5th Grade Fraction Operations Part 2 2015-11-13 www.njctl.org - - PDF document

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5th Grade

Fraction Operations Part 2

2015-11-13 www.njctl.org

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· Multiplying Fractions · Multiplying Fractions and Whole Numbers · Multiplying with Mixed Numbers · Dividing Unit Fractions by Whole Numbers · Dividing Whole Numbers by Unit Fractions

Table of Contents

· Area of fractional side length rectangles · Interpreting Multiplication of Fractions · Line Plots with Fractional Data

click on the topic to go to that section

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Multiplying Fractions

Return to Table of Contents

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Slide over bar below to show equality.

You can use folding to show a fractional part of a

• fraction. (You will need a set of Fraction Pieces.)

Here is how to find of

• 1. Use the fraction piece.
• 2. Fold it in half.
• 3. Compare the folded part with the other fraction

pieces to find a piece that matches. 1 2 1 3 1 3 1 6 1 2 1 3

• f

is

click to reveal

Folding Slide 6 / 130

You can use shading to find a fractional part

• f another fraction.

Here is how to find of

• 1. Divide your whole into fourths.
• 2. Shade a third of
• 3. What fractional part of the whole did you

shade? 1 3 1 4 1 4 1 12

click

Shading

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Problem is from: Click for link for commentary and solution.

Slide 8 / 130 Multiplying Fractions

Multiplication describes events when equal groups of things are combined together. For example, 3 groups of 4 apples can be abbreviated using the number sentence 3 x 4 = ?. Multiplication number sentences should be read as, "What is (variable) groups of (variable) things?" The variable can be a whole number, a rational number, or any expression that represents either a number of groups, or the number of items in a group. We will be working with variables that are fractions. Read this number sentence aloud using the phrase, "What is (variable) groups of (variable) things?"

1 3 x 2 5 =

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Multiplication of two factors can be illustrated using an array called an area model. When you make an array, each factor represents either the vertical or horizontal dimension of a square or a rectangle. With the above example, a rectangle can be constructed with 3 units on one side of the array, and 4 units on the other side. The resulting three by four rectangle contains 12 square units. The result can be read, "12 is 3 groups of 4".

Multiplying Using Area Models

3 4

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How can we turn this model showing 2 x 3 into a model showing 2 x 3.5? 2 3 How many square tiles is it now? What number sentence represents the number of square tiles? What if we add another row? What number sentence will represent the number of square tiles?

Using an Area Model to Multiply Decimals

.5 Click

Review from Decimal Computation Unit

Slide 11 / 130 Lets label this model.

2 3 tenths

2

4 tenths

4

6 tenths 8 tenths

12 hundredths

What is the sum? 4 0.8 0.6 0.12 5.52 +

Click

? ? ?

(Click to remove boxes)

What are these?

Review from Decimal Computation Unit

Slide 12 / 130 Rectangle Model

The array or area model is also very effective when illustrating multiplication of two fractions. When multiplying 1/2 x 5/4, the question can be read, "What is 1/2 of 5/4?" or "What is 1/2 of a group of 5/4?" Lets show a group of 5/4. Now, lets show 1/2 of that. 5/8

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• 1. Use the arrow keys to section off one dimension of a whole into 2

parts, and the other dimension into 4 parts. A 2 by 4 array is a rectangle with 8 units making up a whole group.

• 2. Using the slider bar, highlight 1/2 on the side divided into 2 parts,

and highlight 5/4 on the side divided into 4 parts. The intersection of these shaded parts represents the answer to the question, "What is 1/2 of 5/4?" The answer can be read, "5/8 is 1/2 of 5/4."

(Click for interactive site)

Use the interactive model to show 1/2 of 5/4.

Interactive Area Model Slide 14 / 130

To multiply fractions, multiply the numerators, and then multiply the denominators. Make sure you simplify your answer! 4 5 x 3 4 = 4 x 3 5 x 4 = 12 20 = 3 5

Multiplying Using the Algorithm

Sketch an area model to check your answer.

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7 11 x 2 9 = 7 x 2 11 x 9 = 14 99 3 8 x 4 9 = 3 x 4 8 x 9 = 12 72 = 1 6 8 14 6 7 = 8(6) 14(7) = 48 98 = 24 49

( )

click to reveal click to reveal click to reveal

Examples Sketch an area model to check your answer.

Multiply Slide 16 / 130

1 1 5 x 2 3 =

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2 2 3 x 3 7 =

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3 5 8 x 4 7 =

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4 = 2 11 5 6

( )

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5 = 4 9 3 8

( )

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Problem is from: Click for link for commentary and solution.

5.NF Drinking Juice

Alisa had 1/2 a liter of juice in a bottle. She drank 3/4 of the juice that was in the bottle. How many liters did she drink?

