Grade 6 2023 Virginia SOL - Student Edition by Chris Velis

1 Operations with Fractions

Big ideas

• The properties of real numbers can be applied to many types of expressions.

• Real numbers are either rational or irrational.

• Expressions are the building blocks of algebra. They can be used to represent and interpret real-world situations.

1.01 Review: Simplify fractions and mixed numbers

After this lesson, you will be able to...

• rewrite fractions and mixed numbers in simplest form.

Improper fractions and mixed numbers

Fractions like make up less than one whole. We know this because the numerator is less than the denominator. We call these proper fractions.

What about a fraction like ? Notice that the numerator is greater than the denominator. This means that the fraction is greater than a whole. We call these improper fractions

Each rectangle in this image has been split into ﬁve equal parts, so 5 is the denominator. Eight parts have been shaded, so 8 is the numerator.

Since the number of shaded parts is more than the number of parts in one whole, a complete rectangle and three more parts have been shaded.

We can write this number as 1 , which we call one and three ﬁfths.

Numbers like this are called mixed numbers or mixed numerals. Mixed numbers and improper fractions can also be represented on a number line.

For or 1 , each whole number on the number line is split into 5 equal parts.

For the improper fraction, we would count 8 tick marks from 0, and place the point there.

For the mixed number, we would count 3 tick marks from 1.

What number is plotted on the number line? Give your answer as a mixed number and as an improper fraction.

Create a strategy

For the fraction part, count the number of equal spaces between the two whole numbers, then count how many spaces after the previous whole number the point is located.

Example 1

Apply the idea

The point is located between 3 and 4. It is greater than 3 but less than 4. So, the whole number part is 3.

There are 10 equal spaces between 3 and 4, so each space represents . The point is 9 spaces to the right of 3, so the fraction is

The number plotted on the number line is 3

This can also be written as

Example 2

Rewrite 4 as an improper fraction.

Create a strategy

To ﬁnd the improper fraction, multiply the whole number by the denominator, then add the numerator.

Apply the idea

Multiply the denominator and whole number

Evaluate the multiplication

Evaluate the addition

Reﬂect and check

We could also draw an array for 4 where each part represents We can see that there are 9 shaded parts. So, the improper fraction is

3

Rewrite as a mixed number.

Create a strategy

Divide the numerator by the denominator. The remainder will be the numerator of the mixed fraction.

Apply the idea

11 divided by 3 is 3 remainder 2. So, is made up of 3 wholes and 2 out of 3 remaining.

Example

Idea summary

Fractions where the numerator is greater than the denominator are called improper fractions Improper fractions can be rewritten as mixed numbers

Equivalent fractions

Sometimes different fractions can represent the same amount. Consider the shaded area of the hexagon below.

We can see that 3 of the 6 parts have been shaded in, so the area represents of the whole shape.

But we can also see that of the shape has been shaded in.

Since the same area is shaded for both and , these two fractions must be equal.

These are called equivalent fractions, since they have different numerators and different denominators but are still equal.

Interactive exploration

Explore online to answer the questions mathspace.co

Use the interactive exploration in 1.01 to answer these questions.

1. When multiplying the numerator and denominator of the original fraction by 2, how many pieces is the shaded part of the new fraction separated into?

2. When multiplying the numerator and denominator of the original fraction by 3, how many pieces is the shaded part of the new fraction separated into?

3. Do you think this is the case for any multiple of the numerator and denominator of any fraction?

4. Why do you think we have to multiply both the numerator and denominator by the same number?

We can see equivalent fractions in action by representing them visually and seeing how we can create them. Consider this grid:

This grid is divided into 6 equal parts as shown by the solid grid lines. 1 out of 6 parts is shaded, so we can represent this area as of the grid.

If we separate each of the 6 parts into 3 more equal parts, as shown by the dashed lines, we can create an equivalent fraction.

Notice there are now 18 grid squares, and 3 of them are shaded. This means that we can represent the shaded area with the fraction .

This shows that and are equivalent fractions, which we can express using an equal sign:

By separating each of the original parts into 3 more equal parts, we are multiplying the numerator (the shaded part) and denominator (the entire grid) by 3:

We can always ﬁnd an equivalent fraction by multiplying both the numerator and denominator by the same number.

Example 4

Rewrite with a denominator of 40.

Create a strategy

Multiply the numerator and denominator by the quotient of 40 divided by 8.

Apply the idea

Dividing 40 by 8 gives 40 ÷ 8 = 5.

Multiply the numerator and denominator by 5

Evaluate

Idea summary

When two fractions represent the same amount of a whole, they are equivalent fractions

We can create an equivalent fraction by multiplying the numerator and denominator by the same number.

Simplify fractions

We can also create equivalent fractions by decreasing the denominator. To decrease the denominator, we can remove common factors in both the numerator and denominator. Since the numbers become smaller, we call this simplifying the fraction.

Previously, we saw that

From here, we can remove the common factor of 3:

This shows:

The fraction has no common factors between the numerator and denominator (other than 1), so it is in simplest form. This also means that 3 was the greatest common factor of 3 and 18.

Now, simplify each of the fractions in the answer choices and compare them to :

• Option A

Factor out the GCF of 10

Divide

The fraction is equivalent to because they both simplify to

• Option B

The fraction is already in its simplest form because 7 and 3 have no common factors other than 1. The fraction is not equivalent to because it is greater than and not equal to it.

• Option C

Factor out the GCF of 2

Divide

The fraction is not equivalent to because it is smaller than and not equal to it.

• Option D

The fraction is already in its simplest form and is equivalent to which we already found simpliﬁes to So, options A and D are the correct answers.

Reﬂect and check

Simplifying creates equivalent fractions by making the numerator and denominator smaller. We can also make equivalent fractions by making numerators and denominators larger. is simpliﬁed to . We can also scale up to form another equivalent fraction:

Multiply the numerator and denominator by 10

Evaluate

This conﬁrms our answer, showing that we have correctly identiﬁed the equivalent fractions and

Idea summary

When we ﬁnd an equivalent fraction by removing common factors from the numerator and denominator, we are simplifying the fraction.

When the fraction has no common factors between the numerator and denominator (other than 1), it is in simplest form.

To write a fraction in simplest form, divide both the numerator and denominator by their greatest common factor

Two fractions are equivalent when they can be simpliﬁed to the same fraction.

What do you remember?

1 What does it means when two fractions are equivalent? Use an example to support your answer.

2 Which of these statements are correct? Select all that apply.

A A fraction is in its simplest form if the numerator and denominator have common factors other than 1.

B A fraction is in its simplest form if the numerator and denominator have no common multiple.

C A fraction is in its simplest form if the numerator and denominator have no common factors other than 1.

D A fraction is in its simplest form if the numerator is 1.

E A mixed number is in its simplest form if the whole number is equal to 1.

F An improper fraction is in its simplest form if the numerator and denominator have no common factors other than 1.

3 Which of these fractions are fully simpliﬁed?

4 Complete the solution to simplify the fraction : Let’s practice

5 Complete each equivalent fraction statement:

6 Use the diagrams provided to simplify these fractions:

7 Which fraction is eqivalent to in simplest form?

8 Which fraction is equivalent to in simplest form?

9 Complete the statement using the fractions from the list: ⬚ is equal to ⬚

10 Complete the statement using the mixed numbers from the list: ⬚ is equal to ⬚

11 Select the equivalent fractions from each list:

12 Simplify:

13 Are and equivalent

Explain your reasoning. 14 Which of these pairs of fractions are equivalent? Justify your reasoning. and , and , and 15 Which of these pairs of fractions are equivalent? Justify your reasoning. and , and , and

16 Jane and Timothy disagree on the simplest form of . Determine who is correct and provide an explanation.

• Judy’s Answer:

• Timothy’s Answer:

Let’s extend our thinking

17 Determine an equivalent fraction to that has a denominator of 12. Justify your reasoning.

18 If the numerator and denominator of a fraction are both prime numbers, is the fraction in fully simpliﬁed form? Explain your answer.

19 Maria has a recipe that calls for cup of sugar.

a If she wants to double the recipe, how much sugar will she need?

b Write a fraction that is equivalent to your answer in part (a).

20 A pizza has been cut into 8 pieces, and 3 pieces have been eaten.

a What fraction of the pizza remains?

b Write a fraction with a denominator of 16 that is equivalent to your answer in part (a).

21 Adityah is collecting data on favorite colors. They found that 5 out of 20 students in their class like blue. The class next door has 28 students. If the fraction of students who like blue in Adityah’s class is equivalent to the fraction of students who like blue in the class next door, how many of the students in the classroom next door like blue?

1.02 Add and subtract fractions and mixed numbers

After this lesson, you will be able to...

• add and subtract fractions with like and unlike denominators.

• add and subtract mixed numbers.

Add and subtract fractions with the same denominator

Suppose we want to ﬁnd

Here we have a circle with 2 sevenths shaded and a circle with 3 sevenths shaded.

Notice that the parts of each circle are the same size.

Since the parts are the same size, we can place one circle on top of the other. Now, we can see 5 sevenths of the circle are shaded in.

So, we can conclude that

When the denominators are the same (called like denominators), we are adding quantities of the same size. This means we can add the number of shaded pieces, without needing to change the number of parts in the whole.

Mathematically, we are adding the numerators but keeping the same denominator.

Suppose we want to ﬁnd

Using the same circles, we can take 2 sevenths away from 3 sevenths. The part that remains is 1 seventh of the circle.

So, we can conclude that .

When the denominators are the same, we are subtracting quantities of the same size. Again, the number of shaded pieces is changing, but the number of parts in a whole stays the same.

Mathematically, we are subtracting the numerators but keeping the same denominator.

Example 1

Evaluate and simplify your answer.

Create a strategy

Add the numerators over the same denominator and simplify.

Apply the idea

Add the numerators

Rewrite with a common factor of 2

Evaluate

Idea summary

When two fractions have the same denominator, we can add or subtract them by adding or subtracting the numerators, while keeping the denominator the same.

Add and subtract fractions with different denominators

If the denominators of two fractions are different (called unlike denominators), then we are not adding quantities of the same size.

Interactive exploration

Explore online to answer the questions mathspace.co

Use the interactive exploration in 1.02 to answer these questions.

1. Type the fractions and into the fraction boxes. Then, press enter on your keyboard. Why can’t we add the fractions the way they are?

2. Click “Show common units.” How did the area models change?

3. Explain how we can add the fractions now. What is the sum?

4. Does this work for other fractions with different denominators?

We can use area models to change fractions to equivalent fractions, so the denominators are the same. Then, we can add the equivalent fractions.

Consider . These two fractions look like this:

If we tried to add the pieces together, how could we write the denominator?

We cannot place one circle on top of the other because the number of parts in a whole are different.

Before we can add these two fractions, we need to rewrite them with the same denominator. To do this, we create equivalent fractions.

The denominators here are 9 and 4, so the least common multiple of the denominators is 4 9 = 36. This is sometimes called the least common denominator.

Now, we can rewrite the fractions with a denominator of 36.

The fractions can now be divided into the same number of parts. After shading the correct amount of pieces, the fractions look like this.

Since the sizes of the parts are the same, we can add the fractions together.

This shows that

When the denominators are different, we create equivalent fractions with the same denominator, then add or subtract the numerators.

Example 2

Evaluate

Create a strategy

Find the least common multiple of the two denominators.

Apply the idea

The least common multiple of 4 and 8 is 8.

Since the denominator of is already 8 we only need to ﬁnd the equivalent fraction to that has a denominator of 8.

Multiply the numerator and denominator by 2

Evaluate

Substitute the equivalent fraction

Evaluate

Idea summary

When two fractions have different denominators, we ﬁrst rewrite the fractions with the same denominator. Then, we can add or subtract the numerators and keep the denominator the same.

Add and subtract mixed numbers

Mixed numbers have a whole number part and a fractional part. There are two methods for adding or subtracting mixed numbers:

1. Add or subtract the whole parts, then add or subtract the fractional parts.

• If the fractional parts do not have the same denominator, we need to rewrite them with the same denominator before adding or subtracting.

2. Convert the mixed numbers into improper fractions, then add or subtract the improper fractions.

• If the denominators are different, we need to rewrite them with the same denominator before adding or subtracting.

Example 3

Fill in the boxes to show the work for:

a Rewrite as improper fractions.

Create a strategy

Multiply the whole number by the denominator, then add the numerator.

Apply the idea

b Rewrite with a common denominator.

Multiply the whole number by the denominator

Evaluate the multiplication

Evaluate the addition

Create a strategy

Find the least common multiple (LCM) of the denominators, and multiply each numerator and denominator by the quotient obtained by dividing the LCM by their respective denominators.

Apply the idea

The LCM of 3 and 4 is 12.

So we need to multiply both parts of by 12 ÷ 3 = 4, and multiply both parts of by 12 ÷ 4 = 3.

Multiply by and by

Evaluate

c Evaluate the difference.

Create a strategy

Evaluate the answer from part (b).

Apply the idea

Subtract the numerators

Example 4

For the school fair, Aimee is making a batch of iced tea by mixing liters of water with of a liter of tea concentrate. What is the total volume of the iced tea mixture in liters?

Create a strategy

Add the volume of water to the volume of tea concentrate.

Apply the idea

Write the water and tea concentrate volumes

Convert the mixed number to an improper fraction

Find a common denominator

Add the fractions

Convert back to a mixed number

The total volume of the iced tea mixture is liters.

Idea summary

To add or subtract mixed numbers, we can write them as improper fractions, create equivalent fractions with the same denominator, then add or subtract them.

Another method of adding or subtracting mixed numbers is to rewrite the fractional parts to have the same denominator, then add or subtract the whole parts and add or subtract the fractional parts.

Practice

What do you remember?

1 Find the least common denominator for each set of fractions: a b c

2 The image shows of the rectangle shaded.

a What is the equivalent fraction in tenths?

b We now want to take away What is the answer to ? Write your answer in simplest form.

3 The image shows in blue and in green.

a Write the equation that describes the image if the blue and green tiles would be added.

b What is the total shaded? Express your answer in simplest form.

4 Complete each statement: a

Rewrite with a common denominator

Evaluate

Rewrite as a mixed number

Rewrite with a common denominator

Evaluate

Rewrite as a mixed number

5 Fill in the boxes to complete the working out:

Group the whole number parts and fraction parts

Add the whole number parts and fraction parts

Evaluate

6 For :

a Rewrite as improper fractions.

b Using part (a), rewrite the improper fractions with the same denominator.

c Simplify the expression.

Let’s practice

7 Calculate the following and express in simplest form:

8 Calculate the following and express in simplest form:

9 Jamie solved the following subtraction problem incorrectly:

a Identify the error in Jamie’s calculation.

b Correct the error.

c Explain a strategy that could help Jamie get the correct answer.

10 Which value is closest to the sum of and ?

11 Calculate the following and express in simplest form:

12 Find the value and express in simplest form:

13 Emma adds the mixed numbers and . Show the correct ﬁnal answer and explain the steps to help Emma understand the process.

14 Which value is closest to the difference between and ?

15 What is the closest value to the difference between and ?

16 At a party, Bill makes fruit punch by combining liters of water with liters of juice concentrate. Find the total amount of liters in the punch.

17 During a blizzard, it snowed 10 cm. When the sun came out the next day, cm of snow melted. How much snow was left on the ground?

18 Katrina and Luigi ordered a pizza to share. Katrina ate of the pizza while Luigi ate

a Use a model to estimate what fraction of the pizza Katrina and Luigi ate altogether.

b What fraction of the pizza did they actually eat altogether?

c What fraction of pizza is left?

d Justify your solutions using words.

19 Alex threw a football yards. Morgan threw the football yards. Which statement is true?

A Alex threw the football yards farther than Morgan.

B Morgan threw the football yards farther than Alex.

C Alex threw the football yards farther than Morgan.

D Morgan threw the football yards farther than Alex.

Let’s extend our thinking

20 Skye wants to decorate her window sill with fairy lights. The window sill is ft wide. If she only has ft of fairy lights, how much of the window sill will not be decorated?

a Estimate your solution with a model.

b Determine how much of the window sill won’t be decorated.

21 A rectangular cake measures 9 cm by 8 cm. You cut a piece that measures 4 cm by 3 cm. What fraction of the cake is left after cutting this piece? Explain your reasoning using a diagram.

22 Danielle is studying for three tests. She studies Science for hours, English for of an hour and Visual Arts for of an hour. How many hours does she spend studying? Justify your solution using a model.

23 For :

a Calculate the answer by ﬁrst converting both mixed numbers to improper fractions.

b Calculate the answer by ﬁrst subtracting the whole numbers and then subtracting the fraction parts.

c Which method do you prefer? Explain.

Interactive exploration

Explore online to answer the questions

Use the interactive exploration in 1.03 to answer these questions.

1. What is happening when you slide the slider? How does this represent the multiplication?

2. What does checking the ‘Arrange’ checkbox do? How might this help with understanding the result?

3. Use the applet with a few more multiplication problems. What patterns do you notice between the numbers being multiplied and the result?

4. What can you say about multiplying a whole number by a fraction between 0 and 1?

When a whole number is multiplied by a fraction between 0 and 1, the result is smaller than the original whole number.

Example 1

Each rectangle represents 1 whole.

a Write the product that is represented by the model.

Create a strategy

Look at the shading in each rectangle to ﬁgure out what fraction is shaded. Multiply the shaded fraction of a rectangle by the total number of large rectangles to ﬁnd the product represented by the model.

Apply the idea

In each rectangle, there are 6 equal parts and 4 of them are shaded. We could say “4 out of 6 parts of each rectangle are shaded, of each rectangle is shaded. This simpliﬁes to . There are 3 rectangles, so the product represented by the model is:

3 ⋅ or 3 ⋅

b Evaluate the product using the model.

Apply the idea

Rewrite the whole number as a fraction

Multiply the numerators and denominators

Divide

Another way to evaluate this is we divide out the common factors in the numerator and denominator:

Rewrite the whole number as a fraction

Divide out the common factors

Simplify

So, the product represented by the model is 2.

Reﬂect and check

We can further visualize the result by rearranging the shaded boxes into smaller rectangles and comparing them with two whole rectangles that have 6 parts each.

In the image, we can see that the 12 shaded boxes from the original three rectangles can be rearranged to ﬁll two whole rectangles with 6 parts each. There is one rectangle without any shaded parts. This shows that the shading in the model represents the product 2, as the shaded boxes are equivalent to two whole rectangles.

Example 2

Evaluate ⋅ 35

Create a strategy

Multiply numerators and denominators separately. The denominator of a whole number is always 1.

Apply the idea

Multiply numerators and denominators

Evaluate

Simplify

Idea summary

Finding a fraction of a quantity is the same as multiplying a fraction by a whole number.

To multiply a fraction by a whole number, multiply the numerator by the whole number.

When multiplying any whole number by a fraction between 0 and 1 the result is smaller than the original whole number.

Multiply fractions by fractions

To multiply two fractions together, we’ll start by thinking of the fractions as multiples of unit fractions, and work towards a more eﬃcient strategy.

Let’s take an example of . We can rewrite these fractions as and

We can then multiply the whole parts together:

What can we do with the product of the unit fractions and ?

Well, this is like taking one whole, dividing it into 3 pieces to get thirds.

We can now ﬁnish our multiplication:

Then dividing each of those thirds into 5 pieces.

The result is that the whole has been divided into 15 pieces where we only want 1 piece.

This image represents the fraction

Do you notice the pattern that has happened here?

In a fraction, the denominator tells us the size of the pieces, and the numerator tells us how many pieces there are. When we multiply two fractions, the denominators multiply together to tell us the new size of the pieces, and the numerators also multiply together to tell us how many of the new pieces there are.

That is:

We can multiply with mixed numbers as well, because they’re really just fractions. We just have the added step of converting the mixed number to a fraction ﬁrst. Interactive exploration Explore online to answer the

Use the interactive exploration in 1.03 to answer these questions.

1. What is happening when you slide the slider? How does this represent the multiplication?

2. Use the applet with a few more multiplication problems. What patterns do you notice between the numbers being multiplied and the result?

3. What can you say about multiplying a fraction by a fraction between 0 and 1?

When a fraction is multiplied by a fraction between 0 and 1, the result is smaller than the original fraction.

Example 3

Demonstrate how to multiply and using a number line. Then ﬁnd the product.

Create a strategy

To multiply and using a number line, start by drawing a number line from 0 to . Then, divide each part between 0 and into 3 equal parts. Finally, shade 2 of the 3 parts in each of the sections.

Apply the idea

Draw a number line from 0 to

Divide each part between 0 and into 3 equal parts.

Shade 2 of the 3 parts in each of the sections.

Since 10 pieces are shaded and 12 pieces make up one whole,

Example 4

Find the value of the following: a

Create a strategy

Multiply the numerators and denominators together.

Apply the idea

Multiply the numerators and denominators

Evaluate

Reﬂect and check

We can use an area model to verify our answer. For an area model, we can think of the product as the area of a rectangle with a width of and a length of .

The ﬁrst row of the rectangle represents of its total area and is shaded blue. We further divide each of these thirds into 10 equal pieces, resulting in a total of 30 small rectangles. We then shade 7 columns, each containing three small rectangles, in red. The overlapping purple-shaded area represents the product of and There are 7 purple small rectangles out of 30 small rectangles, which shows that the product of and is

Create a strategy

Multiply numerators and denominators separately.

Apply the idea

Multiply numerators and denominators

Evaluate

Simplify

Apply the idea

Rewrite mixed numbers as improper fractions

Simplify

Evaluate

Reﬂect and check

We can use estimation as a quick way to check if our answer is reasonable. Start by rounding the mixed numbers to numbers that are more familiar and easier to work with. is close to because is close to Next, we can approximate as 5 because it is just less than 5.

For our estimation we are calculating: .

We can easily ﬁnd that 5 ⋅ 5 = 25 and so 5 ⋅ 5 = 25 + 2.5 = 27.5

This is close to our actual answer of 26 so we know our answer is reasonable. However, our estimation is a little too high. This is because we rounded up to 5.

Example 5

Danielle takes minutes to drive from her home to the local shopping center. She spends of this time waiting at traﬃc lights.

Find the number of minutes she spends waiting.

Create a strategy Multiply the given numbers.

Apply the idea

Multiply by

Rewrite as improper fraction

Evaluate

Idea summary

To multiply two fractions, multiply the numerators and the denominators separately.

To multiply mixed numbers by a fraction or by another mixed number, convert the mixed number to an improper fraction ﬁrst. Then, multiply the numerators and denominators separately.

When multiplying any number by a fraction between 0 and 1 the result is smaller than the original number.

Practice

What do you remember?

1 Each circle represents 1 whole. Which product is best represented by the shading shown on this model?

2 Find the value of:

3 What is a reciprocal?

A A fraction where the numerator and denominator are the same.

B A fraction that represents the square root of the original fraction.

C A fraction obtained by interchanging the numerator and the denominator of the original fraction.

D A fraction that represents the sum of the numerator and denominator of the original fraction.

4 Complete the statement:

We can multiply two fractions by multiplying the ﬁrst numerator by the second ⬚ and the ﬁrst ⬚ by the second denominator.

5 Which expression is best represented by this model?

6 The rectangle of blocks represents what happens when is multiplied by

Use the blocks to evaluate

7 The rectangle of blocks represents what happens when is multiplied by .

Use the blocks to evaluate

Let’s practice

8 For each value, state whether the product is greater than, less than, or equivalent to the whole number. a b c d

9 The area model shows . Use the model to ﬁnd the product of

10 The model shows . Use the model to ﬁnd the product of

11 a Demonstrate how to multiply and using the fraction bars. Then ﬁnd the product.

b Is the product of and greater than or less than each of the original fractions?

12 a Demonstrate how to multiply and using a number line. Then ﬁnd the product.

b Is the product of and greater than or less than ?

Demonstrate how to multiply and using pattern blocks. Then ﬁnd the product. = 1 whole

14 In a math problem, Trixie tries to multiply by using a number line. She draws a number line from 0 to 2 and marks the point on it. Then, she divides the segment from 0 to into 3 equal parts and takes 2 of these parts to represent . Based on this method, Trixie concludes that .

Identify and explain the mistake in Trixie’s method. Then, correctly use a number line to ﬁnd the product.

15 Evaluate and express your answer in simplest form:

16 Evaluate and express your answer in simplest form:

18 Evaluate and express your answer in simplest form:

19 Evaluate and express your answer in simplest form:

20 Evaluate and express your answer in simplest form:

Let’s extend our thinking

21 Consider the expression

a Without calculating, determine if the product will be greater than or less than . Explain your reasoning.

b Will the product be greater than or less than ? Explain your reasoning.

22 Find a fraction that, when multiplied by , results in a product that is less than . What conclusion can you draw about multiplying two proper fractions based off your investigation? Explain your thinking.

23 Stephen is years old. He has spent of his life living in Sydney. How many years has he spent living in Sydney?

24 In Wings Station, one batch of chicken wings requires cups of ﬂour. If the Wings Station is making batches this Sunday, how much ﬂour will they need? Draw a picture to justify your thinking.

25 Georgia and Chenny need to follow a recipe which requires of a cup of ﬂour. However, they only want half of the quantity that the recipe is for. Georgia thinks they should divide , but Chenny thinks they should multiply . Who is correct? Use a number line to model your thinking.

26 Every day of the week, Mohamad walks to his school which is of a mile away from his house. On Thursday, he is of the way to school when he realizes that he forgot his English textbook at home and turns back to get it. Explain how to ﬁnd how far Mohamad has traveled in miles when he turns back.

27 Jenna is asked to multiply and she answered right away by canceling the 5s in the numerator and denominator. Explain why this method can be used.

Consider . We cannot ask, “What is 6 divided into groups of the same size” because that does not make sense. Instead, we can say, “How many groups of are in 6?”

2 1 0 3 4 5 6

To answer this question, we can begin by dividing the 6 wholes into thirds. Then, we can make groups of 2 thirds.

If we count, we can see there are 9 equal groups of in 6.

This shows 6 ÷ = 9.

Again, instead of dividing by a fraction, we can multiply by the reciprocal of the fraction:

Interactive exploration

Explore online to answer the questions

Use the interactive exploration in 1.04 to answer these questions.

1. Explain how the model is showing the division.

2. How does the model relate to the final fraction?

3. What is the relationship between the numbers in the original division expression and the numbers in the final fraction?

4. What do you notice about dividing a whole number by a fraction between 0 and 1?

When a whole number is divided by a fraction between 0 and 1, the result is larger than the original whole number.

Example 1

The number line below shows 4 wholes split into sized parts.

a Use the model to evaluate .

Create a strategy

To evaluate , we can use the number line model to count the number of one-third sized parts that make up the whole sections.

Apply the idea

There are 4 whole sections, and each section is divided into 3 parts . There are 12 parts of size in the diagram so 4 ÷ = 12.

b If 4 is divided into parts that are of a whole each, how many parts are there in total?

Create a strategy

We can rewrite the division as multiplication by the reciprocal and solve for the total number of parts. The reciprocal of is or 3.

Apply the idea

Divide the whole number by the unit fraction Multiply by the reciprocal Evaluate

Example 2

Rewrite using multiplication.

Create a strategy

To rewrite the expression using multiplication, remember that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of is

Idea summary

Apply the idea

Multiply by the reciprocal

Dividing a whole number by a fraction is the same as multiplying the whole number by the reciprocal of that fraction.

Divide fractions by whole numbers

When we divided a whole number by a fraction, such as 2 ÷ , we asked the question “how many parts of size ﬁt into 2 wholes?”

Dividing a fraction by a whole number is the reverse of this. Let’s look at ÷ 2 as an example:

We start with of a whole, shown as the shaded area in the image.

We then divide each of these thirds into 2 parts.

How big is the remaining shaded area? Well, there are now 6 parts of equal area and 1 of them is shaded, so this is equal to of the whole.

b What is the size of the part created when is divided by 4?

Create a strategy

We can also count the number of spaces between 0 and 1 on the number line.

Apply the idea 0 1

We can see that there are 12 spaces, so each space (or part) has a size of

Reﬂect and check

We can also divide the unit fraction by the whole number.

Divide the unit fraction by the whole number

Multiply the denominator by the whole number

Evaluate

Example 4

Evaluate the following:

Create a strategy

Rewrite the whole number as a fraction. Then multiply the ﬁrst fraction by the reciprocal of the second fraction.

Apply the idea

Rewrite the whole number as fraction

Multiply by the reciprocal of

Multiply the numerators and denominators

Create a strategy

Convert the mixed number into an improper and the whole number into a fraction. Then rewrite as multiplication by the reciprocal.

Apply the idea

First, convert the mixed number into an improper fraction.

Rewrite the mixed fraction

Evaluate the multiplication

Evaluate

Then convert the whole number 4 into a fraction.

Rewrite the whole number

Rewrite the division as multiplication by the reciprocal.

Divide improper fractions

Multiply by the reciprocal

Multiply the numerators and denominators

Evaluate

Simplify

Idea summary

Dividing a fraction by a whole number is the same as multiplying the fraction by the reciprocal of the whole number.

The reciprocal of a whole number is:

What do you remember?

1 This rectangle is broken into two halves:

a Which rectangle shows that each half has been divided into 3 parts? A B

b Find the size of one piece created when is divided by 3.

2 This rectangle shows 1 whole split into 4 parts of size :

a Which rectangle shows that each quarter has been divided into 3 parts?

b Find the size of the piece created when is divided by 3.

3 This picture represents 6 pizzas.

a Consider the squares shown:

i Divide 10 into 2 groups of equal size.

ii Find the size of each group when 10 is divided by 2.

b Consider the fraction bars shown:

i Divide 1 whole into 2 groups of equal size.

ii Find the size of each group when 1 is divided by 2.

iii Divide into 2 groups of equal size.

iv Find the size of each group when is divided by 2.

5 Rewrite each expression using multiplication:

6 Use the given models to evaluate the expressions.

7 Consider the expression

a Use a model to show 2 wholes split into size pieces.

b How many pieces of size are there in total?

c Is the quotient of 2 ÷ greater than, less than or equivalent to the original whole number?

d Find the number of sized groups in 2 wholes.

e Is the quotient of 2 ÷ greater than, less than or equivalent to the original whole number?

8 Consider the expression

a Use the model to show wholes split into 2 groups of equal size.

b How many pieces of size are in each group?

9 Which expression results in a quotient that is larger than 5?

10 Use the number line to ﬁnd the value of each expression. Express your answer in simplest form.

11 Use the fraction bars to ﬁnd the value of each expression. Express your answer in simplest form.

12 The fraction bar shows divided into 2 equal groups. What fraction of the whole strip makes up one of those groups?

The fraction bar shows 3 wholes, and is shaded.

a Divide the fraction bar into 4 groups of equal size.

b Use your model from part (a) to evaluate

a Use the pattern blocks to show how to divide 3 by , and write the quotient.

b Use the pattern blocks to show how to divide by 3, and write the quotient.

15 Divide each expression. Express your answer in simplest form.

16 Divide each expression. Express your answer in simplest form.

Divide each expression, giving your answer as a mixed number:

Let’s extend our thinking

18 Jordan has gallons of paint. Each wall requires 2 gallons of paint. How many walls can Jordan paint with the available paint? Use a model to solve.

19 A container holds liters of juice. How many containers can be ﬁlled with 18 liters of juice?

20 Sophia is preparing gift bags for a party. She has 48 chocolate bars and wants to distribute them equally among gift bags. If each gift bag is supposed to contain chocolate bars, how many gift bags can Sophia prepare? Use a model to explain your thinking.

21 When 24 is divided by a number, the quotient is 21. Explain how to ﬁnd the number.

22 Mr. Marge gives the following expressions to the class:

Which expression should result in the larger number? Explain your thinking.

23 A bag contains 45 cups of dog food. Robin’s dog eats cups of dog food each day.

a How many days does the bag of dog food last? Explain your answer.

b How many cups of dog food does Robin’s dog eat in months?

c If a bag of dog food costs $12, how much should she pay for dog food that will last for month?

1.05 Divide fractions and mixed numbers

After this lesson, you will be able to...

• divide fractions.

• divide mixed numbers.

• explain the effect of dividing a fraction by a number between 0 and 1.

Divide fractions and mixed numbers

We’ve divided fractions and whole numbers, but what does it look like to divide a fraction by another fraction?

Let’s look at

Let’s start by asking, “How many groups of make up

First we can draw a diagram of .

Then we can split it into groups of

We can see that there are 4 equal groups of in . So

We can also apply the method of rewriting division as multiplication by the reciprocal like we did when dividing with whole numbers.

Multiply by the reciprocal of

Multiply the numerators together and denominators together

Evaluate the multiplication

Divide

Both methods give us the same result.

We can apply the same process to dividing with mixed numbers, because remember a mixed number is just a different form of a fraction. We just have to convert the mixed number into an improper fraction ﬁrst.

Interactive exploration

Explore online to answer the questions

Use the interactive exploration in 1.05 to answer these questions.

1. Explain how the model is showing the division.

2. How does the model relate to the final fraction?

3. What is the relationship between the numbers in the original division expression and the numbers in the final fraction?

4. What do you notice about dividing a fraction or mixed number by a fraction between 0 and 1?

When a fraction or mixed number is divided by a fraction between 0 and 1, the result is larger than the original fraction or mixed number.

Evaluate each expression.

Create a strategy

Divide fractions by multiplying the 1st fraction by the reciprocal of the 2nd.

Apply the idea

Multiply by the reciprocal

Multiply numerators and denominators

Evaluate

Reﬂect and check

We can use an area model to verify the answer:

Overlap the models and count the number of shaded parts in the row and column.

The column has 5 shaded parts and the row has 8 shaded parts. So, the area model shows

Example 1

Create a strategy

Divide fractions by multiplying the 1st fraction by the reciprocal of the 2nd.

Apply the idea

Multiply by the reciprocal

Multiply numerators and denominators

Evaluate

Simplify

Example 2

Evaluate and write your answer in its simplest form.

Create a strategy

Rewrite the mixed numbers as improper fractions, then divide the fractions.

Apply the idea

Rewrite as improper fractions

Multiply by the reciprocal of

Simplify

Multiply numerators and denominators

Create a strategy

Convert the mixed number into an improper fraction, then rewrite as multiplication.

Apply the idea

First, convert the mixed number into its improper fraction form.

Rewrite the mixed number

Evaluate the multiplication

Evaluate the addition

Then rewrite the division as multiplication.

Divide improper fractions

Multiply by the reciprocal

Multiply the numerators and denominators

Evaluate

Example 3

A meter long roll of fabric is to be cut into sections of equal length . How many pieces of fabric will there be?

Create a strategy

Rewrite the mixed numbers as improper fractions then perform division.

Apply the idea

Rewrite as improper fractions

Multiply by the reciprocal of

Write as a single fraction

Divide out the common factor

Evaluate the division

There will be 3 pieces of fabric.

Idea summary

Reﬂect and check

To check the reasonableness of our answer, let’s use estimation. We can round the length of the fabric roll to 8 meters, and the sections we’re cutting to 3 meters long each. Dividing these, we get , which is just less than 3 because . This estimation shows that 3 is a reasonable answer.

To divide one fraction by another, multiply the ﬁrst fraction by the reciprocal of the second.

Practice

What do you remember?

1 Is each statement true or false?

a The reciprocal of a whole number is the whole number over 1.

b The reciprocal of a fraction can be found by swapping the numerator and denominator

c To divide a whole number by a fraction, multiply the whole number by the reciprocal of the fraction.

d To divide one fraction by another, multiply the second fraction by the reciprocal of the ﬁrst.

2 Rewrite each expression using multiplication:

a b c d

3 This picture represents 5 chocolate bars. Exactly how many are in 5?

4 Use the fraction bars to evaluate

How many are in ?

5 The area models show and . Each model has been divided further to create equal sized pieces.

Use the model to divide . Fill in the blanks:

In , there are ⬚ groups of and of another group. Therefore,

Let’s practice

6 Consider the problem

a Use the circle to represent

b Split the model you made in part (a) into groups of

c Use the model to divide by .

d Is the quotient greater than, less than, or equivalent to the ﬁrst fraction?

7 Consider the problem

a Use the fraction bars to represent

b Use the fraction bars to represent .

c Use the model to divide by 1

8 Use the number lines to divide each expression. Express your answer in simplest form.

9 Use the area models to divide each expression. Express your answer in simplest form.

10 a Evaluate each of the expressions:

b Use your answers to select the true statement:

A When dividing any number by a proper fraction, the quotient is always larger than the original number.

B When dividing any number by a proper fraction, the quotient is sometimes larger than the original number.

C When dividing any number by a proper fraction, the quotient is never larger than the original number.

Divide each expression. Express your answer in simplest form.

Divide each expression, giving your answer as a proper fraction or an improper fraction:

Divide each expression, giving your answer as a mixed number:

When is divided by a number, the quotient is . Explain how to ﬁnd the number.

17 Athina thinks that the quotient of is less than the quotient of . Mikki thinks the quotient of both expressions will be equivalent. Who is correct? Explain your thinking.

18 Consider the expression

Without evaluating, will the quotient be greater than, less than, or equivalent to ? Explain your thinking.

19 Evaluate

20 Benjamin wants to engrave his name on a metal plate. There is cm space on the plate.

There are three sizes of letters to choose from: cm wide, cm wide and cm wide.

a Find the maximum number of letters that Benjamin can engrave on the plate using the following sizes: i cm wide ii cm wide iii cm wide

b Which letter sizes could he use for his name?

1.06 Solve problems with fraction operations

After this lesson, you will be able to...

• estimate solutions to real-world problems involving fraction operations.

• solve real-world problems involving fraction operations.

• justify solutions to real-world problems involving fraction operations with clear explanations.

Solve problems with fraction operations

We use fractions to solve many everyday problems. For example, in recipes, ingredients are often measured in fractions of a cup. If we wanted to know the total volume of the ingredients, we could use fraction addition

We can use keywords to help us work out which operation we need to use to solve the problem. Here are the four operations and some common keywords that relate to them:

Addition Subtraction Multiplication Division more less product equally shared add subtract by in each all together how many left times per total difference groups of divided by

Estimation can be a useful strategy for solving real-world problems, especially if the context of the problem doesn’t require us to be exact.

Example 1

At a party, Bill makes a drink by combining of water with juice concentrate.

What is the total amount of the drink?

Create a strategy

Identify the keyword in the story. The word “total” tells us we need to add the amounts for each part of the drink.

Apply the idea

Add the values

Split the mixed numbers into whole and fraction parts

Add the whole parts

Multiply for a common denominator

Evaluate the multiplication

Add the numerators over the common denominator

Evaluate the addition

Rewrite as a mixed number

Reﬂect and check

We could use estimation to see if our exact answer seems reasonable. For example, is close to , and if we add we get 7. So our answer should be a little less than 7, which it is.

Example 2

Jack is making bags for his friends. He has of fabric.

If each bag requires of fabric, how many bags can he make?

Express your answer as an improper fraction.

Create a strategy

Identify the keyword in the story. The word “each” tells us we need to divide the length of fabric into equal sized pieces for each bag.

Apply the idea

Divide the values

Rewrite as multiplication using the reciprocal

Multiply the numerators and denominators separately

Simplify

Example 3

Jamal has of ribbon. After using some for a project, he has left. How much ribbon did he use?

Create a strategy

To ﬁnd out how much ribbon Jamal used, we need to subtract the amount of ribbon he has left from the total amount he started with. This means we will subtract two fractions:

Apply the idea

First, we ﬁnd a common denominator for the two fractions. We can do this by multiplying the denominators together which gives us 6 ⋅ 5 = 30.

Multiply for common denominator

Evaluate the multiplication

Subtract the numerators

Jamal used of ribbon for his project.

Idea summary

Use keywords to help you identify which operation to use: Addition Subtraction Multiplication Division more less product equally shared add subtract by in each all together how many left times per total difference groups of divided by

Practice

What do you remember?

1 Evaluate and simplify:

2 Determine the operation required to solve each contextual problem.

a Oana walks of a mile and runs of a mile to get to school. How many miles has Oana traveled?

b Webster felt dehydrated. He drank one third of a cup. What fraction of a cup does Webster have left?

c Shea has pounds of beads. She shares the beads evenly between 3 friends. How many pounds of beads will each friend receive?

d In a class with 40 students, of the class are boys. How many boys are in the class?

3 A chemistry experiment requires cup of distilled water, but Trace only has cup. Write an expression to represent how much more distilled water Trace needs for the experiment.

4 A food market is open for hours in the morning and hours in the evening. Write an expression to represent the number of hours the food market is open altogether.

5 Selena has gallons of paint. She plans to use of the paint for an art project and the rest for a sciene project. Write an expression to represent the amount of paint Selena plans to use for the science project.

6 A recipe requires of a cup of ﬂour for one batch of cookies. How much ﬂour is needed to make 8 batches of cookies?

A cups of flour B cups of flour C cups of flour D cups of flour

7 A glass holds of a cup of lemonade. If you have 3 cups of lemonade, about how many glasses can be ﬁlled with lemonade?

A 3 glasses B 5 glasses C 2 glasses D 6 glasses

8 Melinda has of a quart of milk in her fridge. After drinking of a quart of milk, how much milk is left in Melinda’s fridge? Express your answer in simplest form.

Let’s practice

9 For each scenario: i Estimate the solution.

ii Find the exact solution in simplest form.

a Eboni has m of fabric. After she cuts off some fabric for a dress, she has m left.

About how much fabric did she use for the dress?

b A bag of oranges weighing lb was divided equally among a group of children. What was the mass of the oranges given to the group?

c For a school bake sale, Farouk prepares a chocolate mix by mixing pounds of chocolate powder with pounds of sugar. Find the total amount of pounds in the mix.

d Helena takes minutes to drive from her home to the local shopping center. She spends of this time waiting at traﬃc lights. Find the number of minutes she spends waiting.

10 A casual cyclist travels at a speed of miles per hour uphill. The uphill route is miles long. Find the total hours the cyclist took to cycle the uphill route.

11 Mayumi works as an accountant and is at the oﬃce for hours each day. She spends hours eating lunch with her coworkers. How much time does she spend actually working in one day?

A hours B hours C hours D 8 hours

12 Paolo is starting his jewelry business. He bought bags of beads. If a single bag weighs ounces, what is the total combined weight of the beads Paolo bought?

13 In a school debate, each speaker was allowed minutes. How many speakers participated if the debate went for minutes?

14 Xia is making bags for her friends. She has yards of fabric. If each bag requires yards of fabric, how many bags can she make?

15 If a hiker has quarts of water and drinks quarts during a break, how much water is left in the hiker’s bottle?

16 of a cake was left after a party. Bella ate of what was left. What fraction of the whole cake did Bella eat?

17 A recipe calls for cup of sugar, but Wei only has cup of sugar. How much more sugar does Wei need to complete the recipe? Is your answer reasonable? Justify your solution.

18 At the beginning of the day, Qing’s 32 oz water bottle was full. She drank some of the water before lunch, which left the bottle only full. After lunch, she drank of what was left, then reﬁlled the bottle.

a How many ounces of water did Qing drink before lunch?

b How many ounces of water did she need to add to the bottle to reﬁll it after lunch?

19 During a school science fair, Quentin needs to complete two experiments. The ﬁrst experiment requires 24 minutes to set up, and the second experiment takes minutes. If Quentin manages to reduce the setup time for each experiment by minutes to save time for analysis, calculate the new total setup time for both experiments.

20 A baker has 960 grams of flour. of the flour is needed to make cookie dough. If times the amount of flour needed for the dough is needed for frosting and decorations, does the baker have enough flour for both the cookie dough and the frosting and decorations?

Explain your answer.

21 Ursula tutors students in mathematics for hours each weekday. On Wednesdays, due to additional commitments, she tutors for only a third of her usual weekday time.

a Calculate the total tutoring hours for Ursula from Monday to Friday, considering her reduced hours on Wednesday.

b If Ursula were to cut her hours in half on Mondays, how many hours would she work in one week?

