
1 8 ÷ 3 5
18. Find the length of the hypotenuse c. 7
4 9 ÷ 1 10
19. Consider the diagram in which two lines and a ray all meet at point C.
a. Which angle relationships will help you solve for x and y?
b. Find the value of y
c. Find the value of x.
Student Edition: Grade 8, Module 1, Topic E, Lesson 24
Name Date
Revisiting Equations with Squares and Cubes
1. Indicate whether the value of x is a solution to the given equations.
Revisiting Equations of the Form x 2 = p
For problems 2–7, solve the equation. Identify all solutions as rational or irrational.
x 2 = 25
x 2 = 35
Revisiting Equations of the Form x 3 = p
For problems 8–13, solve the equation. Identify all solutions as rational or irrational.
6. m 2 = 144
t 2 = 27
8. t 3 = 27
−125 = k 3
10. x 3 = 7
−8 = r 3
12. 25 = w 3
13. 38 = z 3
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. x 3 = −64 x = 3 √ 64
The solutions are −8 and 8.
Solving Two-Step Equations
For problems 15 and 16, solve the equation. Show all steps. 15. 2 x + 1 = 17
x 2 + 1 = 17
For problems 17–20, solve the equation. Identify all solutions as rational or irrational.
Student Edition: Grade 8, Module 1, Topic E, Lesson 24
Name
For problems 1 and 2, solve the equation. Identify all solutions as rational or irrational.
1. x 2 = 13
2. x 3 = 36
Student Edition: Grade 8, Module 1, Topic E, Lesson 24
Name Date RECAP
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
The solution is 3 √100 .
Irrational
3. x 2 + 11 = 36
2 + 11 = 36
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
The solutions are 5 and −5
Rational
Student Edition: Grade 8, Module 1, Topic E, Lesson 24
Name Date
PRACTICE
1. Which numbers are solutions to the equation x 3 = 27? Choose all that apply.
For problems 2–15, solve the equation. Identify all solutions as rational or irrational.
10. 1 8 = t 3
b 2 = −100 12. −5 = x 2 − 7
d 2 + 16 = 36
14. 45 = b 3 + 109
−4 + f 3 = 25
16. Fill in the boxes with any digits 1–9 to create an equation that has the described solutions. Each digit can be used only once in each equation.
a. Two rational solutions
b. One irrational solution
Remember
For problems 17–20, divide.
17. 5 6 ÷ 5 12 18. 15 4 ÷ 3 8
19. 2 3 ÷ 4 7 20. 5 6 ÷ 2 3
21. Approximate the value of √96 by rounding to the tenths place. Explain your answer.
22. A faucet drips water at a constant rate of 2 _ 3 quarts in 15 minutes. What is the rate in quarts per hour?
Student Edition: Grade 8, Module 1, Mixed Practice 1
Mixed Practice 1
1. Use a number line to find the difference of 8 and 5 1 3 .
2. Jonas and Kabir each evaluate the expression 4 + (−12)
a. Model the expression on the vertical number line.
b. Jonas says the answer is −8. Kabir says the answer is −16. Who is correct? Explain.
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?
b. What does the number −1.5 represent in the equation?
c. What does the number 0 represent in the equation?
5. Solve for a
0.5a + 7 = 24.5
6. Vic takes a taxi from the airport to his house. He pays a flat fee of $5.50 plus an additional $1.75 per mile. The total cost of his trip is $28.25. How many miles from the airport is Vic’s house?
7. The total cost for movie tickets is proportional to the number of movie tickets purchased.
a. What is the constant of proportionality?
b. What is the total cost for 10 movie tickets?
For problems 8 and 9, assume that each car travels at a constant rate. If the rate of the car is expressed in miles per hour, what is the unit rate?
8. Car A travels 22 1 2 miles in 3 4 hours.
9. Car B travels 25 1 4 miles in 50 minutes.
Student Edition: Grade 8, Module 1, Mixed Practice 2
Mixed Practice 2
For problems 1 5, write an equivalent expression by using the fewest number of possible terms.
1. 3 x − 5 + 2 x + 9 2. (4 y + 9) − (2 y + 7)
3(5a − 6 a + 2) 4. 6 b + 2 − 5(b − 2)
5. 2 3 (18 x − 21 ) + 11
6. Which expressions are equivalent to 3 5 ÷ 0.5? Choose all that apply.
A. 3 5 ⋅ 1 2
B. 5 3 ⋅ 1 2
C. 5 3 ( 1 2 )
D. 3 5 ( 2)
E. 5 3 ( 2)
F. 3 5 ⋅ 2
3.
7. Find the perimeter of the semicircle. Use π ≈ 3.14.
8. The perimeter of a rectangle measures 48 feet. The length is 4 times as long as the width.
a. What is the width of the rectangle?
b. Make a new rectangle by adding 9 1 2 feet to the length of the original rectangle. What is the length of the new rectangle? 15 cm
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.
b. What is the value of y? Explain your answer.
c. Find the value of x
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?
1 25
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?
b. Are the outcomes of this experiment equally likely? Explain your answer.