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6 At Cliffords’s school, of the students wanted to learn

about the new science museum that was just built. This month, of the students were able to go see it. What fraction of students were able to go see it right away?

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7 A bridge span is of a mile long. Workers have repainted

• f it. How much has been repainted?

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8 The distance between Rosa's house and her school is

mile. She ran of the way to school. How many miles did she run?

Problem is from: Click for link for commentary and solution.

5.NF Running to School

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14 Solve the problem. Simplify prior to multiplying if you can.

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15 Solve the problem. Simplify prior to multiplying if you can.

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16 Solve the problem. Simplify prior to multiplying if you can.

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17 Solve the problem. Simplify prior to multiplying if you can.

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18 Solve the problem. Simplify prior to multiplying if you can.

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19 Solve.

From PARCC EOY sample test #25

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Multiplying Fractions and Whole Numbers

Return to Table of Contents

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Problem is from: Click for link for commentary and solution.

5.NF Connor and Makayla Discuss Multiplication

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To multiply fractions with whole numbers, write the whole number as a fraction (over 1) then multiply the two fractions. W rite your answer in simplest form. 4 9 x = 6 x 4 1 x 9 = 24 9 = 6 9

6

6 1 = 4 9 x

2

= 2 3

2

= 3 5 ( )

7

= 3 5 ( ) 7 1 21 5 = 1 5

4

= 6 1 4 9 x = 2 3

2

6 1 4 9 x 2 3 = 8 3

Alternate Method

• f canceling

components

Whole Number times a Fraction Slide 42 / 130

20 True False x 1 2 =

5

5 1 x 1 2

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21

A

x 4 7

3

B C

3 5 7

D

12 21 12 7

1 5

7

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22

A

x 8 9

12

B C D

32 3 96 9 11 1 3 10 2 3

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23 On Wednesday morning, of Sue’s classmates put blueberries on their cereal. If there are 27 students in Sue’s class, how many put blueberries on their cereal?

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24 Of the 49 kids on the campout, wanted to go to bed right

after the sing-along around the fire. How many kids wanted to go to bed right away?

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Multiplying with Mixed Numbers

Return to Table of Contents

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To multiply fractions with mixed numbers, write the mixed numbers as an improper fractions, then multiply the two fractions. Make sure you write your answer in simplest form. x = 11 x 7 4 x 2 = 77 8 = 5 8

2

11 4 = 7 2 x

9

3 4

3 1

2 = 1 3

( )

1

= 5 1 ( ) 4 3 20 3 = 2 3

6 5

Multiplying Mixed Numbers

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25 True False x = 1 4

3 1

8

6 3

8

2 Slide 50 / 130

26 44 1 2

A

x 1 2

8 5

40 1 2

B C D

88 2 44

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27 15 1 4

A

18 1 8

B

20 3 8

C

19 1 8

D

5 8

( )

5

2 5

(3 )

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Problem is from: Click for link for commentary and solution.

5.NF Half of a Recipe

Kendra is making 1/2 of a recipe. The full recipe calls for 3 1/4 cup of flour. How many cups of flour should Kendra use?

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28 A boat was traveling 12 miles each hour. At that rate, how many miles would it travel in 1 hours?

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29 Riding her bike, Terry averages 9 miles per hour. At that speed, how far could she go in 2 hours?

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Return to Table of Contents

Interpreting Multiplication of Fractions

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You can determine the relative size of the product of a multiplication problem without actually multiplying.

Interpreting Multiplication Slide 57 / 130

When you multiply a given number by a fraction greater than 1, it will result in a product greater than the given number. Examples:

Interpreting Multiplication Slide 58 / 130

30 Which of the following product(s) are greater than 700,000? A B C D

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31 Which of the following product(s) are greater than 876? A B C D

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When you multiply a given number by a fraction less than 1, it will result in a product smaller than the given number. Examples:

Interpreting Multiplication

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32 Which of the following product(s) are less than 555? A B C D

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33 Which of the following product(s) are less than 4,321? A B C D

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Problem is from: Click for link for commentary and solution.

5.NF Reasoning about Multiplication

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Problem is from: Click for link for commentary and solution.

5.NF Calculator Trouble

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Problem is from: Click for link for commentary and solution.

Hint - draw a picture.

5.NF Fundraising

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Problem is from: Click for link for commentary and solution.

5.NF Grass Seedlings

The students in Raul's class were growing grass seedlings in different conditions for a science project. He noticed that Pablo's seedlings were times as tall as his

• wn seedlings. He also saw that Celina's seedlings were

as tall as his own. Which of the seedlings shown below must belong to which student? Explain your reasoning.

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34 Curt and Ian both ran a mile. Curt's time was Ian's

• time. Who ran faster? Explain and draw a picture.