22 Ayako is organizing a 50-minute workout session. The session includes two types of exercises: one lasts minutes and the other lasts minutes.

a Explain how to ﬁnd the total duration for a routine consisting of three sets of the ﬁrst exercise and four sets of the second exercise.

b If the remainder of the time is spent on rest breaks or water breaks, how much time is spent resting or drinking water?

Let’s extend our thinking

23 Xander is planning a fundraising event for his school. He sold of the tickets he had and then bought an additional of the original number of tickets. If the initial number of tickets was 120:

a Find the total number of tickets sold.

b Find the number of tickets he bought.

c After selling and buying tickets, he realized he needed more, so he bought of the current total tickets. How many tickets does he have now?

24 For a project, Akram needs to cut a metal rod so that the longer piece is three times the length of the shorter piece. If the rod is yards long, explain how to ﬁnd the length of the shorter piece.

25 In Fady’s Bakery, one recipe of blueberry muﬃns serves 4 people and requires cups of sugar. If Fady is preparing to serve 22 people for a dinner party, how much sugar will he need?

Myoung is planning a study session for her ﬁnals. The session is divided between two subjects: the ﬁrst requires minutes per review cycle, and the second needs minutes per review cycle.

Myoung does three review cycles for the ﬁrst subject, takes a hour break, does two review cycles for the second subject, then takes a practice test for hours. Calculate the overall duration of her study session including the break and the test.

27 Zainab explains that if each juice box contains cup of juice, then a pack of 6 juice boxes will contain times that amount of juice. She multiplies cup by to ﬁnd the total amount of juice in the pack.

Dwight disagrees and says that 6 juice boxes will contain 6 times that amount of juice. He multiplies cup by 6 to ﬁnd the total amount of juice.

Who is correct? Justify your answer.

28 Create a contextual problem with both multiplication and division, or both addition and subtraction. Use fractions for the quantities.

Fractions, Decimals, & Percents 2

•

2.01 Percents as fractions

After this lesson, you will be able to...

• estimate and write percents from models.

• convert percents to fractions.

• convert fractions to percents.

Percents as fractions

Percent means “one part out of every 100, “per 100” or how many “out of 100”. In other words, 1% is equal to one hundredth or .

Percents can be represented with a 10 × 10 grid which has 100 total squares. Each square represents 1% or 1 square out of 100 total squares.

percent of squares = number of shaded squares%

So, 6 shaded squares represents 6% of the grid.

fraction of squares =

So, the six shaded blue squares represent .

Percents can represent amounts that are less than a whole or greater than a whole, with 100% representing one whole.

• Percents less than a whole will be smaller than 100% and represent part of a whole. They can also be represented by fractions less than 1.

• Percents greater than a whole will be larger than 100% and represent a quantitiy greater than the original. They can also be represented by fractions greater than 1.

Estimating percents can be helpful for quick calculations or when precise measurements are not necessary. When estimating a percent, it is helpful to be comfortable with calculating the benchmark percents of 0%, 25%, 50%, and 100%.

Double number lines can be used with benchmark percents to represent both percents less than 100% and greater than 100%. On this number line, 100% of the quantity is 24 minutes. And we can easily see other percents by looking at the values that line up.

Benchmark percents and their fraction equivalents will be helpful in converting between percents and fractions as well as estimating percents.

To convert any percent to a fraction, remember percent means “per 100”. Create a fraction with the percent quantiity as the numerator and 100 as the denominator. This fraction may be reduced.

To convert any fraction to a percent, there are two methods.

For the ﬁrst method, create an equivalent fraction with 100 as the denominator. The numerator is the percent.

For the second method, divide the numerator by the denominator, and then multiply by 100%.

Example 1

Consider the grid shown.

a How many squares are shaded?

Create a strategy

Count the shaded squares in the row and the number of rows.

Apply the idea

There are 5 rows of 10 squares shaded so there are 50 squares shaded. 5 10 = 50

b Estimate the percent represented.

Create a strategy

Convert the fraction to an approximate percent by considering the benchmark percent that is closest to our fraction.

Apply the idea

Given that our fraction that best represented the model was , we know the 4 whole can be written as 400%.

Visually looking at the last circle representing , this is larger than 25% = and smaller than 50% = . Since the shaded part is near the middle of the two benchmarks, we can estimate this as about 38%. Therefore, the shaded portion is approximately 438%.

Reﬂect and check

When estimating, some variation is expected. Any estimates between 35% and 45% would be appropriate answers.

c What exact percent of one whole circle is shaded?

Create a strategy

Calculate the exact percent by dividing the number of shaded parts by the total parts in one whole circle and multiplying by 100%.

Apply the idea

Number of shaded parts: 22

Total parts: 25

Divide the number of shaded parts by the total parts in one circle

100% Multiply 4.4 by 100% = 440% Evaluate

Reﬂect and check

Our exact percent is very close to our approximation. Consider ways to make an even better estimate.

Write 24% as a fraction.

Create a strategy

To convert the percent into fraction, rewrite the percent as a fraction out of 100.

Apply the idea

Write the percent as a numerator over the denominator 100

Example 3

Reﬂect and check

To simplify the fraction , we ﬁnd the greatest common divisor (GCD) of 24 and 100, which is 4.

Dividing both the numerator and the denominator by 4, we can simplify our fraction:

We can model these fractions on a grid where the denominator is the total number of squares.

On a 10 by 10 grid representing 100%, shading 24 squares shows the original percentage.

We can create a 5 by 5 grid to represent the simpliﬁed fraction

Each square in this grid represents 4 squares from the 10 × 10 grid. Therefore, shading 6 squares corresponds to the same proportion of the whole.

This visualization helps conﬁrm the equivalence of 24% to . Fractions can be simpliﬁed while representing the same value or percent of the whole.

Example 4

Write as a percent.

Create a strategy

We need to ﬁnd an equivalent fraction with the denominator equal to 100. What number can we multiply 5 by to get 100? Then be sure to multiply the numerator and denominator by the same factor.

Apply the idea

What can we multiply 5 by to get a denominator of 100?

Multiply both the numerator and the denominator by 20

Evaluate

40 for every 100 is 40%

Reﬂect and check

When we are unsure how to get a denominator of 100 directly, we can divide 100 by the current denominator to ﬁnd the multiplying factor.

100 ÷ 5 = 20

So, both numerator and denominator should be multiplied by 20 to create an equivalent fraction with a denominator of 100. This method is especially helpful for converting fractions where the denominator is not a simple factor of 100.

Example 5

Write 250% as a mixed number in its simplest form.

Create a strategy

To write a percent as a fraction divide by 100. Then convert the improper fraction into a mixed number.

Apply the idea

Divide by 100

Simplify the fraction

Convert to a mixed number

Reﬂect and check

A double number line is a powerful tool for understanding the conversion between percents and other amounts. Let’s draw a double number line to represent this relationship:

The top number line represents the mixed number amounts. The bottom line of the double number line represents those amounts as percents. This model clearly shows that 250% aligns with , which matches our calculation.

Idea summary

Percent means “per 100” or how many “out of 100”. 1% is equal to one hundredth.

We can convert any percent into a fraction by writing the percent value as the numerator and 100 as the denominator.

We can convert any fraction into a percent by ﬁnding its equivalent fraction that has a denominator of 100. After this, we can write the value in the numerator followed by the % symbol to represent the percent.

What do you remember?

1 What is the meaning of the % symbol when placed after a number?

2 Fill in the blanks to make each statement true:

a 29% is equivalent to 29 out of ⬚.

c ⬚% is equivalent to 73 out of 100.

e 465% is equivalent to 465 out of ⬚

b 53% is equivalent to ⬚ out of 100.

d 150% is equivalent to ⬚ out of 100.

f 0.75% is equivalent to ⬚ out of 100.

3 Each grid is equal to one whole. For each of the grids shown:

i How many squares are shaded?

ii What percentage of the grid or grids is shaded?

iii What fraction of the grid or grids does this percentage represent?

4 15 out of the 25 class members were sick at home. Determine the percentage of students who were sick at home.

5 Out of 400 students, 397 lived locally. Determine the percentage of students who did not live locally.

6 Estimate the percentage represented by the shaded region on the number line.

7 Figure out the percentage represented by the shaded part on the number line.

8 Look at the picture to help you answer the question. A drought affects 8 out of 12 midwestern states. What percentage of the midwestern states are affected by the drought?

9 Mario completed of his workday.

Use the tape diagram to represent what percentage of the workday Mario has worked.

How does your model prove that your answer is correct?

10 For each model:

i Estimate the percent represented.

ii Determine what exact percentage is shaded.

a b c

11 For each model:

i Estimate the percent represented.

ii What exact percentage of one whole circle is shaded?

a b c

12 For each:

i Represent the percentage on the grid.

ii Which fraction in its simpliest form is equivalent to the percent?

a 65% b Michelle eats 40% of a cookie.

13 For each:

i Represent the fraction on the grid.

ii What percent is equivalent to the fraction?

a b Miguel watches of a movie.

14 Write each percentage as a fraction in its simplest form:

68%

94%

15 Write each percentage as a mixed number in its simplest form:

275%

109%

430%

16 Write each percentage as an improper fraction in its simplest form:

125%

120%

17 For each fraction:

175%

i Write an equivalent fraction with a denominator of 100.

ii Write it as a percentage.

18 Write the mixed numbers as percentages:

19 Write the improper fractions as percentages:

20 Represent 40% on the number line and ﬁnd its equivalent fraction.

21 Show 55% on the number line and determine the equivalent fraction for this percentage.

Let’s extend our thinking

22 Consider the fraction

a Explain if has an equivalent fraction with denominator of 100, and a whole number as its numerator.

b Find the closest whole percent to

23 A class has 32 students, and 12 of them are left-handed.

a How many of the students in the class are not left-handed?

b What percentage of the class is left-handed?

c What percentage of the class is not left-handed?

24 Jasper won a match in a table tennis competition. The winner of a match was the ﬁrst player to win 4 games and the winner of each game was the ﬁrst player to win 11 points. The scores were 11 7, 8 11, 11 5, 11 4, 2 11, 11 8.

a What fraction of the games did Jasper win?

b How many points were played in total?

c What fraction of the points did Jasper win?

d What percentage of the points did Jasper lose?

25 In a game, a player scored 24 out of the 30 points scored by her team. Express the points scored by the rest of her team as:

a A percentage b A simpliﬁed fraction

2.02 Percents as decimals

After this lesson, you will be able to...

• convert percents to decimals.

• convert decimals to percents.

Percents as decimals

Let’s use the same visual representations we used for comparing fractions and percents to explore the relationship between decimals and percents.

In a 10 × 10 grid, each box represents 0.01.

Since 14 boxes are shaded, this model represents 14% or 0.14.

We could also write as a fraction:

Double number lines can be helpful to model equivalencies between percents and decimals. Bechmark percents and decimals can make problem solving more eﬃcient.

We can convert between decimals and percentages by taking advantage of the hundredths place value. We know that 1% represents , or 1 hundredth, which we can write in decimal form as 0.01.

We can convert any percentage into a decimal by dividing the percentage value by 100, which is equivalent to decreasing the place value of each digit by two places, and removing the % symbol.

For example, 83% = which can be described as 83 hundredths. This is also 0.83 when written as a decimal.

To convert from a decimal into a percentage, we can just reverse the above steps. We can convert any decimal into a percentage by multiplying the decimal by 100, which is equivalent to increasing the place value of each digit by two places, and attaching a % symbol.

For example, 0.08 is 8 hundredths or = 8%.

A percentage is limited to representing hundredths, so smaller units like thousandths cannot be represented by whole number percentages such as 0.0035 which is 0.35%.

Remember to attach the % symbol to decimal at the same time as increasing the place values.

Example 1

Write 54% as a decimal.

Create a strategy

To convert a percentage as a decimal, we can ﬁrst think of 54% as 54 hundredths.

Apply the idea

54% is or 54 hundredths.

54 hundredths can be written as a decimal as 0.54

Reﬂect and check

Let’s use a 10 × 10 grid to illustrate converting 54% to decimal.

The grid represents a whole. Since there are 100 squares, each square represents 1% or 0.01.

Adding the 54 shaded squares that each represent 0.01 gives us 0.54, which conﬁrms our original conversion of 54% to a fraction.

Example 2

Write 0.314 as a percentage.

Create a strategy

To write a decimal as a percentage multiply by 100 and add the % symbol.

Apply the idea

0.314 100 = 31.4 Multiply by 100 = 31.4% Add the % symbol

Idea summary

We can convert any percentage into a decimal by dividing the percentage value by 100, which is equivalent to decreasing the place value of each digit by two places, and removing the % symbol.

We can convert any decimal into a percentage by multiplying the decimal by 100, which is equivalent to increasing the place value of each digit by two places, and attaching a % symbol.

What do you remember?

1 Draw a picture that represents 0.75. What percentage does your picture represent?

2 What is 10% as a decimal?

3 Plot each number on the number line.

Let’s practice

4 Use the models to convert each percent to an equivalent decimal.

5 Use the diagrams to help you represent an equivalent percent for each decimal.

6 Write each percentage as decimal.

7 Write each decimal as percentage.

8 Which decimal is equivalent to 63%?

9 Which percent is equivalent to 3.87?

10 Write the equivalent decimals for the percentages. Explain how you solved using pictures, numbers, and words.

136% = ⬚

11 Describe and correct the error in converting 0.87 to a percent.

Let’s extend our thinking

12 Plot each number on the number line. Determine any equivalancies.

13 Using this diagram, or otherwise, explain why

14 Explain why is it necessary to multiply by 100% when converting a decimal to a percentage.

15 Explain why is it necessary to divide by 100% when converting from percentage to a decimal.

16 For each, ﬁll in the boxes with the missing numbers:

17 Express each as a decimal, rounding your answer to three decimal places where necessary:

2.03 Convert between fractions, decimals, and percents

After this lesson, you will be able to... • convert between decimals, percents, fractions, and mixed numbers.

Convert between fractions, decimals, and percentages

We now know that decimals, fractions and percentages are just different ways of showing the same value:

1% can be written as or 1 hundredth or 0.01

Interactive exploration

Explore online to answer the questions mathspace.co

Use the interactive exploration in 2.03 to answer these questions.

1. What is the decimal and percentage equivalent of ?

2. A single column of small squares is what fraction of the larger square?

3. What do you notice when the ‘Show equivalent fraction’ checkbox appears?

There are some common conversions that we can remember to help us convert between percentages, fractions and decimals.

Example 1

Write the fraction, decimal, and percent that represents this illustration.

Create a strategy

First, count the number of parts in one whole circle. Then, determine how many parts are shaded. This will give the fraction. Next convert the fraction into a decimal and percent.

Apply the idea

To ﬁnd the fraction:

Number of parts in a whole = 8; Number of shaded parts = 27

Fraction =

To ﬁnd the decimal:

Decimal = 27 ÷ 8

= 3.375

To ﬁnd the percent:

Percent = 3.375 100%

= 337.5%

Example 2

Divide 27 by 8

Evaluate

Multiply the decimal by 100%

Evaluate

Convert between percentages, fractions and decimals to complete the table shown. Write the answers as mixed number percentages and simpliﬁed mixed numbers where necessary.

Create a strategy

To convert fractions and decimals to percentages multiply by 100%. To convert percentages to fractions or decimals, divide by 100.

Apply the idea

Write as a part out of 100 (Fraction)

Convert to a decimal

Multiply by 100

Evaluate

Attach percent symbol

Convert to a fraction

Multiply by 12.5 to make the denominator equal to 100

Evaluate

Convert to percent

Convert to a decimal

Idea summary

Decimals, fractions and percentages can be converted between each other using the relationship: 1% = = One Hundredth = 0.01

Practice

What do you remember?

1 Write as both a decimal and percent.

2 Write 0.65 as both a fraction and percent.

3 Is 4.4 equivalent to ? How do you know?

4 Which four answer choices represent the illustration?

Convert between fractions, decimals, and percents

8 Select all of the numbers represented by the tape diagram below.

9 Convert between percentages, fractions and decimals to complete the table:

10 Convert between percentages, fractions and decimals to complete the table:

11 Select each number that can be placed in the blank to make this statement true. = ⬚

12 Which statement is true?

13 In a charity walk, Alice completes 7 out of the 10 miles of the total event. Express the distance completed by the rest of the participants as:

a A simpliﬁed fraction b A decimal c A percentage Let’s extend our thinking

What is a real life scenario where we may have to convert between fractions, percents, or decimals? 15 Express these as a fraction and either a percent or a decimal:

16 A teacher has 9 pencils. Five of the pencils are red.

a What fraction represents the ratio of red pencils to the total number of pencils?

b What decimal repersents the ratio of red pencils to the total number of pencils?

c What percent represents the ratio of red pencils to the total number of pencils?

17 A group of friends decides to split the cost of a meal, equally. The ﬁrst friend pays of the total bill, while the second friend pays 0.25 of the bill. The third friend pays 37.5% of the bill. Is this a fair split? Explain your answer using pictures, numbers and/or words.

To avoid this, we can compare fractions, decimals and percents algebraically if we convert them to be in the same form. Begin by looking at the given list and deciding which form would be most helpful to make the comparisons.

You may think it makes the most sense to make comparisons as decimals, changing them to be 0.1, 0.6, and 0.5. While someone else might think it makes the most sense to convert them all to percentages, such as 10%, 60%, and 50%. Which form you choose doesn’t matter as long as it is the same for each number in the list.

Example 1

Fill in the ⬚ with < or > to make a correct comparison between the numbers.

a 3.5 ⬚ 35%

Create a strategy

Write the numbers in the same form to easily see which is larger.

Apply the idea

To convert decimal to percentage, we just multiply by 100%

3.5 ⋅ 100% = 350% Multiply by 100

350% > 35% Compare

This means that:

3.5 > 35%

Reﬂect and check

We could have converted the percentage to a decimal and compared the values. To convert a percentage to a decimal, divide by 100.

Divide by 100

Convert to a decimal

Comparing the decimal values, we still ﬁnd that 3.5 is greater than 0.35, conﬁrming our previous solution.

3.5 > 0.35

b 35% ⬚

Create a strategy

Write the numbers in the same form to easily see which is larger.

Apply the idea

To compare the numbers, we ﬁrst need to convert both values to the same form. We can convert both values to fractions with a denominator of 100.

Convert 35% to a fraction with a denominator of 100

Convert to a fraction with a denominator of 100

2.04 Compare and order fractions, decimals, and percents

Now that both values are in the same form, we can easily compare them:

Therefore, 35% < .

Reﬂect and check

We can also compare these numbers using benchmark values. Let’s compare each value to .

We know 35% is less than 50% =

We also know is larger than since 3 is more than half of 5.

Since 35% is less than and is more than , we know 35% <

c ⬚ 0.8

Create a strategy

Write the numbers in the same form to easily see which is larger.

Apply the idea

Multiply by to get a denominator of 100

Convert 0.8 to fraction

Convert to a fraction with a denominator of 100

Compare the two fractions

Therefore, < 0.8

Reﬂect and check

We can also compare these values using a benchmark by seeing how far each is from 1 whole.

We know is less than 1, and 0.8 is 0.2 less than 1.

Since = 0.4 > 0.2, we know that < 0.8.

Example 2

Arrange , 40% and 0.5 in descending order using percentages. a First, convert to a percentage.

Create a strategy

Convert to be a fraction out of 100.

Idea summary

When we need to compare a list of fractions, decimals and percents, converting them to be in the same form can make it easier to compare. Look at the given list and decide which form would be most helpful to make the comparisons in.

Practice

What do you remember?

1 Complete the following using the words ‘divide’ or ‘multiply’.

a To convert a decimal into a percentage you ⬚ by 100%.

b To convert a percentage into a decimal you ⬚ by 100%.

2 Match the following items with their equivalent benchmark fraction, decimal, or percent:

3 Express each decimals as a: i Percentage ii Fraction

4 Express each percentages as a: i Decimal ii Fraction

5 Consider the statement: 69% is less than 0.63 a Convert 0.63 to a percentage. b Now, state whether the statement is true or false.

6 Consider the statement: 1.63 is greater than 16.3% a Convert 1.63 to a percentage. b Now, state whether the statement is true or false.

Let’s practice

7 Consider the shown models:

a Name the percent in each model.

b Write which model represents a greater percent and justify your reasoning.

8 Place the fractions, decimals, and percentages on the number line in the correct location. Explain your thinking. , 1.25, 0.625, 1.2%

9 For each pairs of numbers state which number is greater.

a 0.91 and 82% b 8.9 and 879% c 0.31 and 45% d 0.39 and 26%

e 0.82 and 83% f 0.925 and 88%

Fill in the ⬚ with < or > to make a correct comparison between the numbers.

⬚

11 Determine whether each statements is true or false.

12 Write each list of numbers in ascending order.

, 20%, 1 b , 100%, 0.2

, 25%, 0.5 d , 25%, 1 e , 100%, 0.275 f , 20%, 0.8 g 0.893, , 53% h

13 Write each list of numbers in descending order. a , 70%, 0.8 b , 30%, 0.6 c , 100%, 0.5

, 60%, 0.3

14 Which of these numbers is between and 0.85 on a number line?

15 List the numbers in order from least to greatest.

16 Arrange the numbers in order from greatest to least. 25% 0.40

17 On her ﬁrst Mathematics test, Juliet scored 16 out of 25. On the second Mathematics test, she scored 27 out of 40. On her ﬁnal Mathematics test, Juliet scored 49 out of 50. Arrange her scores in ascending order.

18 Is greater than or less than 90%? Justify your reasoning using benchmarks.

19 Using the models provided, order the following numbers from largest to smallest: , 0.60, 45%. For each number, justify your answer with reference to the models.

Let’s extend our thinking

20 For each list of numbers, ﬁnd:

i The largest value

iii The value closest to 0.5

a 92%, , 0.1, 0.365, 60.1%

c 63%, , 0.6, 0.689, 65.5%

21 For each sets of numbers:

i Plot the numbers on a number line.

ii The smallest value

b 71%, , 0.7, 0.99, 50.8%

d 88%, , 0.9, 0.83, 81.4%

ii Arrange the set in ascending order.

a , 8% b

c 17.5%, , 0.225, 20%, 0.025

22 For each sets of numbers:

i Plot the numbers on a number line.

a 0.351, 0.5, , 25%

d 0.4, 0.175, 4.95%, 1.25%,

ii Arrange the set in descending order.

b 0.15, 0.555, 5%,

23 Joshua makes 75% of his shots, his sister makes of her shots, and his friend makes 0.65 of his shots. Who made the most shots?

24 Finn asked the class presidents of classes A and B how many students of their section will join the school fair.

The president of class A gave the number 0.4. The president of class B gave the number

a Explain how to ﬁnd the number of students that will join in class A given that both sections have total number of students of 20.

b State which section has the higher number of students that will join the school fair.

Model 1

Model 2

Model 3

Integers & Exponents 3

Big ideas

• Integers are a subset of the set of real numbers.

• Real numbers are either rational or irrational.

• Changing the form of an expression or equation can reveal information that was previously unknown.

• Expressions are the building blocks of algebra. They can be used to represent and interpret real-world situations.

There are other ways we can represent integers. Including vertical number lines, integer chips, and pictures.

In this vertical number line, the numbers above zero are positive integers while the numbers below zero are negative integers. The point plotted on the number line represents positive 1.

Integer chips can help us model integer expressions. Each chip represents a single integer unit. Integer chips come in all kinds of colors, in this case the green chips represent +1 and the red chips represent 1.

This grouping has 5 green chips representing +1 each. The total grouping represents +5.

This grouping has 7 red chips representing 1 each. The total grouping represents 7.

Is each number an integer?

Create a strategy

For each number, we can ask ourselves: is it a positive or negative whole number, or 0?

Apply the idea

• 2 is a negative whole number. So, it is an integer.

• is a positive fraction and is not a whole number. So, it is not an integer.

• 0.4 is a positive decimal and is not a whole number. So, it is not an integer.

• 6 is a positive whole number. So, it is an integer.

• is a positive mixed number. So, it is not an integer.

Therefore, 2 and 6 are the only integers from the list of numbers.

Example 1

Example 2

Consider the number 10.

a Represent 10 on a number line.

Create a strategy

We can ask ourselves: is it a positive or negative whole number, or 0? This will help determine where the number belongs on the number lines.

Apply the idea

Since 10 is negative, place it 10 units left of 0 on the number line.

b Represent 10 with integer chips.

Create a strategy

Decide on colors for your positive and negative integer chips. Determine whether you need positive or negative chips to represent 10.

Apply the idea

We need 10 negative integer chips to represent the number 10.

3

For each number line, determine where the point plotted is. a

Create a strategy

Every second tick is labeled with a multiple of 4. Since there are 4 units between every second tick, each tick is 2 units apart.

Apply the idea

The point is 2 units to the left of 4. so it is at 6.

Example

Create a strategy

Remember that numbers above zero are positive numbers and numbers below zero are negative numbers on a vertical number line.

The marked numbers are 0 and 5. The segment between 0 and 5 is divided into 5 parts so each part is equivalent to one unit.

Apply the idea

The point is located above zero on the vertical number line so it is positive.

The point is on the third mark. The integer is positive 3.

Idea summary

An integer is a positive or negative whole number, or 0.

On a number line positive integers are to the right of 0 and negative integers are to the left of 0.

Integers in the real world

Integers are extremely useful numbers because they can be used to describe many things in the real world.

The positive sign (+) and negative sign ( ) tell us important information about a real world situation.

For example, a positive temperature represents a temperature above zero, while a negative temperature represents a temperature below zero.

In some situations, the positive sign indicates an increase, while the negative sign indicates a decrease.

ee Shop Menu

Co ee Shop Menu

For example, let’s say that a local coffee shop is increasing the prices of some of their menu items by $1. We can quickly describe this situation by writing +1. The positive sign indicates an increase, and the 1 indicates going up by 1.

Similarly, a decrease in the price of a menu item by $1 can be represented by the number 1.

Example 4

Write an integer to represent the statement: A loss of $52.

Create a strategy

We will start by identifying whether our integer should be positive or negative based on the situation. Since we are representing a loss of money, we will have a negative integer.

Apply the idea

We know that our integer will be negative since we are representing a loss. Since it is a loss of $52, the integer that represents the statement is 52.

Example 5

Let the location of a city be represented by the integer 0, and let a point 7 km to the east of the city be represented by the integer 7. What integer represents the point 4 km to the west of the city?

Create a strategy

Use a number line to represent the information.

Apply the idea

4 represents the point 4 km to the west of the city.

Idea summary

A rise or increase in value will be represented by a positive integer. A loss or decrease in value will be represented by a negative integer.

Practice

What do you remember?

1 What is an integer?

2 For the point on the number line

a What number is represented by the point on the number line?

b Is the number positive, negative or neither?

c Is the number an integer?

3 Which of these is an integer?

4 Identify each value that represents an integer. Select all that apply.

5 What integer does the point furthest to the left on the number line lie on?

6 Which is the closest location of point B on the number line? 0 3 3

7 Which arrow shows the location of 7 on the number line?

8 Which value represents receiving $33: +33 or 33?

9 Determine the integer that the point is plotted on for each horizontal number lines:

10 Determine the integer that the point is plotted on for each vertical number lines:

14 Write an integer to represent each statement:

a A price rise of $17.

c A temperature drop of 5 °F.

e An elevation of 980 ft.

g 14 more candies today than yesterday

i Descending 9 ﬂoors.

k Depositing $75 in to a bank account.

m A weight gain of 8 kg.

o Withdrawing $145 from a bank account.

b A loss of $176.

d 58 °F above 0 °F.

f Grew by 9 cm last year.

h Deposited $100 to his account.

j A weight loss of 14 kg.

l A distance of 320 ft below sea level.

n Ascending 10 ﬂoors.

p A proﬁt of $650.

15 Using the key below, identify the integer represented in the picture and explain your thinking.

16 When Frankie woke up, the temperature was 4 °C. During the day it rose by 11 °C. Which number line correctly shows this temperature change?

17 What integer represents 40° below zero?

23 The image shows how the location of a miner traveling up and down a mine shaft relates to an integer on the number line.

a What integer represents 5 m below the surface?

b Bill is at a location represented by the integer 3, and he goes up 5 m. Describe Bill’s new location.

24 Using the key below, identify the integer represented in the picture and explain your thinking.

3.02 Compare and order integers

After this lesson, you will be able to...

• compare integers using number lines and inequality symbols.

• use a number line to order integers in ascending or descending order.

Compare and order integers

We can use the following symbols to compare integers:

> means “is greater than”, such as 7 > 2

< means “is less than”, such as 2 < 4

= means “is equal to”, such as 6 = 6

The number that is furthest right on the number line is always greater than the numbers on its left.

4 is the furthest to the left, therefore it is the smallest integer plotted on this number line.

8 is the furthest to the right, therefore it is the largest integer plotted on this number line.

Writing the numbers from smallest to largest is called ascending order: 4, 0, 3, 8.

Writing the numbers from largest to smallest is called descending order: 8, 3, 0, 4.

Interactive exploration

Explore online to answer the questions

Use the interactive exploration in 3.02 to answer this question.

1. What happens to the inequality symbol when the order of the integers is swapped?

Example 1

Which is the largest number marked on the number line? 0 5 10 15 5 10

Create a strategy

Recall that the further an integer is to the right on a number line, the larger the integer is.

Apply the idea

The integer farthest to the right on the number line is 13. So, the largest number is 13.

Example 2

Consider the numbers 3 and 9. a Graph 3 and 9 on the number line.

Create a strategy

We can see that 3 and 9 are both negative and so will be to the left of 0.

Apply the idea

To plot the point 3, start at 0 and count left 3 places. To plot the point 9, we can start at 5 and jump left a further 4 places.

b Insert either < or > to make a true statement.

Create a strategy

If we are comparing two negative numbers, the number closer to zero will be the larger number.

Apply the idea

We can see from the previous number line that 3 is closer to 0 which means it is larger than 9. So, 3 > 9.

Example 3

Arrange the following numbers in ascending order: 11, 25, 19, 15, 28

Create a strategy

Ascending means ordering from smallest to largest. We can do this by going from left to right on the number line.

Apply the idea

Plot the points on the number line:

Arrange the list from least to greatest 25, 15, 11, 19, 28

Example 4

The melting point of krypton is 157 °C. The melting point of radon is 71 °C. Write an inequality comparing the two melting points.

Create a strategy

A negative integer with a smaller number has a greater value than a negative integer with a larger number. It is further to the left of 0 on a number line.

Idea summary

Apply the idea

71 is a negative integer with a smaller number which means it has a greater value than 157.

157 °C < 71 °C

The sizes of integers can be compared using inequality symbols.

The symbol < represents the phrase is less than.

The symbol > represents the phrase is greater than.

The symbol = represents the phrase is equal to.

Practice

What do you remember?

1 Which point on the number line represents the greatest integer?

2 State the greatest number plotted on the following number line:

3 State the smallest number plotted on the following number line:

4 Determine if the following statements are true or false.

a A negative integer is always to the left of a positive integer on a number line.

b Zero is considered a positive integer.

c A negative integer is always smaller than zero.

d On a number line, every integer to the right of another integer is always greater.

e A positive integer is always larger than a negative integer.

f The number zero is always located between the positive and negative integers on a number line.

g Every integer is either positive, negative, or zero.

h An integer’s value decreases as it moves to the right on a number line.

5 For each of the following pairs of numbers: i Plot the numbers on a number line. ii State which number is larger.

a 4 and 8 b 17 and 0

c 2 and 7 d 4 and 9

6 For each of the following pairs of numbers: i Plot the numbers on a number line. ii State which number is smaller.

a 0 and 6 b 7 and 16

Let’s practice

c 8 and 0 d 1 and 3

7 Write a statement using < or >, that compares the two integers plotted on the following number lines:

8 For each of the following pairs of numbers: i Plot the pair of numbers on a number line.

ii Complete each number sentence with < or >, that should appear between each pair to make the statement true.

a 3 ⬚ 7 b 15 ⬚ 4 c 4 ⬚ 7 d 3 ⬚ 9

9 For each of the following statements: i Write each number sentence in words. ii State whether the number sentence is true or false.

a 8 < 4 b 7 > 5

a 9 ⬚ 2 b 6 ⬚ 6

0 > 1 d 23 < 47

10 Identify whether <, >, or = should appear between each pair of numbers to make each statement true:

0 ⬚ 6

7 ⬚ 8 e 32 ⬚ 40 f 29 ⬚ 74

11 State whether the following statements are true or false:

a 14 ≤ 2 b 2 ≥ 14 c 14 > 2

e 9 > 11 f 11 > 9

12 State the greatest number in each set:

9 ≥ 11

2 > 14

9 = 9

a 19, 13, 6 b 16, 0, 12 c 20, 3, 15 d 13, 14, 2

e 7, 17, 12 f 0, 8, 7 g 35, 36, 40 h 99, 87, 71

13 State the smallest number in each set:

a 14, 4, 21 b 7, 24, 10 c 0, 4, 5 d 34, 36, 2

e 12, 0, 2 f 23, 2, 13 g 9, 21, 2 h 16, 25, 27

14 Which of these lists the numbers from greatest to least?

A 15, 4, 59 B 4, 15, 59 C 4, 59, 15 D 59, 4, 15 2 4 6 8 0 2 4 6 8 2 4 6 0 2 4 6

Let’s extend our thinking

15 Arrange the following numbers in ascending order:

a 11, 25, 19, 15, 29 b 13, 7, 0, 4, 8 c 7, 6, 7, 5, 3 d 17, 22, 13, 0, 3

Arrange the following numbers in descending order:

a 2, 11, 7, 4, 18 b 15, 6, 0, 7, 9 c 22, 6, 5, 18, 2 d 27, 12, 0, 8, 16

17 Explain how to complete the following patterns:

a 5, 3, 1, ⬚, ⬚, ⬚

c 11, 6, 1, ⬚, ⬚, ⬚

b 12, 10, 8, ⬚, ⬚, ⬚

d 7, 9, 11, ⬚, ⬚, ⬚

e 25, 22, 19, ⬚, ⬚, ⬚ f 6, 10, 14, ⬚, ⬚, ⬚

g 31, 25, 19, ⬚, ⬚, ⬚ h 15, 8, 1, ⬚, ⬚, ⬚

18 Calgary has a temperature of 5° C while Montreal has a temperature of 11° C. Identify the state which has a higher temperature. Explain your reasoning.

19 For each of the following scenarios, write a sentence to compare the two quantities. Example: Neville has $8 in savings. Iain has $14 saved.

Two possible answers:

• Neville has less savings than Iain.

• Iain has greater savings than Neville.

a Ray’s account balance is $10. Mohamad’s account balance is $13.

b Jenny has made $74 selling ice creams. Irene has made $119 selling lemonade.

c Uther has read 33 books this year. His sister Patricia has read 12 books.

d Tobias has 778 songs on his cell phone. Marge has 525 songs on her tablet.

e Honolulu has a temperature of 98 °F and Ottawa has a temperature of 38 °F.

f The melting point of helium is 458 °F. The melting point of xenon is 169 °F.

g Charlie has made $38.92 this week from his part time job. Quiana has made $80.37 from her online craft store.

3.03 Introduction to exponents

After this lesson, you will be able to...

• identify and explain the meaning of the base and exponent of a power.

• represent repeated multiplication using exponents.

• find the value of a number raised to an exponent.

• recognize and represent patterns with bases and exponents that are whole numbers.

Introduction to exponents

An exponent (or power) is a small number placed in the upper right hand corner of another number to note how many times a base is being multiplied by itself.

For example, in the expression 103 the number 10 is the base term and the number 3 is the exponent (or index or power). The expression 103 is the same as 10 ⋅ 10 ⋅ 10, or the number 10 multiplied 3 times.

Exponent/power Base Multiplied 3 times 10 10 ⋅ 10 ⋅ 10 3

In the above expression, we call 103 the exponential form and 10 10 10 the expanded form of the expression.

We often encounter a power of 2 when measuring area. Consider the area of a square, for example, which is given by side length times side length. A number, e.g. 5 with an exponent (or power) of 2, can be expressed as 52, and can be read as “5 to the power of 2” or “ﬁve squared”.

A number, e.g. 10 to the power of 3, can be expressed as 103, and can be read as “ten cubed”. A power of 3 is involved in calculations like measuring the volume of a cube.

A base to the power of any other number, e.g. 34, can be read as “three to the power of four”, and means that the base number is multiplied by itself the number of times shown in the exponent.

34 = 3 3 3 3

To evaluate or simplify an exponential expression, the only step we need to take is completing the multiplication.

34 = 3 3 3 3 = 81 Simplify the multiplication

Exploration

Complete the following table of values using a pattern:

1. Describe the pattern you used to complete the table.

2. What do you notice about 21 ?

3. What do you notice about 20 ?

4. Test this observation by ﬁlling in a new table with a different base. Do you notice the same thing?

5. Now try to complete the entire table if the base is 1. What do you notice?

Any number raised to the power of 1 is equal to the original number. And 1 raised to any power is still 1 because 1 times itself any number of times will always be 1.

Any number raised to the power of 0 is 1. Though there is debate among mathematicians about whether 00 = 1 or is undeﬁned.

Example 1

Identify the base of 32

Create a strategy

Use the base and exponent deﬁnition: baseexponent

Example 2

Identify the exponent of 46

Create a strategy

Use the base and exponent deﬁnition: baseexponent

Apply the idea

baseexponent = 32

The base of the expression is 3.

Apply the idea baseexponent = 46

The exponent of the expression is 6.

Example 3

Write 75 64 in expanded form.

Create a strategy

Use the exponent to know how many times the base should be multiplied by itself.

Apply the idea

Multiply each of the bases by themselves the number of times indicated by the exponent

Example 4

Write 8 ⋅ 8 ⋅ 8 ⋅ 8 ⋅ 8 in exponential form.

Create a strategy

To write repeated multiplication of the same number in exponential form, count how many times the number is multiplied by itself. This will be the exponent. The base will be the number that is multiplied repeatedly.

Example 5

Given the table of values:

a Complete the table of values.

Create a strategy

The expanded form shows the base being multiplied by itself repeatedly, the number of times equivalent to the exponent. To evaluate, we calculate the result of this multiplication.

Apply the idea

The number 8 is multiplied by itself 5 times, so in exponential form, this is written as 85

Apply the idea

b What do you notice about the numbers in the “Evaluate” column?

Create a strategy

Observe the pattern formed by the numbers in the “Evaluate” column to identify any relationships or sequences.

Apply the idea

Each number in the “Evaluate” column is four times the number before it. This pattern reﬂects the fact that as the exponent grows larger by 1 we are multiplying by 4 an additional time.

Reﬂect and

check

We will continue to explore this concept of exponential growth throughout our mathematics courses. It demonstrates how quickly values can grow as the exponent increases, which has many real world applications.

Idea summary

An exponent (or power) notes how many times a base is being multiplied by itself. A base to the power of any other number means that the base number is multiplied by itself the number of times shown in the exponent.

Practice

What do you remember?

1 Identify the base of 29.

2 Identify the exponent of 106.

3 Write each exponential form with base 9:

4 Write each expression in expanded form: a 35 b 34 c 32 d 37

5 Show why 20 = 1.

Let’s practice

6 Which of the following is equivalent to 53?

7 Which of the following is equivalent to 11 11 11?

8 Write two squared in exponential form.

9 Write each expression in expanded form: a 52 b 55 c 102 d 73 e 61 f 84

10 Write each expanded form in exponential form:

11 Complete the table shown:

Six to the power of four

Eight cubed

Eleven to the power of three

Twenty three to the power of ﬁve

Seven squared

Fifteen to the power of four

12 Explain how to evaluate 45

13 Evaluate:

a 31 b 13 c 26 d 63

e 83 f 50 g 82 h 122

14 Claudia wrote the values of the powers of 4 that she knew:

41 = 4

42 = 16

43 = 64

44 = 256

45 = ?

What is the value of 45? A 20 B 260 C 1024

15 For each table of values:

i Complete the table of values.

ii What do you notice about the numbers in the “Evaluate” column?

16 Which is larger 123 or 127 and explain why without calculating.

17 Ethan and Soﬁa want to write 182 in expanded form. Ethan wrote 18 2, and Soﬁa wrote 18 18.

a Who wrote it correctly?

b If they solved the expanded form, would they get the same answer? Explain.

Let’s extend our thinking

18 Using the concept of exponents, explain how to ﬁnd the missing base in the equation ⬚4 = 16.

19 Using the concept of exponents, explain how to ﬁnd the missing exponent in the equation 5⬚ = 125.

20 If a population of 3 rabbits triples every hour, represent the population after 4 hours.

21 Write each expression in expanded form: a 54 96 b 43 82

e 46 92 f

22 Write each expression in exponential form:

23 Aaliyah calculated the ﬁrst 6 powers of 3 and put the results in the table: 31 32 33 34 35 36 3 9 27 81 243 729

a What would Aaliyah get if she continued and tried to evalute 37?

b What is the last digit Aaliyah would get after evaluating 39?

c Aaliyah used a calculator to evaluate 311. After seeing 19 683 as answer from the calculator, she realized that she made an error in putting the input to the calculator. Is Aaliyah’s statement correct? Explain your answer.

24 Sam is wanting to solve a Tower of Hanoi puzzle with 8 disks. He can calculate the minimum number of moves to ﬁnish the puzzle by solving the expression 28 1.

a Evaluate 28.

b Now, ﬁnd the minimum number of moves to ﬁnish Tower of Hanoi with 8 disks.

If we build out a 14 × 14 grid, we get 196 squares so we need to add 4 more squares to get to 200. This is not enough units to create a 15 × 15 grid so 200 is not a perfect square.

Reﬂect and check

Building out grids can be helpful ways to determine and justify whether a number is a perfect square. It is important to consider whether we have enough leftover units to build a larger grid, so it’s always a good idea to try building a grid with a 1 unit increase in length and width.

Idea summary

Perfect squares are numbers raised to the power of two or can be obtained by multiplying an integer by itself.

Practice

What do you remember?

1 Complete the table, one column has been done for you.

2 Complete the table below:

3 Where do all of the perfect squares lie on this multiplication chart? Why? Multiplication table

4 Explain how you know whether a number is a perfect square.

5 Write two squared in exponential form.

Let’s practice

6 Determine

7 Tania was playing around with pennies and made 4 different arrangements. Which arrangement of pennies makes a perfect square? Explain.

6 pennies

8 Explain why 0 is a perfect square.

9 For the values below, determine if the number is a perfect square. Justify your answer using a diagram.

10 For the values below, determine if the number is a perfect square. Justify your answer mathematically or with words.

11 What is the area of the square traﬃc sign in square inches?

12 A square painting measures 4 feet on each side. What is the area of the painting?

13 Write three perfect squares that have values greater than 120 and less than 200.

14 A gardener is planning to arrange tiles in a square pattern in a section of the garden. If she uses 100 tiles for the pattern, how many tiles will be in each row?

15 Which is a perfect square between 81 and 144?

A 84 B 121

C 90 D 114

16 A square patio has an area of 225 m2. Find the dimensions of the patio.

17 Fill in the missing numbers in the pattern:

a 4, 9, 16, ⬚, 36, 49, ⬚

c 25, 36, ⬚, 64, 81, ⬚, 121, 144, ⬚, 196

18 How many blocks do you need to add to:

a Square 6 to get Square 7?

b Square 9 to get Square 10?

c Square 15 to get Square 16?

Let’s extend our thinking

b 81, ⬚, 121, 144, ⬚, 196, 225, ⬚, 289, 324

19 I am a two-digit perfect square number. The sum of my digits is 13. What perfect square number am I?

20 Consider the equation 72 = ⬚2. The missing number is between what two whole numbers?

21 Barney was considering the pattern of square numbers and thought he spotted a pattern existed when calculating the differences between each pair of square numbers.

a Calculate the differences between consecutive square numbers. i 12 – 02 ii 22 – 12 iii 32 – 22 iv 42 32

b What pattern did Barney identify when he looked at the answers from part (a)?

c Barney isn’t sure if the pattern will continue. Explain whether you think it will or will not continue forever.