Student Edition: Grade 8, Module 1, Topic E, Lesson 21
Apply the properties and definitions of exponents to write an equivalent expression with a single base. Let all variables represent nonzero numbers.
ANumber Correct:
Apply the properties and definitions of exponents to write an equivalent expression with a single base. Let all variables represent nonzero numbers.
1. 10 6 10 3
2. 10 6 ⋅ 10 4
16. t 8 t 2 17. r 3 r 5
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.
1. 10 4 ⋅ 10 3
2. 10 4 ⋅ 10 4
Student Edition: Grade 8, Module 1, Sprint: 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. 1. 8 −3 ⋅ 8 7 2. 8 −3 ⋅ 8 −4
ANumber Correct:
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.
3.
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. 1.
Write the expression with an exponent. Let all variables represent nonzero numbers.
1. 5 · 5 · 5
2. k · k · k · k
Number Correct:
Write the expression with an exponent. Let all variables represent nonzero numbers.
Number Correct: Improvement:
Write the expression with an exponent. Let all variables represent nonzero numbers. 1. 20 ·
Student 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.
Adriana Akers, Amanda Aleksiak, Tiah Alphonso, Lisa Babcock, Christopher Barbee, Reshma P. Bell, Chris Black, Erik Brandon, Beth Brown, Amanda H. Carter, Leah Childers, David Choukalas, Mary Christensen-Cooper, Cheri DeBusk, Jill Diniz, Mary Drayer, Karen Eckberg, Dane Ehlert, Samantha Falkner, Scott Farrar, Kelli Ferko, Krysta Gibbs, Winnie Gilbert, Danielle Goedel, Julie Grove, Marvin E. Harrell, Stefanie Hassan, Robert Hollister, Rachel Hylton, Travis Jones, Kathy Kehrli, Raena King, Emily Koesters, Liz Krisher, Alonso Llerena, Gabrielle Mathiesen, Maureen McNamara Jones, Pia Mohsen, Bruce Myers, Marya Myers, Kati O’Neill, Ben Orlin, April Picard, John Reynolds, Bonnie Sanders, Aly Schooley, Erika Silva, Hester Sofranko, Bridget Soumeillan, Ashley Spencer, Danielle Stantoznik, Tara Stewart, James Tanton, Cathy Terwilliger, Cody Waters, Valerie Weage, Allison Witcraft, Caroline Yang
Trevor Barnes, Brianna Bemel, Adam Cardais, Christina Cooper, Natasha Curtis, Jessica Dahl, Brandon Dawley, Delsena Draper, Sandy Engelman, Tamara Estrada, Soudea Forbes, Jen Forbus, Reba Frederics, Liz Gabbard, Diana Ghazzawi, Lisa Giddens-White, Laurie Gonsoulin, Nathan Hall, Cassie Hart, Marcela Hernandez, Rachel Hirsh, Abbi Hoerst, Libby Howard, Amy Kanjuka, Ashley Kelley, Lisa King, Sarah Kopec, Drew Krepp, Crystal Love, Maya Márquez, Siena Mazero, Cindy Medici, Ivonne Mercado, Sandra Mercado, Brian Methe, Patricia Mickelberry, Mary-Lise Nazaire, Corinne Newbegin, Max Oosterbaan, Tamara Otto, Christine Palmtag, Andy Peterson, Lizette Porras, Karen Rollhauser, Neela Roy, Gina Schenck, Amy Schoon, Aaron Shields, Leigh Sterten, Mary Sudul, Lisa Sweeney, Samuel Weyand, Dave White, Charmaine Whitman, Nicole Williams, Glenda Wisenburn-Burke, Howard Yaffe
Student Edition: Grade 8, Module 1, Talking Tool
Talking Tool
Share Your Thinking
I know . . . . I did it this way because . . . . The answer is because . . . . My drawing shows . . . .
Agree or Disagree
Ask for Reasoning
I agree because . . . . That is true because . . . . I disagree because . . . .
That is not true because . . . .
Do you agree or disagree with ? Why?
Why did you . . . ? Can you explain . . . ? What can we do first? How is related to ?
Say It Again
I heard you say . . . . said . . . .
Another way to say that is . . . . What does that mean?
Thinking Tool
When I solve a problem or work on a task, I ask myself
Before
Have I done something like this before? What strategy will I use?
Do I need any tools?
During Is my strategy working?
Should I try something else? Does this make sense?
After
What worked well?
What will I do differently next time?
At the end of each class, I ask myself
What did I learn?
What do I have a question about?
MATH IS EVERYWHERE
Do you want to compare how fast you and your friends can run?
Or estimate how many bees are in a hive?
Or calculate your batting average?
Math lies behind so many of life’s wonders, puzzles, and plans.
From ancient times to today, we have used math to construct pyramids, sail the seas, build skyscrapers—and even send spacecraft to Mars.
Fueled by your curiosity to understand the world, math will propel you down any path you choose.
Ready to get started?
Module 1
Scientific Notation, Exponents, and Irrational Numbers
Module 2
Rigid Motions and Congruent Figures
Module 3
Dilations and Similar Figures
Module 4
Linear Equations in One and Two Variables
Module 5
Systems of Linear Equations
Module 6
Functions and Bivariate Statistics
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.