A Curt B Ian

Problem is from: Click for link for commentary and solution.

5.NF Running a Mile

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35 Three friends are comparing their hair length. Abby's hair is the length of Beth's hair, and Carol's hair is the length of Beth's hair. Who's hair is the shortest? Explain. A Abby B Beth C Carol

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36 Four friends have started collecting coins. Chris has the amount Dan has. Ben has times what Chris has. Alex has the amount that Dan has. Who has the most coins? A Alex B Ben C Chris D Dan

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37 Select a phrase to correctly complete each sentence. The product of and 4 is ________ 4. A less than B equal to C greater than

From PARCC EOY sample test #29

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38 Select a phrase to correctly complete each sentence. The product of and 2 is ________ 2. A less than B equal to C greater than

From PARCC EOY sample test #29

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39 Select a phrase to correctly complete each sentence. The product of and is ________ . A less than B equal to C greater than

From PARCC EOY sample test #29

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Return to Table of Contents

Area of Fractional Side Length Rectangles

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Find the area of rectangles with mixed number side lengths.

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On grid paper, make a rectangle that has sides

• f 3 units by 2 units.

How many unit squares would you need to cover the square?

unit square

REVIEW

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40 How many unit squares will it take to cover a rectangle that is 3 units long and units wide?

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41 A tablecloth has dimensions of feet by 6 feet. What is the area of the tablecloth in square feet?

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42 A banner is being made to hang in the gym It needs to be 5 meters by meters. What is the area of the banner?

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43 A rectangular patio design is shown below. What is the area of this patio?

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Find the area of rectangles with fractional side lengths.

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A field measures a mile wide by mile long. What is the area in square miles of the field? Draw a Square - Steps: · Divide the left edge into 3 equal parts and label one part 1/3. · Divide the top edge into 2 equal parts and label one part 1/2. · Draw lines across the square for the 1/3 mark and the 1/2 marks to subdivide the square into small rectangles. · How many rectangles are there? · If the area of the large square has an area

• f 1 square mile, what is the area of one of

the small rectangles? · Can the area of one of the small rectangles be found by multiplying the lengths of its sides?

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A field measures a mile wide by mile long. What is the area in square miles of the field? Steps: · Divide the left edge into 3 equal parts and label one part 1/3. · Divide the top edge into 2 equal parts and label one part 1/2. · Draw lines across the square for the 1/3 mark and the 1/2 marks to subdivide the square into small rectangles. · How many rectangles are there? · If the area of the large square has an area

• f 1 square mile, what is the area of one of

the small rectangles? · Can the area of one of the small rectangles be found by multiplying the lengths of its sides?

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A field measures a mile wide by mile long. What is the area in square miles of the field? Steps: · Divide the left edge into 3 equal parts and label one part 1/3. · Divide the top edge into 2 equal parts and label one part 1/2. · Draw lines across the square for the 1/3 mark and the 1/2 marks to subdivide the square into small rectangles. · How many rectangles are there? 6 · If the area of the large square has an area of 1 square mile, what is the area of one of the small rectangles? 1/6 · Can the area of one of the small rectangles be found by multiplying the lengths of its sides? yes

• f a

square mile

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A foot by foot rectangular piece of wood is cut from a 3 foot by 3 foot piece. Steps: · Use a 3 by 3 square on grid paper. · Subdivide it and label as shown. · The area is the sum of the labeled parts. · Can the area also be found by computing the products?

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44 How many unit squares will it take to cover a square that has units long side lengths?

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45 How many square meters of carpet will it take to cover a rectanglular room that is meters by meters?

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46 Find the area of the rectangle below.

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47 Mr. Fernandez is building a small table. He has a rectangular piece of wood for the top that is 8 feet by 4 feet. If he cuts the piece for the table top to be feet by feet, how many square feet of wood will be have left?

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48 Jen makes a rectangular banner. It is yard long and yard wide. What is the area, in square yards, of the banner?

From PARCC EOY sample test #19

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49 Kurt drew a rectangular maze with a length of foot and a width of foot. What is the area, in square feet, of Kurt's maze?

From PARCC EOY sample test #23

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Dividing Unit Fractions by Whole Numbers

Return to Table of Contents

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Four students are sitting together. They are given

• f a cake to share equally among themselves. How much will each

student get of the cake if they share the of it equally? whole cake cake they would each get 1/12 of the whole cake

Visual Fraction Model Slide 96 / 130

You have one half of a bag of popcorn to share evenly among 3 people. How much of the bag does each person get? 1/2 bag 1/2 bag 1/3 1/3 1/3 This darker shaded part shows each person will get 1/6 of the bag of popcorn.