22 Consider the ﬁgure:

First Di erence Second Di erence

a Complete the ﬁgure by ﬁlling in the boxes with the differences between each pair of numbers above.

b Describe the pattern that exists in the ﬁrst difference row.

c Describe the pattern that exists in the second difference row.

23 Neil was working out 22 52 and thought that he could simplify the expression using the fact that 2 5 = 10

a Evaluate 22 ⋅ 52 by ﬁrst evaluating each square.

b Now, using the fact 2 ⋅ 5 = 10 evaluate 102

c Is 22 52 = (2 5)2 a true statement?

3.05 Powers of 10 and place value

After this lesson, you will be able to...

• identify powers of 10.

• use patterns in place value to write, represent and evaluate powers of 10.

Powers of 10 and place value

Recall that an exponent (or power) tells us the number of times to multiply a certain number by itself. We just looked at special properties of the exponent 2, now let’s look at some special properties of 10.

Interactive exploration

Explore online to answer the questions

mathspace.co

Use the interactive exploration in 3.05 to answer these questions.

1. What patterns do you notice in the numbers?

2. Why do you think 100 = 1?

For any power of ten, the expanded form will have the same number of tens as the power. The number that it evaluates to will have the same number of zeros as the exponent.

The following table demonstrates another way to think of some of the powers of ten.

Power of Ten Meaning Value (basic numeral) In Words

We can see that the exponent relates to the place value of the 1. The larger the exponent, the larger the place value.

Example 1

If you have a 1 in the hundred thousands place, what power of 10 does this represent?

Create a strategy

To determine the power of 10 that a 1 in the hundred thousands place represents, we can use a place value table. Each place to the right of 1 increases the power of 10 by 1.

Apply the idea

Hundred Thousands Ten Thousands Thousands Hundreds Tens Ones

1 0 0 0 0 0

There are ﬁve places to the right of 1, from ten thousands to ones. So, 1 in the hundred thousands place is 105

Reﬂect and check

The power of 10 that a 1 in the hundred thousands place represents is 105 because 100, 000 = 105. This is because moving each place to the left in the place value table multiplies the value by 10, starting from 100 = 1 for the ones place.

Example 2

A library has exactly 1, 000, 000 books. Write this number of books as a power of 10.

Create a strategy

To write 1, 000, 000 as a power of 10, we will count the number of zeros to the left of the decimal point. This number will be the exponent for 10.

Apply the idea

The number 1, 000, 000 has six zeros to the left of the decimal point, indicating that the library’s book collection can be represented as 106 books.

Example 3

Find the missing exponent.

Reﬂect and check

Writing large numbers as a power of 10 makes them easier to read and understand. In this scenario, 106 easily shows that the library has a very large collection of books.

10⬚ = 10, 000, 000

Create a strategy

To ﬁnd the missing exponent, we can count the number of zeros in the number 10, 000, 000. This number is the value of the missing exponent for 10.

Apply the idea

The number 10, 000, 000 has seven zeros. Therefore, the missing exponent that makes 10⬚ = 10, 000, 000 true is 7.

107 = 10, 000, 000

Idea summary

For a power of ten, the number of zeros after the 1 is the same as the exponent. The power of ten changes the place value of the 1.

What do you remember?

1 Match each power of 10 to its correct numerical value.

a 106 i 100

b 105 ii 1, 000, 000

c 104 iii 1, 000

d 103 iv 100, 000

e 102 v 10, 000

2 How should 105 written in a place value chart?

3 If you have a 1 in the millions place, what power of 10 does this represent?

4 Select all that are powers of 10.

5 Which is smaller: 105 or 107? Explain why.

Let’s practice

6 How many zeros should we use when writing 109 as a whole number? Explain your answer.

7 If a city has a population of 1, 000, 000 express this population count as a power of 10.

8 A certain bacteria culture grows to 100, 000 cells in one day. Express this number as a power of 10.

9 Xian placed the numeral 1, 000 in the place value chart.

What is 1, 000 written in powers of 10?

10 What is the value of 107?

11 Find the missing exponent.

10⬚ = 10, 000

12 Find the missing exponent.

10⬚ = 100, 000, 000, 000

13 Find the missing exponent.

10⬚ = 100, 000, 000

14 Express 6 × 107 as a whole number.

Let’s extend our thinking

15 Complete the statement: 4, 000 = ⬚ × 103

16 The observable universe is estimated to be about 93, 000, 000, 000 light-years in diameter. Write this number as a power of 10.

17 Write 45, 000 as a power of 10.

18 Write 1, 090, 000 as two different powers of 10.

19 Write as whole numbers:

20 The distance between two stars is approximately 9 × 107 meters. Express the distance as a whole number.

Operations with Integers 4

Big ideas

• The properties of real numbers can be applied to many types of expressions.

• Expressions are the building blocks of algebra. They can be used to represent and interpret real-world situations.

• Integers are a subset of the set of real numbers.

Chapter outline

4.01 Add and subtract integers

After this lesson, you will be able to...

• represent addition and subtraction of integers using models and pictures.

• add and subtract two integers.

• find and justify the solultion to real-world problems involving addition and subtraction of integers.

Add integers

We know that integers can be either positive, negative, or 0 (which is neither positive or negative). This is indicated by the sign on an integer. Integers with + (or no sign at all) are positive. Integers with are negative.

The sign of an integer gives it a direction. We can imagine that for every integer on the number line there is an arrow going from 0 to that integer.

For a number line with the positive direction to the right, the positive integers have arrows that point to the right, and the negative integers have arrows that point to the left.

Interactive exploration

Explore online to answer the questions

Use the interactive exploration in 4.01 to answer these questions.

1. What direction represents adding a positive integer?

2. What direction represents adding a negative integer?

3. What are the ways you can create a sum that is positive?

4. What are the ways you can create a sum that is negative?

5. What are the ways you can create a sum that is 0?

6. What happens when 0 is one of the numbers being added?

The addition of integers can be represented by adding their arrows on the number line. When we combine the lengths and directions of two arrows, we get a third arrow whose length and direction corresponds to the sum. This is because, adding two (or more) integers always results in another integer.

The image shows how 6 + 2 = 8 is represented using the addition of arrows on the number line. Can you see how the order of addition does not affect the result?

What if we want to add a negative integer? We use the same approach, the only difference being that the arrows are pointing in different directions. The image shows that 4 + ( 9) = 5, which is the same result that we get from ( 9) + 4.

To think of this process more simply, we can plot the ﬁrst integer on the number line and move the direction and number of spaces indicated by the second integer.

When we add a positive number we move to the right.

When we add a negative number we move to the left.

Example 1

Find the value of 7 + 13.

Create a strategy

Draw a model with arrows using a number line.

Apply the idea

We start by drawing an arrow for 7 in the number line.

Adding a positive integer means we move to the right. So, to draw the arrow that represents adding 13, we need to count 13 units to the right starting at 7.

Finally, we draw a third arrow which starts at 0 and ends at the tip of the second arrow. This third arrow represents the sum of 7 and 13 which is 6.

Reﬂect and check

What if we start by drawing ﬁrst the arrow that represents 13 then the arrow that represents adding 7?

Notice that when we draw the third arrow that represents the sum, we get the same answer.

This shows that 7 + 13 is the same as 13 + ( 7). The order of drawing the arrows does not affect the sum.

Find the value of 11 + ( 6).

Create a strategy

Adding a negative integer means we will move to the left on the number line.

Apply the idea Plot 11 on the number line: 2 1 0 1 2 3

From 11 we want to move 6 units to the left.

Reﬂect and check

We can combine the adjacent signs by writing: 11 + ( 6) = 11 6 Adding a negative 6 is the same as subtracting 6 = 5 Subtract

Find the value of 12 + ( 8).

Create a strategy

Adding a negative integer means we move to the left on the number line.

Example 2

Example 3

Apply the idea

Plot 12 on a number line:

From 12, move 8 units to the left.

Reﬂect and check

12 + ( 8) = 12 8

Idea summary

Adding a negative is the same as subtracting a positive

When ﬁnding the sum of two integers, we can use a number line.

The sum of:

• two positive integers is another positive integer

• a positive integer and a negative integer may be positive or negative

• two negative integers is another negative integer

• an integer and its opposite is 0

Subtract integers

We have looked at how to model addition with number lines. Now, we will explore integer chips. Integer chips are another way to model integers and use the fact that the sum of two opposite integers is 0.

Interactive exploration

Use the interactive exploration in 4.01 to answer these questions.

1. Which problems were the simplest?

2. Why did we have to add zero pairs for some problems and not others?

3. If we are subtracting two negative numbers, how do the integer chips work?

4. If we are subtracting one negative and one positive integer, how do the integer chips work?

Now that we’ve explored subtraction with integer chips, let’s explore subtraction on a number line. When we added integers on a number line we went in the direction indicated by the sign of the number. But the subtraction operation tells us to reverse the direction of the integer that follows.

When we subtract a positive number we move to the left, because we’re reversing the direction indicated by positive 4.

= 6

When we subtract a negative number we move to the right, because we’re reversing the direction indicated by 3.

Let’s look at 3 5. We can see from the number line that this is actually the same as 3 + ( 5). In other words, subtracting 5 is the same as adding the opposite of 5.

Let’s look at 7 ( 2). Starting at 7, we move to the right 2 because we’re reversing the direction of 2. From the number line we can see this gives us the same result as adding 7 + 2.

Example 4

Find the value of 8 7.

Create a strategy

Subtracting a negative integer means we will move to the left on the number line.

Apply the idea

Plot 8 on a number line:

From 8, move 7 units to the left.

5 Use the plotted point shown to determine each of the following values. 10 5 0 5 10

a 6 units left of the plotted point

c 2 units right of the plotted point

6 Evaluate:

b 5 units right of the plotted point

d 4 units left of the plotted point

Subtraction can be rewritten as addition. Example: 3 4 = 3 + ( 4)

Rewrite each subtraction problem as addition.

a 3 7 b 4 11

c 11 ( 6) d 13 ( 2)

8 Write an integer to represent each of the following statements:

a A price rise of $17

c A temperature drop of 5 °F

e An elevation of 980 ft

g Descending 9 ﬂoors

i A distance of 320 ft below sea level

k Withdrawing $145 from a bank account

Let’s practice

b A loss of $176

d 58 °F above 0 °F

f Grew by 3 in last year

h Depositing $75 in to a bank account

j Ascending 10 ﬂoors

l A proﬁt of $650

9 Write and solve the addition problem represented by the model. a b c

10 Use the plotted point on the number line to add each pair of integers.

11 Use the plotted point on the number line to subtract each pair of integers.

a 1 4 b 1 ( 5)

12 Evaluate:

+ 7

13 Evaluate:

a 12 3

36 53

5 ( 198) d 288 677 e 3000 1495

14 Evaluate:

a 7 + ( 5)

51 + 23 c 120 ( 52) d 95 ( 95) e ( 17) + ( 17) + ( 18) f ( 18) ( 11) ( 17)

76 29 + 112 h 175 + ( 345) + 28

15 Felix is trying to determine the smaller result between 27 + 38 and 41 + ( 32). He decides to use a number line to ﬁnd each sum and then plot the resulting points on a number line to determine the smaller sum.

a Describe how Felix will ﬁnd each sum using the number line, including the directions he will move and how many units he will move in each sum.

b Plot each sum on a number line and state which sum is smaller.

16 Use a number line or integer chips to describe how both 6 7 and 5 11 both result in negative answers.

17 Determine whether the given situation would represent addition or subtraction.

a The total length of 3 pencils with lengths 6 in, 7 in, and 9 in.

b Total volume of water if one can holds 8 oz and the other contains 12 oz.

c The change for $10 bill if you buy a notebook worth $2.

d The change in a plant’s height if it measured 14 inches last month and measures 17 inches this month.

e The difference in temperatures from a high of 45 °F and a low of 12 °F.

f The balance in a bank account after a deposit of $108 and a second deposit of $62.

18 Amy solved a subtraction problem and wrote her work as:

53 ( 26) = 53 + ( 26) = 79

Describe and correct the error in her steps.

19 Valentina dives off a platform that is 8 m above the water. She descends 14 m before returning to the surface.

a Plot the integer that represents Valentina’s initial position on the diving platform on a number line.

b Write the integer that represents her lowest position throughout the dive.

c Write the greatest depth that Valentina reaches during the dive.

20 One evening in Moscow the temperature fell to 3° C. By midday the next day, the temperature was forecast to be 15° C.

a Plot the temperature during the evening on a number line.

b Plot the temperature the next day on the same number line.

c Find the expected rise in temperature.

21 Liquid nitrogen can be used to make ice cream. The ice cream mixture starts at 26° C, then rapidly cools to 185° C after liquid nitrogen is added to it. State the integer that represents this change in temperature.

Let’s extend our thinking

22 Each letter has been assigned to an integer. A word must be formed using these letters and each letter can only be used once. The table shows the assignment of the integers:

The score of the word is the sum of the integers assigned to its letters. Find the score of each word: a NET b EAR c RAIN d CARES

23 Each bottle of orange juice is supposed to contain 250 mL. Nine bottles are inspected by the quality assurance department. The results of the inspection are shown in the table:

below or above the required

a Find the actual volume of: i Bottle 1 ii Bottle 2 iii

b Calculate the total volume contained in the 10 bottles.

24 The table shows the proﬁt of Eiichiro’s Ramen from August to December. The goal of Eiichiro’s Ramen was to have a total proﬁt of $10 from August to December.

August September October November December $13 $10 $11 $3 $14

Did Eiichiro’s Ramen reach their goal? Explain your answer using number lines.

25 Write in order from greatest to least: 5 ( 8), 7 + ( 12), 10 18, 16 + 21, 31 + ( 31)

26 A dolphin is 12 feet below the surface of the water. It swims up and jumps out of the water to a height of 6 feet above the surface. Find the vertical distance the dolphin travels. Explain your answer using number lines.

27 Consider the integer 18:

a Write two integers with different signs that have a sum of 18.

b Write two integers with the same sign that have a sum of 18.

28 Consider the diagram at the right:

Place the integers 3, 2, 1, 0, 2, 3, 4 and 5 in the circles so that any three circles in a straight line add up to 3.

29 Consider the diagram at the right:

Place the integers 5, 4, 3, 2, 1, 0, 1, 2 and 3 in the circles so that any three circles in a straight line add up to 3.

4.02 Multiply and divide integers

After this lesson, you will be able to...

• represent multiplication and division of integers using models.

• multiply and divide two integers.

• find and justify the solution to real-world problems involving multiplication and division of integers.

Multiply integers

Interactive exploration

Explore online to answer the questions

Use the interactive exploration in 4.02 to answer these questions.

1. What is the sign of the product of two positive integers?

2. If you are looking at the product 3 ( 4), how many tiles are in 3 groups of 4 tiles?

3. What is the sign of the product of one positive and one negative integer?

4. Is the product of 3 ( 4) the same as 4 3? Check this for other products.

5. What is the sign of the product of two negative integers?

Unlike adding and subtracting integers, where we can use the number line or counters, multiplication comes down to looking at the sign of factors.

We have seen that the product of two positive integers is a positive integer.

The product of a positive integer and a negative integer is a negative integer.

The product of two negative integers is a positive integer.

Example 1

Find the value of:

Create a strategy

Determine the sign of the product, then multiply 4 and 5 and apply the sign.

Apply the idea

The sign of the product is negative because we are multiplying a negative and a positive integer.

4 ⋅ 5 = 20

Evaluate

We determined that the sign of the product should be negative, so the answer is 20.

Reﬂect and check

Let’s visualize the multiplication of a negative and a positive integer using an array. In this case, we are multiplying 4 and 5.

In the array, we have 4 rows and 5 columns, which represent the multiplication of 4 and 5. Each row has 5 orange tiles, representing negative integers. When we count the total number of orange tiles, we have 20 orange tiles, which represent the product of 4 and 5. Since orange tiles represent negative integers, our product is 20 which matches our previous answer.

Find the value of:

Create a strategy

We have the product of two negative integers, so the product will be positive.

Apply the idea

⋅ ( 5) = 35

Reﬂect and check

Evaluate

Let’s visualize the multiplication of the two negative integers by using an array representation with blue tiles for positive integers and orange tiles for negative integers.

7 ⋅ ( 5) = 35

In the array, there are 35 blue tiles, which represent positive integers. This shows that when we multiply two negative integers, like 7 and 5, the product is a positive integer, in this case, 35.

Example 2

Example 3

A submarine dives 22 m each minute for 16 minutes. What integer represents the total depth of the dive after 16 minutes?

Create a strategy

Diving 22 meters is represented by the integer 22. We can think of this as repeated addition because an extra 22 m is added for each minute the diver is under. That means we can also think of this as multiplication.

Apply the idea

Depth of the dive = 22 ⋅ 16 Set up the equation = 352 m Evaluate

Reﬂect and check

We can check the reasonableness of our answer by thinking about the signs. We knew diving 22 m was a negative integer because the diver is going under water. And time was positive because time is always moving forward.

Multiplying a positive and negative integer results in a negative integer, which is what we got. This makes sense because the diver ends up deeper underwater.

Idea summary

We have seen that the product of two positive integers is a positive integer.

The product of a positive integer and a negative integer is a negative integer.

The product of two negative integers is a positive integer.

Divide integers

The same principles that help us to multiply integers also apply to divide.

Interactive exploration

Explore online to answer the questions mathspace.co

Use the interactive exploration in 4.02 to answer these questions.

1. What is the sign of the quotient of two positive integers?

2. What is the sign of the quotient of two negative integers?

We have seen that the quotient of two positive integers is a positive number.

The quotient of a positive integer and a negative integer is a negative number.

The quotient of two negative integers is a positive number.

The same sign properties apply to both multipication and division. However, unlike multiplication, division of two integers does not always result in another integer.

Example 4

Find the value of:

Create a strategy

48 ÷ ( 6)

We have the quotient of one positive and one negative number, so the quotient will be negative.

Apply the idea

48 ÷ ( 6) = 8 Evaluate

Example 5

Evaluate:

Reﬂect and check

We can check if we have determined the correct sign, by using multiplication. Would 8 times 6 equal 48?

Create a strategy

We have the quotient of two negative integers, so the quotient will be positive.

Apply the idea

Evaluate

Idea summary

Reﬂect and check

We can check if we have determined the correct sign, by using multiplication. Would 6 times 10 equal 60?

We have seen that the quotient of two positive integers is a positive number.

The quotient of a positive integer and a negative integer is a negative number.

The quotient of two negative integers is a positive number.

What do you remember?

1 Determine whether the following is true or false:

a The number which divides the other number is known as the divisor.

b The number to be divided into is known as the quotient.

c The product of the divisor and quotient is the dividend.

d If the quotient has no remainder, then the quotient and divisor are factors of the dividend.

2 Detemine whether multiplication or division should be used to ﬁnd each of the following. Do not solve.

a The total length of 5 ribbons, each measuring 7 inches long

b Amount to pay if you buy 14 packs of notebook paper and each pack costs $2 each.

c The length of a strip of paper 36 inches long that will be cut into strips of 6 inches in length

d The number of number of glasses needed if a pitcher which holds a half gallon, or 64 fl oz, if each glass holds 8 fl oz.

3 Evaluate:

4 State whether the following expressions will be positive or negative:

a The product of two negative numbers

b The quotient of a positive number and negative number

c ( 147) 183 d ( 11) ( 85) e 157 ( 14) f 35 8 g 85 ÷ (

5 Write an equation representing the multiplication on the number line below:

6 Consider the equation 6 4 = 24, to ﬁnd the value of the following: a ( 6) ⋅ 4 b 6 ⋅ ( 4)

7 Consider the equation 42 ÷ 7 = 6, to ﬁnd the value of the following:

( 42) ÷ 7

Let’s practice

42 ÷ ( 7)

( 42) ÷ ( 7)

8 The counters below represent the integer 20.

a Use the counters to demonstrate the operation 20 ÷ 10, and include the solution in your answer.

b Use your demonstration from part a to explain why a negative divided by a positive is negative.

9 Identify the expressions that have equivalent solutions:

10 Evaluate:

11 Evaluate:

12 Complete the following equations:

13 Renee made 6 withdrawals of $95 each from her bank account.

a Renee wants to estimate the total withdrawn her bank account without checking her balance. How could she estimate this total, and what value would it be?

b Write an integer expression that represents how to determine the actual total withdrawn from Renee’s bank account.

c Determine the total withdrawn in Renee’s bank account as both an integer, then describing this change in context.

14 Zach’s score in a video game was changed by 105 points because he missed some goals. He got 15 points for each missed goal.

a Write an expression that Zach could use to determine the number of missed goals.

b Determine the number of missed goals.

15 Find the integer in the following statements:

a An integer when multiplied by 12, gives 108.

b An integer when divided by 7, gives 13.

Let’s extend our thinking

16 Evaluate:

18 Evaluate:

19 Cooper is trying out a new city design program that will build towers of blocks in yellow and red. The program uses multiplication to duplicate patterns, but the sign of the original values indicates color. He tries multiplying his original pattern by both 2 and 2 and the patterns are shown below:

Pattern 1

Pattern 2

a Which pattern uses 2 as a multiplier? Which pattern uses 2 as a multiplier? Explain your thinking.

b Cooper loads a city that a friend has designed. This city shows at town square with the design shown. Write a series of expressions using → between each new operation that would create this sequence of blocks. Choose where you would like to begin.

20 Josie answers questions in two online trivia games about animals. In each quiz, she loses points when she gives a wrong answer.

The table shows the number of points lost in each quiz for incorrect answers and the total number of points Josie lost per game.

a In which game did she have more incorrect answers?

b Josie is competing against Lily to see who loses less points at the end of both trivia games. Lily ﬁnished both games and lost a total of 80 points. Josie has ﬁnished the farm animals game and is still playing the zoo animal game. How many more questions can Josie miss and still win?

21 Explain why there is no integer n so that n n = 1.

22 The product of three distinct integers is 21. Determine all of the possible integers.

Example 4

Evaluate ∣7 11∣ and represent the solution on a number line.

Create a strategy

We need to follow Order of Operations, so we will need to do the subtraction inside the absolute value bars ﬁrst.

Apply the idea

7 11

Subtract the inside of absolute value bars = 4 The distance between 4 and 0 is 4 on our number line

Reﬂect and check

We can also think about this as the distance between 7 and 11 on a number line which is 4.

Example 5

Evaluate and represent the absolute value step on a number line.

Create a strategy

We need to follow Order of Operations, so we will take the absolute value ﬁrst.

Apply the idea

We can see that 12 is 12 units away from 0 on our number line. So, ∣12

= 12.

Find the absolute of 12 using number line above

Divide 12 by 2

Reﬂect and check

Would the answer be different if we changed the problem to ?

3 Which value does not belong?

4 For each of the following pairs of numbers:

i Plot the pair of numbers on a number line.

ii Complete each number sentence with < or >, that should appear between each pair to make the statement true.

5 Identify whether <, >, or = should appear between each pair of numbers to make each statement true:

2 ⬚ 0

⬚ 104

6 Evaluate each expression.

Let’s practice

7 Which two points on the number line shown have the same absolute value? Justify your reasoning using the deﬁnition of absolute value.

8 Evaluate and state what additional integer has an equivalent absolute value.

9 Find the possible values of x for each value:

10 Identify the sign, <, > or =, that should appear between each pair of numbers to make each statement true:

11 Evaluate each value.

12 Evaluate each of these numbers and order the results, from smallest to largest.

13 Identify each value smaller than 14 and plot on the number line provided.

14 Today, Santiago has a peak temperature of 37 °C, while Minneapolis has a peak temperature of 11 °C. Using the concept of number lines, explain which city had the least peak temperature.

15 Amelia has a bank account balance of $18. Marge has a bank account balance of $55. Using the concepts of absolute values, explain who has a bigger debt.

16 Aurelia has a bank account balance of $13. Then she spent $54. Using the concepts of absolute values, calculate how much debt she has. Represent your answer on a number line.

Let’s extend our thinking

17 Michael was solving the problem ∣ 8 12∣. His work and reasoning is shown below:

8 12

= ∣ 8

12∣

Find the absolute value of each term = 8 12 Subtract = 4

He used a number line to evaluate 8 12 and realized he made a mistake. Explain what the number line would show, and correct his error in thinking.

18 If ∣a + 8∣ = 20, what values could a be equal to? Explain your answer.

19 Evaluate.

20 Serena is setting up automatic deposits and withdrawals for her bank account that will occur on a regular basis. She wants to set up deposits of $150 each month and withdrawals of $50 each week. She will only do this if the absolute value of the deposits is greater than the absolute value of the withdrawals.

Reminder: there are 12 months and 52 weeks in a year.

a Calculate how much money Serena will deposit and withdraw in a year, writing each as an integer.

b Compare the absolute values found in the previous part. Does this plan make sense for Serena’s budget?

21 A palindrome is a word or sentence that reads the same forwards and backwards. Using the number line shown, follow the instructions to graph and label the points to discover the mystery palindrome. What is the mystery term?

• E: ∣ 4∣ 6

• R: 2 ∣ 4∣ and ∣ 2 4∣

• C: Any integer with an absolute value of 4

• A: and an integer the same distance away from 0

4.04 Real-world problems with integers

After this lesson, you will be able to...

• estimate solutions to real-world problems involving operations with integers.

• solve one and two-step real-world problems involving operations with integers.

• justify solutions to problems involving operations with integers.

Real-world problems with integers

We can use our knowledge of addition and subtraction on the number line to describe how the real world quantities change.

We can also talk about changes in the quantity we are representing using integer operations. Given a starting temperature and some change in a certain direction, what is the ﬁnal temperature? Given a starting balance and an ending balance of money in an account, what has been the amount and sign of the change?

When using integers to represent real-world situations, it is important to correctly identify any positive and negative integers as well as the correct operations.

Table of common key words to look for:

Term

Key Words:

Positive excess, proﬁt, above zero temperature

Negative debt, owes, below sea level

Addition plus, more than, total, sum, combined

Subtraction minus, difference, less than, decreased by Multiplication times, product of, double/triple

Division quotient, per, split, ratio, into

Estimation can be a good strategy for solving real-world problems if the context doesn’t require an exact solution.

Example 1

Tara is waiting for the next ﬂight to Los Angeles, which was scheduled to be in 64 minutes, but there is a 34-minute delay. She takes a nap and wakes up 23 minutes later. How much longer does Tara have to wait before the plane departs?

Create a strategy

Add the delay and subtract her sleep time from her wait time.

Apply the idea

Waiting time = 64 + 34 23 Set up the equation = 98 23 Perform 64 + 34 = 75 minutes Evaluate

Example 2

A science club has a budget of $600 for a new project. The club members decide to spend $450 on laboratory equipment and the rest on protective gear. If each set of protective gear costs $25, how many sets of protective gear can the club buy with the remaining budget?

Create a strategy

Our goal here is to ﬁgure out how many sets of protective gear the science club can buy after spending part of their budget on laboratory equipment.

We will subtract the amount of money spent on lab equipment from the total budget. This will let us know how much they have left to spend on protective gear. We will divide this amount by the cost of protective gear to ﬁnd how many they can purchase.

Apply the idea

The total budget for the project is $600. The club decided to spend $450 of that budget on laboratory equipment. We can ﬁnd how much money is left by subtracting the amount spent on equipment from the total budget.

Money leftover = 600 450 Set up the equation = 150 dollars Subtract

We have now ﬁgured out that we have $150 left in the club’s budget. We know that each set of protective gear costs $25, so to ﬁnd how many sets of protective gear we can purchase we will divide our remaining money, $150, by the cost of each set of protective gear, $25

Number of sets of gear = 150 ÷ 25

Set up the equation = 6 sets of gear Divide

The science club can purchase 6 sets of protective gear with the remaining budget.

Idea summary

Answers that are integers can be positive or negative. When solving a problem, the sign of the integer determines the location of a thing or person, or whether we have a proﬁt or loss, or savings or debt.

Term

Key Words:

Positive excess, proﬁt, above zero temperature

Negative debt, owes, below sea level

Addition plus, more than, total, sum, combined

Subtraction minus, difference, less than, decreased by Multiplication times, product of, double/triple

Division quotient, per, split, ratio, into

Practice

What do you remember?

1 Write the integer that represents each situation.

a 15° degrees below zero

c 11 feet below sea level

e A withdrawal of $50

b A deposit of $74

d A gain of 39 yards

f A loss of 3 feet

2 Match each verbal situation to its integer operation.

a A diver swimming at 8 feet below sea level descends 15 more feet

b A loss of 8 yards followed by a gain of 15 yards

c A hiker at an elevation of 8 feet ascends an additional 15 feet

d The total change in a bank account balance for a $15 withdrawal each day for 8 days

3 Evaluate each integer operation.

i 8 + 15

ii 15 ⋅ 8

iii 8 + 15

iv 8 15

4 Simplify each expression.

5 Determine whether the ﬁnal answer will be positive or negative. Do not solve.

a The local movie theater reported losses of $475 each day for three days. What was the loss for the three days?

b An elevator is on the twelfth floor. It goes down 8 floors and then up 2 floors. What floor is the elevator on now?

c On a test, Ruby scores 25 points for the questions she answered correctly and gets 2 points for each incorrect answer. Ruby answered 4 questions incorrectly. How many points did Ruby score?

d The elevation of Mt. Everest is 29 028 feet. The elevation of the Dead Sea is 485 feet. What is the difference in the elevation between Mt. Everest and the Dead Sea?

e Jessie has $58 left on her checking account. If she writes a check for $65, what will Jessie’s balance be?

f Joseph reported that the coldest day on record for his town was ﬁve times colder than yesterday’s temperature, 4 °C. What was the temperature of the coldest day on record in Joseph’s town?

Let’s practice

6 Valentina and Maria want to meet at the cinema to watch a movie. Valentina lives 3 km west of the cinema, and Maria lives 5 km east of the cinema.

a In the number lines below, the distance between each tick is 1 km. Which of the following number lines show the location of Valentina and Maria in relation to the cinema?

Valentina’s House Maria’s House

b Draw a new number line that shows the location of Valentina and Maria using integers, where 0 is the cinema, east is the positive direction and west is the negative direction of the number plane.

c Calculate how far Maria would have to travel to get to Valentina’s house.

16 Overnight, the temperature decreased 4 °F each hour for 6 hours then rose 6 degrees each hour for 3 hours as the sun rose.

Which operation and answer show the correct integer operation that represents the overall change in temperature?

A female brown bear weights 600 pounds. After hibernating for 6 months, she weighs only 420 pounds. Determine her monthly change in weight if she lost the same amount of weight each month.

18 Ms. Jones has an account balance of $75. She earns $104 for babysitting and tutoring. She works 4 hours as a babysitter and 3 hours as a tutor.

a If she deposits her money into her account, what would her new account balance be?

b If she earned $56 for babysitting, how much does she earn per hour? How much does she earn for tutoring per hour?

c If she babysits for 2 more hours and tutors for 1 more hour, what will her new bank account balance be after depositing the money?

Let’s extend our thinking

19 A clothing store was selling jackets for $68 each. The store changed the price by these amounts over 3 months:

+$27, $12, $11

a What is a reasonable estimate for the price of the jacket? Explain your reasoning.

b What is the actual current price for a jacket?

20 Three contestants are heading into the ﬁnal round of a game show. Eileen has 39 points and Roxanne has 3 points.

If Oprah has 14 more points than Eileen, calculate how many points separate Roxanne and Oprah.

21 Victoria’s bank account has a balance of $350 at the beginning of the year. She earns $1100 a month, and deposits it into her bank account. She also withdraws $200 a month to pay for rent.

a Write an expression to represent how to calculate the change to Victoria’s bank account over the course of a 12 month year, explaining your choice in what operations to include.

b Use your expression from the previous part to calculate the total change in Victoria’s bank account, writing your answer as both an integer and describing the integer in context of the context.

c Find the account balance at the end of the year.

22 In golf, your score for each hole is determined by comparing it to the par (or expected) number of shots for that hole. Taking one less shot than expected will give a score of 1, one more shot will give +1, and so on.

Ivan played 18 holes of golf, and the frequency of each score he obtained is shown in the adjacent table:

a Find Ivan’s total score for his 6 scores of +2.

b Find Ivan’s total score for his 5 scores of 1.

c Find Ivan’s total score for the total 18 holes.

23 The motion of a robot to the left or right along a straight line is determined by whether the robot is facing left or right, and whether it is stepping forwards or backwards.

Use a number line where 1 unit represents 1 step and the positive direction is to the right to answer the following questions:

a For each of the following, ﬁnd the integer that represents the ﬁnal position:

i The robot starts at 0, facing to the right, and takes 5 steps forward.

ii The robot starts at 0, facing to the left, and takes 13 steps forward.

iii The robot starts at 0, facing to the left, and takes 11 steps backwards.

b Create a set of instructions for the robot to have a ﬁnal position of: i 8 ii 15 iii 23

Facing Stepping Moving Right Forward Right

Right Backward Left

Left Forward Left

Left Backward Right

24 Jenny is going to buy 8 notebooks, 4 folders, 1 pack of pencils, and 1 lunch box for school. She will use a $10 gift card, then pay the rest of the total.

a As a way to estimate, Jenny rounded the price of pencils, folders, and notebooks to $5 and the lunch box to $10. Jenny calculates her total as 8 $5 + 4 $5 + $5 + $10 = $75.

After subtracting $10 from her estimate for the gift card amount, she would pay $65. Is this a reasonable estimate? Explain your thinking.

b Calculate the actual amount Jenny will need to pay for this purchase after using the $10 gift card.

c How different were the estimate and the actual cost? Did Jenny choose an accurate method to estimate?

Back-to-School Savings

6 pk. Pencils Spiral Notebook Folder

Lunch Box

The coordinates of a point are given in relation to the origin. In the image, we can see that the cat is 6 units to the right of the origin, and 2 units above the origin. So we can say the cat has the coordinates (6, 2). The x-coordinate is 6 and the y-coordinate is 2.

Notice that the order of the numbers is important. It would be incorrect to say the cat has the coordinates (2, 6).

These coordinates refer to the point 2 units to the right of the origin, and 6 units above the origin.

Coordinates are always written with parentheses in the form (x, y) where the ﬁrst number, x, is the x-coordinate and the second number, y is the y-coordinate.

Notice that the x-coordinate also tells us how far a point is from the y-axis and the y-coordinate tells us how far a point is from the x-axis.

In the image of the cat, the coordinates (6, 2) tell us the cat is 6 units from the y-axis and 2 units from the x-axis.

Example 1

Consider the coordinate plane shown:

a What object has coordinates (1, 4)?

Create a strategy

Use the numbers on the axes in locating the coordinates.

b What object has coordinates (10, 1)?

Create a strategy

Use the numbers on the axes in locating the coordinates.

Apply the idea

Start at (0, 0). Move 1 space to the right, then 4 spaces up.

The object with coordinates (1, 4) is a star.

Apply the idea

Start at (0, 0). Move 10 space to the right, then 1 space up.

The object with coordinates (10, 1) is beach ball.

c What are the coordinates of the bicycle?

Create a strategy

Follow the grid line up to the horizontal axis to identify the x-coordinate. Then, follow the grid line across the vertical axis to identify the y-coordinate.

Example 2

Plot the point (6, 3) onto the coordinate plane.

Create a strategy

Use the numbers on the axes to move on the coordinate plane.

Apply the idea

The number on the horizontal axis directly below the bicycle is 3 and across the vertical axis is 6. So, the coordinates are (3, 6).

Example 3

Consider the point with coordinates (9, 4).

a How far is the point from the x-axis?

Create a strategy

To ﬁnd the distance of a point from the x-axis, consider the y-coordinate of the point. The distance to the x-axis is measured vertically, making it equal to the absolute value of the y-coordinate.

b How far is the point from the y-axis?

Create a strategy

To determine the distance of a point from the y-axis, examine the x-coordinate. This distance is measured horizontally, making it equal to the absolute value of the x-coordinate.

Apply the idea

Start at (0, 0). Plot the point 6 space to the right, then 3 spaces up.

This will be the point on the plane described by (6, 3).

Apply the idea

For the point (9, 4), the y-coordinate is 4. Thus, the distance from the x-axis is ∣4∣ = 4 units.

Apply the idea

Given the point (9, 4), the x-coordinate is 9. Therefore, the distance from the y-axis is ∣9∣ = 9 units.

Example 4

Write the coordinates of the point that is 5 units to the right of (9, 6).

Create a strategy

Plot the given coordinates, then move horizontally by the required number of units.

Apply the idea

(9, 6) (14, 6)

Plot (9, 6) on the coordinate plane and move 5 units to the right. The new coordinates are (14, 6).

Reﬂect and check

Another way to ﬁnd the coordinates of the new point is by realizing that moving right will increase the x-coordinate so we need to add 5 to the x-coordinate.

New coordinates = (9 + 5, 6) Add 5 to 9 = (14, 6) Evaluate

Example 5

Point A has the coordinates (3, 6), and point B has the coordinates (8, 6). What is the distance between A and B?

Create a strategy

Plot the points on the coordinate plane and then count the horizontal units from point A to point B

Apply the idea

Plot the point A and point B. The distance between them is 5 units.

Reﬂect and check

Another way to ﬁnd the distance between the two points is by realizing that point A and point B have the same y-coordinate. Therefore, the distance between them is the difference in their x-coordinate.

Distance = (8 3) Subtract 3 from 8 = 5 Evaluate

Idea summary

The coordinate plane is used to describe the location of actual points called coordinates in a two-dimensional space.

The coordinates are pair of numbers that are in the form of (x, y)

x is the ﬁrst number which is found in the x-axis y is the second number which is found in the y-axis

Quadrants in the coordinate plane

Now that we know how to graph points with positive coordinates, let’s see what happens if we extend the axes of a coordinate plane in both directions.

Interactive exploration

Explore online to answer the questions

Use the interactive exploration in 4.05 to answer these questions.

1. Drag P into the section labeled 1st quadrant. What do you notice about the coordiantes? Is that true for every point in the 1st quadrant?

2. Repeat for the other 3 quadrants. What do you notice about the points in each quadrant? Are your observations true for every point in that quadrant?

3. Drag P along the y-axis. What do you notice about the coordinates? Is that true for every point on the y-axis?

4. Drag P along the x-axis. What do you notice about the coordinates? What do you notice about the coordinates? Is that true for every point on the y-axis?

The coordinate plane is divided into four distinct regions, called quadrants. The 1st quadrant is on the top right. The x-coordinate and y-coordinate of a point in the 1st quadrant are both positive.

The quadrants are numbered in an counterclockwise direction:

• 2nd quadrant: x-coordinates are negative, y-coordinates are positive

• 3rd quadrant: both coordinates are negative

• 4th quadrant: x-coordinates are positive, y-coordinates are negative

Points that lie on an axis, like ( 5, 0) or (0, 4), are not in any quadrant.

Points on the x-axis have a y-coordinate of 0.

Points on the y-axis have an x-coordinate of 0.

y-axis

Quadrant 2

Quadrant 1

Origin x-axis

Quadrant 3

Quadrant 4

Example 6

What are the coordinates of the point shown in the coordinate plane?

Create a strategy

Follow the grid line up to the horizontal axis to identify the x-coordinate. Then, follow the grid line across the vertical axis to identify the y-coordinate.

Apply the idea

The number on the horizontal axis directly above the point is 4 and across the y-axis is 6. So, the coordinates are (4, 6).

Example 7

What are the coordinates of the point shown in the coordinate plane?

Give the coordinates in the form (x, y).

Create a strategy

Count the number of horizontal and vertical units required to move away from the origin and determine if it is in the positive or negative direction.

Apply the idea

The point is located 2 spaces to the left, then 1 space down. So, the coordinates are ( 2, 1).

Example 8

Plot the point ( 9, 3) on the coordinate plane.

Create a strategy

The ﬁrst coordinate tells us how far to the right (positive) or left (negative) the point is from the origin.

The second coordinate tells us how far above (positive) or below (negative) the point is from the origin.

Example 9

In which quadrant does the point (3, 2) lie?

Create a strategy

Recall the characteristic of each quadrants:

• 1st quadrant: positive x and positive y

• 2nd quadrant: negative x and positive y

• 3rd quadrant: negative x and negative y

• 4th quadrant: positive x and negative y

Apply the idea

Since the coordinates have positive x and negative y, the point (3, 2) lies in 4th quadrant.

Apply the idea

Starting from the origin, go 9 units in the left direction and then 3 units in the upward direction.

Reﬂect and check

We can plot the point (3, 2) to see which quadrant it is in:

Example 10

What is the distance between A (6, 8) and B (6, 4)?

Create a strategy

Since the x-coordinates are the same, ﬁnd the difference of the y-coordinates.

Apply the idea

Distance = 8 ( 4)

= 8 + 4

= 12 units

Subtract 4 from 8

Combine the adjacent signs

Evaluate

Reﬂect and check

We can check the distance between the two points by graphing:

Example 11

On which axis does point (0, 4) lie?

Create a strategy

Plot the point (0, 4) on a coordinate plane to determine which axis it lies on.

Apply the idea

We can observe that the point (0, 4) lies on the y-axis.

Reﬂect and check

If one of the coordinates is 0, then the point lies on the axis of the coordinate that is not 0.

For point (0, 4), the x-coordinate is 0 and the y-coordinate is 4. Therefore, the point must lie on the y-axis.

Idea summary

The coordinate plane is divided into 4 quadrants

y-axis

Quadrant 2

Quadrant 1

Origin x-axis

Quadrant 3

Quadrant 4

Points that lie on an axis, like ( 5, 0) or (0, 4), are not in any quadrant.

Points on the x-axis have a y-coordinate of 0.

Points on the y-axis have an x-coordinate of 0.

The point (0, 0) is the origin.

Practice

What do you remember?

Tell whether the integers below would be placed to the left or to the right of 0 on a number line. a 4 b 7 c 16 d 11 e 2

2 State whether each of the following statements is true or false.

a The coordinate plane is used to describe the location of actual points, not regions, in a two-dimensional space.

b The number line laying down horizontally on the plane is the y-axis.

c The vertical distance along the y-axis is the y-coordinate.

d The coordinates of the point at which the x-axis and the y-axis intercept are (0, 0).

e The x-coordinate and y-coordinate of any point in the ﬁrst quadrant are both positive.

3 Select the coordinate plane that correctly labels the origin, axes, and 4 quadrants.

Quadrant IV Quadrant I

Quadrant II

Quadrant I

Quadrant III

Quadrant IV

4 State the quadrant(s) that have:

a Points with a negative y-coordinate.

b Points with a negative x-coordinate.

Quadrant II

(0, 0)

Quadrant I Quadrant III

Quadrant I

Quadrant IV

Quadrant II

Quadrant IV

c Points where the x-coordinate and y-coordinate have the same sign.

d Points with a negative x-coordinate and a positive y-coordinate.

e Points with a negative x-coordinate and a negative y-coordinate.

5 Match each coordinate point to the quadrant it belongs to:

a (7, 8) i Quadrant III

b ( 3, 4) ii Quadrant IV

c (2, 1) iii Quadrant I

d ( 5, 6) iv Quadrant II

6 On which axis do the following points lie?

Quadrant III

a (0, 9) b (10, 0) c (0, 6) d ( 1, 0)

7 Which ordered pair best represents point A on the grid?

A ( 5, 4)

B (3, 7) C ( 6, 2) D (4, 5)

13 Consider the given points on the number plane:

Which graphed point is best represented by:

a (9, 3)

b ( 4, 7)

c (2, 4)

d ( 8, 8)

14 Rubiel placed the point of his marker on the origin of a regular coordinate plane. He marked a point after moving his marker 3 units to the right and 5 units down.

Which ordered pair identiﬁes where Rubiel marked his point?

A (3, 5)

B (3, 5) C ( 3, 5) D ( 3, 5)

15 Using the coordinate plane below, answer the following questions:

a Describe point P in terms of distance from the x- and y-axis.

b Describe point M in terms of distance from the x- and y-axis.

c Describe the distance from point Q to point S on their shared horizontal line.

d Describe the distance from point N to point L on their shared vertical line.

16 For each description below:

i Find the coordinate of the point.

a The point 9 units below the origin.

c The point 4 units to the left of ( 3, 6).

ii Graph the point in a coordinate plane.

b The point 3 units to the left of the origin.

d The point 7 units to the right of ( 1, 2).

e The point 2 units to the right and 2 units below the point (2, 5).

f The point 6 units to the left and 5 units above the point (4, 4).

Let’s extend our thinking

17 Graph the ordered pairs on the coordinate plane shown.

a ( 3.5, 6.5)

b (0, 4.7)

c (1, 8.2)

d ( 5.9, 9)

18 For each of the following:

i Determine the coordinate of the ordered pair based on the description provided.

ii Plot the point on the coordinate plane.

a The ordered pair is 8 units away from the y-axis.

b The ordered pair is located in the third quadrant.

c The y-value of the ordered pair is equal the x-value plus 5.

d To get from the ordered pair to the coordinates (10, 3), you would move 18 units to the right.

e The ordered pair is 3 units away from the x-axis.

Describe in words, how to get to the point ( 9, 4) from the point (10, 6) in the coordinate plane.

20 The locations of different places around Linda’s town are represented in the coordinate plane.

Which place is the farthest vertically from the origin?

Explain how you know.

21 A student graphed the following ordered pairs in the coordinate plane:

a (5, 7)

b ( 6, 4)

c (8, 1)

d ( 2, 0)

Find, correct, and explain the mistake the student made when graphing the ordered pairs.

Ratios & Proportional Relationships

Big ideas

5.01 Introduction to ratios

After this lesson, you will be able to...

• represent relationships as ratios using a/b, a:b and a to b notations.

• make part to part, part to whole, and whole to whole comparisons.

• write real-world situations to represent a ratio expressed symbolically.

The language of ratios

Exploration

Consider the following image that shows ingredients for a batch of smoothies:

1. What kind of comparisons can you make between the number of strawberries and the number of bananas in the image?

A ratio compares the relationship between two values. It tells us how much there is of one thing compared to another. In other words, a ratio is an association between two or more quantities.

The ratio of strawberries to bananas is 6 to 4. The ratio of bananas to strawberries is 4 to 6.

We can also associate the two quantities by saying for every 6 strawberries, there are 4 bananas. For every 4 bananas, there are 6 strawberries.

We can also write ratios in the form a : b which is read as “a to b”.

If we want to describe the relationship between the number strawberries to the number of bananas, we could write the ratio as 6 : 4.

Finally, ratios can be represented as a fraction.

We can describe the ratio of bananas to strawberries as the fraction

The order that the words are written correspond to the order of the values in the ratio, so it is important that pay attention to the order.

Example 1

Write the ratio of butterﬂies to ladybugs in three ways.

Create a strategy

Consider all the ways we talked about how to compare quantities using both words and symbols.

Apply the idea

The ratio of butterﬂies to ladybugs is 3 to 8. Using ratio notation, we can also express the ratio as either 3 : 8 or

Reﬂect and check

Consider whether 8 : 3 would describe the desired ratio. This would mean that there are 8 butterﬂies for every 3 ladybugs. This does not match the image since there should be more ladybugs than butterﬂies. The order of the numbers in a ratio is important.

Example 2

Lily describes the ratio between two quantities at the beach as . Use words to describe the relationship that the ratio may represent and draw a picture to represent the description of your ratio.

Create a strategy

Remember that ratios describe the relationships between two quantities. In a fraction, the numerator describes one quantitiy and the denominator describes the other.

Apply the idea

We have 5 of one quantity and 4 of another. At the beach, she could be noticing the number of sea shells and the number of sand dollars.

We could say there are 5 seashells for every 4 sand dollars; or the ratio of seashells to sand dollars is or 5 : 4.

Idea summary

A ratio compares the relationship between two quantities. It tells us how much there is of one thing compared to another.

We can also write ratios in the form a : b which is read as “a to b” or write them as the fraction

Ratios as comparisons

We can also use compare quantities by using part to part ratios, part to whole ratios, and whole to whole ratios These ratios can describe different relationships in the same scenario. Recall this recipe for smoothies:

A part to part ratio describes the ratio between two parts of a whole. In this recipe, the number of strawberries and the number of bananas are both parts of the whole group of fruit. The ratio of strawberries to bananas is 6 : 4.

A part to whole ratio describes the relationship between one quantitiy and the total group of quantities. For example, if we wanted to describe the ratio of bananas to the total amount of fruit in this recipe, we could say “for every 4 bananas, we need 10 total fruit for this batch of smoothies”. We could write it as 4 : 10.

We now want to decide between two recipes for a batch of smoothies. Consider these two recipes:

Recipe 1

Recipe 2

A whole to whole ratio is a ratio that compares the total of one quantity to the total of another. In this example, Recipe A has 10 total fruit, and Recipe B has 6 total fruit. So the ratio of the total fruit in Recipe A to the total fruit in Recipe B is 10 : 6.

A part to part ratio can also compare between parts of one group to parts of another group. For example, for every 6 strawberries needed in Recipe A, we need 4 strawberries for Recipe B. This ratio can be written as 6 : 4 or .

Apply the idea

In the tape diagram, Emma had 5 parts. We established that each part was worth 3 miles.

In total, Emma’s weekly mileage is 5 3 = 15 miles.

Example 5

The ratio 3 : 4 represents a comparison between items at a grocery store.

a Create a context using this ratio in a part to part comparison.

Create a strategy

A part to part ratio describes the ratio between two parts of a whole. Consider any two quantities that are “part” of the grocery store.

Apply the idea

The ratio of oranges to apples in the grocery store is 3 : 4. This means for every 3 oranges, there are 4 apples in the grocery store.

Reﬂect and check

Remember that the ratio 3 : 4 does not mean there are only 3 oranges and 4 apples in the grocery store.

This group would also represent the ratio 3 : 4 because for every 3 oranges, there are 4 apples.

b Create a context using this ratio in a part to whole comparison.

Create a strategy

A part to whole ratio describes the relationship between one quantity and the total group of quantities.

Apply the idea

We could compare the amount of white bread to the total amount of bread sold at the grocery store.

The ratio 3 : 4 would mean that for every 3 white breads sold, there are 4 total breads sold.

c Create a context using this ratio in a whole to whole comparison.

Create a strategy

A whole to whole ratio is a ratio that compares the total of one quantity to the total of another.

Apply the idea

We could compare the total number of frozen vegetables to the total number of fresh vegetables.

The ratio of 3 : 4 would mean that for every 3 frozen vegetables, there are 4 fresh vegetables.

Idea summary

A ratio can represent different comparisons within the same quantity or between different quantities.

• part to whole - compares part of a whole to the entire whole

• part to part - compares part of a whole to another part of the same whole

• whole to whole - compares all of one whole to all of another whole

Practice

What do you remember?

1 The ratio of vowels to consonants in a word is 4 to 9. Are there more vowels or consonants in the word? Explain.

2 Given a ratio of 3 cats to 5 dogs.

a What does the ﬁrst number in the ratio represent?

b What does the second number in the ratio represent?

c What is the ratio between 3 cats and 5 dogs?

3 You are comparing apples to oranges in a fruit bowl. Three-fourths of the fruit are apples. Is the fraction the same as ? Explain.

4 Which of the following represent the ratio of cats to dogs? Select all that apply.

9 cats to 5

5 Which ratio does not belong with the other three? Explain your reasoning. 3 parts to 7 parts 3 out of every 7 3 for each 7 3 for every 7

Let’s practice

6 Express the following ratios as word statements: a 4 : 3

7 Create a set of items with squares and stars. The set of items should represent a 2 : 3 ratio for the number of squares to the number of stars.

8 For each of the following diagrams:

i Write the ratio of circles to triangles.

ii Write the ratio of triangles to circles.

For each of the following diagrams:

i Write the ratio of shaded squares to unshaded squares.

ii Write the ratio of shaded squares to total squares.

For each of the following diagrams:

i Write the ratio of shaded sections to unshaded sections.

ii Write the ratio of shaded sections to total sections.

Consider each ratio written as a fraction.

In each fraction, the numerator represents one quantitiy and the denominator represents another quantity. Write a scenario that could match each ratio.

a b c d

Consider each scenario.

i Express each pair of quantities as a ratio written in the form a : b.

ii Express each pair of quantities as a ratio written as a fraction .

a There are 11 cats to 6 dogs.

b In a classroom there are 12 girls to 5 boys.

c When baking a cake, you need 4 cups of sugar for every 5 cups of ﬂour.

d There are 3 red marbles to 2 blue marbles.

e The Great Dane weighed twice as much as the Labrador.

f In a working day, a bricklayer lays 50 bricks every hour.

13 A student has two bags of sour candies. They separate them by color.

Bag 1: Bag 1 has 24 total candies. There are 4 green candies, 6 blue candies, 12 red candies and 2 yellow candies.

Bag 2: Bag 2 has 20 total candies. There are 4 green candies, 3 blue candies, 9 red candies and 4 yellow candies.

Find each of the ratios:

a Green to red in Bag 1

c Total in Bag 1 to total in Bag 2

14 For each of the following diagrams:

b Blue to red in Bag 2

d Total yellow to the total in Bag 1 & Bag 2

i Find the fraction of the shaded region of chickens to the unshaded region of chickens.

ii Find the fraction of the shaded region of chickens to the total number of chickens.

15 When making pasta, you need 5 cups of plain ﬂour to 8 eggs.

a Write down the ratio of eggs to ﬂour.

b Write down the ratio of ﬂour to eggs.

16 Use the table to write the ratio. Explain what the ratio means.

a dramas to movies

b comedies to movies

c movies : action

d movies : dramas

Movie Number

Drama 5

Comedy 6

Action 4

17 A student describes the relationship between two groups of objects at school with the ratio 2 : 11. Use words to describe what the student may be describing.

18 In a bag, there are 20 marbles. If there are 5 green marbles, 6 blue marbles, 2 red marbles, and 7 yellow marbles, ﬁnd the ratios of the colored marbles below:

a Green to red.

c Green to blue.

e Blue to green.

g Yellow to the total number of marbles.

b Blue to red.

d Red to blue.

f Green to the total number of marbles.

h Red to the total number of marbles.

Let’s extend our thinking

19 The ratio of your monthly allowance to your friend’s monthly allowance is 6 : 4. The monthly allowances total $80. The ratio 6 : 4 is diplayed using a tape diagram.

You Your friend

i How much money does each part represent? Justify your reasoning.

ii What is your allowance? Justify your reasoning.

iii What is your friend’s allowance? Justify your reasoning.

20 Marisol is comparing 2 different cookie recipes. The first recipe call for 3 cups of flour. The second recipe calls 2 cups of flour. She has 15 cups of flour and wants to make the same number of batches of each recipe.

a Draw a tape diagram that could represent this scenario.

b How many batches of each recipe can she make? Justify your reasoning.

21 A fruit salad contains 4 apples, 6 oranges, and 8 strawberries. Explain what the ratio 6 : 18 would represent.

22 Identify all of the ratios that could be used to represent the number of lawns that could be mowed in each number of hours.

Eddie can mow 4 lawns in 7 hours 4 : 7 4 to 7 8 to 14 2 : 3 1 : 2 12 : 21

23 A rectangular pool has a length of 12 meters and a width of 6 meters. Lucidia states that the ratio of the length to the width of the pool is 12 : 6. Annalee states that the ratio of the width to the length of the pool is 6 : 12. Determine who is correct and explain your reasoning.

24 A bridge is made up of 3 lengths, , , and , as shown in the ﬁgure:

a Find the total length of the bridge.

b Write the lengths of to as a ratio.

c Write the lengths of to as a ratio.

d Write the lengths of to to as a ratio.

25 James and Emma scored goals in their basketball game in the ratio 4 : 3.

a Find the fraction of the goals scored by James.

b Find the fraction of the goals scored by Emma.

26 Harris and Bella divided chocolates between them in the ratio 5 : 4.

a Find the fraction of chocolates that Bella gets.

b Find the fraction of chocolates that Harris gets.

c If there are 27 chocolates, describe how you might ﬁnd how many chocolates Harris will get?

A special type of equivalent ratio is a simpliﬁed ratio.

Simpliﬁed ratio

A ratio that has no equivalent ratio with smaller integer values

The two integers in the ratio have a greatest common factor of 1. Therefore, all ratios equivalent to the simpliﬁed ratio have to be multiples of it.

To simplify a ratio, we can identify the greatest common factor (or the largest integer that can evenly divide both numbers in the ratio) and divide both parts of the ratio by it.

Let’s say we have a recipe for 5 cakes using 25 cups of ﬂour. The ratio of the number of cakes to the number of cups of ﬂour is 5 : 25.

To simplify the ratio we can see that both 5 and 25 can be evenly divided by 5.

The simpliﬁed ratio tells us that one cake requires 5 cups of ﬂour. This is very useful information for planning ingredients for lots of cakes.

Example 1

The ratio of tables to chairs is 1 : 2. If there are 14 chairs, how many tables are there?

Create a strategy

Multiply the both sides of ratio by a number to get the equivalent ratio.

Apply the idea

The ratio 1 : 2 says that each table has two chairs.

1 : 2 = ⬚ : 14 Rewrite the equivalent ratio

1 : 2 = 1 ⬚ : 2 ⬚ What number should be multiplied to 2 to become 14

1 : 2 = 1 7 : 2 7 Multiply by 7 = 7 : 14 Evaulate

So, If there are 14 chairs, then we will need 7 tables.

Example 2

The ratio of players to teams is 60 : 10. If there are only 12 students present, how many teams can be made?

Create a strategy

Divide both sides of the ratio by a number to get the equivalent ratio.

Apply the idea

60 : 10 = 60 ÷ 5 : 10 ÷ 5 Divide by 5

= 12 : 2 Evaulate

So, If there are 12 students, then there will be 2 teams.

Example 3

Simplify the ratio 10 : 24.

Create a strategy

Simplify the ratio by dividing each part by a common factor to ﬁnd the equivalent ratio.

Apply the idea

Divide by a common factor

Evaluate

Example 4

Write 54 cents to $3.00 as a fully simpliﬁed ratio.

Create a strategy

Convert the dollar value into cents, then divide by a common factor to simplify.

Apply the idea

$1.00 = 100 cents

Convert dollar value into cents

Divide by 6

Evaluate

Idea summary

Two ratios are equivalent if one of the ratios can be increased or decreased by some multiple to be equal to the other ratio.

A ratio is a simpliﬁed ratio if there is no equivalent ratio with smaller integer values.

Ratio tables

We can use a ratio table to represent a series of equivalent ratios. For example, if a pie recipe calls for 1 tablespoon of brown sugar for every 2 cups of ﬂour, we could write this as a ratio: 2 : 1.

In a ratio table we have:

Sugar 2 4 6 8

Flour 1 2 3 4

We can also use a ratio table to help us determine unknown values. For example, if we wanted to ﬁnd out how much ﬂour is needed when we use 12 tablespoons of brown sugar, we have the following:

Sugar 2 4 6 8 12

Flour 1 2 3 4 ⬚

We can determine the corresponding amount of ﬂour to 12 tablespoons of brown sugar by ﬁnding equivalent ratios.

6 : 3 = 12 : ⬚ Equivalent ratios

6 : 3 = 6 2 : 3 2 Multiply both parts of the ratio by 2

= 12 : 6

Therefore, for every 12 tablespoons of brown sugar, we can use 6 cups of ﬂour. You may have noticed that there was another way to ﬁnd this using the table.

Sugar 2 4 6 8 12

Flour 1 2 3 4 ⬚

We can see in the table that for 4 cups of ﬂour we need 2 tablespoons of brown sugar, and for 8 cups of ﬂour we need 4 tablespoons of brown sugar.

We know that 4 + 8 = 12 so we could have added 2 + 4 to get the 6 tablespoons of brown sugar.

Example 5

The table shows the ratio of dogs to cats:

Dogs to Cats

9 : 5

18 : 10

27 :

45 : : 50 a Complete the table of equivalent ratios.

Create a strategy

We can ﬁnd the equivalent ratios by multiplying or dividing both sides of a ratio by the same value.

Apply the idea

9 : 5 = 9 3 : 5 3

= 27 : 15

Multiply by 3

Evaluate

= 9 5 : 5 5 Multiply by 5

= 45 : 25

Evaluate

= 9 ⋅ 10 : 5 ⋅ 10 Multiply by 10

= 90 : 50

Dogs to Cats

9 : 5

18 : 10

27 : 15

45 : 25

90 : 50

Evaluate

9 Complete the pairs of equivalent ratios:

b 20 to 5 = ⬚ to 10

30 : 115 = 6 : ⬚ d e 256 to ⬚ = 16 to 3

10 Simplify each ratio:

a 4 : 10

396 : ⬚ = 18 : 12

8 : 4 c 20 : 40

e 10 : 20 f 15 : 40

100 : 2000

18 : 22 h 36 : 48 i 16 : 24 j 45 : 35

35 : 5 l 450 : 350

11 Simplify each ratio:

12 Write as simpliﬁed ratios:

a 540c to $3.00

c 120 cm to 5 m

b 5 km to 420 m

d 11 years to 28 months

e 20 days to 5 weeks f 10 minutes to 120 seconds

g 15 000 mL to 5 L h 3 L to 39 000 mL

13 Complete the equivalent ratios:

14 The ratio of students to teachers competing in a charity race is 9 : 4. If 54 students take part in the race, how many teachers took part?

15 Tricia and Luigi invest money into a business in the ratio . If Tricia has invested $2400, how much has Luigi invested?

16 At a sporting event, the ratio of security guards to spectators is 1 to 120. If 8400 spectators attend the event, how many security guards will be required?

17 There are 6 fiction books and 18 nonfiction books on a shelf. Which three ratios represent the relationship of the number of fiction books to the total number of books on the shelf?

18 If each ratios are equivalent to 5 : 28, ﬁnd the value of x:

a 50 : x b 60 : x c 80 : x d x : 84

e 110 : x f x : 168 g x : 252 h x : 504

19 Complete the ratio table for the ratio 4 : 6.

20 Complete the ratio table for the ratio .

21 The table shows the ratio of dogs to cats:

a Complete the table of equivalent ratios.

b If there are 270 dogs, how many cats would there be?

c Simplify the ratio of the number of dogs to cats from part (b).

22 Sarah created the following ratio tables but made a mistake in each. Describe and correct her error in creating the ratio tables.

a b

23 Every 10 pens cost $4.

No. of pens 10 20 30 40 50 Cost ($)

a Complete the ratio table.

c How much would you expect to pay for 5 pens?

b Find the cost of buying 90 pens.

24 To convert US dollars, USD ($), to Philippine Peso, PHP, we can use the ratio table below. Every USD($) is worth 56 PHP.

USD ($) 1 2 3 4 5 PHP

a Complete the ratio table.

b How many pesos will you be able to buy with $11?

c Hector has just returned from holiday with 840 pesos. How many US dollars can he exchange this for?

Let’s extend our thinking

25 Two kinds of pine trees, Bristlecone and Aleppo, are planted in rows. In each row, the ratio of Bristlecone to Aleppo is 10 : 9. Altogether, 600 Bristlecone pine trees are planted. Explain how to ﬁnd the number of planted Aleppo pines.

26 A painter wants to create a certain color by mixing two different colors of paint, Vespa and Nitro, in the ratio 4 : 1. He uses 4 liters of the Nitro color.

a How many liters of the Vespa color must he use?

b How many liters of paint will he have altogether once the two colors are combined?

27 The ratio of y : x is 5 : 3. Find two more points that could fall on this line. x 0 3 9 y 0 5 15

28 Ryan and Valerie are preparing for a party. Ryan blows up 12 balloons in 15 minutes. Valerie blows up 24 balloons in 28 minutes. Assume that both keeps blowing up balloons at a constant rate.

a Complete the table for the number of balloons Ryan blows up for each time period:

Time (minutes) 0 15 45 60 No. of balloons 12

b Complete the table for the number of balloons Valerie blows up for each time period:

Time (minutes) 0 28 56 84 No. of balloons 96

c Who is blowing up balloons faster?

29 David and Justin both travel to school by riding scooters. David travels 8 km in 4 minutes. Justin travels 35 km in 16 minutes. Assume that both travel at a constant speed.

a Complete the table for the distance traveled by David for each time period: Time (minutes) 4 8 16

(kilometers) 8 24 32

b Complete the table for the distance traveled by Justin for each time period:

c Who travels faster? Explain your answer.

30 To convert US dollars, USD ($), to Japanese yen, JPY (¥), we can use the ratio table below: USD ($) 1 2 3 4 5

a Complete the ratio table.

b Oriana has $40.00, and wants to buy a dress that costs ¥4080. Explain whether she can afford the dress.

Since he can run 10 meters in 1 second, we say the unit rate is 10 meters per second or 10 m/s.

We can calculate how far the sprinter runs in 1 second by dividing the 100 meters evenly between the 10 seconds. Let’s represent that as a rate:

Sprinter’s speed =

Sprinter’s speed = 10 m/s

This calculation tells us that the sprinter runs 10 meters in one second.

A unit rate can also be found from a ratio. If a dance game at the arcade cost $1.25 to play for 5 minutes, the ratio of cost to time played is 1.25 : 5. To ﬁnd the unit rate, divide both quantities of the ratio by 5 to ﬁnd the ratio for a single minute:

The unit rate is $0.25 per minute played. To determine how much it would cost to play the dance game for 20 minutes, we can multiply our unit rate by 20:

It would cost $5.00 to play the dance game for 20 minutes.

Example 1

A tap ﬁlls up a 240-liter tub in 4 hours.

a Which is the compound unit for the rate of water ﬂow?

Create a strategy

The rate of water ﬂow represents the number of liters that ﬂows from the tap each hour.

b What is the ﬂow of the water as a unit rate?

Create a strategy

Apply the idea

The unit is liters per hour, L/hr.

We can ﬁnd the rate of water ﬂow in liters per hour by dividing the capacity of the tub in liters by the number of hours passed.

Apply the idea

Evaluate the division.

Example 2

A car travels 320 km in 4 hours.

a Complete the table of values.

Time taken (hours) 4 2 1 Distance traveled (kilometers) 320

Create a strategy

Since the time is being divided by 2 each time, divide the distance traveled by 2 for each new distance in the table.

Apply the idea

b What is the speed of the car as a unit rate?

Create a strategy

Use the table from part (a) to ﬁnd what the distance was after 1 hour.

Example 3

Henry bikes 45 miles in 3 hours.

a What is the speed of the bike in miles per hour?

Create a strategy

Divide 320 by 2

Evaluate

Divide 160 by 2

Evaluate

This means that the tables of values is given by:

Time taken (hours) 4 2 1 Distance traveled (kilometers) 320 160 80

Apply the idea

Divide 80 km by 1 hr

Evaluate

Miles per hour is the unit rate for speed. We need to ﬁnd the distance traveled in 1 hour.

Apply the idea

Substitute the values of the distance and time in the expression

Consider what factor to divide 3 by to get 1

Divide by to get the unit rate

Evaluate

Speed as unit rate

b If Henry travels at this constant rate, what distance will Henry travel in 2 hours?

Create a strategy

Set up a table from the ratio original and create equivalent ratios in the table to solve the problem.

Apply the idea

We can set up a table from the ratio 45 : 3.

miles 45 hours 3

Divide both values by 3 to ﬁnd the distance for 1 hour.

miles 15 45

hours 1 3

Use that unit rate to solve for the remaining missing values.

miles 15 30 45 60

hours 1 2 3 4

From the table, we can see that in 2 hours, Henry can travel 30 miles.

Example 4

Iain feels like buying some ice-cream for himself and his friends.

• A box of 6 Cornettos costs $7.20

• A box of 4 Paddle pops costs $6.40

a How much does each Cornetto cost?

Create a strategy

Find the unit price for each Cornetto.

Apply the idea

Cornettos cost $1.20 per piece.

Divide the total cost by the number of Cornettos

Divide the numerator and denominator by 6 to get the unit price

Evaluate

b How much does each Paddle pop cost?

Create a strategy

Find the unit price of Paddle pop.

Apply the idea

Paddle pops cost $1.60 per piece.

Divide the total cost by the number of Paddle pops

Divide the numerator and denominator by 4 to get the unit price

Evaluate

c Which type of ice cream is the better buy?

Create a strategy

The better buy is the ice cream is the one with a lower unit price.

Apply the idea

Compare the unit prices of Cornettos and Paddle pops and choose the lower unit price.

$1.40 < $1.60

Cornettos are a better buy than Paddle Pops.

Idea summary

A rate is a measure of how quickly one measurement changes with respect to another. When rates are expressed as a quantity with a denominator of 1, such as 2 feet per second or 5 miles per hour, they are called unit rates.

Practice

What do you remember?

1 Select the rate in each set:

2 The speed of a car is usually measured using the units mi/hr. Write suitable units for each rate:

a The price of gasoline

c Heart rate

3 Which rates are examples of unit rates?

b Goals scored in a soccer game

d Pay rate at a part-time job

4 Consider each scenario.

i Describe each situation in your own words.

ii Write the rate indicated by each situation. What are the units?

February 2019 Length: 108 in February 2020 Length: 120 in

5 Complete the following pairs of equivalent ratios: a

6 A sprinter runs 100 m in 10 seconds. Determine whether the statements are true:

a The quantities compared are distance and time. b The unit of speed can be expressed as s/m

c The unit rate is 10 m/s d In 10 seconds, the runner runs 1 m.

Let’s practice

7 Justin earns $420 in 6 hours.

a What two quantities are being compared?

b What is the ratio for the two quantities of Justin’s earnings?

c What is the unit rate?

8 A three episode TV series had a total run time of 72 minutes. Find the rate of run time per episode.

9 A carton of 10 eggs cost $6.50. Calculate the cost of 1 egg in cents.

10 A 0.75 L bottle of milk costs $1.05. Calculate the cost of milk per liter.

11 Lynette can read 35 pages of books in 20 minutes.

a How many pages of books can she read in 1 minute?

b Create a table of values for the situation.

Population in City

12 Write the rate as a unit rate:

a 1040 books per 13 libraries

c 234 kilometers per 4.5 hours

13 For each table, ﬁnd the unit rate.

14 For each scenario:

i Find the unit rate.

a Tricia reads 20 pages of her book in 2 hours.

b 85 people per 5 buses

d 9 cakes per 72 guests

ii Complete the table

b A car travels 200 mi in 4 hours.

c A toy manufacturer makes 720 toys in 9 hours.

d Tina burns 640 calories by jogging 8 km.

15 Create a scenario that matches each set of double number lines.

16 Myrtle drives the same number of miles to and from work each day, as shown on the graph below.

Miles

to Work 2 days 3 days 4 days 5 days

Based on the graph, what is the unit rate of miles driven per day?

Explain your thinking.

17 The seventh grade choir sold pizzas as a fundraiser. The choir teacher created the graph below for the students.

Pizza Sale Proﬁts x

4 Pizzas 8 Pizzas

Based on the graph, what is the unit rate of proﬁt for the pizzas? A $0.56 per pizza B $4.00 per pizza C 18 pizzas per $100 D $1.80 per pizza

18 Tina counts 26 heart beats over a 20-second interval. Find Tina’s heart beat in one minute.

19 A cyclist travels 70 yards per 10 seconds.

a Find the distance the cyclist travels in 1 minute.

b Create a table of values for the situation.

20 A 1.35 kg pack of hamburger costs $15. If 1 kg = 1000 g how many grams of hamburger does each dollar buy?

21 Luke writes 2400 words in 20 minutes.

a Determine Luke’s writing speed in words per second.

b Create a table of values for the situation.

22 Examine the table.

Is the rate between time spent on a cell phone and the cost of Wi-Fi constant? Explain your reasoning.

23 Jackson and Carlos were running around a track. They started running at the same time. When Jackson had run 5 laps, Carlos had run 2 laps. The table shows the laps Jackson and Carlos completed.

Jackson Carlos 5 2

Is the ratio between Jackson’s and Carlos’ number of laps consistent? Explain your reasoning.

24 A supermarket sells two different brands of eggs, Happy Hens and Sunny Side Up.

• Happy Hens eggs cost $6.00 for 12 eggs.

• Sunny Side Up eggs cost $1.50 for 5 eggs.

a Calculate the cost in cents per egg from Happy Hens.

b Calculate the cost in cents per egg from Sunny Side Up.

c Which brand sells its eggs at a cheaper price?

25 Jade and Blake create videos at a constant rate each week. The proportional relationship between the number of videos created each week by Jade and Blake is represented in the tables below.

a Complete the missing value in each ratio table.

b Determine who is making videos at a faster rate.

Jade Blake

26 A store has two different brands of trash bags. Which brand of trash bags is the better deal? Explain your thinking.

Let’s extend our thinking

27 Explain how to convert 120 km/hr into meters per minute.

28 Neville is deciding between four cars to buy. He wants to buy the most fuel eﬃcient one.

a Complete the table:

b Which car is the most fuel eﬃcient?

29 Iain feels like buying some ice-cream for himself and his friends.

• A box of 8 Cornettos costs $18.00

• A box of 5 Magnums costs $7.25

Which type of ice-cream is cheaper individually? By how much?

30 Isabelle is buying juice for her nephew’s birthday party.

• A 3.2 L bottle of apple juice costs $13.76

• A 2.1 L bottle of orange juice costs $6.30

Which juice is the best buy? Justify your reasoning.

31 A tap ﬁlls up a 240 L tub in 4 hours.

a Find the rate of water ﬂow of the tap.

b Complete the table of values: Time (hours) 4 3 2 Liters of water 240

c How many liters of water can a tap ﬁll up in hours? Explain your reasoning.

32 Avril is running from John in a game of tag. Avril runs 30 yd every 5 seconds and John runs 21 yd every 3 seconds.

Who is faster? Explain your answer.

33 Adam really likes apples and eats 4 per day.

a Find the number of apples that Adam eats in one week.

b If Adam buys 44 apples, how many days will that last him?

34 A worm takes 15 seconds to travel 20 cm.

a Find the unit rate of the worm’s distance traveled in meters per second.

b Find the rate of the worm’s time spent in seconds per meter distance traveled.

c Explain why these two numbers are different.

35 Consider the scenarios:

• Han paid $124.62 for 67 L of petrol in Humbleton.

• Amelia paid $82.08 for 57 L of petrol in Dunkilderry.

a Calculate the cost per liter of petrol in Humbleton.

b Calculate the cost per liter of petrol in Dunkilderry.

c In which suburb is petrol cheaper?

d Han buys 30 L in Statesota, and this petrol station matched the best price between Humbleton and Dunkilderry. How much did he pay?

5.04 Proportional relationships

After this lesson, you will be able to...

• identify a proportional relationship when given a table of values, graph or context.

• find the unit rate and create (or complete) a table of values or graph for a proportional relationship representing a real-world context.

• compare multiple representations of the same proportional relationship using verbal descriptions, ratio tables, and graphs.

Proportional relationships

We say that two quantities have a proportional relationship if the values are always represented by the same ratio. When two quantities are proportional, we can use a ratio table to show equivalent ratios and ﬁnd unknown values. If a cookie recipe calls for 2 cups of sugar for every 4 cups of ﬂour, we could write this as the ratio 4 : 2.

Putting this in a ratio table, we have:

This relationship is proportional because the ratio, , of ﬂour to sugar is constant:

Let’s consider another scenario. The cost to rent a scooter and time rented are shown in the table below:

This relationship is not proportional because the ratios between cost and time are not constant:

We see what each relationship looks like in a graph by turning each column from the table into an ordered pair (x, y).

Graph shows the proportional relationship between cups of ﬂour and cups of sugar.

Graph shows the non-proportional relationship between cost of a scooter and time.

In both graphs, we connected the points with a line because the values in between the points make sense for the context. Fore example, we could make a recipe with 3.5 cups of sugar or (depending on the renting rules) we could probably rent a scooter for 7.25 minutes.

If the contexts were changed so the values in between the points did not make sense, we would not connect them. For example, if we were comparing the ratio of flour : eggs in a recipe we would probably not use a fraction of an egg. Or if the scooter rental only allowed us to rent for speciﬁc amounts of time, then it would not make sense to calculate cost times outside of that.

We call the relationship continuous if it would make sense to include the values between points and discrete if only the points make sense in the context.

Two different graphs can be represented by the data depending on the order we choose. The above graph shows the relationship between cups of ﬂour and cups of sugar where y represents the cups of ﬂour and x represents the cups of sugar. This graph represents the ratio of y to x as 4 : 2. 2 1 3 4 5 6 7 8 9 10 11

We can also create a graph for the ratio of y to x as 2 : 4 where y represents the cups of sugar and x represents the cups of ﬂour. Notice the similarities and differences between the two graphs. They both pass through (0, 0) but the ratio 4 : 2 is much steeper than the ratio 2 : 4.

It is important to state which quantity is represented by x and which quantity is represented by y

A graph is proportional if the graph is linear, meaning it looks like a straight line, and it passes through the origin, (0, 0).

The graph of a proportional relationship always includes (0, 0) because we can create the equivalent ratio 0 : 0 by multiplying both parts of the ratio by 0. For the sugar and ﬂour example, this would mean that a recipe that calls for 0 cups of ﬂour would need 0 cups of sugar.

Example 1

If the ratio of y to x is represented by 3 : 1, plot the ratio as a single point on the coordinate plane.

Create a strategy

A ratio of the form y : x, means that the x represents the horizontal position and y the vertical position of the point.

Apply the idea

The horizontal position is 1, and the vertical position is 3.

Example 2

Consider the given graph:

a What ratio of y : x has been plotted?

Create a strategy

Use the graph to ﬁnd the ratio, y : x. Often, we choose the point where x = 1 if y is a whole number as well. This will give us the most simpliﬁed ratio.

Apply the idea

The point (1, 2) lies on the line. At this point x = 1 and y = 2. So, the ratio is 2 : 1.

Reﬂect and check

We could have chosen any point on the line in our graph. They all represent equivalent ratios. For example, if we chose the point (5, 10), we know:

So, the ratio 10 : 5 is an equivalent ratio to 2 : 1.

Apply the idea

Fractions are equivalent if we can multiply the numerator and denominator by the same number. Let’s determine if the ratio of each coordinate pair is equivalent to the unit rate:

All of the ratios are equal because we can create each fraction by multiplying by a constant. Since the ratio of y : x is constant, this table represents a proportional relationship.

Create a strategy

In a proportional relationship, the coordinates will fall on a straight line that goes through the origin.

Create a strategy

In a proportional relationship, the coordinates will fall on a straight line which goes through the origin.

Apply the idea

The graph shows a straight line that goes through the origin. Therefore, this graph represents a proportional relationship.

Apply the idea

The graph shows a straight line that does not go through the origin. Therefore, this graph does not represent a proportional relationship.

e Mark has $500 in savings and the amount in his savings account doubles every year.

Create a strategy

In a proportional relationship, a situation with increase or decrease at the same rate. It will also make sense to have 0 of both quantities.

Apply the idea

At the start (time 0), Mark had saved $500. If the amount of money doubles every year, this is not a constant rate. We can create a table of values to conﬁrm this:

The ﬁrst year, the account grows by $500. The next year, the account grows by $1000. Since the rate of increase is not constant and there are $500 in his account at the beginning, this situation is not proportional.

Reﬂect and check

We could have determined this relationship was not proportional from the start. The point (0, 0) could not ﬁt this relationship, because Mark has $500 at the start which means this relationship begins at (0, 500).

f It just started raining, and it rains half an inch every hour all day.

Create a strategy

In a proportional relationship, a situation with increase or decrease at the same rate. It will also ﬁt the pattern to have 0 of both quantities at some point.

Apply the idea

At time 0, there were 0 inches of rain. Then, the rain fell the same amount each hour.

Since the rate of increase is constant and there are 0 inches of rain at the beginning, this situation is proportional.

Example 4

The ratio table represents the proportional relationship between number of pens purchased and cost. a Complete the table:

Create a strategy

In a ratio table, every column must represent an equivalent ratio. We can multiply or divide the number of pens by any value as long as we do the same to the cost in that column.

Apply the idea

Since 20 pens cost $11.60, 10 pens (half of 20) cost $11.60 ÷ 2 = $5.80.

30 pens cost 3 times more than 10 pens.

40 pens cost 4 times more than 10 pens.

58 = 29 2, so the number of pens is 100.

Reﬂect and check

We could have also found the unit rate by dividing the cost by the number of pens.

We could then multiply $0.58 by any number of pens to ﬁnd the total cost.

b Calculate the cost of buying 90 pens.

Create a strategy

We know the cost of of 10 pens is $5.80. Consider how many groups of 10 pens we have.

c How much would you expect to pay for 5 pens?

Create a strategy

Apply the idea

Cost = 9 ⋅ 5.80

5 pens is half the amount of 10 pens so they will cost half as much.

Apply the idea

Cost = 5.80 ÷ 2 Halve the cost of 10 pens = $2.90 Evaluate

Example 5

Zoe eats 6 sour candies every minute.

a Does this situation represent a proportional relationship?

Create a strategy

In a proportional relationship, a situation will increase or decrease at the same rate. It will also ﬁt the pattern to have 0 of both quantities at some point.

Find the cost of 9 lots of 10 pens

= $52.20 Evaluate

Reﬂect and check

We could also use the unit rate we found in part (a). The unit rate was $0.58. For 5 pens:

$0.58 5 = $2.90

This is the same result as our other strategy.

Apply the idea

If 0 minutes has passed, Zoe will have eaten 0 candies. The ratio of candy eaten to time is always 6 : 1 so this situation represents a proportional relationship.

b Create a table and a graph that represent this situation.

Create a strategy

Let y be the number of candies Zoe eats and x be the time in minutes. We can use the ratio of y : x, which is 6 : 1, to create both a table and a graph.

Apply the idea

The ratio of candies to time is always 6 : 1 in this table: x (Time)

To graph, we will assume that Zoe is continuously eating the candy. We will use our ratio to graph and connect our points with a line:

Reﬂect and check

y (1, 6)

In this problem, we assumed Zoe was eating the candy continuously, so we drew the line on the graph to connect the points. If we instead knew that she was eating each piece instantaneously at each minute we would not connect the points because the values in between would have no meaning since she does not eat additional candy between each minute.

c Compare the characteristics that are easier or harder to see from the written context, the table, and the graph.

Create a strategy

Consider the different parts of a proportional relationship that are important. Use the representation from parts (a) and (b) to answer the question.

Apply the idea

The context makes it easy to see the connection between the ratio and real life. We can easily see the unit rate of 6 candies per minute. It is harder to see that the situation includes the point (0, 0).

The table highlights that every ratio of y : x is equivalent and we can see how many candies were eaten for each of the different minutes. However, the table won’t have all of the possible values.

The graph makes it easy to see that the relationship goes through the origin. It is easy to see that the relationship increases at a constant rate because it is linear, but it takes a little more work to see what the rate actually is.

Idea summary

We can use ratio tables to determine unknown values by multiplying or dividing. All of the values in the table will be equivalent ratios.

The graph of a proportional relationship (if extended far enough) is a straight line that passess through the origin (0, 0).

Practice

What do you remember?

1 For each of the following representations, how can you tell if a relationship is proportional? i Graph ii Table iii Context

2 Find the missing value in each case, given that the two quantities are in proportion: a b c d

3 If the following ratios are equivalent to 5 : 8, ﬁnd the missing value:

4 Given that the relationship is proportional, explain how to use unit rate to ﬁnd the value missing in the table.

x 5 7 10 ⬚ 15

y 15 21 30 36 45

5 Which of the following represents a proportional relationship?

A The number of pages in a book and the color of the book cover.

B The weight of a bag of apples and the number of apples in the bag.

C The height of a building and the number of doors in the building.

D The amount of gas in a car’s tank and the color of the car.

6 Sarah drove 180 miles in 3 hours to reach her destination. What was her average speed for the trip?

B C D

7 Consider the given table:

x 0 1 2 3

y 0 3 6 9

a Graph the relationship shown by the table.

b Does the relationship include (0, 0)?

c Is the ratio between the values of y and corresponding values of x consistent? If so, determine the ratio of y : x

d State whether the relationship is proportional.

8 The number of cupcakes eaten by guests at a party is shown on the graph:

a How many cupcakes are eaten by 3 guests?

b How many cupcakes are eaten by 2 guests?

c Is the ratio between the number of guests and the corresponding values of cupcakes eaten consistent? If so, determine the ratio of x : y that has been plotted.

d State whether the graph represents a proportional relationship.

9 Match each table with its missing value.

Let’s practice

10 Consider the following tables of proportional relationships.

i Find the unit rate.

ii Complete the missing values to show a proportional relationship between x and y

11 A trail mix recipe states that 2 cups of peanuts should be mixed with a of a cup of raisins. Complete the given table.

12 It requires approximately 8 mL of polyethylene to create 20 plastic bags.

a Find the unit rate of polyethylene used per plastic bag.

b Complete the following table to demonstrate this relationship.

c How much polyethylene would be required to create 5 plastic bags?

13 Ivan can wash 75 cars in 5 days.

a Create a graph that matches this scenario.

b At this rate, how many cars can Ivan wash in 11 days?

14 A bus travels at a speed of 80 km/hr. How far does the bus travel in 5 hours?

15 State whether each situation is proportional. Explain your reasoning.

a Lisa is trying to save money. She starts with no money in her account. After 3 weeks, she had saved $234.21. After 8 weeks, she had saved $624.56.

b The amount of money a restaurant makes doubles every hour.

c Two games cost $15. Five games cost $37.50.

d Two chickens laid 7 eggs. Five chickens 13.

16 For each of the following table of values, state whether or not they represent a proportional relationship.

a b

c d e f

17 Examine the table.

Does a proportional relationship exist between the time spent on a cell phone and the cost of Wi-Fi? Explain your reasoning.

18 The cost of eggs is shown in the table below.

a Find the cost per egg for each quantity: i 12 eggs ii 36 eggs

b Is the cost proportional to the number of eggs?

19 A tap ﬁlls up a 240 L tub in 4 hours.

a Find the rate of water ﬂow of the tap.

b Complete the table of values:

c How many liters of water can a tap ﬁll up in hours?

20 Harry and Carl love reading. They both read at a constant rate. Harry reads 16 books every 12 weeks. Carl has kept a table of his reading habits which is shown below:

Number of weeks 12 24 36 48 Number of books read 20 40 60 80

a Complete the following table for Harry:

b Determine who reads more quickly.

21 The tables represent a proportional relationship with a constant unit rate of change of y with respect to x.

Table 1: Table 2:

Which table has a greater unit rate of change of y with respect to x?

22 State whether each of the following graphs represents a proportional relationship. a b

23 For each of the following graphs:

i Determine the ratio y : x

ii Describe the ratio as a unit rate for the given values.

a The graph shows the amount of time it takes Kate to make beaded bracelets.

b The graph shows the number of liters of gas used by a ﬁghter jet per second.

c The graph shows the number of liters of ice cream per tub.

d The graph shows the distance Natalia swam per minute.

24 For each graph:

i Is the relationship shown proportional?

ii Explain your reasoning?

25 The ratio of y : x is 6 : 2. By ﬁnding two points that represent equivalent ratios to 6 : 2, graph the ratio on a coordinate plane.

26 The table and the graph represent a proportional relationship with a constant unit rate of change of y with respect to x.

Which describes a greater unit rate of change of y with respect to x, the table or the graph?

Let’s extend our thinking

27 The coordinate plane shows the parking fee for various numbers of hours parked:

a Does the graph represents a proportional relationship? Explain your reasoning.

b What should have been the hourly cost of parking if the ﬁrst three ordered pairs are only considered in the graph?

c If the relationship is proportional, the parking fee of $18 should be for how many hours?

28 The graph shows the cost, in dollars, of printing x digital photos:

a Is the cost of printing digital photos proportional to the number of photos printed?

b Determine the ratio of the number of photos to the cost of printing digital photos.

c At this rate, how much would it cost to print 70 digital photos?

29 The table and the graph represent a proportional relationship with a constant unit rate of change of y with respect to x

Murphy says that the table has a greater unit rate than the graph because the unit rate of the table is 3 while the graph is only 2.2.

Raven said that the table and the graph has the same unit rate which is 2.2. Which student is incorrect? Explain.

30 The cost y of buying x pounds of fruit is shown on the graph. One graph shows the cost of buying x pounds of apples, and the other shows the cost of buying x pounds of bananas.

a Which graph has the greater unit rate?

b Which fruit costs more per pound?

31 Consider the scenarios:

• Han paid $124.62 for 67 L of petrol in Humbleton.

• Amelia paid $82.08 for 57 L of petrol in Dunkilderry.

a Calculate the cost per liter of petrol in Humbleton.

b Calculate the cost per liter of petrol in Dunkilderry.

c In which suburb is petrol cheaper?

d Han buys 30 L in Statesota, and this petrol station matched the best price between Humbleton and Dunkilderry. How much did he pay?

Equations & Inequalities

Big ideas

• Expressions are the building blocks of algebra. They can be used to represent and interpret real-world situations.

• The properties of real numbers can be applied to many types of expressions.

• An equals sign indicates an equivalent relationship between two expressions.

• A solution set is the collection of all values that make an equation or inequality true.

6.01 Algebraic expressions

After this lesson, you will be able to...

• identify examples of an equation, variable, expression, term, and coefficient.

• write examples of an equation, variable, expression, term, and coefficient.

• represent algebraic expressions with visual models.

Translate algebraic expressions

We use algebraic expressions when we want to write a number sentence but we don’t know one of the numbers involved.

For example: What is the total weight of a cat and a 10 lb weight?

In this case, let’s use c for the weight of the cat.

c + 10 is called an algebraic expression which is an expression that contains at least one variable.

c is called a variable. This is a symbol used to represent an unknown quantity.

Coeﬃcients are the numerical factor in a term and are used to show how many variables we have. The variable u with a coeﬃcient of 3 is written as 3u which means 3 u.

3 the coeﬃcient u the variable

Terms are a number, variable, product, and/or quotient in an expression. They are the building blocks of an expression. Terms are separated by + or signs.

Consider the expression:

• This is an expression with 2 terms.

• The term has a coeﬃcient of . The negative belongs with the coeﬃcient.

• The term 5 has no variable. It is called a constant term.

Exploration

In order to write an expression that can be used to model the total cost of a home renovation project, Ms. Chen deﬁnes the variables:

Let w represent the cost replacing a window, and p represent the cost of painting a room.

1. What could these expressions represent in this context?

2. In this context, what do the coeﬃcients describe?

3. What expressions could we write that wouldn’t make sense in this context?

Expressions and parts of expressions, like factors and coeﬃcients, all have unique meanings in a given context. Viewing expressions in parts and as a whole while paying attention to the quantities represented by the variables can explain the relationships described by the expressions.

Example 1

For the algebraic expression 4x + 23:

a Determine the number of terms.

Create a strategy

Terms are separated by plus or minus signs in the expression.

b Identify the coeﬃcient of the ﬁrst term.

Create a strategy

The coeﬃcient of a term is the number that is multiplied by the variable in the term.

c Identify the constant term.

Create a strategy

The constant term in an algebraic expression is the term that does not contain any variable.

Example 2

Apply the idea

The algebraic expression 4x + 23 contains two terms: 4x and 23.

Apply the idea

The first term is 4x, so the coefficient of the first term is 4.

Apply the idea

In the expression 4x + 23, the constant term is 23.

A local fruit stand charges $3 per pineapple. Write an algebraic expression for the total cost of purchasing p pineapples.

Create a strategy

The total cost changes based on the number of pineapples purchased.

Apply the idea

The total cost is $3 times the number of pineapples purchased. This can be represented by the algebraic expression of 3p

Example 3

Write an algebraic expression for the phrase “seven more than twice x”.

Create a strategy

Translate the terms into mathematical symbols and operations.

Apply the idea

The phrase “seven more than” indicates that we need to add 7. The “twice” means multiply by 2, so “twice x” is 2x

We can combine the whole description into a single expression:

2x + 7

Example 4

The perimeter of a square can be written as 4s. Explain what each part of the expression represents.

Create a strategy

First, we need to identify the two parts of the expression. The coeﬃcient is 4 and the variable is s

We know that the perimeter of an object is the distance around the outside edges and a square has 4 sides of equal length.

Apply the idea

Perimeter = 4s

We can see from the perimeter formula that there are 4 of an unknown quantity s

The coeﬃcient 4 represents the 4 equal length sides of the square.

For 4s to be the perimeter, s must represent the length of one side of the square.

Idea summary

Reﬂect and check

Another way to represent the perimeter of a square is s + s + s + s. This shows that to ﬁnd the perimeter of a square, we just need to add the side length to itself 4 times.

Expressions can be used to represent mathematical relationships. In an expression, sums often represent totals and coeﬃcients and factors represent multiplication. When interpreting an expression in context, we can use the units to help understand the meaning.

Represent algebraic expressions

We can use algebra tiles to help us visualize algebraic expressions. The tile x represents an unknown number. The tile +1 represents adding one unit and 1 represents subtracting one unit.

This table demonstrates how expressions can be built using the tiles:

Word Expression Algebraic Expression Representative with Algebra Tiles

three more than x x + 3

three less than x x 3

x +1 +1 +1

x 1 1 1

the product of x and three x × 3

Algebra tiles can also help us identify the terms of the equivalent algebraic expression. Let’s break down the algebra tiles below.

Notice that there are two different types of algebra tiles. These represent the two terms in the expression.

The ﬁrst term in blue, are the two tiles with the +x. This represents the term 2x where the coeﬃcient is the 2 and the variable is the x.

The second term in green, are the ﬁve tiles with the +1. This represents the term 5.

When we add them together, we get the algebraic expression 2x + 5.

Example 5

Write an equivalent algebraic expression and identify each term for the following:

Create a strategy

There are many ways to write expressions that are algebraically equivalent by rearranging the terms and combining like terms, but for simplicity, we’ll directly reﬂect the layout shown by the tiles.

Apply the idea

From the image, we have one positive variable tile and four positive unit tiles. To express this algebraically we can write:

x + 1 + 1 + 1 + 1

Another way to write the expression is to count up the +1 tiles and show that we have 4 in total: x + 4

There are two terms in this expression and they are the x and the 4. These terms are seperated by the + sign.

Example 6

Represent the expression 2x 5 using algebra tiles.

Create a strategy

We can use negative variable tiles and negative unit tiles to represent the expression.

Apply the idea

Idea summary

We can represent algebraic expressions with visual models to better understand them. We can rearrange models of algebraic expressions to generate equivalent expressions.

Practice

What do you remember?

1 What is a variable used for in algebra?

2 Farmer Jones has sheep, cows, goats and chickens on his farm. Use variables to write an expression for the total number of animals on the farm.

3 Match the following terms with the correct example from the options: i Variable ii Coeﬃcient iii Expression iv Constant

a 12 in the term 12x b b in the term 5b c 10

4 Consider the following diagram:

a An algebraic expression using addition to represent the diagram is ⬚ + ⬚ + ⬚ + ⬚ + ⬚

b An algebraic expression using multiplication to represent the diagram is 5 ⋅ ⬚

c Write the answer to part (b) in another way.

5 Use algebra tiles to draw a diagram that represents each expression:

Let’s practice

6 Examine the given diagram.

a Use the diagram to label each part with the correct algebraic term. Variable Coeﬃcient Expression Constant

b What algebraic term describes the entire mathematical statement x + 8 = 16?

c How many terms are in x + 8?

7 Write down the number of terms in the following expressions:

8 What is the coeﬃcient in the following terms?

9 What is the constant term in the following expressions?

10 What is the constant term in the following equations?

11 What is the difference between an expression and an equation? Provide an example of each in your explanation.

12 Identify the following as equation or expression.

13 In the equation 5xy 3y = 45x + 6, list all of the: a Coeﬃcients b Terms c Expressions d Variables

14 Write a simpliﬁed algebraic expression for the following diagrams: a Each single box represents p. b Each single box represents b. c Each single box represents y. d Each purple box represents p and each green box represents g

15 Write a simpliﬁed algebraic expression for the following diagrams:

16 Provide an example of each of the following vocabulary terms.

Equation

Variable

Expression

Term

Coefficient

17 Describe and correct the error in identifying the number of terms, coeﬃcients, and constants in the algebraic expression

• 3 Coeﬃcients: , x, and y

• 1 Term: xy

• 0 Constants

Let’s extend our thinking

18 Write an algebraic expression for each of the following statements without using a multiplication or division sign:

a Seven multiplied by x

c Three less than 8y

e The quotient of 8u and 9

g 7 less than the quotient of 5 and x

b Eight more than 5x

d The product of 9 and b

f Four times y take away 9

h The sum of a and 4, divided by 6

19 Write a word statement for each of the following expressions:

a 2b + 18 b 3x 5

20 A teacher asked his students to write an expression to represent the number of red and yellow marbles in the jar.

Delaney used the expression r + y to represent the total number of red and yellow marbles. Dalton used 9r + 6y to represent the the total number of red and yellow marbles.

Who is correct? Explain your thinking using algebraic vocabulary.

21 A recipe for a cake calls for 2 cups of ﬂour for every 3 eggs. Write an algebraic expression for the number of cups of ﬂour needed for n eggs.

22 The sum of the ages of a mother and her daughter is 42 years. If the mother’s age is q years, write an algebraic expression for the daughter’s age.

23 Write down an algebraic expression for:

a The total value of m coins, where each coin is worth p cents.

b The length of string remaining when two pieces of length K cm are cut from a piece of string originally L cm long.

24 Describe and correct the error in writing the phrase as an expression.

a The quotient of 8 and a number y:

b 16 decreased by a number:

6.02 Properties of real numbers

After this lesson, you will be able to...

• identify and apply the associative, commutative, inverse, and identity properties of addition and multiplication.

Commutative properties

Interactive exploration

Explore online to answer the questions

mathspace.co

Use the interactive exploration in 6.02 to answer these questions.

1. What do you notice when you change the order of the addition?

2. What do you notice when you change the order of the multiplication?

The commutative properties of real numbers are:

Property

Symbols Example

Commutative property of addition a + b = b + a 3 + 6 = 6 + ( 3)

Commutative property of multiplication a b = b a 6 ( 3) = ( 3) 6

The commutative property is the reason that we can add numbers in any order or multiply numbers in any order. Keep in mind that while addition and multipication are commutative, subtraction and division are not.

Let’s see why the commutative property applied with subtraction does not work.

7 4 ≠ 4 7 Apply the commutative property with subtraction

3 ≠ 3

Simplify

Notice that if we applied the commutative property with subtraction, the left and right side are not equal. We can adjust this by turning the subtraction into a addition of a negative.

7 4 = 7 + ( 4) Rewrite subtraction as addition

7 4 = 4 + 7 Apply the commutative property of addition

3 = 3

Simplify

We can see that when we convert the subtraction operation to an addition operation, we can still apply the commutative property of addition.

Let’s see why the commutative property applied with division does not work. Apply the commutative property with division

Simplify

Notice that if we applied the commutative property with division, the left and right side are not equal. We can adjust this by turning the division into a multiplication of it’s reciprocal.

Rewrite division as multiplication

Apply the commutative property of multiplication

Simplify

We can see that when we convert the division operation to an multiplication operation, we can still apply the commutative property of multiplication.

The commutative property can be applied to help us evaluate expressions more easily.

Example 1

Find the value of:

+ ( 5) + 5

Create a strategy

Since we can add numbers in any order, let’s add 5 and 5 ﬁrst to make the calculation easier.

Apply the idea

6 + ( 5) + 5 = 5 + 5 + 6

= 0 + 6

= 6

Reﬂect and check

Rewrite using the commutative property of addition

Evaluate 5 + 5

Evaluate

We could apply a different property to get the same result:

6 + ( 5) + 5 = 6 + (( 5) + 5) Rewrite using the associative property of addition

= 6 + 0

= 6

Evaluate 5 + 5

Evaluate

2

Use the commutative property of addition to ﬁll in the missing number:

19 + ( 15) = ( 15) + ⬚

Create a strategy

The commutative property of addition means that when we add two numbers, it does not matter what order we add them.

Apply the idea

We want to write 19 + ( 15) the opposite way around.

19 + ( 15) = ( 15) + 19

Example

Associative properties

The associative properties of real numbers are:

Property

Symbols

Example

Associative property of addition a + (b + c) = (a + b) + c 6 + (( 3) + 2) = (6 + ( 3)) + 2

Associative property of multiplication a (b c) = (a b) c 6 ( 3 2) = (6 ( 3)) 2

The associative property is the reason that we can group sums or products of numbers differently and the result remains the same. While addition and multiplication are associative, subtraction and division are not.

Let’s see why the associative property applied with subtraction does not work.

(7 4) + 3 ≠ 7 (4 + 3)

3 + 3 ≠ 7 7

6 ≠ 0

Apply the associative property with subtraction

Simplify inside of the parentheses

Simplify

Notice that if we applied the associative property with subtraction, the left and right not equal. We can adjust this by turning the subtraction into addition by the opposite.

(7 4) + 3 = (7 + ( 4)) + 3

(7 4) + 3 = 7 + ( 4 + 3)

3 + 3 = 7 + ( 1)

6 = 6

Rewrite subtraction as addition

Apply the associative property of addition

Simplify inside the parentheses

Simplify

We can see that when we convert the subtraction operation to an addition operation, we can still apply the associative property of addition.

Let’s see why the associative property applied with division does not work.

Apply the associative property with division

Simplify inside of the parentheses

Simplify

Notice that if we applied the associative property with division, the left and right side are not equal. We can adjust this by turning the division into a multiplication of it’s reciprocal

Rewrite division in the parentheses as multiplication

Rewrite division outside the parentheses as multiplication

Apply the associative property of multiplication

Simplify inside of the parentheses

Simplify

We can apply the associative properties to evaluate and simplify expressions.

Idea summary Property

Identity property of addition

Identity property of multiplication

Inverse property of addition

Inverse property of multiplication

Multiplicative property of zero

Practice

What do you remember?

1 Match the name of the property to its description:

a Commutative property of addition

b Commutative property of multiplication

c Identity property of addition

d Identity property of multiplication

e Inverse property of addition

f Inverse property of multiplication

g Multiplicative property of zero

2 Determine whether each statement is true or false:

a The additive identity is 1 because any number added to 1 is equal to itself.

b Zero has no multiplicative inverse.

c 5 + ( 5) = 0 and 5 + 5 = 0

3 Evaluate:

a 13

4 Complete each statement by ﬁlling in the blank(s):

a Using the commutative property of addition:

b Using the commutative property of multiplication:

4 = 4 and 4 = 4

5 Choose the property that is demonstrated by each of the following statements:

• Associative property

• Commutative property

• Inverse property

• Identity property

a 0 + 34 = 34

c 5 + 14 = 14 + 5

e ( 2 + 4) + 6 = 6 + ( 2 + 4)

g 176 + ( 176) = 0

6 State whether the following equations are true or false:

a 13 12 = 13 3 4

c 20 ⋅ 75 = 20 ⋅ 3 ⋅ 25

e 14 ⋅ 18 = 18 ⋅ 14

b 3 = 1

d 56 ⋅ 1 = 56

f 4 (3 6) = (4 3) 6

h 5 7 = 7 5

b 4 1005 6 = 24 1005

d 2 ⋅ 94 ⋅ 6 = 94 ⋅ ( 12)

f 6 + 6 ( 7) = 0 + 7

g 90 20 = 20 90 h 90 ÷ 6 ÷ ( 3) = 90 ÷ (6 ÷ ( 3))

7 Find the missing numbers in the following equations:

a 16 15 13 = 15 ⬚ 16

c 6 250 11 = ⬚ 250

b 24 50 = 24 ⬚ 10

d 21 ( 250) = 21 ⬚ 50

e 3 ⋅ 88 ⋅ ⬚ = 88 ⋅ 36 f 1230 ⋅ 3 = 123 ⋅ ⬚ ⋅ 3

g 17 ( 4) 19 = 68 ⬚ h 3 (8 5) = 3 ⬚

8 Find the missing number in each equation:

a 43 + ⬚ = 0 b ⬚ + ( 32) = 0 c 11 + ⬚ = 11

⬚

9 Evaluate: a 41 14 + 0 b 3 (1 ÷ 3) c 53 + ( 37) + 53 d 15 1 + ( 15) 1

10 Choose the operation that would make these statements true.

11 Select all expressions that are equivalent to 20 + 5:

12 Write an example equation that demonstrates each property:

a Additive Identity Property b Multiplicative Identity Property

c Additive Inverse Property d Multiplicative Inverse Property

13 Answer the following questions:

a Find (32 ÷ 4) ÷ 2 by dividing 32 by 4 then dividing the result by 2.

b Find 32 ÷ (4 ÷ 2) by dividing 4 by 2 then dividing 32 by the result.

c Is the associative property applicable to division?

14 Consider the equation:

a Determine whether the statement is true or false.

b Does the commutative property apply to subtraction? Explain your answer.

15 A student claims that division is commutative and gives the example shown:

Is the student correct? Explain your answer.

Let’s extend our thinking

16 Consider the following statement:

Explain whether the statement is true for all, true for some, or never true for all integers a, b and c.

17 Imagine you have 10 apples and you want to share them equally, but you are told to share them with zero people. Using this scenario, explain why dividing by zero does not work and what it means for an operation to be “undeﬁned.”

18 Evaluate using the properties of real numbers:

19 Explain how the properties of real numbers can help you evaluate the following expression using mental math.

20 Ruﬁno is given the expression :

Ruﬁno says that he can use the associative property of multiplication to add the 6 and the 4, and then add ( 22) + ( 22), as it would be easier to multiply 44 10.

Is Ruﬁno correct? Explain your reasoning using the properties of real numbers.

6.03 One-step equations with addition and subtraction

After this lesson, you will be able to...

• represent one-step linear equations involving addition or subtraction using a variety of concrete manipulatives and pictorial representations.

• solve one-step linear equations involving addition or subtraction using a variety of concrete manipulatives and pictorial representations.

• apply properties of real numbers and properties of equality to solve one-step equations involving addition or subtraction.

• write a one-step linear equation involving addition or subtraction to represent a verbal or contextual situation.

• use a variety of concrete manipulatives and pictorial representations to verify solutions to equations involving addition or subtraction.

• write a verbal situation in context when given a one-step linear equation involving addition or subtraction.

Model equations

An algebraic equation is a mathematical statement that says two expressions are equal. There are many ways to represent an algebraic equation. Some of the ways are:

• balance scales • algebra tiles • pictorial models

Balance scales are beneficial because they show that the left and right sides of an equation are equal and so the equation is balanced.

x 5 2

The equation shown on this scale is: x 5 = 2

Another way to represent an algebraic equation is to use Algebra tiles. These tiles represent the variables and units on each side of the equation.

We can use the key above, to represent an equation with algebra tiles. The equation x 5 = 2 can be created with this combination of tiles:

Pictorial models can be really helpful for visualizing equations that represent real-world situations. These models can use an image of any object to represent the variables and units of an equation.

Here is a pictorial model of x 5 = 2 where the variable x is represented by a backpack and the units are represented by a pencil. The gray pencils are meant to show they have been removed from the situation.

Balance scales, algebra tiles, and pictorial modles are three different ways that we can better understand and represent algebra equations.

Example 1

Write the equation represented by the algebra tiles.

Create a strategy

Count the number of +x tiles. Then count the number of +1 tiles

Apply the idea

There is 1 + x tile and 4 + 1 tiles on the left side. There are 5 + 1 tiles on the right side. So the equation is:

Represent x 3 = 5 using algebra tiles. Do not solve the equation.

Create a strategy

Use a positive variable tile to represent x and 1 tiles to represent the negative constants.

Apply the idea

On the left side of the equation, there is an x which we can represent with one +x tile and a 3 which we can make by using three 1 tiles. On the right side of the equation the 5 can be shown using ﬁve 1 tiles.

Example 2

Example 3

Write an equation that represents the given balance scale.

Create a strategy

A balance scale represents an equation that is equal or balanced. The left side of the scale corresponds to the left side of the equation, and the right side of the scale corresponds to the right side of the equation.

Apply the idea

On the left side of the balance there is an x + 4 and on the right side of the balance there is an 2. We set the left and right side of the balance to equal:

Example 4

Represent x + 1 = 6 using a pictorial model.

Create a strategy

Choose objects that could represent the variable x and the constants 1 and 6. The objects should be related for it to make sense.

Apply the idea

The coin purse represents x and the coins represent 1 and 6. So the total number of coins in the purse, x, plus 1 coin is equal to 6 total coins.

Idea summary

Algebraic equation: mathematical statement that says two expressions are equal. We can represent algebraic equations pictorially with:

balance scales

algebra tiles

pictorial models

Solve equations

Interactive exploration

Explore online to answer the questions

Use the interactive exploration in 6.03 to answer these questions.

1. What kinds of things can you do that keep the scale balanced?

2. What kinds of things can you do to make the scale unbalanced?

3. Test this with different types of tiles. Are these observations always true?

When working with equations, we must keep the equation balanced or it will no longer be a true statement.

Adding or subtracting the same amount to both sides keeps the equations balanced. These are two of the properties of equality.

Addition property of equality: adding the same number to both sides of an equation creates an equivalent equation. Example:

We can also visualize this with a scale. If

If we add the same amount to both sides of the scale it stays balanced.

Subtraction property of equality: subtracting the same number from both sides of an equation creates an equivalent equation.

Example:

If x + 5 = 7

Then x + 5 5 = 7 5

We can also visualize this with a scale.

If we take away the same amount from each side of the equation, it will remain balanced.

Notice for these equations the number we chose to add or subtract was the opposite of a number in the original equation. This strategy helps us solve equations by utilizing inverse operations and the additive inverse property.

Addition and subtraction are opposite or inverse operations that undo one another. If we choose the numbers we add or subtract carefully we can use this to eliminate extra numbers from an equation.

For example, with the equation x 5 = 17 we want to isolate x which requires getting rid of the constant term 5. To do this we can use the inverse operation by adding 5. x 5 + 5 = 17 + 5

Now we can see the additive inverse in action: x + 0 = 17 + 5

Now applying the additive identity: x = 17 + 5

Then we can simplify the right side of the equation: x = 22

We don’t always write out all of these steps, but it is still important to know what is happening algebraically.

Once we have solved an equation, we can verify the solution using the substitution property of equality.

Substitution property

If a = b, then b can be substituted for a in any expression, equation, or inequality

Consider the solution of x = 22 for the equation x 5 = 17.

Reﬂect and check

Let’s verify the answer using a balance scale model.

We start with twenty one +1 tiles on the left side of the scale, then one +x and thirteen +1 tiles on the right side to represent 21 = x + 13.

Based on our solution, 8 = x, we can replace the +x tile with 8 of the +1 tiles.

The balance scale is now balanced with both sides having the same number of positive unit tiles (+1), verifying that our solution is correct.

Example 7

Solve: x 1 = 7

Create a strategy

The inverse of subtraction is addition, so we need to add to both sides of the equation.

Apply the idea

x 1 = 7

x 1 + 1 = 7 + 1

x = 7 + 1

x = 8

Reﬂect and check

Original equation

Addition property of equality

Additive inverse

Evaluate the addition

Let’s verify the answer using a balance scale model.

We start with a +x tile and a 1 tile one the left side of the scale, then 7 of the +1 tiles on the right to represent the equation x 1 = 7.

Based on our solution, x = 8 we can replace the +x tile with 8 of the +1 tiles.

Remove zero pairs. The scale stays balanced.

The balance scale is now balanced with both sides having the same number of positive unit tiles, verifying that our solution is correct.

Example 8

4 is the solution to the equation 8 + x = 12.

a Verify using substitution.

Create a strategy

Substitute 4 into the equation. The left side of the equation will be equal to the right side if 4 is a solution.

Apply the idea

8 + ( 4) = 12

12 = 12

Substitute x = 4

Evaluate the addition

The left side of the equation is equal to the right side of the equation, verifying that 4 is a solution.

Reﬂect and check

We can also use a pictorial model to verify if 4 is the solution to 8 + x = 12. This pictorial model represents the equation 8 + x = 12, where each ice cube represents 1 and the cooler represents x. = 1 = +x

We then replace the cooler with four ice cubes, representing x = 4.

We now have twelve ice cubes on both sides of the equation, verifying that 4 is a solution to the equation.

b Verify using a model.

Create a strategy

Represent the equation using algebra tiles on a scale. Make the left side of the scale to have the same number of unit tiles as the right side by replacing the variable tile with the appropriate tiles.

Apply the idea

The equation 8 + x = 12 can be represented by:

The given solution is 4, so we replace the variable tile with four negative unit tiles.

Both sides of the scale now have twelve negative unit tiles, verifying that 4 is a solution to the equation.

A box of matches contains 500 matches. The match box falls to the ground and you count 78 matches on the ground. The rest of the matches are still in the box.

a Write an equation that shows the relationship of the number of matches.

Create a strategy

The amount of matches that are still in the box is the unknown value.

Apply the idea

Let m represent the number of matches still in the box.

m + 78 = 500

Example 9

b Solve the equation and interpret the solution.

Create a strategy

The inverse of addition is subtraction, so we need to subtract to both sides of the equation.

Apply the idea

m + 78 = 500

m + 78 78 = 500 78

m = 422

Original equation

Subtraction property of equality

This means that there are 422 matches that did not fell out of the box.

Example 10

Write a situation that could represent the equation x + 5 = 25.

Create a strategy

Think of a situation where there is an unknown value, x added to 5 which would equal 25.

Apply the idea

The amount of money in James’ wallet, x, plus the $5 in his hand is equal to the $25 he received for his birthday.

Idea summary

Addition property of equality If

Subtraction property of equality If

Inverse property of

Practice

What do you remember?

1 Consider this set of tiles:

a Write an expression to represent the tiles at the left side of the equal symbol.

b Write an expression to represent the tiles at the right side of the equal symbol.

c Now, write an equation to represent the set of tiles.

2 Write the algebraic equation that represents each set of tiles:

25 Wendy and Astrid are solving the equation x 2 = 4. The following are the steps of their solutions: Wendy’s solution: Astrid’s solution:

Step 1: x 2 = 4

Step 2: x 2 + 2 = 4 2

Step 3: x = 2

Which soution is incorrect? Explain your answer.

Step 1: x 2 = 4

Step 2: x 2 + 2 = 4 + 2

Step 3: x = 6

26 John has a debt that he’s paying off. After making a payment of $30, he still owes $ 150. Let d represent the original debt amount before the payment.

a Find the amount of money John originally owed.

b Use substitution to verify your solution.

27 Alex gives Lisa 20 books from his library. Alex now has only 30 books in his library. How many books (b) did Alex have initially? Justify your solution.

6.04 One-step equations with multiplication and division

After this lesson, you will be able to...

• represent one-step linear equations involving multiplication or division using a variety of concrete manipulatives and pictorial representations.

• solve one-step linear equations involving multiplication or division using a variety of concrete manipulatives and pictorial representations.

• apply properties of real numbers and properties of equality to solve one-step equations involving multiplication or division.

• write a one-step linear equation involving multiplication or division to represent a verbal or contextual situation.

• use a variety of concrete manipulatives and pictorial representations to verify solutions to equations involving multiplication or division.

• write a verbal situation in context when given a one-step linear equation involving multiplication or division.

One-step equations with multiplication and division

Just like we saw with addition and subtraction, multiplication and division are also inverse operations. For example, multiplying a number by two is the opposite of dividing it by two.

Looking at a balance scale model again, we can see how we can multiply or divide by the same number on both sides of an equation to keep it balanced.

Multiplication property of equality: Multiplying each side of an equation by the same number produces an equivalent equation. Example:

We can visualize this with a scale.

The scale tells us that x = 3. We can double the values on both sides and the scale will still be balanced.

We multiplied the tiles on both sides of the scale by 2 and it is still balanced. We could have also multiplied both sides of the original scale by 3 to create 3 equal groups on each side of the balance and it would still be balanced.

If we treat each side of the scale like a group that aligns with the other side, we can keep applying the multiplication property of equality.

Division property of equality: Dividing each side of an equation by the same number produces an equivalent equation. Example: If 4x = 8 Then =

We can visualize this with a scale.

Notice that the groups on the left and right align where every 2x aligns with 4 unit tiles. We can divide both sides by 2 (removing half of the tiles) and the scale will still be balanced.

We can also divide both sides of the original scale by 4, since we can see there are 4 groupings where every x tile is equal to 2 unit tiles.

Apply the idea

We must add three +1 tiles on the right side of the scale for every new positive tile on the left side. So, we need to add a total of six +1 tiles to the right side of scale 1.

We need 9 positive 1 tiles in place of the question mark to balance scale 2, option D.

Reﬂect and check

We can write this algebraically as:

x = 3

3x = 3 3

3x = 9

Solve 3x = 18

Create a strategy

Write the equation

Multiply both sides by 3

Evaluate the multiplication

To undo multiplication, we can divide both sides of the equation.

Example 3

Solve:

Create a strategy

To undo division, we can multiply both sides of the equation.

Apply the idea

5 is a solution to the equation 8x = 40.

a Verify using substitution.

Create a strategy

Substitute 5 into the equation. The left side of the equation should be equal to the right side if 5 is a solution.

Apply the idea

Write the original equation

Divide both sides by 3

Evaluate the division

Apply the idea

8 (5) = 40

40 = 40

Write the original equation

Multiply both sides by 8

Evaluate the multiplication

Substitute x = 5

Evaluate the multiplication

The left side of the equation is equal to the right side of the equation, verifying that 5 is a solution.

Example

b Verify using a model.

Create a strategy

Apply the idea

The equation 8x = 40 can be represented by:

The given solution if 5, so we replace each variable tile with 5 unit tiles. Notice that the variable tile is positive, so the resulting 5 unit tiles are positive.

Both sides of the scale now have 40 positive unit tiles, verifying that 5 is a solution to the equation.

Example 5

At a beach fruit stand, fresh squeezed juice is sold at $6 per quart. You and your friends spent a total of $84 on quarts of juice. Write an equation that shows the relationship between the total cost and the number of quarts of juice purchased.

Create a strategy

The number of quarts of juice purchased is the unknown value.

Example 6

Write a situation that could represent the equation = 52.

Create a strategy

Think of a situation where there is an unknown value, x, divided into 4 equal sized groups and each group is size 52.

Apply the idea

Let q represent the number of quarts of juice purchased. 6q = 84

Apply the idea

At a craft store, you buy 4 packs of markers. Each pack contains the same number of markers. In total, you have 52 markers. Write an equation that shows the relationship between the number of markers in a pack and the total number of markers you bought.

Idea summary

Multiplication property of equality If a = b then a

Division property of equality If a = b, and c ≠ 0, then

Inverse property of multiplication and Identity property of multiplication a ⋅ 1 = a and 1 ⋅ a = a

Substitution property If a = b, then b can be substituted for a in any expression, equation, or inequality.

Practice

What do you remember?

1 Write the algebraic equation that represents each set of tiles:

2 Consider the scale shown. Which image represents the scale after each side is divided by 2?

3 State the operation needed to solve the following equations: a 2x = 5 b

12y = 36 d Let’s practice

4 Solve the equation represented by the following a

5 For each pair of scales, Scale 1 has been changed in some way to get Scale 2 so that both scales are balanced. Draw what could go in place of the question mark on Scale 2. Then, use the properties of equality to justify that both scales are balanced.

a Scale 1:

Scale 2:

19 Consider the following equation: 3x = 42

a Solve the equation for x.

b Create a real-world scenario where this equation could be applied.

20 Write a real-world world context that each equation could be used to represent.

a 8x = 32 b c 5y = 60 d

Let’s extend our thinking

21 The price of a kilogram of apples is p dollars. If John bought 5 kilograms of apples for $30, ﬁnd the cost of 1 kilogram of apples.

22 Solve each equation. Justify your work using a property of equality.

a 8k = 56 b c d 12n = 60

23 Find the unknown number for each of the following:

a When a number is multiplied by 10, the result is 60.

b When a number is divided by 8, the result is 9.

c The product of 3 and a number is 18.

d The quotient of a number and 8 is 12.

24 Show how you can solve the equation 4x = 12 using the multiplication property of equality.

25 Bryon is solving the equation = 6. The following are his steps:

Step 1: = 6

Step 2: 2 = 6 ÷ 2

Step 3: x = 3

In which step did Bryon made a mistake? Explain your answer.

If we switch the order so that there are three cats on the left and ﬁve cats on the right, we can say that three cats is less than ﬁve cats. We can write this as an inequality 3 < 5.

Both images mean the same thing but are stated differently. If we switch the order of an inequality, we have to change the inequality sign. This is also true with algebraic inequalities.

For example, x > 10 means the same thing as 10 < x. In other words, “x is greater than ten” is the same as “ten is less than x”.

For example, the expressions x > 5 and 5 > x represent different sets of numbers, while x > 5 and 5 < x represent the same set of numbers.

We can use this understanding of inequality symbols to write inequalities that represent real world situations. Let’s write an inequality to represent the statement: “a student needs to score at least 75 points to pass an exam.”

Let s represent the student’s score. The key phrase “at least 75 points” tells us that the lowest passing score is 75. So the student will pass the exam if they score 75 points or if they score more than 75 points. If we use s to represent the score we can write the inequality s ≥ 75.

Here are some common phrases and examples used for the different inequality symbols.

Inequality

Symbol

Vocabulary/Representations

Example

< less than, fewer than, under “The speed limit is less than 60 mph.” translates to S < 60 > greater than, exceeds, more than “The temperature is greater than 30 °C” translates to T > 30

≤ less than or equal to, at most, no more than, up to “You can spend up to 50 dollars.” translates to C ≤ 50

≥ greater than or equal to, at least, no less than “You need at least 8 hours of sleep.” translates to H ≥ 8

Example 1

For the number sentence ⬚ 0.3.

a Choose the mathematical symbol that makes the number sentence true.

Create a strategy

Convert the fraction to a decimal to compare values easily.

Apply the idea

Convert to decimal

Compare the decimals

So we can see that > 0.3.

Reﬂect and check

Another true inequality with the same numbers is 0.3 < .

b Write the statement in words.

Create a strategy

Replace the inequality symbol > with a vocabulary that has the same meaning.

Example 2

Apply the idea “ is greater than 0.3” or “ is more than 0.3”

Write an inequality to represent each of the following situations. a n is greater than 9

Create a strategy

The phrase “greater than” tells us which inequality symbol to use.

Apply the idea n > 9

b The weight of the package is under 5 kg. Let w be the weight of the package.

Create a strategy

The word “under” means less than.

Apply the idea w < 5

c You must be at least 18 years old to vote in the United States. Let a be the age of the voters.

Create a strategy

The phrase “at least 18” means you can vote in the US if you are exactly 18 years old or if you are older.

Apply the idea a ≥ 18

d The maximum height for the ride is 120 cm. Let h be the height of the riders.

Create a strategy

The word “maximum” means highest. So, to ride you can be that height or shorter but you cannot be taller.

Example 3

Write a real-world scenario for each inequality. a x ≤ 20

Create a strategy

We can think of x as an object that can contain up to 20 pieces of another object.

Apply the idea h ≤ 120

Apply the idea

A sample scenario would be “A group of students is preparing boxes for toy donation. Each box can hold up to 20 small toys”.

Reﬂect and check

Let’s check s ≤ 4 on a number line.

The number line shows that all values to the left of 4, and including 4 are all possible values of s

However, in the given context, s represents the number of garments, which means that s should have a minimum value of 0, because it is not possible to try on a negative number of garments.

Idea summary

Inequalities are mathematical sentences where two expressions are not necessarily equal, indicated by the symbols: <, >, ≤, and ≥ Symbol

< less than, fewer than, under 3 < 6

> greater than, exceeds, more than 6 > 3

≤ less than or equal to, at most, no more than, up to 4 ≤ 6

≥ greater than or equal to, at least, no less than 6 ≥ 5

Practice

What do you remember?

1 Write the symbol (<, >, ≤, ≥ ) that best matches the description. Symbols may be used more than once.

a Is at least b Is more than c Is fewer than d Is at most

e Is no less than f Is below g Does not exceed h Is not less than

2 For each description, determine if the scenario represents an equation or an inequality. Explain your answer.

a A theme park ride requires riders to be at least 48 inches tall.

b To qualify for a marathon, participants need to complete a qualifying race in under 4 hours.

c Jamie is baking cookies and the recipe calls for 2 cups of sugar.

d A soccer goalie aims to allow at most 10 goals scored in a season.

e A science project requires a bean plant to grow to exactly 15 cm

Let’s practice

3 Write the following inequalities in words:

b < 2.23 f m ≥ 24

4 Write the inequality described by the following statements:

a n is greater than 9.

c n is less than 10.

e a is positive.

b n is greater than or equal to 9.

d n is less than or equal to 6.

5 A movie theater only allows entry if there are fewer than 120 tickets sold to ensure everyone has a seat. Which inequality represents this situation?

6 Jaime wants to buy a bike that costs $200. He must save at least this amount to purchase it. Which inequality represents this situation?

7 Write an inequality to represent each of the following situations:

a The width of a particular road is 5 meters or greater. Let w be the width of the road.

b The elevator in a building should carry less than 13 people at one time. Let p be the number of people in the elevator.

c Michael lives where it takes him at most 15 minutes to get to work. Let x be the time Michael takes to get to work.

d In an amusement park, children below 5 years old can enter free of charge. Let c be the age of the a child that can enter the park free of charge.

e Frasier is expecting that at least 50 guests will attend his party. Let g be the number of guests in Frasier’s party.

f In a gift exchange party, the attendees must spend at most $50 for their gifts. Let x be the amount of a gift in the gift exchange party.

8 Write in a real-world scenario for each inequality:

x ≤ 5

y ≥ 12 c a < 33

9 The library has a rule on how many books you can borrow. The rule can be represented by n ≤ 5, where n represents the number of books one can borrow. Which options show the number of books you can borrow? Select all correct options.

10 Mitch and Nicole are having a mathematical debate. Mitch states that x > 4 is the same relationship as 4 > x Nicole states that x > 4 is the same relationship as 4 < x. Who is correct? Explain why.

Let’s extend our thinking

11 Write the inequality described by the following statements:

a The product of 5 and x is less than or equal to 20.

b Seven more than the value of x is at least 9.

c Eight is greater than the result of taking 7 away from x.

12 Jack is saving up to buy a smartphone that is selling for $510. He has in his bank account and expects some money for his birthday next week.

If the amount he is about to receive for his birthday is represented by x, write an inequality that models the situation where he is able to afford the smartphone.

Example 5

Create a scenario that could ﬁt the following inequality: x < 48

Create a strategy

Think about a scenario where something has to be under 48.

Idea summary

Apply the idea

A local carnival has rides based off of heights. Some rides are designed for children under the height of 48 inches. So, you must be under 48 inches to ride the kids train ride at the carnival.

To graph an inequality, start by determining which direction the line will be shaded, right or left. This can be determined by making sure that the shaded line covers all the values that make the inequality true.

The end point of the line will be an open circle if the inequality has a < or >

The end point of the line will be a closed circle if the inequality has a ≤ or ≥.

The solution set of an inequality is made up of all values that make the inequality true. Solutions lie in the shaded region on a number line.

Practice

What do you remember?

1 Match the inequality sign with the correct description. i < a Less than ii > b Less than or equal to iii ≤ c Greater than iv ≥ d Greater than or equal to

2 When writing an inequality from a graph, which symbol(s) do we use for the following endpoints: a Closed b Open

3 Write the inequality described by the following statements:

a You must be at least 60 inches tall to ride a carnival ride. Let h represent the height.

b Gianna has more than 6 books. Let b represent books.

c The class has at most 30 students. Let s represent students.

4 Consider the given equation and inequality:

• x = 5

• x ≤ 5

a Select all values that make the equation x = 5 true: A 5 B 5

b Select all values that make the inequality x ≤ 5 true:

5 B 5

c Describe the difference between the solutions of equations and inequalities.

5 Determine whether or not each of the following is a solution of x > 5:

x = 1

= 10

6 Determine whether or not each of the following is a solution of k ≤ 17:

Let’s practice

7 Consider the inequality x ≥ 4.

a Determine whether the following are solutions to the inequality:

= 1 ii

b What is the smallest possible value of x?

c Will the graph have a closed or open endpoint?

d Now, graph the inequality on a number line.

8 Select the values that are part of the solution set of x ≤ 2.

9 Which inequality represents the number line graph? Select all correct answers.

10 Write an inequality for x that is represented on the following number lines:

11 Match the inequalities to the number lines.

12 Use the number line graph to justify whether each value is a solution of the inequality x > 5:

13 Identify two inequality statements that represent the graph.

14 Write an inequality that represents the graph in two different ways.

15 Select the number lines for which x = 3 is a solution.

16 Graph the following inequalities on a number line:

17 Consider the following statement: x is no more than 3

a Write an inequality that represents the statement.

b Is x = 4 in the solution set? Use a number line graph to justify your answer.

18 Cadence needs to make at least 4 baskets in her next basketball game to break her record.

a Write an inequality that represents the number of baskets she needs to score to break her record. Let b represent baskets made or points scored.

b Graph the inequality on a number line.

19 A building must be more than 160 meters tall to be considered a skyscraper. Let h be the height of the building.

a Write an inequality representing this situation.

b Would a building 152 meters tall be considered a skyscraper?

20 The temperature inside a freezer is always below 5 °F. Let T be the temperature inside the freezer.

a Write an inequality representing this situation.

b Will the temperature inside the freezer ever be 2 °F?

Let’s extend our thinking

21 For each of the following, ﬁnd the largest whole number value q can have.

22 For each of the following, ﬁnd the smallest whole number value n can have.

23 Olivia and Lucy graph the inequality that represents the condition “the largest value of x that does not satisfy the inequality is 5.5”. Their plots are shown:

Olivia’s plot

a Identify the error in Olivia’s plot.

b Identify the error in Lucy’s plot.

c Now, graph the solution to the inequality.

Lucy’s plot

24 For a food to be labeled high protein, it must contain at least 20 grams of protein per serving.

a Write and graph an inequality to represent the amount of protein in a high-protein serving.

b Write and graph an inequality to represent the amount of protein in a serving that does not qualify as high protein.

c The Nutrition Facts label provided shows a serving of Chicken Nuggets with 14.3 g of protein. Does this food qualify as a high-protein food? Explain.

25 Select the number line that shows the solution set of the inequality 2x < 18.

26 Xander is on a hiking trip and is allowed to pick at most 2 types of wildﬂowers. Each type of wildﬂower must have no less than 10 petals.

a Write and graph an inequality to represent the number of types of wildﬂowers Xander is allowed to pick.

b Write and graph an inequality to represent the number of petals on each type of wildﬂower Xander is allowed to pick.

c Is Xander allowed to pick a sunflower, coneflower, and blanket flower that each have more than 10 petals? Explain your answer.

27 Lila needs to save at least $75 for a school trip. She can save $15 per week. This can be represented by the inequality 15x ≥ 75. Is 4 weeks enough time for Lila to save enough money for the trip?

Polygons 7

Big ideas

• The relationships between the sides, angles, and diagonals of a polygon can be used to classify the polygon and solve problems.

• Physical objects can be modeled with 2D and 3D geometric figures whose properties can be applied to solve real-world problems.

• The properties of polygons can be applied to solve problems involving other polygons.

• The position in space of a geometric figure can be represented in the coordinate plane. Using coordinate algebra, the properties of that figure can be uncovered and applied to solve problems.

Chapter outline

7.01 Congruent figures

After this lesson, you will be able to...

• determine the congruence of segments, angles, and polygons given their properties.

• write congruence statements for segments, angles, and polygons given their properties.

• state whether polygons are congruent or noncongruent using the measures of their sides and angles.

Congruent segments

Segment

A segment starts at one point and stops at the other.

This segment can be named using its endpoints. We call this . The line over the letters means segment, so we say “segment AB”.

We can also use the reverse order to name the same segment. or “segment BA”.

When talking about the length of the segment or the distance between points A and B we can say AB without the line over top.

It’s important that when discussing a segment, we use the line over the two letters to indicate segment. For example, or is a segment, while EF or FE is a distance or length.

Interactive exploration

Explore online to answer the questions

mathspace.co

Use the interactive exploration in 7.01 to answer these questions.

1. Set the sliders for each segment to be the same number then overlap them. What do you notice?

2. Set the sliders for each segment to be different numbers then overlap them. What do you notice?

3. Repeat both with new numbers. Do you notice the same things each time?

Congruent segments

Line segments that have the same length. The symbol ≅ is used to represent congruence.

We place small markings on segments when we want to show that they are equal in length.

A C 2 in 2 in

The small identical markings on each segment, called ‘hash’ or ‘hatch’ marks, tell us they are equal in length, or congruent.

We also see has the same length, 2 in as . We can say AB = AC when discussing length, but we must say segment is congruent to segment .

We can write a congruency statement using our congruence symbol:

Lengths and distances are said to be equal while segments are congruent.

This does not mean that the two segments are made up of the same points - only that they have the same length. Sometimes we will use more than one kind of marking to show that some segments are equal to others. In this diagram we use both single hatch markings and double hatch markings. The segments with single hatch markings are congruent to each other. The segments with double hatch markings are congruent to each other.

The length of AB is equal to the length of BC, so is congruent to

The length of CD is equal to the length of DE, so is congruent to

Example 1

Use the diagram to answer the following:

a What segment is congruent to ?

Create a strategy

Choose the segment with the same marking as .

Apply the idea

Segment has the same marking as This shows us that is congruent to .

b Write a congruence statement

Create a strategy

Reﬂect and check

Since segments are named using their endpoints, the order of letters does not matter. We could have also said segment is congruent to

Use the congruence symbol ≅ to write a conguence statement for the two segments.

Apply the idea

Reﬂect and check

The congruence statement is also true.

Example 2

Identify any congruent segments in the ﬁgure and write a congruence statement for each.

Create a strategy

Remember, congruent segments are equal in length.

Idea summary

Apply the idea

Since the length of is equal to the length of , we can say .

Since the length of is equal to the length of , we can say

• Congruent segments are equal in length.

• We place small markings to show the segments that are equal in length.

• We use our congruency symbol ≅ to write congruency statements. and

Congruent angles

Angle

An angle is formed when two rays, lines, or segments, are joined at their endpoints. Angles are measured in degrees.

Whenever two lines, rays, or segments pass through the same point, we can describe the amount of turn or rotation it takes to get from one to the other using an angle

Here are two rays drawn from the same point forming an angle. The vertex of an angle is the angle formed by two lines or rays that intersect at a point.

In this image, our vertex angle is ∠A. We use the letter at the vertex to name the angle.

Let’s name the three angles of the triangle:

Now, let’s identify the measure of each angle:

Interactive exploration

Explore online to answer the questions

Use the interactive exploration in 7.01 to answer these questions.

1. Set the sliders for each angle to be the same degree measure then overlap them. What do you notice?

2. Set the sliders for each angle to be different degree measures then overlap them. What do you notice?

3. Repeat both with new numbers. Do you notice the same things each time?

Similar to segments, we must be careful with our notation when discussing angles. Congruent angles have the same angle measure.

Since the measure of ∠B is equal to the measure of ∠Q, we know ∠B is congruent to ∠Q

Congruence Statement: ∠B ≅ ∠Q

Since ∠B and ∠Q have the same angle marking, this also tells us they are equal in measure and therefore congruent.

Just like with segments, we can use additional markings to show that two angles are congruent. We draw multiple arcs to show that different angles are congruent to each other.

In this diagram the two angles drawn with double arcs have equal measure. Since the measure of the angles is equal, we know the angles are congruent.

Congruence Statement:

Example 3

Use the image to answer the questions.

∠1 ≅ ∠4

There are no angles that are congruent to ∠2 and ∠3 because there are no other angles with the same number of markings.

a Which angles are congruent?

Create a strategy

Look for angles with identical angle markings showing the angle measures are equal.

Apply the idea

The two angles marked as having equal measure are ∠C and ∠A. So, we know ∠C is congruent to ∠A.

b Write a congruence statement

Create a strategy

Use the congruence symbol ≅ to write a congruence statement for the angles we identiﬁed as congruent in part (a).

Apply the idea

∠C ≅ ∠A

Reﬂect and check

The congrunce statement ∠A ≅ ∠C is also true.

Example 4

Use the image to answer the questions.

a Which angles are congruent?

Create a strategy

Identify any angles that have the same degree measure.

Apply the idea

The two angles with an equal measure are ∠G and ∠S since they both measure 42°. So, we know that ∠G is congruent to ∠S.

b Write a congruence statement

Create a strategy

Use the congruence symbol ≅ to write a congruence statement for the angles we identiﬁed as congruent in part (a).

Apply the idea

Reﬂect and check

The

Idea summary

• Congruent angles have the same measure.

• Congruent angles are marked using an arc or multiple arcs to show congruence.

• An angle is named using ∠ symbol followed by letter of vertex angle. We use our congruence symbol, ≅ to write a congruence statement. ∠B ≅ ∠Q means angle B is congruent to angle Q.

Congruent polygons

Polygon

A polygon is a closed plane ﬁgure composed of at least three line segments that do not cross.

When we are naming a polygon, we use the labels on its vertices.

Use the interactive exploration in 7.01 to answer these questions.

1. Overlap several different pairs of triangles. What do you notice about the triangles that overlap perfectly?

2. What do you notice about the triangles that do not overlap perfectly?

Congruent polygons

Polygons are congruent if they have an equal number of sides, and all the corresponding sides and angles are congruent.

Corresponding angles are a pair of matching angles that are in the same spot in two different shapes. Corresponding sides are a pair of matching sides that are in the same spot in two different shapes.

Here we have an image with two triangles.

Since the corresponding angles and corresponding sides are equal, we know these two triangles are congruent.

Here we have an image of two polygons.

The corresponding angles are congruent, however, the corresponding sides are not congruent.

This means that these two polygons are not congruent.

If two polygons are congruent, we can show that with a congruence statement. When writing congruence statements for polygons, the letters must be in the correct order according to the corresponding angles and sides.

Here is an example of two congruent triangles. Let’s identify our corresponding angles and sides.

For the angles, the markings on image tell us:

For the sides, we see from the labeled lengths that:

This shows us that vertex G corresponds with vertex R, S with K and P with F so we can write the congruence satement: △

GSP ≅ △RKF

The statements △SPG ≅ △KFR and △GPS ≅ △RFK are also true because they match the corresponding angles. There are several more true congruence statements we could write as long as we make sure the order matches up the congruent angles.

Example 5

Determine if the two polygons are congruent.

Create a strategy

Polygons are congruent if they have an equal number of sides, and all corresponding sides and angles are congruent. We see these triangles have an equal number of sides. So, we need to check all corresponding sides and angles to determine if they are congruent.

Apply the idea

Let’s ﬁrst start by identifying the corresponding angles. One arc marking is used with ∠E and ∠L, which tells us ∠E ≅ ∠L. A double arc marking is used for ∠F and ∠M, showing that ∠F ≅ ∠M. A triple arc marking is used for ∠G and ∠N, telling us that ∠G ≅ ∠N.

All corresponding angles are congruent. Now let’s check the corresponding sides. Side EF is marked with one hatch mark and so is side LM. This tells us Both side EG and side LN are marked with two hatch marks, showing us Three hatch marks are used for side FG and side MN, showing us .

Since all corresponding sides and angles are congruent, these two polygons are congruent.

Reﬂect and check

We can write a congruency statement:

EFG ≅ △LMN

Example 6

Determine if the two polygons are congruent.

Create a strategy

Polygons are congruent if they have an equal number of sides, and all corresponding sides and angles are congruent. These polygons both have 4 sides, so we need to check all corresponding sides and angles to determine if they are congruent.

Apply the idea

Since all of the angles are marked as right (90°) angles, we know all corresponding angles are congruent.

Let’s check if the corresponding sides are congruent. Side JI and side OP, are both 26 mi long, so . Side JG in the first figure has a length of 44 miles and has no corresponding side of equal length in the second figure. These two polygons are not congruent.

Reﬂect and check

Notice that we did not need to check all pairs of corresponding sides. As soon as we found one side that did not have a corresponding side of equal length we were able to say the ﬁgures are not congruent.

Idea summary

Congruent polygons: Polygons are congruent if they have an equal number of sides, and all the corresponding sides and angles are congruent.

When writing a congruency statement, we must make sure we are putting the letters in the correct order according to corresponding angles and sides.

Practice

What do you remember?

1 Describe each of the following geometric objects, using the points A and B: a b c

2 Look at the rhombus shown. Which side is parallel to ?

3 Look at the rectangle shown. Which pair of lines are perpendicular to ?

4 Consider the following triangle:

a Name the angle that is directly opposite the segment

b Name the segment that is directly opposite ∠1.

5 For each diagram, name the sides and angles that have been marked congruent: a b c d e f

6 Select all of the following that must be true for two polygons to be congruent.

A All congruent sides

C Same number of sides

Let’s practice

B Unequal number of sides

D All interior angles are congruent

7 In the following diagrams, name the congruent segments: a b c

8 AB = 4.5 in. If , what is the length of ?

9 Determine whether or not the two angles are congruent given their measurements. a b

10 Use the triangles below and complete the statements:

• ∠M ≅ ∠⬚

• • ∠Q ≅ ∠⬚

• Are the two triangles congruent? Explain your reasoning.

11 Which two parallelograms are congruent?

Parallelogram 1

Parallelogram 3

Parallelogram 2

Parallelogram 4

12 For each of the following polygons, state whether they are congruent or noncongruent. Explain your reasoning. a b c

13 Are the two ﬁgures congruent? Explain your reasoning.

14 Explain why these two ﬁgures are not congruent.

15 Alex and Jamie are observing the following polygons:

• Alex claims that Polygon X and Polygon Y are congruent because they have the same side lengths and their angles are also the same, just arranged in a different order.

• Jamie says that even though Polygon X and Polygon Y have sides and angles that are the same size, they’re not exactly the same shape. He thinks it’s because the order of the sides and angles makes the shapes look different.

Who is correct? Explain your thinking.

16 Statement: Polygon I and Polygon J are congruent polygons.

Analyze the descriptions to determine if this statement is true or false:

a Polygon I: A hexagon with sides of 4 cm each, and all angles measuring 120°.

Polygon J: A hexagon with sides of 4 cm, 4 cm, 4 cm, 4 cm, 3 cm, and 4 cm, and angles of 120°, 120°, 120°, 120°, 110°, and 130°.

b Polygon I: A square with sides of 5 cm each, and all angles measuring 90°.

Polygon J: A square with sides of 10 cm each, and all angles measuring 90°.

c Polygon I: A triangle with sides of 3 cm, 4 cm, and 5 cm, and angles that correspond to a right triangle.

Polygon J: A triangle with sides of 5 cm, 3 cm, and 4 cm, and angles that correspond to a right triangle.

d Polygon I: An octagon with sides of 2 in each, and all angles measuring 135°.

Polygon J: An octagon with seven sides of 2 in each, but one side is 3 in.

e Polygon I: A rectangle with sides of 6 cm and 4 cm, and angles of 90°.

Polygon J: A rectangle with sides of 4 cm and 6 cm, and angles of 90°.

Polygon X

Polygon Y

17 Given that PQRST ≅ ABCDE: What are the measures of ∠P and ∠R?

18 For each ﬁgure below, identify all congruent sides:

Let’s extend our thinking

19 Which two triangles are congruent?

20 Consider a park garden.

a The dimensions of the garden are 22 ft, 18 ft, 22 ft, and 18 ft. All angles are congruent and have measures of 90°. What is the shape of the garden?

b If a fountain in the garden is in the shape of an equilateral triangle where one side is 75 inches, what are the lengths of the other sides? Explain your thinking.

21 If a bookshelf within the community library is designed as an isosceles triangle, will there be any congruent sides or angles? Explain your thinking.

22 Consider two 4-sided polygons. ABCD ≅ RSTU.

a Which segment in ABCD is congruent to ?

b Which angle in RSTU is congruent to ∠C?

23 Look at the quadrilateral:

Construct a shape that is congruent to the quadrilateral.

Explain how you made sure your constructed shape was congruent to this quadrilateral.

7.02 Regular polygons and symmetry

After this lesson, you will be able to...

• identify regular polygons.

• draw lines of symmetry to divide regular polygons into two congruent parts.

Regular polygons

Regular Polygon

A regular polygon has congruent sides and congruent interior angles.

Here are some examples of regular polygons.

All squares are regular polygons. They have 4 congruent sides and all angles measure 90°.

6 cm

Markings showing all angles are congruent and sides are congruent for this regular pentagon.

Equilateral triangles are regular polygons because all sides and angles are congruent.

This is a regular hexagon.

Now, let’s take a look at some examples that are not regular polygons, but rather irregular polygons.

This polygon has sides and angles of different measures, so it is irregular.

In this rectangle, all angles are congruent but the sides are not, so it is irregular.

A line of symmetry divides a ﬁgure into two congruent parts. Notice in the diagram, the side and angle markings show us that the two halves of the ﬁgure are congruent.

A shape can have 0, 1 or many lines of symmetry. A square has four lines of symmetry.

For a regular polygon, the number of lines of symmetry is equal to the number of sides.

One way to picture lines of symmetry is with reﬂections. Another way is to picture folding the shape along a line. Whether reﬂecting or folding, the resulting shape should perfectly overlap the original.

A shape that has no lines of symmetry is called asymmetric. Neither of these shapes have a line of symmetry.

Use the image shown to answer the following:

a How many lines of symmetry does this regular polygon have?

Create a strategy

In a regular polygon, we know the number of sides is equal to the number of lines of symmetry.

Apply the idea

Since this regular polygon has 10 sides, we know it has 10 lines of symmetry.

Example 2

b Draw the lines of symmetry

Create a strategy

We know from above part that this polygon has 10 lines of symmetry.

Apply the idea

Draw in the lines of symmetry:

Idea summary

A line that reﬂects a shape onto itself is called a line of symmetry. Asymmetric shapes are shapes without lines of symmetry.

In a regular polygon, the number of sides is equal to the number of lines of symmetry.

Practice

What do you remember?

1 Match the following terms with their deﬁnitions:

a Regular polygon

b Irregular polygon

c Congruent ﬁgures

d Line of symmetry

e Congruent parts

i Sides are not all congruent and interior angles are not all congruent

ii Has all congruent sides and congruent interior angles

iii Divides a ﬁgure into two congruent parts, each of which are mirror images of each other

iv Parts of a ﬁgure that are identical in shape and size

v Have exactly the same shape and size

2 Arrange the polygons in ascending order according the number of sides they have.

• Pentagon

• Triangle

• Octagon

3 Which image shows a line of symmetry?

The image that shows a line of symmetry is image ⬚

• Heptagon

• Quadrilateral

• Hexagon

4 Which image shows a line that does not divide the polygon into two congruent parts?

5 Do all shapes have a line symmetry? Give some examples to explain why or why not.

6 Use geometric markings to mark the congruent angles and sides on the regular polygons. Let’s practice

7 Identify the regular pentagon.

8 Determine whether the polygons are regular or irregular:

9 Sort the shapes below as regular or irregular polygons. Regular polygon Irregular polygon

10 Circle all the shapes that are regular polygons.

Explain how you determined which shapes to circle.

11 Determine whether each description is of a regular polygon. Explain.

a A rhombus with all sides equal to 7 inches, 2 angles equal to 40 degrees, and 2 angles equal to 140 degrees.

b A rectangle with sides measuring 4 cm and 6 cm, and all angles measuring 90 degrees.

c A right triangle.

18 For each polygon, determine whether the line divides the ﬁgure into two congruent parts, where each part is the mirror image of the other. Explain your reasoning.

a b c d

e f g h

Let’s extend our thinking

19 Henrique says the line drawn on the parallelogram is a line of symmetry because it divides the polygon into two congruent triangles.

Jacqueline says it is not a line of symmetry because the triangles are not mirror images of each other. Who is correct? Explain your answer.

20 Complete the following tasks:

a Create your own regular polygon.

b Draw all the lines of symmetry it has.

c Label the sides and angles to show congruency.

21 Consider a rhombus. The points B, D, F and H represent the midpoints of the sides of the rhombus.

a Write down the pairs of points that can be joined to form a line of symmetry for the rhombus.

b Is a rhombus a regular polygon? Explain.

22 If possible, draw an example for each of these polygons. If it is not possible, explain why not.

a A regular polygon with no lines of symmetry.

b A regular polygon with 1 line of symmetry.

c A regular polygon with 2 lines of symmetry.

23 A new community center is being designed in the shape of a regular polygon. Inside the building, there will be a circular-shaped auditorium. The outside of the building needs to have multiple, evenly spaced entrances, but the shape of the building has not yet been designed.

a Propose a solution for designing the shape of the outside of the building using a regular polygon.

b Explain your choice based on the properties of symmetry and congruence, and how it would contribute to the eﬃciency and beauty of the construction.

7.03 Perimeter of triangles and parallelograms

After this lesson, you will be able to...

• solve mathematical and contextual problems involving the perimeter of triangles and parallelograms.

Perimeter of triangles and parallelograms

Perimeter

The measure of the distance around a ﬁgure.

Interactive exploration

Explore online to answer the questions

Use the interactive exploration in 7.03 to answer these questions.

1. How are the perimeter formulas similar? How are they different?

2. Which formulas could be simplified? Why do you think that is?

To ﬁnd the perimeter of any shape we can add up all of the side lengths, however the properties of some shapes result in special perimeter formulas.

The perimeter, P, of a triangle is given by P = a + b + c where a, b, and c are the lengths of each side.

A parallelogram is a quadrilateral with two pairs of parallel sides, its opposite sides are congruent.

We will use the base, b, and slant height, a, to ﬁnd its perimeter.

Perimeter = b + a + b + a

Perimeter = 2a + 2b = 2 (a + b)

l w

Consider a rectangle, a special type of parallelogram. Its opposite sides are congruent.

If the width is w and the length is l, the perimeter is:

Perimeter = l + w + l + w

Perimeter = 2l + 2w = 2 (l + w)

Example 1

Consider a square, a special type of rectangle. All of its sides are congruent.

If each side has length l, the perimeter is:

Perimeter = l + l + l + l

Perimeter = 4l

Find the perimeter of the isosceles triangle shown.

Create a strategy

Use the perimeter of the triangle formula.

Apply the idea

Since the triangle is isosceles, we can let the opposite sides a = 12 and b = 12. The base is c = 8.

Perimeter = a + b + c Write the formula

= 12 + 12 + 8

= 32 cm

Example 2

Substitute a = 12, b = 12, c = 8

Evaluate

Find the perimeter of an equilateral triangle with a side length of 5 mm.

Create a strategy

All 3 sides in an equilateral triangle are congruent. If we call the side length s, we can write the perimeter forumula:

P = s + s + s or P = 3s

Apply the idea

We are given that side length = 5.

Perimeter = 3s Write the formula

= 3 5

= 15 mm

Substitute s = 5

Evaluate the multiplication

Example 3

Find the side length indicated on the diagram if the perimeter of the shape is 69 cm.

Create a strategy

Use the formula for the perimeter of a triangle and substitute the known values.

Apply the idea

The triangle has P = 69. We can let the side lengths a = 23, b = 19, and the missing side c P = a + b + c Write the formula

69 = 23 + 19 + c Substitute P = 69, a = 23, b = 19

69 = 42 + c Evaluate the addition

27 = c Subtract 42 from both sides of the equation c = 27 cm Rewrite with c on the left

Example 4

In the rectangle ABCD, side AB has a length of 6 units.

a State the other side of ABCD which must have a length of 6 units.

Create a strategy

The opposite sides of a rectangle are congruent.

Apply the idea

Since has a length of 6 units, properties of rectangles tells us the opposite side, , will also have a length of 6 units.

b If another side of the rectangle measures 5 units, ﬁnd the perimeter.

Create a strategy

Use the formula for the perimeter of a rectangle, P = 2 (l + w). The fourth side must also be 5 units since the opposite sides of a rectangle are congruent.

Apply the idea

The length is 6 units and the width is 5 units.

Perimeter = 2(l + w)

Perimeter = 2(6 + 5)

= 2 11

= 22 units

Example 5

Formula for perimeter of a rectangle

Substitute the length and width

Evaluate the addition

Evaluate the multiplication

Jacob is helping his father build a garden with a fence around it, and they decide to make the garden in the shape of a parallelogram. The garden has one side that is 15 feet long and the side adjacent to it is 10 feet long. What is the total perimeter of the garden fence that Jacob and his father need to prepare?

Create a strategy

Opposite sides of a parallelogram are equal in length. Use formula for perimeter of a parallelogram: P = 2 (a + b).

Apply the idea

The base is 15 feet and the slant height is 10 feet. Using the formula to ﬁnd perimeter.

Perimeter = 2(a + b)

Perimeter = 2(10 + 15)

= 2 ⋅ 25

= 50 feet

Reﬂect and check

Formula for perimeter of a parallelogram

Substitute the slant height and base

Evaluate the addition

Evaluate the multiplication

If we couldn’t remember the formula we could have just written down all of the side lengths and added them together:

10 + 10 + 15 + 15 = 50 feet

Idea summary

To ﬁnd the perimeter of a ﬁgure, add up all of its side lengths.

The perimeter of a triangle, with sides a, b, and c has the formula:

Ptriangle = a + b + c

The perimeter of a parallelogram, with slant height, a, and base, b, has the formula:

Pparallelogram = 2 (a + b)

The perimeter of a rectangle, with length, l, and width, w, has the formula:

Prectangle = 2 (l + w)

The perimeter of a square, with side length, l, has the formula: Psquare = 4l

What do you remember?

1 Find the perimeter of each of these rectangles: a b c d

2 Identify the following triangles as either scalene, isosceles, or equilateral:

3 Complete the steps to ﬁnd the perimeter of this triangle: P = 55 + ⬚ + ⬚ = ⬚ cm

4 Complete the steps to ﬁnd the perimeter of this parallelogram:

5 Are these statements always, sometimes, or never true?

a Adjacent sides of a rectangle are equal in length.

b The diagonals of a rectangle are congruent.

c A rectangle with two equal connected sides is a square.

d A rhombus is a parallelogram.

e A rectangle with adjacent sides perpendicular is a square.

f All sides of a square are different lengths.

6 In the parallelogram ABCD, side AB has a length of 6 units.

a Find the other side of ABCD which must have a length of 6 units.

b If another side has a length of 4 units, what is the perimeter of the parallelogram?

Let’s practice

7 Find the perimeter of a parallelogram given the dimensions:

a Base: 20.5 ft; Slant Height: 7 ft

8 Find the perimeter of each triangle:

b Base: 2 in; Slant Height: 13 in

a A scalene triangle with side lengths 13 in, 17 in and 18 in.

b An equilateral triangle with a side length of 4 yd.

c An isosceles triangle where the 2 equal side lengths are 13 ft each and the third side measures 4 ft.

9 Find the perimeter of each triangle:

10 Find the perimeter of each quadrilateral. Explain your reasoning. a Base: 4 cm b

11 Which parallelogram has the greater perimeter? Explain how you know.

Parallelogram A

Parallelogram B

12 Your local community park is creating a triangular picnic area. Find the perimeter of the picnic area.

13 Nadia wants to build a thin wire frame for a photo that is in the shape of a parallelogram. The frame has a base of 13 cm and a slant height of 6 cm. How much wire will she need to go around the entire photo?

14 A triangular garden bed has side lengths of inches, inches, and 20 inches. What is the perimeter of the garden bed?

15 You are planning to add a decorative border around your new oﬃce desk, which has a parallelogram shape. The desk has a base measuring 5 ft and a slant height of 3.5 ft. Calculate the total length of the border needed to go around the desk.

16 The Grand Park in Centerville forms a triangular area with its community center, water fountain, and playground serving as vertices.

The distance between the community center and the water fountain is approximately 3.25 miles. The distance between the water fountain and the playground is about 4.80 miles. The distance between the playground and the community center is roughly 6.25 miles.

Estimate the perimeter of the triangular park area.

17 The client wants the four kitchen windows to be framed with a special molding. If the windows are 18 inches by 20 inches, how many inches of molding are needed for the kitchen window?

18 Hebah is tiling her bathroom with equilateral triangle tiles. She knows the perimeter of each tile, but she wants to ﬁnd the length of each side. If one tile has a perimeter of 36 cm, how long is each side?

Describe and correct the error in ﬁnding the perimeter of the polygon.

Perimeter = 12 yd 18 yd 24 yd = 5184 yd

Let’s extend our thinking

20 A square has the same perimeter as an equilateral triangle. If the triangle has sides of length 8 yd, ﬁnd the side length of the square.

21 Given the perimeter, P, of the each triangle, ﬁnd the value of the variable:

22 Find the side length of a triangle that has a perimeter of 18 ft and two side lengths that both measure to be 5 ft.

23 Find the side length of one of the equal sides of an isosceles triangle that has a perimeter of 22 in and a third side length that measures to be 4 in.

24 Find the value of x if the perimeter of this parallelogram is 48 m.

25 A quilt is made by sewing together 4 identical parallelograms as shown ﬁgure: If the total perimeter of the quilt is 194 cm calculate the slant height of each parallelogram piece.

We know the area of a rectangle is length times width.

We can take the rectangle, cut off one side, and move it to the other side to create a parallelogram. This does not change the size of the ﬁgure so both the rectangle and the parallelogram have the same area.

In this case, both the rectangle and the parallelogram have an area of:

7 cm 4 cm = 28 cm2

We can use the base and perpendicular height of the parallelogram to ﬁnd its area, just like we do for a rectangle.

The area of a parallelogram is found by

= b ⋅ h

b is the base h is the height height height base base

The height is always measured perpendicular to the base (at a right angle).

Every parallelogram has two base height pairs.

Since we are ﬁnding the product of two lengths, area is always measured in square units.

Example 1

Find the area of this parallelogram.

Create a strategy

Use the formula for the area of a parallelogram: A = bh

Apply the idea

We know b = 13 and h = 8. A = b h

= 13 8

= 104 m2

Formula for area of a parallelogram

Substitute b = 13 and h = 8

Evaluate 13 m 8 m

Area of parallelograms

Example 2

Find the area of this parallelogram.

Create a strategy

Use the formula for the area of a parallelogram: A = bh

Apply the idea

We have b = 3 and h = 11.

A = b h

= 3 11

= 33 mm2

Example 3

Formula for area of a parallelogram

Substitute b = 3 and h = 11

Evaluate

A school is adding a new athletic field that is parallelogram-shaped. The longest side of the field measures 100 yards and the shortest distance (height) from this side to its opposite side is 60 yards. Calculate the area of the athletic field.

Create a strategy

Start by drawing a diagram and then use the formula for the area of a parallelogram: A = bh

Apply the idea

We know b = 100 and h = 60.

A = b ⋅ h

= 100 60

= 6000 yards2

60 yards

100 yards

Formula for area of a parallelogram

Substitute b = 100 and h = 60

Evaluate

Idea summary

The area of a parallelogram is found by:

A is the area of a parallelogram b is the base of a parallelogram h is the height of a parallelogram

The height is measured perpendicular to the base.

Practice

What do you remember?

1 Draw the perpendicular height and mark the right angle for each parallelogram: a b

2 Are these statements true or false?

a The area of a parallelogram can be found by multiplying the base by the parallel height.

b Parallel sides of a parallelogram are always equal in length.

c A rectangle and a parallelogram with the same base and height measurements will always have the same area.

3 Identify the base and the height needed to ﬁnd the area of each paralleogram:

4 Complete the steps to ﬁnd the area of this parallelogram:

Let’s practice

5 This parallelogram with a base of 9 cm and a height of 6 cm is formed into a rectangle:

a Find the length of the rectangle.

b Find the width of the rectangle.

c Is the area of the rectangle greater than, less than or equal to the area of the parallelogram? Explain your thinking.

6 a Redraw this parallelogram as a rectangle.

b Find the area of the rectangle.

c Find the area of the parallelogram.

7 a If the parallelogram is formed into a rectangle, determine:

i The length of the rectangle.

ii The width of the rectangle.

iii The area of the rectangle.

b Find the area of the parallelogram.

8 a State the length of the base.

b State the length of the height.

c What is the size of the angle formed by the base and the height?

d Find the area of the parallelogram.

9 Find the area of each parallelogram:

10 Find the area of these parallelograms with these dimensions:

a The base is 5 ft and the perpendicular height is 2 ft.

b The base is 15 in and the perpendicular height is 7 in.

11 Determine whether these could be the dimensions of a parallelogram with an area of 28 mm2:

a Base = 1 mm, height = 28 mm

c Base = 4 mm, height = 7 mm

b Base = 7 mm, height = 4 mm

d Base = 2 mm, height = 28 mm

12 Find the value of the base or height in each of these parallelograms: a Area = 36 cm2 b Area = 35 m2

13 Find the base length of a parallelogram whose area is 108 ft2 and perpendicular height is 9 ft.

14 Complete this table of base and height measurements for three different parallelograms with an area of 60 in2:

Base (in) 10 30

Height (in) 12

Area (in2) 60 60 60

15 A parallelogram-shaped sign is being designed for a park with a base of 4.5 feet and a height of 2 feet.

Find the area of the sign in square feet.

16 The ﬂoor of a room measures 8 meters by 10 meters. What is the area of the ﬂoor in square meters if the room is in the shape of a parallelogram?

17 Kwame is building a ramp for the local skateboard park. The ramp is in the shape of a parallelogram, with a base of 12 feet and a height of 6 feet. What is the area of the ramp?

Let’s extend our thinking

18 A parallelogram has parallel sides of length 7 cm and 5 cm. A rectangle also has parallel sides of length 7 cm and 5 cm. Draw the two quadrilaterals and explain which one will have the larger area.

19 If you are given two parallelograms with the same base but different heights, explain how you would determine which parallelogram has the greater area.

20 A parallelogram has a base of 10 cm and a height of 20 cm.

a If 2 cm is added to the base and the height, is the area increased by 2 cm2?

b If the base and height are doubled, is the area doubled?

c If the base and height are doubled, by what factor has the area increased?

d Explain why this number is the increase factor.

21 Two parallelograms have the same area. If one parallelogram has a base of 8 cm and a height of 4 cm, what are the possible dimensions of the other parallelogram, given the base and height are whole numbers?

22 A roof in the shape of a parallelogram is being designed for a new house. The roof will be covered with tiles, which cost $10 per square meter. The roof has a base length of 20 meters and a height of 8 meters.

a Find how many square meters of tiles are needed to cover the roof.

b How much will it cost to tile the roof?

23 A jewelry maker is creating parallelogram-shaped pendants for a new collection. Each pendant has a base length of 3 cm and a height of 40 mm. The pendants will be covered with a thin layer of gold that costs $0.50 per square centimeter. If the jewelery maker needs to make 500 pendants, ﬁnd the total cost of covering them with gold.

Example 1

Find the area of the triangle shown.

Create a strategy

Use the area of a triangle formula.

Apply the idea

Use the formula

Substitute b = 10 and h = 7

Evaluate

Example 2

Alex is making a kite shaped like a triangle. He plans to use a stick that is 15 cm long for the height, and he wants the base of the kite to be 20 cm. What is the minimum amount of material Alex needs to buy?

Create a strategy

We need to calculate the area of the kite.

Apply the idea

Formula for area of a triangle

Substitute b = 20 and h = 15

Evaluate the multiplication

Idea summary

The formula for the area of a triangle is:

A Area of the triangle

b Base of the triangle

h Height of the triangle

6 Find the area of the following triangles:

7 Find the area of a triangle which has the following dimensions:

a Height = 8 in and base = 9 in.

c Height = 7 mm and base = 8 mm.

b Height = 4 yd and base = 6 yd.

d Height = 3 cm and base = 9 cm.

8 Determine whether the following could be the dimensions of a triangle with an area of 20 in2:

a Base = 8 in, height = 5 in.

c Base = 5 in, height = 8 in.

b Base = 1 in, height = 20 in.

d Base = 2 in, height = 20 in.

9 A farmer wants to install a triangular shade sail in her paddock to protect her sheep from the sun. The base of the shade sail is 12 ft long, and the height of the triangle is 8 ft. What is the area of the triangular shade sail in square feet?

10 A civil engineer needs to design a triangular traﬃc island for a new road intersection. The base of the triangle is 20 ft long, and the height of the triangle is 15 ft. Calculate the area of the triangular traﬃc island in square feet.

11 An artist is painting a triangular canvas with a base of 80 cm and a height of 50 cm. Calculate the area of the canvas that needs to be painted.

12 Ambrose is retiling half of his kitchen ﬂoor which is in the shape of a rectangle. The remaining part of the ﬂoor is roughly in the shape of a triangle, as shown.

The ﬂoor is 22 ft long, and 12 ft wide. Each square tile is 1 ft × 1 ft. How many square feet of ﬂooring does Ambrose need to buy?

13 Describe and correct the error in calculating the area of the triangle shown.

14 For each of the following triangles, ﬁnd the value of b or h given the area: a Area = 20 mm2 b Area = 120 mm2

15 Complete this table of base and height measurements for four different triangles, which all have an area of 30 m2:

Base (m) 10 15

Height (m) 12 2

Area (m2) 30 30 30 30

16 A gutter running along the roof of a house has a cross-section in the shape of a triangle as shown: If the area of the cross-section is 40 cm2, and the length of the base of the gutter is 10 cm, ﬁnd the perpendicular height h of the gutter.

17 Explain how the area of a triangle is connected to the area of a rectangle using this diagram:

Let’s extend our thinking

18 The area of a triangle measures 45 m2. Find all the possible dimensions of the triangle.

19 You know the height and perimeter of an equilateral triangle. Explain how to ﬁnd the area of the triangle. Draw a diagram to support your reasoning.

20 If the base of a triangle is 16 cm and the height is two times the base, what is the area of the triangle?

21 A triangle has a height of 32 cm and a base that is as long as the height. What is the area of the triangle?

22 A triangle has a base of 45 cm and a height that is as long as the base. What is the area of the triangle?

23 Edith wants to place a triangular ﬂower garden in the front of the house to increase curb appeal.

a What could be the dimensions of the garden if she has a space of 16 square feet to work with?

b The longest side in the triangular ﬂower garden will be against the house. Edith wants to place decorative bricks around the other two sides of the garden. How many feet of brick does she need to purchase? h

7.06 Polygons in the coordinate plane

After this lesson, you will be able to...

• solve mathematical and contextual problems involving the perimeter and area of parallelograms in the coordinate plane.

• draw polygons in the coordinate plane given coordinates for the vertices.

• use coordinates to determine the length of a side joining points with the same first coordinate or the same second coordinate.

• solve real-world and mathematical problems involving finding the distance between two points and drawing polygons on the coordinate plane

Polygons in the coordinate plane

Remember that the coordinate plane can be used to describe the location of points in a 2D space.

By connecting 3 or more points on the coordinate plane with line segments, we can plot polygons. Plotting polygons on the coordinate plane will allow us to easily determine lengths and distances without needing a ruler.

Using the points A( 1, 1), B(3, 1), C(3, 3), and D( 1, 3) we can draw quadrilateral ABCD. We can calculate the side lengths of the ABCD using the ordered pairs.

The length of , which is horizontal, can be found by subtracting the x-coordinates of A and B. Because distance is always positive, we will take the absolute value.

= ∣3 ( 1)∣ = 4

The length of , which is vertical, can be found by subtracting the y-coordinates of B and C and taking the absolute value.

= ∣3 1∣ = 2

We can use these side lengths to calculate perimeter and area from polygons on the coordinate plane.

Recall, the perimeter of a rectangle can be found by adding up all of the side lengths, or using the formula P = 2l + 2w and here l = 4 units and w = 2 units. So:

P = 2(4) + 2(2) = 8 + 4 = 12 units

b What is the length of AB?

Create a strategy

For points with the same x-coordinates, we can ﬁnd the distance by subtracting the absolute values of the y-coordinates.

Apply the idea

The y-coordinate of A is 7, and the y-coordinate of B is 9.

AB = ∣7 ( 9)∣ Find the difference of the two y-coordinates

= ∣16∣

= 16

Reﬂect and check

Evaluate the subtraction

Evaluate the absolute value

AB = 16 units

Another strategy is to count the number of spaces between two points on the coordinate plane. For points with the same x-coordinates, we can count the vertical spaces between the points.

For points with the same y-coordinates, we can count the horizontal spaces between the points.

c Find the length of BC

Create a strategy

For points with the same y-coordinates, we can ﬁnd the distance by subtracting the absolute values of the x-coordinates.

Apply the idea

The x-coordinate of B is 7, and the x-coordinate of C is 6.

BC = ∣7 ( 6)∣ Find the difference of the two x-coordinates

= ∣13∣

= 13

d Find the area of △ABC

Create a strategy

Evaluate the subtraction

Evaluate the absolute value

BC = 13 units

Use BC for the base and the length of AB for the height of the triangle. Then, use the area of a triangle formula

A = b h

Apply the idea

b = BC = 13

h = AB = 16

Formula for area of a triangle

Substitute b = 13 and h = 16

Evaluate the multiplication

Example 3

Consider the square LMNO

a Find the perimeter of LMNO.

Create a strategy

Use the lengths LM, MN, OR, or LO to ﬁnd the side length of the square. Then either add up the four sides or use the perimeter of a square formula P = 4l

Apply the idea

Let’s use MN to ﬁnd the length of the side of the square LMNO. This is vertical line segment so we need to ﬁnd the absolute value of the difference between the y-coordinates.

Now, we can calculate the perimeter of the square.

P = 4 l

= 4 5

= 20

b Find the area of LMNO.

Create a strategy

Use the perimeter formula for a square

Substitute l = 5

Evaluate

Use the side length of the square found in part (a) with the formula for area of a rectangle A = bh or the formula speciﬁcally for a square A = s2, where s is the side length.

Apply the idea

In part (a) we found the side of the square is 5 units in length. Now, we can calculate the area of the square.

A = s 2

= 52

= 25

Formula for area of a square

Substitute s = 5

Evaluate the exponent

Idea summary

To ﬁnd the distance between two points with the same x-coordinates, subtract the y-coordinates and then ﬁnd the absolute value of the difference.

The same is true for points with the same y-coordinates. Subtract the x-coordinates and then ﬁnd the absolute value of the difference.

We can also ﬁnd the distance between points that share an x- or y-coordinate, by counting the number of spaces between them on the coordinate plane.

Practice

What do you remember?

1 Match each description to the correct quadrant:

a The x-value is negative and the y-value is positive.

b The x-value and y-value are both positive.

c The x-value is positive and the y-value is negative.

d The x-value and y-value are both negative.

2 Describe how the point P moves on the coordinate plane from the origin. a b

i Quadrant I

ii Quadrant II

iii Quadrant III

iv Quadrant IV

3 Describe how you would move on the coordinate plane from ( 1, 1) to plot the point (9, 10).

10 Find the distance between the following pairs of points:

a A ( 5, 8) and B ( 2, 8) b A (7, 3) and B ( 1, 3) c A (6, 5) and B (6, 1) d A ( 6, 2) and B ( 6, 7)

e A ( 5, 7) and B ( 2, 7) f A ( 9, 3) and B ( 1, 3) g A ( 5, 6) and B (4, 6) h A ( 8, 5) and B (1, 5)

11 Consider the point plotted on the coordinate plane.

a Plot a point on the graph that has the same x-coordinate as point F. Label the point G.

b Plot another point on graph that has the same y-coordinate as point F. Label the point H

c Describe the distance between point F and point G

d Describe the distance between point F and point H

e Which two points are along the same horizontal line?

f Which two points are along the same vertical line?

12 Consider the points: A ( 4, 8), B ( 7, 8) and C ( 7, 1)

a Plot the points on a coordinate plane. b Draw lines to connect the vertices of each point.

c Find the length of

d Find the length of

13 Jaime travels from Beach A to Beach C through Beach B. The coordinates are plotted on the following coordinate plane:

If each unit represents 10 meters, ﬁnd the total distance that Jaime traveled.

14 Look at the points graphed in the coordinate plane. Describe the distance between:

a Point A and Point F

b Point C and Point D

c Point B and Point E

15 A rectangular garden has vertices with the coordinates of ( 6, 7), (7, 7), (7, 9) and ( 6, 9).

a Plot the rectangular garden on a coordinate plane.

b If each unit represents 1 foot, ﬁnd the perimeter of the garden.

16 A triangle has points A (1, 2), B ( 2, 3) and C (6, 3).

a Plot the triangle ABC on a coordinate plane.

b Find the perpendicular height of the triangle if is the base.

c Find the length of base

17 Consider the quadrilateral with the points A ( 3, 4), B (5, 4), C (5, 4) and D ( 3, 4).

a Plot the quadrilateral ABCD on a coordinate plane.

b Find the length of the following sides: i ii iii iv

c State the type of the quadrilateral ABCD. Explain your thinking.

Let’s extend our thinking

18 The points given represent three vertices of a parallelogram. Find the coordinates of the fourth vertex if it is known to be in the second quadrant.

19 The points given represent two of the three vertices of a triangle.

a State three possible coordinates of point C if the three points form an isosceles triangle.

b State four possible coordinates of point C if the three points form a right-angled triangle.

20 The coordinates of the three vertices of a rectangular farm in a topographic map are (4, 5), ( 3, 5) and ( 3, 4).

a Find the coordinates of the missing vertex.

b Plot the farm on a coordinate plane.

c The coordinates are measured in kilometers. Find the area of the farm.

d The farmer is going to seed the farm with alfalfa grass. Each bag of seed will cover 9 square kilometers, how many bags of seed will the farmer need?

e If each bag of seed costs $32.97, what is the total cost to seed the farm with alfalfa grass?

21 You are a landscape architect designing a small neighborhood park. This park will be situated on a plot of land represented by a coordinate plane, with a total area of 2000 square units. Your design plan must include:

• A square ﬂower garden that occupies an area of 200 square units.

• A triangular bird-watching area with an area of 300 square units.

• A rectangular relaxation zone covering an area of 400 square units.

In the coordinate plane, outline each section with labels. Determine the area and perimeter for the ﬂower garden and relaxation zone.

As you create your design, take into account both the functional use of space and the visual appeal of the park.

Circles 8

Big ideas

For this circle:

The circumference of a circle is the distance around its boundary. We can think of the circumference as the perimeter of the circle.

A chord is a line segment connecting any two points on a circle. A chord may or may not go through the center of a circle. The diameter is the longest chord of a circle.

a What is the diameter of the circle?

Create a strategy

Find out how long the line is that goes through the middle of a circle and touches the edge at two points that are opposite each other.

Apply the idea

Look for a straight line that goes from one side of the circle to the other, passing through the middle. This line is called a diameter, and its length is 10 cm.

Diameter = 10 cm

b What is the radius of the circle?

Create a strategy

Remember the radius is half the length of the diameter.

Apply the idea

In part (a), the value of the diameter is provided as 10 cm.

Radius = Divide 10 by 2 = 5 cm Evaluate

Example 1

Idea summary

A circle has many parts:

The diameter of a circle is double the radius:

d is the diameter r is the radius

Practice

What do you remember?

1 Describe the difference between a line and a line segment.

2 Match the following terms with their deﬁnitions:

a Chord i A line segment passing through the center of the circle with its endpoints on the circle

b Diameter ii A line segment with both endpoints on the circle

c Radius iii A line segment with one endpoint at the center of the circle and the other on the circle

d Circumference iv The space inside the circle

e Area v The distance around the edge of the circle

3 Name the indicated part of these circles: a b c d e f

4 You want to calculate the distance a bicycle covers when its wheel completes one full turn (360-degree rotation). Which of the following best describes this distance?

A The radius of the wheel

C The circumference of the wheel

B The diameter of the wheel

D The area of the wheel

5 Use the three circles to draw and label radius, chord, and diameter: a b c

6 Describe a real-world example of each: a Radius b Diameter c Circumference d Area e Chord

7 If given the radius of a circle, explain how to ﬁnd the diameter.

8 State the diameter of these circles:

9 State the radius of these circles:

10 For each pair, determine if the radius and diameter measurements could be from the same circle. Explain your reasoning.

a Radius: 11 cm; Diameter: 22 cm

c Radius: 60 cm; Diameter: 30 cm

e Radius: 220 cm; Diameter: 110 cm

b Radius: 17 cm; Diameter: 38 cm

d Radius: 41 cm; Diameter: 82 cm

f Radius: 66 cm; Diameter: 132 cm

11 Explain the concept of a chord and give two real-life examples where a chord is relevant.

12 M is the center of the circle, and the length of CG is 12 cm.

a Name a segment that is a radius. What is the length of the radius?

b Name a segment that is a diameter. What is the length of the diameter?

c Write a comparative statement about the diameter and a radius.

d Write a comparative statement about a radius and a diameter.

13 The radius of the smaller circle in the ﬁgure is 16 cm.

If CD = 4 cm, ﬁnd AD, the diameter of the larger circle.

14 Here is a circle with the center point P and some line segments.

a Identify all examples of diameters, radii, and chords and explain your reasoning. Measure the line segments to use as part of your justiﬁcations.

b Explain why the diameter is the longest chord in a circle.

Example 3

Find the circumference of the circle shown, correct to two decimal places.

13 cm

Create a strategy

The circumference of the circle can be found using the formula: C = πd

Apply the idea

C = π ⋅ 13

Substitute d = 13

= 40.84 cm Evaluate

Example 4

If the radius of a circle is equal to 17 cm ﬁnd its circumference correct to one decimal place.

Create a strategy

The circumference of the circle can be found using the formula: C = 2πr.

Apply the idea

C = 2 π 17

Substitute r = 17

= 106.8 cm Evaluate

Example 5

Lisa is cleaning the leaves out of the pool in her backyard. The pool is a circular shape and has a radius of 5 m

What distance does Lisa cover if she walks all the way around the pool? Give your answer to one decimal place.

Create a strategy

The distance around the outside of a circle is its circumference.

Apply the idea

C = 2πr

= 2 π 5

= 31.4 m

Write the formula

Substitute r = 5

Evaluate

Lisa will walk 31.4 m around the pool.

Example 6

Carl is performing an experiment by spinning a metal weight around on the end of a nylon thread. How far does the metal weight travel if it completes 40 revolutions on the end of a 0.65 m thread? Give your answer correct to one decimal place.

Create a strategy

The total distance traveled by metal weight can be found using the formula:

Total distance traveled = circumference number of revolutions

The radius is equivalent to the length of thread.

Apply the idea

Find for circumference:

C = 2πr

= 2 π 0.65

= 1.30π m

Total distance traveled = 1.30π 40

= 163.4 m

The metal weight traveled a total of 163.4 m

Idea summary

Write the formula for circumference

Substitute r = 0.65

Evaluate

Substitute circumference and number of revolutions

Evaluate

π is the ratio between the circumference and diameter, which we approximate as 3.14. The formula for circumference of a circle is :

= πd

C Circumference d Diameter and because the diameter is twice the radius, we can also write the formula as

C = 2πr

C Circumference r Radius

What do you remember?

1 What term refers to the perimeter of the circle?

2 For this circle, identify the letter that corresponds to:

a Radius

b Diameter

c Circumference

d Chord

e Area

3 Look at the table below. For each circle, analyze the relationship between the radius and circumference. Then answer true or false for each of the following questions.

a The radius of a circle is equal to two times π times the circumference.

b The circumference of a circle is equal to two times π times the radius.

c The circumference of a circle is double the radius.

d The circumference of a circle is equal to double π times the radius.

e The radius of a circle is equal to double π times the circumference.

4 Examine the table below. Analyze the relationship between the diameter and circumference for each circle. Then, determine whether each of the following statements is true or false.

a The diameter of a circle is approximately one-third of the circumference.

b The circumference of a circle is approximately π times the diameter.

c The circumference of a circle is triple the diameter.

d The circumference of a circle is 2π times the radius, which is half the diameter.

e The diameter of a circle is equal to the circumference divided by π

5 Determine whether these statements are true or false:

a π is a whole number.

b π is a number that cannot be written as a fraction using whole numbers and is irrational.

c π is a number that goes on forever and can’t be written exactly.

d π is exactly equal to

e π is the ratio between the diameter and the radius of a circle.

Let’s practice

6 Find the circumference of each circle. Use 3.14 for π and round your answers to two decimal places: a

7 Find the circumference of the following circles. Use 3.14 for π and round your answers to two decimal places.

a A circle of radius 7 ft

c A circle with a diameter of units

b A circle of radius 11 in

d A circle with a diameter of 9.5 cm

8 If the radius of a circular swimming pool is 6.7 yards, how far would you travel if you swam around the pool’s edge exactly once? Explain your reasoning. Use 3.14 to approximate π

9 If the diameter of a wheel is 18 units, how far will the wheel roll in one full revolution? Explain your thinking. Use 3.14 for π

10 A medium pizza has a circumference of 40 in.

a Describe what the radius of the pizza would be, rounded to two decimal places.

b Describe what the diameter of the pizza would be, rounded to two decimal places.

c Explain your thinking.

11 A trampoline has a circumference of 44.5 ft.

a Describe what the radius of the trampoline would be, rounded to two decimal places.

b Describe what the diameter of the trampoline would be, rounded to two decimal places.

c Explain your thinking.

12 For each pair, determine if the diameter and circumference measurements could be from the same circle. Explain your reasoning.

a Diameter: 14 cm; Circumference: 44 cm b Diameter: cm; Circumference: π cm

c Diameter: 30 cm; Circumference: 90 cm d Diameter: 20 cm; Circumference: 62.8 cm

e Diameter: 16 cm; Circumference: cm f Diameter: 8 cm; Circumference: 25.12 cm

13 Samantha measured a circular table and noted that d, the diameter, was 12 inches and C, the circumference, was approximately 38 inches.

C = 38 in

d = 12 in

Write an expression using the values of the circumference and diameter that represents an approximate value for π

14 After measuring several circles, a student recorded the following circumferences and diameters:

Calculate the ratio of circumference to diameter for each circle.

15 For each circle in the previous question, what do you notice when circumference is divided by the diameter? What conclusions can you draw about the approximate measurement of π from this data?

Write down the equation for the circumference of each circle if:

a you know the radius of the circle is equal to 27 cm

b you know the diameter of the circle is equal to 12.5 cm

17 Alex says, “To ﬁnd the circumference of a circle, you just multiply the radius by π. So, if the radius is 4 inches, the circumference is 4 × π inches.”

a Identify the error in Alex’s statement and correct it by providing the correct formula.

b Explain the relationship between the diameter, radius, and circumference in simple terms.

18 Caitlin and David calculate the circumference of this circle using different formulas.

David uses the formula C = 2πr to calculate the circumference, where r is the radius.

Caitlin uses the formula C = πd to calculate the circumference.

Explain why they will get the same result.

19 A pie has a radius of 12 inches. Which is closest to the circumference of this pie? Use 3.14 for π to calculate the circumference.

A 37.68 in B 75.36 in C 24.56 in D 50.24 in

20 Jordan has a circular pool with a diameter of 15 ft. Jordan plans to install a safety barrier around the pool. Complete the statement about Jordan’s circular pool.

To install the safety barrier around the pool, Jordan needs a minimum of ⬚ feet of barrier.

21 The bottom of a ﬂower pot has a radius of 16 cm. What is the circumference of the bottom of the ﬂower pot? Use 3.14 for π and round your answers to one decimal place.

22 A scooter tire has a diameter of 34 cm. What is the circumference of the tire? Use 3.14 for π and round your answer to one decimal place.

23 What is the length of the strip of seaweed around the outside of the sushi? Use 3.14 for π and round your answers to one decimal place.

24 Find the circumference of the Ferris wheel: Use 3.14 for π and round your answers to one decimal place.

Let’s extend our thinking

25 Describe what happens to the circumference when the radius of a circle is doubled. Use examples to explain your thinking.

26 A shallow circular wading pool has a diameter of 4 yards. A deeper circular swimming pool has a radius of 16 yards. How many times greater is the circumference of the swimming pool than the wading pool?

27 A circular running track has a diameter of 23 m. How many laps must be completed to run 1600 m? Round your answer to one decimal place.

28 Think of a real-life situation that involves the distance around a circle. Write a story problem about it. Use what you know about circumference to ﬁnd out something interesting in your story. Solve the problem and show your solution.

8.03 Area of a circle

After this lesson, you will be able to...

• solve mathematical and contextual problems involving the area of a circle when given the length of the diameter or radius.

Area of a circle

We already know that area is the space inside a 2D shape. We can ﬁnd the area of a circle, but we will need a special rule.

Let’s look at what happens when we unravel segments of a circle.

Interactive exploration

Explore online to answer the questions

mathspace.co

Use the interactive exploration in 8.03 to answer these questions.

1. Explain how the width of the shape relates to the circumference of the circle.

2. What figure is formed? Explain how the area of this figure relates to the area of the circle.

We can calculate the area of a circle using the formula:

A Area of the circle r Radius of the circle

Example 1

Find the area of the circle shown, correct to one decimal place.

Create a strategy

The area of a circle can be found using the formula:

Apply the idea

Example 2

If the diameter of the circle is 24 cm, ﬁnd its area correct to one decimal place.

Create a strategy

Remember that the radius of a circle is half its diameter, r =

Apply the idea

r = Divide the diameter by 2

= 12

A = π 122

Evaluate

Substitute r = 452.4 cm2

Evaluate

Example 3

Carlo and his friends ordered a pizza on a Saturday night. Each slice was 10 cm in length. Find the area of the pizza that Carlo and his friends ordered. Use 3.14 to approximate π

Create a strategy

The length of the pizza is the radius of the pizza. Use the formula of the area of the circle to ﬁnd the area of the pizza.

Apply the idea

A = 3.14(10)2

Substitute the value of the π and the radius = 314

Evaluate

The area of the pizza is 314 cm2

Idea summary

Area of a circle = πr 2 r Radius of the circle

Practice

What do you remember?

1 Describe the difference between area and circumference of a circle.

2 What does each part in the formula A = πr 2 stand for?

16 A cat is leashed to the pole in the middle of a yard. If the leash is 5 ft long, how much area can the cat roam? Explain how you found your answer using the formula for the area of a circle.

Let’s extend our thinking

17 Jake has a bicycle. The area of its wheel is 144π in2. Find the radius, x, of the wheel.

18 Find the radius of each circle, correct to two decimal places. Use 3.14 for π

a A circle with area of 64π cm 2

c A circle with area of 36π cm 2 .

b A circle with area of 36 cm2

d A circle with area of 25 mm2.

19 Find the diameter of each circle, correct to two decimal places. Use 3.14 for π.

a A circle has an area of 81 ft2

b A circle has an area of 144 mm2

20 A ﬂowerpot has a circular base with an area of 88.25 in2. Find the radius of the base, correct to two decimal places.

21 A wind turbine has blades that are R m long which are attached to a tower 60 m high. When a blade is at its lowest point (pointing straight down), the distance between the tip of the blade and the ground is 20 m.

a Calculate the value of R

b Find the distance traveled by the tip of the blade during one full revolution, correct to two decimal places.

c A factor in the design of wind turbines is the amount of area covered by their blades. The larger the area covered, the more air can pass through the blades.

Find the area inside the circle deﬁned by the rotation of the blade tips, correct to two decimal places.

Data & Statistics 9

Big ideas

• Collecting and analyzing data can inform predictions and decisions, as long as the data is based on a valid sample.

• Different representations of data highlight different characteristics of the data.

• Many patterns can be found in sets of data. These patterns can be useful in making inferences but any inferences made from a set of data should not be taken as fact.

This column graph shows the fruits that students have in their lunch one day.

Notice that we have four categories Banana, Apple, Mandarin, and Pineapple and the graph helps us count how many items fall in each category.

Discrete numerical data

Data that can only take certain values and has a limited range of values. It can be displayed in line plots, step-and-leaf-plots, and line graphs.

Example: Shoe size or number of siblings

This line plot shows the shoe sizes of a group of students.

Notice that responses are restricted to possible shoe sizes, which are discrete numerical. We can see how many people wear each size from the line plot.

A clear question helps us know what kind of data to gather and who to collect it from. The type of question we ask can lead us to collect different data.

The group we are hoping to answer the question about is called the population

When we write a question, it should be about the population we want to learn about and have more than one possible answer.

Non-examples of questions

Examples of well formulated questions

How old is my neighbor? What ages are the people in my neighborhood? What brand is my computer? What is the most popular brand of computer among my classmates? What is your favorite color? What colors are preferred by students in my neighborhood?

It needs to be clear which attributes we are exploring with our question. An attribute is a speciﬁc characteristic or feature of a given subject.

For example, if we want to learn more about pets in our class, we need to be clear which attribute we are interested in. These could include:

• Number of pets

• Type of pets

• Size of pets

• Age of pets

Example 2

Is each question well formulated for the data cycle? Explain why or why not.

a Who was the ﬁrst president?

Create a strategy

A well formulated question should have a variety of possible answers and relate to a speciﬁc population.

Apply the idea

There is one answer and no clear population, so this is not a well formulated question for the data cycle.

Reﬂect and check

A related question we could use the data cycle for is “What is the most common term length for a US president?”

b How do the shoe sizes of 5th and 6th graders at my school compare?

Create a strategy

A well formulated question should have a variety of possible answers and relate to a speciﬁc population.

Apply the idea

This is a well formulated question as shoe size is a clear attribute with different possible answers.

Reﬂect and check

This data would be discrete numerical.

c How much money do professional athletes in the US make?

Create a strategy

There is a clear population of professional athletes in the US. Now we need to check if data could be collected to give a variety of answers.

Idea summary

Apply the idea

Different athletes will make different amounts, so this something we could ﬁnd data on. This is a well formulated question.

We use the data cycle to formulate questions, then collect, show, and explain information. Depending on the question being asked, the data may be categorical data or numerical data

A well formulated question should have more than one possible answer and relate to a population.

Collect data

When we have questions, we use different ways to collect data to ﬁnd answers:

• Observation: Watching and noting things as they happen.

• For example, watching birds at a feeder to see which type comes most often.

• Measurement: Using tools to ﬁnd out how much, how long, or how heavy something is.

• For example, using a ruler to measure the growth of a plant over several weeks.

• Survey: Asking people questions to get information.

• For example, asking classmates about their favorite school subject and recording the answers.

Example 3

Aditya wants to investigate the social media habits of the students in her grade.

a Formulate a question to help her complete this investigation.

Create a strategy

She could choose to explore which platforms are used, how much they are used, reasons for usage, and the impact on their academic and social lives.

Apply the idea

A potential question could be, “How many hours per week do students in my grade spend on social media?”

Reﬂect and check

The question can be answered by collecting data, allows for a variety of answers, and could be organized in a data display like a line plot answers are rounded to the nearest hour, so is a well formulated question.

b What attributes would you need to measure to answer the question?

Create a strategy

This question from part (a) is well written as it clearly states the attribute.

Apply the idea

For each person, she would need to identify the time they spend across all social media platforms per day.

Reﬂect and check

When collecting the data, Aditya could collect the data rounded to the nearest hour to make it discrete or could leave it open-ended for more options when displaying or summarizing data.

c Should she use observation, measurement, survey, or experiment to collect the data? Explain.

Create a strategy

She should choose a method that is practical, ethical, and will give reliable results.

Apply the idea

A survey would be the most appropriate method for this investigation.

Reﬂect and check

The population is relatively small, so it would be possible to ask them all a single question about their social media use.

Observing her classmates all day to make conclusions about their social media use would likely not be ethical or easy to do.

Example 4

Collect data that can be used to answer the question “How many ﬁrst cousins do students in my school have?”

Create a strategy

This situation would require a survey. Depending on the size of your school, it could be done with a sample or you could survey the whole population. It is not possible to do observation, measurement, or an experiment for this question.

2 When formulating a question, is each statement true or false?

a The question is expected to have similar answer from all responders.

b The answers to the question may vary from one person to another.

c The answers to the question must have numerical values.

3 State whether or not each of the following questions are well formulated questions.

a How old is your principal?

b How tall are the basketball players in your school?

c Do you prefer burger or pizza?

d Which city is the capital of the Philippines?

e How does the proportion of the students at your school that prefer Math compare to the proportion that prefer Science?

4 Match each procedure with the type of method that was used.

a Observation b Measurement c Survey d Experiment

i Isaiah asks each student in his dance class questions about their food preferences.

ii Harper uses a ruler to record data on the foot lengths of some of the students at her school.

iii Faisal asks all of the teachers in his school to complete some tasks and counts how many they can get done. He plays music while some are doing the tasks and not for others and compares the differences between listening to music and not.

iv Lucas watches and counts how many people go into different stores at different times of day.

5 Hannah has chosen to collect information using a sample instead of surveying the whole population.

State whether each statement is an advantage to doing a sample.

a It is cheaper to conduct.

c It’s more accurate.

b We don’t need to worry about who we survey.

d It takes less time.

6 Determine whether the following uses a sample or a population:

a Lucy has asked everyone in her oﬃce what snacks should be provided in the oﬃce.

b James asks a few of his friends how they did in the test to see if he is above average in his class.

c Joanne ﬁnds the height of the entire class to try to ﬁnd the average height of 15-year old students in America.

d Justin has determined the age of 10% of houses from each suburb in Virginia.

e Valentina tests every engine that the factory produces.

f Oprah checks every dog brought to her vet to assess the treatment of dogs in the city.

g Asking all the teachers at your school whether they approve of a new class timetable.

h A taste test of a large batch of cookies Michael just baked.

7 A study is being done on physical activity in a small town.

a Would this characteristic be important to consider when selecting the sample?

i Favorite color ii Gender

iii Age

v Preferred method of transportation

b Does the number of people in the sample matter?

iv Favorite sport

vi Name

c Does how we select the people who will be the surveyed matter?

8 Classify each data set as discrete numerical or categorical.

a Types of vegetables

c Daily UV index

e Number of siblings

b Brands of tablets

d Types of dogs

f Number of languages spoken at home

9 For each well formulated question, will the collected data be categorical data, discrete numerical data, or neither?

a What sports are played by students at my school?

b Precisely how long do runners take to complete a 5K race?

c What attendance can be expected at a local baseball game?

d What types of cats are the most popular?

e How many televisions to people have in their homes?

10 Is each question well formulated for the data cycle? Explain why or why not.

a What are the favorite movies of sixth graders in my school?

b How tall are the science teachers in your school?

c Is your mom taller than your dad?

d How far away is the moon from the Earth, right now?

e How much do puppies weigh?

f Have you ever visited Paris, France?

g How old are US presidents on the day they are elected?

11 For each scenario, formulate a question to help them complete their investigation.

a Morgan wants to investigate the physical activity habits of the students in his grade.

b Gunnar wants to investigate the impact of listening to music while doing homework.

c Cadence wants to investigate the modes of transportation of the adults at her school.

12 A math class wants to answer the following question: “Are the 12-year olds in the class taller than the 11-year olds in the class?”

a Is this a well formulated question?

b Which of the following attributes are needed to answer the question?

• Height

• Age

• Name

• Physical activity level

c Which of the following units could be used to measure someone’s height?

• Decibels

• Degrees

• Feet

• Inches

13 Consider the question: “What is the expected cost of owning a dog? Does it vary by breed?”

a Is this a well formulated question?

b What attributes would you need to measure to answer the question?

14 Determine whether the following attributes are represented by the given graph:

a Number of children who travel to school and their mode of transport

b Number of people that prefer each mode of transport

c Number of teachers who travel to school and their mode of transport

Key 1 graphic = 54 children

15 For each of the following samples, give an example of a population that the sample could have been chosen from. The samples may not be well selected.

a A sample containing 50 people who drive white cars.

b A sample of 50 people drawn from a population. In this sample, the youngest is 4 years old, and the oldest is 18.

c A sample containing the ﬁrst 50 people to enter a train station on a given day.

16 Beth is interested in which students from her school use public transportation. Determine whether or not the following sampling methods would result in samples that are representative of the population.

a Selecting every 10th person on the bus she takes.

b Selecting every 10th person on the student list.

c Selecting the ﬁrst 50 students that arrive in the morning.

d Selecting by having a computer randomly choose student numbers.

17 Petra is interested in the types of ﬂowers that are grown in her neighborhood. She formulates the question “What types of ﬂowers are most popular in my neighborhood?”

a Design a simple plan to collect data for the question using each method. i Observation ii Measurement iii Survey

b Formulate a question involving ﬂowers that would require an experiment to collect data.

Let’s extend our thinking

18 Change the following questions to make them well formulated questions for the data cycle.

a How many books does your teacher have?

b How many points did the grade school basketball team score in its last game?

c What is your grade in Mathematics during the ﬁrst term?

19 Nuala’s mom has started playing pickleball and Nuala is interested in learning more about different sports, especially ones she hasn’t heard of before.

a Formulate a question that Nuala could use to explore popularity of different sports.

b Identify which attributes she would need to collect data on.

c Identify the population.

d Describe how she could collect data to help answer the question.

20 Georgia wants to know how the people in America are going to vote in an upcoming election. She selects 50 random people from her city to interview.

Suggest reasons why this sampling might give poor results.

21 The government is thinking about increasing the minimum retirement age, but they want feedback on the change, so they plan a survey. The sample they selected was mostly people who are already retired because they were free during oﬃce hours of 9 am to 4 pm.

Explain why this sample is not representative of the population and describe a better sample.

22 A political polling company calls 1000 people at home between 4 pm and 7 pm on weeknights to ﬁnd out who they are most likely to vote for in an upcoming election. They publish their numbers based on the responses of only the 410 people who answered their call.

a How could they have used random sampling to choose the 1000 people to call?

b Explain why the company’s published results will not be accurate.

23 A random sample of 98 students were asked about their favorite sports:

a Using the sample data, which sport can we conclude is liked most by the students?

b If there are 240 students at the school, about how many would you expect to have swimming as their favorite sport?

24 There is a large apartment tower going up in Roxie’s neighborhood. She is curious about what types of homes like detached houses, semi-detached townhouses, and apartments are the most common now compared to 10 years ago.

a Formulate a question that relates to types of housing in the US.

b Collect secondary data from a reliable source.

c Create a data display, such as a bar graph, to organize the data.

d Compare the displays for now and 10 years ago.

e Make a conclusion about the types of housing that is available.

A circle graph is different from a bar chart or line plot, because it does not show the count or frequency of each category. Instead, it shows the proportion of the data that is in a category as parts of a whole.

A circle graph is sometimes called a pie chart because each sector could be a piece of pie.

Fish colors in a tank

A circle graph:

• Is a circle broken into pieces called sectors

• Has a key or labels to show what each sector represents

• Has a title that tells you what the graph is about

• Will have sectors that are proportional to the percentage of the data that is in each category. For example, half of the circle is red, so 50% of the ﬁsh in the tank would be red. Fish colors in tank

If we look at how much of the circle each sector takes up, we can identify what percentage of the total ﬁsh are of each color.

The sum of the percentages should always be 100%, because they represent parts of a whole.

We may not always be able determine the percentages just by looking at it.

We often label the percentages on each sector, so that we can compare more easily and do calculations.

For example:

Fruits purchased from the grocery

Sometimes, we will show the number labels instead of percentages, for example, if 300 people were surveyed, this circle graph shows the same information as the one before:

Fruits purchased from the grocery

Orange Red Blue Yellow

It is important that we always check that the percentages on the graph add up to 100% since a circle graph always represents the whole of the data points.

Circle graphs are not helpful for representing data with large numbers of categories because they get hard to read with too many sectors.

We can use circle graphs in the “Organize and Represent” stage of the data cycle. They can be helpful for questions that ask about a relationship of the parts of a whole.

Circle graphs can show us the probability of the events they represent. Recall that

Probability of an event =

In a circle graph the favorable outcomes are represented by sectors of the graph and the total outcomes are the entire circle. So, the percentage of the circle that the sector(s) makes up is the probability of that event occurring. Probabilities of an event can be described as:

• Impossible - if no sectors represent that event

• Unlikely - if the sector(s) for that event make up much less than half of circle

• Equally likely - if the sector(s) for the event make up half of the circle

• Likely - if the sector(s) for the event make up much more than half of the circle

• Certain - if the sector(s) for the event make up the entire circle

Reason for being at the arena

For example, this circle graph represents the reasons why people are at the arena. If we randomly select one person from the arena, the probability is:

• Impossible that they are there for soccer

• Unlikely that they are working

• Equally likely that they are a spectator

• Likely that they are not playing ringette

• Certain that they are in the arena

Example 1

For each of the following questions for the data cycle, determine if the data can be well represented using a circle graph. If yes, explain why. If not, suggest a different type of display and explain why you chose it.

a How much time do students spend on homework, in hours?

Create a strategy

If we are comparing parts of a whole then a circle graph would be a good choice. We don’t to use a circle graph if there are a large number of categories or possible answers. Decide if this question would lead to seeing how big one part is compared to the others.

Apply the idea

No, a circle graph isn’t the best choice. There are too many possible answers and no need to compare parts of a whole.

A bar graph would be a good choice as it can have more categories for the different possible answers.

Reﬂect and check

A line graph shows changes over time, so total amount of time in a day would not be displayed well in a line graph.

Idea summary

Circle graphs represent the data as parts of a whole. Each sector of a circle graph represents a different category. The larger the sector, the larger the percentage of data in that category.

A circle graph should include:

• A title to explain what the graph is about

• A key to explain how to read the graph

• Percents or number labels for each category

Circle graphs are good for representing categorical or countable numerical data with only a few categories.

Create circle graphs

Circle graphs can be created by hand or using technology. Some programs like Excel or Google Sheets refer to circle graphs as pie charts.

Using technology, we can:

• Enter the data as a list or a frequency table

• Highlight the data

• Insert a chart and select pie chart or circle graph

Season Number of students Winter 5 Spring 15 Summer 30 Fall 10

Suppose we formulate the question “What proportion of students at my school prefer each season?” and collect this data from a sample of 60 students.

We can convert each category to a fraction or a percentage.

Then we can divide up the circle by ﬁrst cutting it in half, then splitting one half into two quarters, then splitting one quarter into twelfth.

Summer

Summer Spring

One half of the circle represents summer

One quarter of the circle represents spring

Create a strategy

Determine the percentage of market share for each manufacturer by dividing the number of responses for each manufacturer by the total number of responses and multiply the decimal by 100.

Apply the idea

Let’s start by looking at the market share of Brand A.

51% is more than half of 4000, so the sector corresponding to Brand A should take up more than half the circle.

For Brand B, we have

which is a little under a quarter of a circle.

For Brand C,

For others we have

So, the sector for Brand C and the others should be about the same size.

Smartphone market share

Here is the circle graph that most accurately represents the data would look like.

So, the correct answer is option C.

Brand A Brand B Brand C Other

Reﬂect and check

We could create the circle graph using technology to check.

b Write a conclusion that the marketing company could make based on the data and representation.

Create a strategy

Circle graphs show the proportion of each category, so the conclusion will be about which brands are the most or least popular.

Apply the idea

Brand A has the majority of the market share with more than 50% of those surveyed using Brand A.

Brand B has about a quarter of the market share, so a similar number of people use Brand B as all other brands, not including Brand A.

Reﬂect and check

We are assuming that this sample is representative of the population. If the sample was not well selected, then this conclusion would only be valid for the sample and not the population.

Idea summary

To create a circle graph, we can use technology, or create it by hand. To make it by hand, we can:

• Create a table with the categories, their count, fraction, and percentage

• Draw a circle

• Divide the circle into segments that match the proportions for each category

• Check that the proportions add up to 1 or 100%

Practice

What do you remember?

1 Recently the Northern Lights, Aurora Borealis, were visible further south than usual. A survey was done of people who saw them and they were asked what color of Northern Lights was their favorite.

Complete the table.

Number of people Fraction of people Percentage of people Purple 40 Green 30 Pink 10

2 For each of the following questions for the data cycle, determine whether or not the data can be well represented using a circle graph.

a What writing utensils, like pencils or markers, do students prefer?

b How many days do students typically spend at camps during the summer break?

c What clothing material is preferred by students?

3 A snack company is looking to add a new item to their product line. They took a small sample and found the following preferences:

• Breaded veggie chips: 156

• Soft oat bars: 19

• Gelatin based product: 32

• Gluten free and vegan option: 74

Which sector represents Gelatin based products?

4 The given circle graph shows the results of a class survey where students were asked to choose their favorite food:

a Which was the most popular food?

b Which two foods were equally popular?

5 Are the following statements true or false?

a The larger the sector, the smaller the percentage of data in that category.

b The percentages on a circle graph should always add up to 100%.

c Circle graphs are used to display categorical data.

6 The owner of a business wants to have a company-wide retreat in Virginia. He narrowed the list down to ﬁve cities, but he cannot decide which city to choose. He sends a survey to all 180 of his employees, asking them to choose the city in which the retreat should be held.

Would the following data displays be appropriate for the owner to use to analyze the results of the survey?

a Dot plot (Line plot) b Bar graph c Pictograph d Circle graph

7 At different times of day, a sample of people were asked what they were doing at the park. Describe the probability that someone was there to go for a run using one of these words.

• Impossible

• Unlikely

• Equally likely

• Likely

• Certain

a At 5 a.m.

b At 10 a.m.

c At 4 p.m. d At 7 p.m.

8 A ﬂorist recorded the roses of different colors sold in his store during the week of Valentine’s day. This is displayed in the circle graph. Order the colors of roses from most to least popular.

9 Edward surveyed a large group of people and asked for their favorite milkshake flavor. The results of the survey are in a circle graph.

a What fraction of people chose chocolate?

b What percentage of people chose chocolate?

c What fraction of people chose strawberry?

d What percentage of people chose strawberry?

The following circle graph shows the results of a survey where 100 children were asked for their favorite color:

a Which was the least popular color?

b How many children chose yellow as their favorite color?

c How many children chose blue as their favorite color?

11 The following circle graph represents the results of 11th grade class president elections for the year 2024:

a Who won the elections?

b What is the percentage of the votes obtained by Jess?

Pink Red Orange White Milkshake ﬂavors

Vanilla Strawberry

12 The results of a senatorial election in a European country, where 4000 people voted, are shown in the given circle graph:

a Who won the election?

b What fraction of people voted for Zorg?

c After the election, the winner decides to send promotional packages to everyone they suspect voted for Zorg. How many packages should they prepare?

13 At a dance studio, there are students who take tap, jazz, ballet, acro, and hip-hop. This circle graph shows the proportion in each class.

One student wins a prize every month.

What is the probability that the winner this month is in:

a Ballet?

b Hip-hop or Jazz?

c Acro?

14 Roald organizes the break-down of his income on this circle graph.

a What does most of this money go towards?

b What does he spend the least money on?

c If his total income is $2200 per month, how much does he spend on clothes?

d Mason has an income of $1200 and puts $300 into savings. Does Mason or Roald save a higher percentage of the monthly income?

15 Lucille formulates the question “What dairy products are the most popular for Americans?”. She collects data from her classmates and makes Circle Graph A. Her friend DeShaun found data from a national survey of 10 000 people and summarized the results in Circle Graph B.

Circle Graph A Circle Graph B

a Was Lucille’s sample representative of the population?

b Whose data would be more reliable?

c Draw a conclusion to answer Lucille’s question.

Milk Yogurt Cheese Cottage cheese

16 Manuela is an international student and is feeling homesick. She formulates the question, “How long are international students usually homesick for?” She does a survey of everyone currently on exchange through her program. She organizes the data in this table.

Construct a circle graph to represent this data.

17 A math teacher formulates the question “Which other subject would students like to see incorporated into math class more often?” She surveys 220 students to ﬁnd their preference and needs to construct a circle graph to display the data. The survey results are shown in the following table:

a Complete the table to ﬁnd the percentage of students that prefer each subject.

b Construct a circle graph to represent this data with or without technology.

c Make a recommendation based on the circle graph.

18 A physiotherapist is doing a study with 1080 volunteers who have knee pain. He is trying a new technique and after six weeks asking the patients how they are feeling.

Result Number of volunteers

a Create a circle graph to represent this data using technology.

b Draw a conclusion about whether or not the physiotherapist should continue with this new technique.

19 Bob asked his students to choose their favorite toy. 35% students picked cars, 10% of the students picked planes, 30% picked balls and 25% picked video games.

a Decide the best type of display for the data and justify your choice.

b Construct the display chosen in part (a).

Let’s extend our thinking

20 The following circle graphs show the number of students registered for different types of courses at a university.

a Which course is the most popular according to the circle graphs?

b Which two courses have around half of the students registered that are not in Arts or Engineering?

c How many courses have more than 100 students registered?

d Elise thinks these circle graphs are confusing and says a bar graph would be better. Do you agree or disagree with Elise? Explain your answer.

21 This question is being explored with the data cycle: “What activities do students do after school? Does it vary by school level?”

Random samples of students were taken at several elementary schools, middle schools, and high schools. Students were asked about the ﬁrst activity they do after school.

These circle graphs summarize the data:

a What conclusions can we make that are true for all three groups?

b What are some of the differences between the different groups?

c If all of the data was grouped together, we get this circle graph. Is this a good overall summary, or is it better to have the graphs separate?

d Formulate another question that is related to after school activities.

22 The given circle graph about the school election was displayed on poster: a Describe two mathematical errors with this circle graph.

Would vote for Flanders

b Which of the following images better represents the same information? A B

Would vote for Flanders Would vote for Grylls

Would vote for McGill

Would vote for Grylls

Would vote for McGill

Would vote for Flanders

c After looking at the data, a student concludes that the majority of people would vote for each of the three candidates. Which of the following is not an explanation for how this is possible?

A The three candidates are running for different positions in government.

B Three different samples were surveyed.

C The survey questions said, “Would you vote for Grylls if they ran for class president? Would you vote for McGill if they ran for class president? Would you vote for Flanders if they ran for class president?

D The survey question said, “If Grylls, McGill, and Flanders run for class president, who would you vote for?” Would vote for Grylls

23 Emil was at his auntie’s house and noticed that she used a desktop computer. He had only ever seen gamers use them. He is now curious about what proportion of internet traﬃc comes from desktops/laptops, mobile phones, tablets, and consoles.

a Formulate a question that Emil could use to explore the proportion of different device types used on the internet using the data cycle.

b Could he use observation, measurement, survey, experiment, or acquire secondary sources? Explain.

c Explain how Emil could collect data that could be used to answer his question from part (a).

d Suppose the given data shows the amount of time 30 of his relatives spend on different types of devices. Organize and represent the data using at least one circle graph.

e Analyze the data to draw a conclusion about internet usage on different devices.

f Formulate another statistical question that could be used to explore device types.

Exploration

This table tells us how many students have birthdays in the different seasons.

The data was also organized into these two displays.

1. Which season had the most birthdays? Which display did you use, the table, circle graph, or bar graph?

2. Which season had the fewest birthdays? Which display did you use?

3. How do winter and fall compare? Which display did you use?

Let’s look at when different displays are useful. A circle graph is useful for showing proportions and parts of a whole.

Advantages:

• We can easily compare the proportions of different categories visually. They are commonly used.

Disadvantages

• It can be diﬃcult to compare similar groups if they are not labeled. We may lose the original totals if we just show the percentages.

A bar graph is useful for showing the count or frequency of different categories.

b In which display is it easier to see the tire brand that was sold the most?

Create a strategy

In the bar graph, we need to look for the bar with the greatest height, and for the circle graph, we need to look for the biggest section.

Apply the idea

We can see this information easily from both graphs. So either A or B is a correct answer.

c In which display is it easier to see the tire brands that made up half of the sales?

Create a strategy

In the bar graph, we can see the totals, and for the circle graph, we can see the portions of a whole.

Apply the idea

The circle graph makes it easier to see fractions or portions, so the correct answer is option B.

Example 2

Reﬂect and check

Brands C and E make up one half of the sales and Brands A, B, and D make up the other half.

Five teams are completing a walking challenge. A circle graph and dot plot are used to show the distances each team walked.

EaglesWombatsHyenas

Roosters

Lions

a Which team walked the furthest?

Create a strategy

We need to identify the team with the greatest distance on the graphs. Select the graph where we can easily identify the highest number.

Apply the idea

The Wombats team walked the furthest as clearly shown by the dot plot.

Reﬂect and check

We can also identify the team that walked the furthest using the circle graph, but using the dot plot is ideal as it better displays the data we need to answer the question.

The part of the circle graph with the biggest section or highest percentage is the team that walked the furthest.

b Which display would you use to compare the Eagles and Lions? Explain.

Create a strategy

The dot plot shows us the number, but we need to count the dots. The circle graph shows the proportions.

Apply the idea

In the dot plot, we can see that there are two more points for Eagles compared to the Lions, but it is a bit diﬃcult because they are not beside each other. The circle graph shows the proportion for each team, and the Eagles and Lions are beside one another which is nice.

It may depend on the question we are looking to answer, but in general using the circle graph to compare the Eagles and Lions, would be a good choice.

c Complete this table that would show the same information as the two displays.

Team Distance (miles) Percentage of miles

Eagles

Wombats

Hyenas

Roosters

Lions

Create a strategy

We can get the distance from the dot plot and the percentage from the circle graph.

Apply the idea

Fill in the distance column using the data from the dot plot and the percentage column using the data from the circle graph. Team

Reﬂect and check

We should always double check that the percentages add up to 100%.

From this we can also see that the total distance walked by all teams was 50 miles.

6 The table shows the team points earned by four different houses at their swimming competition: Team Yellow Blue Red Green Points 15 25 40 60

Select the best display to represent the given set of data:

A Stem-and-leaf plot B Pictograph C Circle graph D Line plot

7 The table shows the number of books students read in one month.

Students Isla Ozair Mia Mehrab Ivy Number of books 3 6 5 8 7

a Select the displays which could be appropriate.

A Circle graph B Line graph C Pictograph D Bar graph E Stem-and-leaf plot

b What display would you use to represent this data? Explain your choice.

8 The table shows the number of books Deana read in each month.

Month Number of books Month Number of books

January 5 July 8

February 2 August 7

March 3 September 6

April 0 October 5

May 4 November 2

June 3 December 1

What display would you use to represent this data? Explain your choice.

9 The data in the table shows summarizes the scores that students got on a quiz out of 15.

a What is the best way to display this data? Explain your choice.

A Circle graph B Line plot C Pictograph D Bar graph

b Construct the display chosen in part (a).

10 Mr. Rodriguez recorded the number of pets of his students. He found that 15 students had no pets, 19 students had only one pet, 3 students had two pets and 8 students had three pets.

a What is the best way to display this data? Explain your choice.

A Circle graph B Line graph C Pictograph D Bar graph E Stem-and-leaf plot

b Construct the display chosen in part (a).

11 An online booking service allows customers give a rating out of 5 at the end of their transaction. Over the last month the service has been tracking the feedback.

0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5

Determine if each plot type could be used to display this data and describe the advantages or disadvantages of each.

a Line plot (dot plot)

b Circle graph

15 The principal at Sunny Meadows Middle School is asked “How has student enrollment changed over the years?”

Which representation should they show to answer the question?

Student enrollment at Sunny Meadows Middle School

Number of students

16 This bar graph and circle graph show the number of times that each type of sport is played.

a Which sport was played the most? State which graph you used to answer.

b What are the beneﬁts and drawbacks of the circle graph?

17 This line plot and circle graph represent the data that was collected to answer the question “How do sales of tomato seeds vary by type of tomato?”.

a Describe the advantages and disadvantages of each display.

b Create a circle graph that shows the number of seed packets sold on the display.

18 Mr. Smith asked his students to choose their favorite animals. He plotted the results in the bar graph.

a Complete this table using the bar graph.

Animal Number of students

Cats

Dogs

Hamsters

Birds

Total

b How does the proportion of students that prefer dogs compare to those that prefer birds?

c Convert this display to a circle graph and explain which graph makes it easier to compare between animals.

Let’s extend our thinking

19 The circle graph represents the popularity of each chip brand amongst high schoolers.

a Interpret the graph. List all information you know to be true.

b Here is a dot plot for the same data set. Explain what we can see more clearly from the dot plot.

c A snack brand which is preferred by more than 25% of the population does not need to worry about marketing to parents, which display can we more easily identify this from? Explain.

20 The sales of different products are shown in the following horizontal bar graph:

a If you wanted it to be clear that Product D is the most popular which type of display would you use. Explain.

b If you wanted it to be unclear that Product D is the most popular which type of display would you use. Explain.

Tortillos Hot Cheezos Snakis Wrinkles

21 An amusement park recorded the number of people that visited during a week of the summer break and displayed the data in the graph shown.

to Adventure Land

a Convert this bar graph to a circle graph.

b What is easier to visualize from the bar graph?

c What is easier to visualize from the circle graph?

22 Explain why you can convert a bar graph into a circle graph, but may not be able to convert a circle graph to a bar graph.

23 The 6th graders need to choose a class captain who will go to student council meetings. Felicia was asked to predict who might win. She did a poll and asked an anonymous sample who they are planning to vote for. The results are shown in these two displays.

a If you were Constantina which display would you use? Explain.

b If you were Alanna which display would you use? Explain.

24 Lashanda’s mom just bought an air fryer and she now uses it all the time. This makes her curious about what people use as their main cooking appliance.

a Formulate a question that Lashanda could use to explore the types of appliances that people use using the data cycle.

b Could she use observation, measurement, survey, experiment, or acquire secondary sources? Explain.

c Explain how Lashanda could collect data that could be used to answer her question from part (a).

d Suppose the given data shows the preferred appliance for 40 people who she knows do the cooking. Organize and represent the data using at least one circle graph. Person

e Analyze the data to draw a conclusion about preferred appliances.

f Her friend Michael said his dad’s countertop toaster oven stopped working, so he is wants to make a recommendation to his dad about what he should get to replace it. Michael makes this bar chart to show the same data.

AirfryerCountertoptoasteroven Barbeque

Compare the conclusions that are easy to see from the circle graph compared to those with the bar chart.

g Formulate another statistical question that could be used to explore appliance popularity.

9.04 Review: measures of center and spread

After this lesson, you will be able to...

• calculate the mean of a data set.

• identify the median of a data set.

• identify the mode of a data set.

• calculate the range of a data set.

• use mean, median, mode and range to solve problems.

Mean

The mean is the average of the values in the data set. It is a measure of center, meaning it is an approximation of where the middle of a data set is.

Let’s think about a situation where three friends are planning a trip to Palm Springs. They plan to fly there, and learn that the airline has a rule: each person can only bring 35 lbs of stuff in their bags. On the night before the flight they weigh their luggage and find that their luggage weights from this data set: 29, 32, 37

One of them has packed too much. They decide to share their luggage around so that they all carry the same amount. How much does each person carry now?

Thinking about it using more mathematical language, we are sharing the total luggage equally among three groups. As a mathematical expression, we ﬁnd:

Each person carries 32.67 lbs. This amount is the mean of the data set.

If we replace every number in a numerical data set with the mean, the sum of the numbers in the data set will be the same. To calculate the mean, use the formula:

Example 1

Find the mean of the scores: 6, 14, 10, 13, 5, 9, 14, 15

Give your answer as a decimal.

Create a strategy

Use the formula Mean =

Apply the idea

Use the formula

Add the numbers in the numerator

Perform the division

Idea summary

Mean =

Mean is the average of a data set.

Median

The median is the middle of the data set when ordered least to greatest. It is also a measure of center. Let’s say seven people were asked about their weekly income, and their responses form this data set:

$300, $400, $400, $430, $470, $490, $2900

The mean of this data set is = $770, but this amount doesn’t represent the data set very well. Six out of seven people earn much less than this.

Instead, we can select the median, which is the middle income. We remove the biggest and the smallest incomes to get:

$400, $400, $430, $470, $490

Then the next biggest and the next smallest to get:

$400, $430, $470

Then the next biggest and the next smallest to get:

$430

There is only one number left, and this is the median - so for this data set the median is $430. This weekly income is much closer to the other scores in the data set, and summarizes the set better.

The median is the number in the middle of a numerical data set.

• If the list has an odd number of data points, the median is the one right in the center.

• If the list has an even number of data points, the median is the number halfway between (or the average of) the two middle ones.

Half the numbers in the list will be bigger than the median, and half will be smaller. Sets ordered smallest to largest

Median

Odd number of scores:

Even number of scores:

Median: + 2

Example 2

Find the median of the scores:

3, 18, 10, 19, 12, 5, 6, 20, 7

Create a strategy

We need to put the scores in order and ﬁnd the middle score.

Apply the idea

The scores in order are: 3, 5, 6, 7, 10, 12, 18, 19, 20

The middle score is 10 because it has 4 scores above it and 4 scores below it.

The median of the scores is 10.

Idea summary

The median of a numerical data set is the data value in the middle when the data is ordered from least to greatest.

To ﬁnd the median of a data set:

• If the list has an odd number of data points, the median is the one right in the center.

• If the list has an even number of data points, the median is the number halfway between (the average of) the two middle ones.

Mode

The mode of a data set is the result with the greatest frequency, or the data value that appears most often in the data set. If there are multiple results that share the greatest frequency then there will be more than one mode. Yvonne asks 15 of her friends what their favorite color is. She writes down their answers. Here is what she wrote down:

Blue, Pink, Blue, Yellow, Green, Pink, Pink, Yellow, Green, Blue, Yellow, Pink, Yellow, Pink, Pink

She then counts the number of colors to see which is the most picked.

Number of Friends

The mode of the data is pink.

Example 3

Thomas conducted a survey on the average number of hours his classmates exercised per day and displayed his data in a table.

What is the mode of the data?

Create a strategy

Choose the result with the greatest frequency in the data. Apply the idea 1 hour of exercise is the mode because it has the greatest frequency.

Idea summary

The mode of a data set is the result with the greatest frequency. If there are multiple results that share the greatest frequency then there will be more than one mode.

Range

The range is a measure of the spread of a data set from the highest value to the lowest.

Two bus drivers, Kenji and Bjorn, track how many passengers board their buses each day for a week. Their results are displayed in this table:

Kenji 10 13 14 16 11

Bjorn 2 27 13 5 17

Both data sets have the same median and the same mean, but the sets are quite different. To calculate the range, we start by ﬁnding the highest and lowest number of passengers for each driver:

Kenji 16 10

Bjorn 27 2

Now we subtract the lowest from the highest to ﬁnd the difference, which is the range:

Range

Kenji 16 10 = 6

Bjorn 27 2 = 25

Notice how Kenji’s range is quite small, at least compared to Bjorn’s. We might say that Kenji’s route is more predictable and that Bjorn’s route is much more variable (is more likely to change).

The range of a numerical data set is the difference between the highest and the lowest data point.

Range = Highest data point Lowest data point

Example 4

Find the range of the following scores: 10, 7, 2, 14, 13, 15, 11, 4

Create a strategy

Use the formula Range = Highest score Lowest score.

Apply the idea

The highest score is 15 and the lowest score is 2.

Range = 15 – 2 Subtract 2 from 15 = 13 Perform the subtraction

Idea summary

Range = Highest data point Lowest data point

Range is a measure of how spread apart a data set is from its highest to lowest value.

Practice

What do you remember?

1 Describe what the mean measures for a set of values.

2 Describe what the median measures for a set of values.

3 Find the mean of the data sets: a 6, 4 b 9, 7, 11, 4 c 4, 10, 2, 9, 5 d 8, 15, 6, 27, 3, 19

4 Yvonne asked a number of her friends what their favorite color is, and writes down their reponses: Blue, Pink, Blue, Yellow, Green, Pink, Pink, Yellow, Green, Blue, Yellow, Pink, Yellow, Pink, Pink

Which color is the mode?

5 A diver measures how long she can hold her breath underwater over several dives. The median time is 3.9 minutes. Are these statements true or false?

a The longest she held her breath was 7.8 minutes.

b Most of the time, she held her breath for less than 3.9 minutes.

c For half the dives, she was able to hold her breath longer than 3.9 minutes.

d The shortest time she held her breath was 1.95 minutes.

e Most of the time, she held her breath for longer than 3.9 minutes.

6 Why might the mean not accurately represent the data set {1, 2, 2, 100}?

7 List the steps to ﬁnd the median of any odd-sized data set.

Sample Data Set:

Let’s practice

8 Find the mean of the data sets:

{23, 47, 58, 61, 34, 82, 82, 19, 7, 92}

a 6, 14, 10, 13, 5, 9, 14, 15 b , , , , , , ,

c 22.4, 25.4, 19.1, 24.3, 7.4

9 Which two data sets have the same mean?

• Data Set A: 11, 5, 14, 18, 15

• Data Set B: 8, 14, 24, 7, 12

d 14, 0, 2, 18, 0, 15, 1

• Data Set C: 13, 27, 9, 6, 8

• Data Set D: 13, 15, 7, 17, 9

10 Mrs. Tran asks her students how many bedrooms they have in their house and writes their responses below:

3, 3, 4, 4, 3, 3, 4, 5, 2, 2, 4, 4

Determine the mode number of bedrooms.

11 Victoria has scored 12 goals this season, with a mean of 2 goals per game. How many games did she play?

12 Consider the scores:

a What is the largest number?

c Find the range.

13 Find the median of the data sets:

a 6, 9, 3, 5, 4

3, 6, 8, 10, 2, 11

b What is the smallest number?

b 1, 4, 6, 8

c , , , d , , ,

e 13, 1, 10, 29, 12, 5, 16, 20, 6

g 5.3, 8.4, 2.6, 4.7, 3.5

14 Calculate the range of the scores:

15 For each data sets, calculate:

i The mode

f 7, 19, 6, 14, 18, 11

h 49.4, 25.4, 47.2, 34.2, 28.4, 32.2

10, 16, 6, 18, 17, 11, 9, 15, 14

ii The range

a b Score 11 14 19 22 29

Frequency 10 13 4 7 12

Score 14 19 22 25 32

Frequency 9 14 12 1 10

16 A phone support center tracks the length of calls made each day in minutes. During a shift, one employee made calls of lengths:

2, 3, 3, 2, 2, 4, 4, 2, 5, 13

a How many calls did the employee make during their shift?

b Calculate the length of calls made: i Mean ii Median iii Mode iv Range

17 Which set of data has the largest range?

• Set A: 101, 105, 118, 129, 136

• Set B: 19, 23, 25, 28, 29

• Set C: , , , ,

• Set D: 104.15, 107.05, 113.24, 128.33, 141.57

18 A group of ﬁve friends tracks the number of books they read over the summer for a reading competition. Their totals are as follows: 15, 18, 20, 22, and 25 books.

a Calculate the mean number of books read.

b A new friend joins their group, who read 45 books over the summer.

Explain what would happen to the mean number of books read by the group if this new friend’s total is added.

19 Determine whether the statements are always, sometimes, or never true. Explain your reasoning.

a The median of a data set is a value in the set.

b The mean of a data set is larger than all values in the set.

c The range describes how far the lowest data values lies from the highest data value.

20 A local amateur photographers’ club that meets monthly to share their best shots. The number of photos taken by seven club members in a month is as follows: 50, 55, 60, 65, 70, 75, and 80

a Calculate the median number of photos taken by the members.

b The following month, inspired by a spectacular meteor shower, the one member who previously took 80 photos manages to take 120 photos. The other members took the same amount of photos.

Explain how the median number of photos taken by the club members changed for the second month.

21 Consider this data set that represents the number of apps on six people’s phones: 110, 113, 117, 121, 127, 132

Explain what would happen to the range if each person downloads another 9 apps.

22 The list shows the number of points scored by a basketball team in each game of their previous season: 59, 67, 73, 82, 91, 58, 79, 88, 69, 84, 55, 80, 98, 64, 82

a Find the range and explain what it represents in context.

b Explain why the range might be important for the team’s coach to know.

23 Consider the performance data from two groups of athletes, Group A and Group B, over a series of events. The data represent the number of goals scored by each athlete in a season. For Group A, the goals scored are as follows: 5, 7, 7, 8, 10, and 12. For Group B, the goals scored are: 3, 5, 7, 7, 7, 11, and 14.

a Calculate the mean number of goals scored for each group.

b Determine the median number of goals scored for each group.

c Identify the mode of goals scored for each group.

d Compute the range of goals scored for each group.

e Based on these calculations, compare the performance of the two groups of athletes. Consider which group demonstrates greater consistency in performance and which group has a higher variability in the number of goals scored.

24 A school nurse is investigating the number of sick days taken by students in a class, and constructs a table:

Determine the most common number of sick days for the class.

25 CA retail company has compiled its weekly sales ﬁgures (in thousands of dollars) for the past twelve weeks as follows:

52, 55, 55, 60, 62, 65, 65, 70, 70, 75, 80, 85

a Calculate the mean, median, mode and range of the weekly sales ﬁgures.

b How consistent is the sales ﬁgures week over week?

c What could cause the variability in the sales ﬁgures week over week?

26 A student received a data set containing the following numbers: , 4, 6, 6, 7, 7, 7, , 11, and 13.

The assignment was to calculate the mean, median, mode, and range of the data set.

The student’s reported answers were as follows:

• Mean: 7

• Median: 7

Identify and correct the errors in the student’s calculations.

Let’s extend our thinking

• Mode: 6

• Range: 10.5

27 Create a data set where the mean is 7, median is 8, mode is 8, and range is 10.

28 Consider the yearly high temperatures (in degrees Fahrenheit) recorded in Springﬁeld City over the last twelve months: 58.24, 60, 65, 67.15, 70, 72.02, 75, 75, 78, 80.38, 82, and 85.46. As part of the city’s environmental study, you’re tasked with analyzing these temperatures to understand the city’s climate better.

a Calculate the mean, median, mode and range for the temperatures.

b Based on your calculations, decide which measure (mean, median, mode, or range) best represents the city’s climate for the year and explain your reasoning.

29 The mean of four values is 21. If three of the values are 17, 3 and 8, ﬁnd the fourth value.

30 A set of 34 scores is arranged in increasing order. Between which two scores does the median score lie?

31 In a competition, a contestant must complete 12 challenges earning as many points as possible. Her scores for the ﬁrst 11 challenges are:

66, 105, 38, 108, 67, 82, 92, 43, 119, 45, 102

Determine her score in the 12th round if the median of all of her 12 scores is 83.5.

32 Counting numbers in increasing order such as 7, 8, 9 are called consecutive numbers. Write down ﬁve consecutive numbers whose median is 6.

9.05 Mean as a balance point

After this lesson, you will be able to...

• use line plots to find and interpret the mean of a set of data.

• represent the mean of a set of data graphically as the balance point represented in a line plot/dot plot. .

Mean as a balance point

Interactive exploration

Explore online to answer the questions mathspace.co

Use the interactive exploration in 9.05 to answer these questions.

1. Can you make the mean the same value as one (or more) of the data points? How many ways? What do you notice?

2. Can you make more than one data set with a mean of 4?

3. Will the mean ever be outside of the data set?

4. Set up the points to 4, 6 and 11. How far is each value from the mean? Use negative values for below the mean and positive values for above the mean. What is the sum of these values?

The mean is the balance point of a data set. This means that the sum of the distances from the mean of all of the points below the mean is equal to the sum of the distances from the mean of all of the points above the mean.

When given a data set the balance point can be found by plotting the points on a line plot.

The points to the left are a total of 3 units away from the mean, 6.

The point on the right is 3 units away from the mean, 6.

This creates a balanced distance of 3 on either side of the mean.

To ﬁnd the balance point, when it is not given, points can be moved one-by-one towards the middle.

Step 1: Start by plotting the data set on the line plot. Step 2: Start with the farthest left point and move it one unit towards the center. 0 1 2 6 3

Step 3: Then go to the farthest right point and move it one unit towards the center.

Step 4: Repeat on the left.

Step 5: Repeat on the right.

0 1 2 6 3 5 4 7 8 9 10 11

Step 7: Repeat on the right.

Step 6: Repeat on the left.

1 2 6 3 5 4 7 8 9 10 11

Mean “balance point”

Step 8: Stop when all points are stacked over the same value. The balance point for this data is 6.

If the balance point is located between two values, we can ﬁnd the halfway point between those values by averaging the numbers.

The balance point for the data set is located halfway between 7 and 8.

0 1 2 6 3 5 4 7 8 9 10

Balance point

We can calculate the halfway point by ﬁnding the sum of the two values and dividing by two.

Adding or removing a data point might throw off the balance of the data set resulting in a new balance point.

Example 1

A classroom recorded the number of pets for each student. The results for the class are represented in the given line plot.

0 1 2 6 3 5 4

Number of pets

a What was the total number of pets for the entire class?

Create a strategy

Multiply the total number of each number of pets by the number of students that have that many pets.

Apply the idea

Total

Multiply each number of pets by its frequency =

= 26

Evaluate the values in the parentheses

Find the sum

b What is the mean number of pets per student?

Create a strategy

Create a line plot of the data set. Alternate moving each far left and far right point towards the center until ﬁnding the balance point of the data set, which is the mean.

Apply the idea

Start on the far left and move one of the points at 0 one unit to the right.

Alternate to the far right and move the point at 6 one unit to the left.

Alternate back to the far left and move another point at 0 one unit to the right.

Alternate back to the far right and move the point at 5 one unit to the left.

Alternate back to the far left and move the point at 0 one unit to the right.

Alternate back to the far right and move one of the points at 4 one unit to the left.

Continuing this process, we end up with all points on 2.

This creates a balance point at 2. The mean is 2 pets per student in the class.

Reﬂect and check

Alternatively, we can calculate the mean by ﬁnding the sum of all the points and dividing by the total number of data points.

Multiply each number of pets by its frequency and divide by the total number of students

Evaluate the multiplication and addition

Evaluate the quotient for the mean

Example 2

During a ﬁtness challenge, Alex recorded the number of push-ups completed each day. The table shows the number of push-ups Alex did.

Push-ups 15 20 15 10 15 20 10 15 20 20

a Create a line plot of the data.

Create a strategy

The number line should include values in between 10 and 20 based upon the given data. Then, count the number of times that Alex completed each number of push-ups, this number will tell us how many dots to put on the number line.

Apply the idea

For this problem, we do not need to know which days Alex completed the number of push-ups, only how many days he completed them.

Alex completed 10 push-ups 2 days, so we will place 2 points above 10.

Alex completed 15 push-ups 4 days, so we will place 4 points above 15.

Number of push-ups 10 11 12 13 14 15 16 17 18 19 20

Alex completed 20 push-ups 4 days, so we will place 4 points above 20.

b What was the total number of push-ups completed?

Create a strategy

Multiply the number of completed push-ups by the frequency.

Apply the idea

Total number of push-ups = (10 2) + (15 4) + (20 4)

Multiply each number of push-ups by its frequency = 20 + 60 + 80

Evaluate the values in the parentheses = 160

Find the sum

Alex completed a total of 160 push-ups over the 10 days.

c What is the balance point of push-ups completed each day?

Create a strategy

Use the line plot created in part (a) to alternate moving each far left and far right point towards the center until ﬁnd the balance point of the data set.

Apply the idea

Start on the far left and move one of the points at 10 one unit to the right.

Alternate to the far right and move one of the points at 20 one unit to the left.

Alternate back to the far left and move the point at 10 one unit to the right.

Alternate back to the far right and move one of the points at 20 one unit to the left.

Alternate back to the far left and move one of the points at 11 one unit to the right.

Alternate back to the far right and move one of the points at 20 one unit to the left.

Continue this process until all the points have been balanced at 16.

The mean number of push-ups completed per day is 16 each day.

Reﬂect and check

Number of push-ups 10 11 12 13 14 15 16 17 18 19 20 6 1 4

Each point on the far left and right sides alternates and get moved one place closer towards the center until all the points are at the balance point.

The 2 points at 10 move 6 units to the right.

The 4 points at 20 move 4 units to the left.

The 4 points at 15 move 1 unit to the right.

The points on the left side of the mean moved a total of 16 units.

The points on the right side of the mean moved a total of 16 units.

Example 3

Use the given line plot to identify the mean of a group of friends’ shoe sizes.

Create a strategy

Use the line plot to alternate moving each far left and far right point towards the center until you ﬁnd the balance point of the data set.

Apply the idea

Start on the far left of the line plot and move the point at 4 one unit to the right.

Alternate to the far right side of the line plot and move the point at 9 one unit to the left.

Alternate back to far left side of the line plot and move one of the points at 5 one unit to the right.

Alternate back to the far right side of the line plot and move one of the points at 8 one unit to the left.

Alternate back to the far left side of the line plot and move the point at 5 one unit to the right.

Alternate back to the far right side of the line plot and move the point at 8 one unit to the left.

The balance point for this data is the value halfway between 6 and 7.

To ﬁnd the value halfway between 6 and 7 we can ﬁnd the sum of these values and divide by 2.

Halfway = Find the halfway value between 6 and 7

= Evaluate the sum

= 6.5 Evaluate the quotient

The mean of the shoe sizes for this group of friends is 6.5.

Reﬂect and check

Alternatively, we can ﬁnd the mean by ﬁnding the sum of all the shoe sizes and divide by the number of friends. Multiply each shoe size by its frequency and divide by the total number of friends

Evaluate the multiplication and addition

Evaluate the quotient for the mean

Idea summary

The mean is the balance point of a data set and is best when there are not any values which are far away from the rest.

To ﬁnd the balance point from a line plot we alternate moving each point on the far left side and far right side one unit closer to the center of the line plot. Eventually, all the points should be at balance point, which is the mean of the data set.

The points located to the left of the balance point are the same distance away from the balance point as the points located on the right side.

What do you remember?

1 Deﬁne “mean”. Explain why it is an important measure of center in data analysis.

2 Which of the following data displays shows a line plot?

3 Find the mean of each set of scores:

a 8, 15, 6, 27, 3

b 56, 89, 95, 71, 75, 84, 65, 83 c 22.4, 25.4, 19.1, 24.3, 7.4 d 14, 0, 2, 18, 8, 0, 15, 1

4 Does each set of numbers have a mean of 3? a 8, 4, 2, 3, 1 b 3, 2, 5, 1, 4 c 1, 3, 7, 5, 2 d 2, 4, 5, 4, 3

5 The mean of this set of number is 5 : 2, 4, 5, 6, 8. If we add another 5 to the data set, what is the new mean?

6 Which line plot shows a set of data with a balance point of 37?

7 Find the balance point of each given data set.

8 In each game of the season, a basketball team recorded the number of ‘three-point shots’ they scored.

The results for the season are represented in the line plot:

a How many games did they play during the season?

b If the team scored 4 three-point shots in a game, how many points are scored from three-point shots?

c What was the total number of points they scored from three-point shots during the season?

d Use the dot plot to ﬁnd the mean number of points they scored per game from the three-point shots.

9 Orly created a line plot to represent his ﬁrst four test grades in Geometry. Where would he place the balance point to represent the mean of his four grades?

Each dot represents one of his test grades.

10 Sephira completed eight laps during her daily cycling practice and recorded the time taken for each lap, as shown in the table.

She wants to represent the mean as a balance point for this data.

a Use the data to create a line plot.

b Find the balance point for the data.

c Explain how you found the balance point using the line plot.

Sephira’s Time Record for Each Lap

Mr. Rowan made a line plot showing how many hours each of his students spends at the library during the week. He organized the data into the line plot shown (each dot represents one student). What is the balance point for the data? Explain your reasoning.

Let’s extend our thinking

12 Cecille recorded the time it took for seedlings to grow into full-grown plants, in weeks, in three different planting beds. She made a line plot of the data for each type of planting bed and provided this summary:

• In the ﬁrst planting bed, seedlings are growing faster compared to those in the second and third planting beds.

• The growth times of the seedlings in the third planting bed are more similar to each other than those in the ﬁrst and second planting beds.

Use Cecille’s summary to match each line plot to the correct planting bed.

a First planting bed

b Second planting bed

c Third planting bed i ii iii

These six numbers have a mean of 107: 102, 103, 111, 107, 108, 111

a If a new number is added that is larger than 111, will the mean will be higher or lower?

b Create a line plot to represent the data set after a number larger than 111 is added.

14 At a local fair, three different carnival games were set up side by side. Each game had its own line of participants eager to try their luck. The line plots represent the ages of the people who played each game during the afternoon. Carnival game A

a Among these carnival games, one has no age restriction. Do you think it was carnival game A, B, or C? Explain your reasoning.

b Which carnival game’s line plot shows ages centered around 30 years old?

c What is a typical age for the people who were at carnival game A?

9.06 Changing data values and measures of center

After this lesson, you will be able to...

• determine the effect on measures of center when a single value of a data set is added, removed, or changed.

Changing data values and measures of center

We have previously learned about various measures of center:

• mean - also called average, is the sum of values divided by the number of values.

• median - is the middle value when the values are sorted.

• mode - the value that occurs most often.

Interactive exploration

Explore online to answer the questions

mathspace.co

Use the interactive exploration in 9.06 to answer these questions.

1. What happens to the measures of center when the blue point is removed?

2. What happens to the measures of center when the blue point is added back?

3. What happens to the measures of center when the blue point is changed?

4. Repeat with new data sets to see how the measures of center change as the blue point is added, removed, or changed. Do your previous observations continue to be true?

Recall we can calculate the mean by ﬁnding the ‘average’ of the data set:

Mean =

Since every data value in the set is a part of the sum, adding, removing, or changing a value can change the numerator signiﬁcantly, depending on what that value is. While the denominator will only increase or decrease by 1 (or not at all if we’ve only changed an existing value). This is why the mean is so easily affected by changing the data.

To ﬁnd the median, we list all the numbers in order from smallest to largest and ﬁnd the middle value. Adding, removing, or changing a value in a data set can often change its median. Though this change will not be major because the data values are ordered numerically and changing a single value will only cause it to shift to a nearby value.

The mode is the value with the highest frequency (the one that appears most often). When we add, remove, or change a value in a data set, it may affect the mode by causing a new number to become the mode, or the mode may remain the same.

Example 1

Consider the data:

Scores: {39, 39, 39, 39, 39, 39, 40, 40, 40, 40, 41, 41, 42, 42, 43, 43, 43, 43}

a Find the total number of scores.

Create a strategy

Add the total number of scores.

Apply the idea

Total number of scores = 6 + 4 + 2 + 2 + 4

Add the total number of frequencies = 18

Evaluate the addition

There are 18 total scores.

b Approximate the sum of the scores.

Create a strategy

Add all the scores together.

Apply the idea

Sum of the scores = 39 6 + 40 4 + 41 2 + 42 2 + 43 4

Multiply each score by its frequency = 234 + 160 + 82 + 84 + 172

Evaluate the multiplication = 732

Evaluate the addition

The sum of all the scores is 732

c Find the mean, median, and mode of the scores, correct to two decimal places.

Create a strategy

For the mean, we can use the formula: Mean =

The mode is the most repeated score. The median is the middle score.

Apply the idea

Mean = Divide the sum of the scores by the total number of scores = 40.67 Evaluate to two decimal places

Mode:

The mode is the score with the highest frequency which is 39.

Median:

There are 18 scores. The median score should be the average of the 9th and 10th score. The 9th and 10th score are both 40, so the median is 40.

d A new score of 10 is added. Find the new mean, median, and mode of the scores, correct to two decimal places.

Create a strategy

For the mean, we can use the formula: Mean =

The mode is the most repeated score. The median is the middle score.

Apply the idea

Mean = Divide the sum of the scores by the total number of scores = 39.05

Evaluate to two decimal places

Mode:

The mode is the score with the highest frequency which is still 39.

Median:

There are 19 scores. The median score should be the average of the 10th score. The 10th score is 40, so the median is 40.

Reﬂect and check

Notice that the mode and median did not change. When much smaller or much larger data is added, it is more likely to have a greater impact on the mean since it measures the distance of each point from the balance point.

Example 2

A data set consists of ﬁve numbers 11, 13, 9, 13, 9.

a The data set has a current mean of 11. If the data set changes to 11, 15, 9, 13, 9 will the mean be higher, lower, or remain the same?

Create a strategy

Identify what has changed in the data set. One of the values of 13 has been increased to 15. Think about what increasing the sum of the data will do to the mean.

Apply the idea

Remember the mean is calculated by dividing the sum of all values by the number of values in the set.

Increasing a data value will increase the sum (numerator) but the number of data values (denominator) will stay the same. When we divide a larger numerator by the same denominator, the result we get will be larger than the original. The mean will increase.

Reﬂect and check

Find the sum of the data set

Divide the sum of the values by the total number of values

Evaluate to two decimal places

The mean will be higher because the balance point is pulled up towards the changed, larger data value.

b The data set has a current median of 11. If a new number is added that is larger than 13, will the median be higher, lower, or remain the same?

Create a strategy

The median is the middle value in a data set. So adding a value could affect the location of the middle of the data set.

Apply the idea

If we arrange the original data set in ascending order, the data set looks like this: 9, 9, 11, 13, 13.

The median of this ordered set, which is the middle value when all the numbers are listed from smallest to largest, is 11.

If we add a number larger than 13, it is added to the right side of the ordered list of data and would look like: 9, 9, 11, 13, 13, ⬚

We can see the new median will fall between 11 and 13.

Without calculating the value of the median, we can say that it will increase.

Reﬂect and check

To ﬁnd the middle, organize the data from smallest to largest.

9, 9, 11, 13, 13, ⬚

There are now 6 data values so the middle falls between the 3rd and 4th value. To ﬁnd the middle average 11 and 13.

Find the sum of the the values being averaged

Find the quotient

Evaluate

c The current data set has two modes of 9 and 13. If the data set changes to 11, 9, 13, 9 will the modes remain the same?

Create a strategy

Remember the mode represents the data value occurring with the highest frequency.

Apply the idea

In the original data set, the values of 9 and 13 occurred twice. The new data set removed one of the values of 13, so it now only occurs once and is no longer a mode. This means, 9 is the only mode of the new data set.

3

25 students took an assessment. Their scores are shown below. 58, 60, 60, 60, 61, 62, 62, 63, 63, 63, 64, 64, 64, 64, 64, 65, 65, 65, 66, 66, 66, 67, 68, 70, 70

a A teacher calculated the mean of 25 students’ scores to be 64. A student who later completed the assessment got a score of 55. Find the new mean of the class, correct to two decimal places.

Create a strategy

We can use the formula: Mean =

Example

Apply the idea

Substitute the values

Evaluate the addition and multiplication

Evaluate to two decimal places

The new mean of the class is 63.65 which is 0.35 lower than the original mean of 64. This makes sense because the new score of 55 being added is a value far away from the previous mean, causing the mean score to drop.

Reﬂect and check

Notice the sum of all the scores was not recalculated because the old mean was already calculated. The old mean can be used in calculating the new mean as long as none of those previous data values change. The new score needs to be included in the new sum and the total number of scores needs to be increased by 1 to show that there are now a total of 26 scores in the data set.

b Find the median of the class before and after the ﬁnal student took the assessment. Did the median change?

Create a strategy

The median is the middle value of a data set when ordered from least to greatest. So, adding a value could affect the location of the middle of the data set.

Apply the idea

There were originally 25 students who took the assessment, making the median the 13th score. The median of the original data set, before the ﬁnal student took the assessment, is 64.

We can list out the new list of scores with the ﬁnal student’s score included. 55, 58, 60, 60, 60, 61, 62, 62, 63, 63, 63, 64, 64, 64 64, 64, 65, 65, 65, 66, 66, 66, 67, 68, 70, 70

Adding another score to the data set makes 26 scores so the median falls between the 12th and 13th score. Both the 12th and 13th score is 64, making the median 64.

In this case the median did not change because the score of the middle value remained 64.

c Find the mode of the class before and after the ﬁnal student took the assessment. Did the mode change?

Create a strategy

The mode represents the data value with the highest frequency. So, we will need to count out the common scores and ﬁnd the score that occurs most frequently.

Apply the idea

The new score of 55 only occurs 1 time, so it does not affect the mode before or after the ﬁnal student took the assessment.

The scores of 60, 63, 65, and 66 have a frequency of 3.

The score of 64 has a frequency of 5 making it the score with the highest frequency. Therefore, the mode of the class before and after the ﬁnal student took the assessment is 64.

Idea summary

Adding a new data point can signiﬁcantly impact the measures of center, which include the mean, median, and mode.

Mean:

• The most sensitive to changing the data set

• Increases if an existing value is increased, a value smaller than the mean is removed, or if a value that is larger than the mean is added to the set

• Decreases if an existing value is decreased, if a value larger than the mean is removed, or if a value that is smaller than the mean is added to the set

• Stays the same if a value equal to the mean is added or removed

Median and mode are not signiﬁcantly impacted by changing a value in the data set.

Practice

What do you remember?

1 Match each term with the correct deﬁnition:

a Mean i The average of all the values in the data set

b Median ii The most common value in the data set

c Mode iii The difference between the highest and lowest values in the data set d Range iv The middle value of the data set when in descending order

2 Which two data sets have the same mean?

• Data Set A: 13, 9, 14, 16, 11

• Data Set B: 8, 24, 14, 8, 17

• Data Set C: 12, 28, 10, 6, 7

• Data Set D: 13, 15, 7, 18, 8

3 The mean of a set of scores is 38.6 and the sum of the scores is 694.8. Calculate the number of scores.

4 A set of ﬁve numbers has a mean of 10. Two of the numbers are 6 and 13. Determine whether the three other numbers could be in the set: a 15, 11, 8 b 10, 13, 8 c 13, 5, 6 d 10, 6, 18

5 Given the data set {4, 7, 5, 9}, calculate: a Mean b Median c Mode d Range

Let’s practice

6 The ﬁve numbers 16, 16, 17, 24, 17 have a mean of 18. If a new number is added that is bigger than 24, will the mean be higher or lower?

7 The ﬁve numbers 9, 13, 9, 11, 8 have a mean of 10. If a number is removed that is smaller than 9, will the mean be higher or lower?

8 Five numbers have a mean of 7. If 4 of the numbers are 10, 10, 8 and 7 and the last number is x, ﬁnd the value of x.

9 The mean of four scores is 21. If three of the scores are 17, 3 and 8, ﬁnd the fourth score.

10 The mean of a set of 41 scores is 18.6. If a score of 71.8 is removed from the set, ﬁnd the new mean. Round your answer to two decimal places.

11 A teacher calculated the mean of 25 students’ marks to be 64. A student who later completed the assessment got a mark of 55. What is the new mean of the class, to two decimal places?

12 Change the value 6 to 9 in the data set {3, 5, 6, 7} and recalculate the mean, median, and mode. Compare the results.

13 Given the data set {4, 5, 6, 7}, perform three operations: Add 10, remove 5, change 6 to 8. Discuss how each operation affects the mean, median, and mode.

14 Moris runs every morning and records the minutes in his tracker.

Monday Tuesday Wednesday Thursday Friday 20 20 30 10 20

a What is his average or mean number of minutes for these ﬁve days?

b If Moris skips his Friday run, what is the average number of minutes for the other days?

c What happens to Moris’ mean number of minutes if he doesn’t run on Friday?

15 Last week, Alberto earned $18, $14, $16, $19, and $17 by helping at his uncle’s bakery.

a Calculate the measures of center: mean ⬚, median ⬚, mode ⬚

b This week, Alberto attended a sports clinic and only earned money for helping on four days. He earned $18, $16, $19, and $17. Discuss how you think each measure of center will be affected by earning for four days instead of ﬁve days.

Predictions: mean ⬚ median ⬚ mode ⬚

c Calculate the new measures of center and compare to your predictions. mean ⬚, median ⬚, mode ⬚

Let’s extend our thinking

16 The table shows the monthly rent for various residential properties.

Monthly Rents (dollars) 1950 1670 1870

a Find the mean, median, and mode of the data.

b Rent on each of the residential properties raised by 5%. Find the mean, median, and mode of the data with the raise. How does this increase affect the mean, median, and mode of the data?

c Use the original monthly rents to calculate the annual rents. Find the mean, median, and mode of the annual rents. How are these values related to the mean, median, and mode of the monthly rents?

17 You are part of a research team that analyzing the annual crop yields (in tons) from farms situated across diverse agricultural regions in your country, aiming to understand agricultural productivity ﬂuctuations and identify factors inﬂuencing harvest outcomes. The data set provided to you contains the annual total yield recorded by 15 farms over the last year. Here is the data:

a Calculate the mean, median, and mode of the annual yield for the 15 farms. Brieﬂy discuss your ﬁndings.

b Inspect the data and identify any value(s) that seem unusually high or low compared to the rest. Explain why you consider this value(s) an outlier.

c Remove the value you identiﬁed in part (b) from the data set and recalculate the mean, median, and mode of the annual yield for the remaining farms.

d Discuss how the removal of the value you choose affected the mean, median, and mode.

18 The head coach of your school’s cycling team is analyzing the times (in minutes) of the 30-mile race results from the most recent sports meet to evaluate the team’s performance. The data set consists of the following times:

a Calculate the range of the times for all of the athletes. What does this measure tell you about the team’s performance variability?

b A new athlete’s time of 70.4 minutes is added to the data set. How did adding this point affect the range?

c The time of athlete 9 is determined to be incorrect due to a timing error and is removed from the data set. How did the removal of athlete 9’s time affect the range?

d Why is the range a valuable measure to consider when analyzing data?

19 City Wheels, a local pre-owned car dealer, is analyzing the sale prices of pre-owned cars to understand the current market and set pricing guidelines for similar vehicles. The data set provided to you contains the sale prices (in thousands of dollars) of pre-owned cars sold in the last month.

a Calculate the mean, median, and mode of the pre-owned car sale prices. Brieﬂy discuss your initial observations about the central tendency of this data.

b Discuss why the mean might not be a reliable measure of center for this data set.

c Explain how the presence of the outlier (Car 5) affects the mean compared to the median and mode.

20 The table shows the scores of Student A and Student B in ﬁve separate tests:

a Find the mean score for Student A.

b Find the mean score for Student B.

c What is the combined mean of the scores of the two students.

d What is the highest score overall? Which student obtained that score?

e What is the lowest score overall? Which student obtained that score?

9.07 Outliers

After this lesson, you will be able to...

• identify outliers in a given data set.

• describe the effect of outliers on the mean, median, mode and range of a data set.

Outliers

In statistics, we tend to assume that our data will ﬁt some kind of trend and that most things will ﬁt into a “normal” range. This is why we look at measures of center, such as the mean, median and mode.

A measure of center is a way to describe where the center of a set of data is. However, not all measures describe the center in the same way and some measures are heavily affected by extreme data values, or outliers.

Outlier

A data value that is an abnormal distance from the other data values in the set (much larger or much smaller)

Interactive exploration

Explore online to answer the questions

Use the interactive exploration in 9.07 to answer these questions.

1. What do you notice about the mean as you move the position of the blue point to be much larger than the data?

2. What do you notice about the mean as you move the position of the blue point to be much smaller than the data?

3. What do you notice about the median as you move the position of the blue point to be much larger than the data?

4. What do you notice about the median as you move the position of the blue point to be much smaller than the data?

Outliers are data points that lie far outside the majority of a data set and can signiﬁcantly affect the measures of center (mean, median, and mode) as well as the range.

• The mean is most affected by outliers. Extreme data values cause the mean to increase or decrease signiﬁcantly.

• The median is less affected by outliers because it only shifts based on how many data values are added or removed from the set, their values do not matter.

• The mode is least affected by an outlier because an outlier should be far away from the rest of the data so it is unlikely to impact the mode which is the value that appears most often in the set.

• The range is extremely affected by outliers because an outlier greatly increases the distance between the largest and smallest data value.

b Identify the outlier.

Create a strategy

Identify the game score that is much greater or smaller than most of the scores.

Apply the idea

We can see that the dot for 300 is far away from the rest of the dots.

Outlier = 300

c State the mean and median of the data without the outlier.

Create a strategy

To ﬁnd the mean, use the formula: Mean =

To ﬁnd the median, ﬁnd the middle score.

Apply the idea

Add all the scores and divide by the total 19

Evaluate the division

Since the number of scores is odd, then the middle score is 49.

Median = 49

Reﬂect and check

Notice that the mean changed from 63 to 50.53 and the median stayed the same. This is because the mean is extremely affected by outliers while the median is much less affected.

Example 4

The data set 6, 8, 10, 10, 12 has measures of:

• Mean = 9.2

• Median = 10

• Mode = 10

• Range = 6

Suppose we add the number 20 to the data set. Predict how the addition of this outlier will affect the mean, median, mode, and range of the new data set.

a Will the mean be higher, lower, or remain the same? Explain.

Create a strategy

The mean is the average of all the data values. Values signiﬁcantly higher or lower than the mean will cause the mean to shift in the direction of that high or low value.

Apply the idea

The mean will be higher.

The mean is calculated as the sum of all values divided by the number of values. Adding a number as high as 20 signiﬁcantly increases the sum of the data set (numerator) while only increasing the number of values (denominator) by 1. Since 20 is much higher than the original mean, 9.2, the new mean must be higher.

Reﬂect and check

Let’s check our prediction by calculating the new mean.

To ﬁnd the mean, use the formula: Mean =

The sum of all values divided by the total number of values

Evaluate the addition

Evaluate the division

The new mean, 11, is higher than the original mean, 9.2.

b Will the median be higher, lower, or remain the same? Explain

Create a strategy

The median of a data set is the middle value when the data is organized from least to greatest. Adding a value could cause a shift in the location of the middle of the data.

Apply the idea

Originally, the median was 10, which is the middle value when the numbers are ordered.

After adding the number 20, the ordered data set becomes 6, 8, 10, 10, 12, 20. The median will now between, 10 and 10.

The median remains the same.

Reﬂect and check

Let’s check our prediction by calculating the new median.

First, we list the data set with the new value added.

6, 8, 10, 10, 12, 20

There are now 6 values in the data set, so the median will fall between the 3rd and 4th terms.

The 3rd and 4th terms are both 10. To ﬁnd the average these values we need to ﬁnd the sum of the values and divide by two.

The median remains the same at 10.

c Will the mode be higher, lower, or remain the same? Explain.

Create a strategy

The mode is the most frequently occurring value in the data set.

Apply the idea

The original mode is 10, and since the added number 20 was not in the original data set, it will not appear more frequently than 10.

The mode will remain the same.

d Will the range be higher, lower, or remain the same? Explain.

Create a strategy

The range is the difference between the largest and smallest values in the data set.

Apply the idea

Originally, the range was from 6 to 12, which is 6 units. By adding 20, the new range will be from 6 to 20, which is covering a larger spread of values than the original set.

The range will be higher.

Reﬂect and check

Let’s check our prediction by calculating the new range. The range is calculated by ﬁnding the difference of the highest and lowest values.

20 6 = 14

The new range of 14 is higher than the original range of 6.

Idea summary

An outlier is a data point that varies signiﬁcantly from the rest of the data. An outlier will be a value that is either signiﬁcantly larger or smaller than other observations.

Removing outliers will have the following effects on the summary statistics:

A really low outlier

A really high outlier

The range will decrease The range will decrease

The median might increase The median might decrease

The mean will increase The mean will decrease

The mode will not change The mode will not change

Practice

What do you remember?

1 What is an outlier?

2 Identify the outlier for each data set:

a 9, 10, 12, 9, 1

c 104, 115, 275, 109, 118, 121

e f

b 54, 52, 99, 50, 57

d 4.0, 3.5, 5.1, 2.4, 1.6, 3.9, 3.5, 3.1

3 Below is a set of data representing the number of books read by students in a 6th grade class during summer break: 5, 3, 7, 5, 4, 5, 6, 4

a Calculate the mean number of books read.

b Find the median number of books read.

c Determine the mode of the dataset.

d Calculate the range of the dataset. 2 1 0 3 4 5 6 7 8 9 10

4 When an outlier is removed from a data set, describe the effect on these statistical measures: a Mode b Range

5 For each scenario, an outlier was removed. Was the outlier smaller or larger than the values that remain?

a The mean decreased after the outlier was removed.

b The mean increased after the outlier was removed.

c The median decreased after the outlier was removed.

d The median increased after the outlier was removed.

Let’s practice

6 The weight of ﬁsh caught in a “weigh and release” ﬁshing competition, in kilograms are given: 12.5, 15.1, 13, 14.2, 14.5, 14.9, 12.5, 14.3

a Find the mean weight.

b Find the median weight.

c Recalculate the mean weight if the value of 1.5 is added to the data set.

d Recalculate the median weight if the value of 1.5 is added to the data set.

7 Consider the given frequency table:

a What is the mode?

b Which weight is the outlier?

c If the outlier is removed, what is the new mode?

d Compare the mode before and after removing the outlier.

8 Consider the given frequency table:

a Which weight is an outlier?

b If the outlier is removed, will the new mean be higher or lower than the current mean?

c Calculate the new mean, if the outlier is removed. Round your answer to one decimal place if needed.

9 For each scenario, decide if the given measure would increase, decrease, or stay the same: i Mean ii Median iii Mode iv Range

a If 25 is added to the data set: 1, 1, 2, 3, 3, 3, 4

b If 10 is removed from the data set: 10, 42, 55, 60, 65, 70

c If 5 and 27 are removed from the data set: 5, 12, 12, 13, 15, 15, 27

d If 1 is added to the data set: 24, 25, 25, 26

10 Consider the data set shown in the frequency table:

Suppose one score of 8 is changed to a 15. Would any statistical measure change?

a Mean b Median c Mode d Range

11 Look at data sets a, b, and c below. For each set of data:

i Find the mean, median, mode, and range. Round your answers to two decimal places where necessary.

ii Which data value is an outlier?

iii Describe how each of the four statistics may change if the outlier is removed.

iv Remove the outlier from the set and recalculate the values found in part (i).

a 27, 50, 24, 37, 47, 41, 27, 126, 44, 27

b 4.7, 2.8, 1.9, 0.9, 0.9, 2.2, 2.2, 1.2, 1.5, 0.9

c 4700, 4700, 4700, 4500, 5300, 4900, 5200, 4800, 1500, 5100

12 The ﬁve numbers 16, 16, 17, 24, 17 have a mean of 18 and a median of 17. Describe the effect on the mean and median if a new number is added that is larger than 24.

13 The number of three-pointers scored in different basketball games by a single team are shown in the table.

Describe the effect on each statistical measure if the outlier is removed:

a Mean b Median c Mode d Range

Let’s extend our thinking

14 Tanya loves bird-watching and keeps a record of the number of different bird species she spots each week. One month, the mean number of different bird species she spotted per week was 21. However, three out of these four weeks, she spotted 17, 3, and 8 species respectively.

a Find the number of species Tanya spotted in the fourth week.

b Describe how the outlier value of bird species spotted in the fourth week affects the mean and median of Tanya’s four-week bird-watching data.

15 Below is a set of data showing the scores of 5 students on a math test: 72, 76, 80, 84, 88

a Calculate the original mean and range of the scores.

b Choose an outlier to add to this data set.

c After adding the outlier, calculate the new mean and range.

16 Ted is an employee who works for a company and currently gets paid $26 000 annually. Here is the breakdown of all the employees’ salaries at the company he works for in a frequency table:

a Would the owner of the company use the mean, median, or mode to describe the company’s annual pay? Why?

b Would Ted use the mean, median, or mode to describe the company’s annual pay? Why?

c Would a mathematician who is not involved with the company use the mean, median, or mode to describe the company’s annual pay? Why?