Visual Fraction Model

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50 One-half of a room is painted. Each of four people did the same amount of painting. How much of the room did each person paint?

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51 After the barbecue there is of a watermelon left. If 5 people evenly share it, how much of the whole watermelon will each person get?

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52 Each table gets one third of a bottle of paint in Art Class. If there are 3 people at a table, how much of the bottle will each of them get?

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53 Two brothers want to evenly share the of the apple pie that is left. How much will each of them get?

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Dividing Whole Numbers by Unit Fractions

Return to Table of Contents

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When dividing fractions - problems will answer one

• f two questions.
• 1. How many groups?
• r
• 2. How many in each group?

Dividing Fractions

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There are 6 cups of raisins in a box. Each serving is one fourth of a

• cup. How many servings are in the box?

To solve the problem we need to find out how many servings (groups) are in the box. Draw a picture to solve. 4 servings 3 servings 2 servings 1 serving There are 4 servings in 1 cup, how many servings in 6 cups?

• 1. How many groups?

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54 A package contains 4 cups of oatmeal. There is cup of

• atmeal in each serving. How many servings of oatmeal

are there in the package? Draw a picture to illustrate your solution.

Problem is from: Click for link for commentary and solution.

5.NF How many servings of oatmeal?

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55 Julius has 4 blue marbles. If one third of Julius' marbles are blue, how many marbles does Julius have? Draw a picture to illustrate your solution.

Problem is from: Click for link for commentary and solution.

5.NF How many marbles?

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How are the pictures that represent the last two problems different?

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56 There were a total of 2 pounds of apples to make the apple treats. Each treat contained 1/5 of a pound of

• apples. How many treats were made?

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57 Eva’s classmates bought 5 pizzas to help celebrate all their successes in math. Each student received a slice that was 1/8 of a pizza. How many slices did they cut up?

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58 There are 5 red soccer balls in the school gym. The rest

• f the balls are multi-colored. If the red balls represent

1/4 of the total soccer balls, how many are in the gym?

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59 Jim uses ribbon to make bookmarks. Jim has 9 feet of

• ribbon. He uses 1/3 foot of ribbon to make each
• bookmark. What is the total number of bookmarks Jim

makes with 9 feet of ribbon?

From PARCC EOY sample test #1

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60 Mr. Edwards is making sandwiches. He has 4 pounds of

• cheese. He puts 1/8 pound of cheese in each sandwich.

What is the total number of sandwiches Mr. Edwards makes using all 4 pounds of cheese?

From PARCC EOY sample test #30

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Line Plots With Fractional Data

Return to Table of Contents

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When placing numbers and/or fractions it is important to see how the number line is divided. They will not always be the same! 1 1 2 2 3 3 5 4 What number would be between the whole numbers? What number would be between these whole numbers? Before we begin working with line plots, lets review number lines.

Number Lines

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61 What number is missing?

3

?

2 1

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62 Which number represents the red dot correctly? A B C

1 2

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63 Which number represents the red dot correctly? A B C

1 4 2 3

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1 2 Place these fractions on the number line 1 8 3 4

3

8

1

2 1

2

1 3

4

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A line plot is a number line with marks that show the frequency of data. Example:

1 2

x x x x x x x x x x x x x x x x x x x x x x x x Length of Girls Hair in Inches The count of "x" marks above each score represents the number of girls who have that length hair.

Line Plot Slide 120 / 130

What is the length represented by the red dot? How many girls have that length of hair?

1 2

x x x x x x x x x x x x x x x x x x x x x x x x Length of Girls Hair in Inches

Line Plot

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64 What is the hair length that is represented by the red dot?

1 2

x x x x x x x x x x x x x x x x x x x x x x x x Length of Girls Hair in Inches

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65 How many girls have hair that is inches in length?

1 2

x x x x x x x x x x x x x x x x x x x x x x x x Length of Girls Hair in Inches

1

3 4

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66 What length of hair do the most girls have?

1 2

x x x x x x x x x x x x x x x x x x x x x x x x Length of Girls Hair in Inches

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Make and label a line plot to display the data.

Amount of Rainfall (In Inches)

1 1/8 1 1/4 1 3/8 1 1/2 1 5/8 1 3/4 2

Number of times

2 3 4 2 1

Rainfall was collected, and measured in inches, over several days. The table below shows the number of times each amount was collected.

Line Plot Slide 125 / 130

1 Number of Hours x x x x x x The line plot above shows the number of hours that Jeremy spent doing homework each night last week. You will use this line plot to answer the next 5 questions.

Line Plot Slide 126 / 130

67 What is the difference between the greatest number of hours that Jeremy spent on his homework and the number of hours Jeremy spent on his homework most frequently?

1 Number of Hours x x x x x x

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68 What is the difference between the greatest number of hours that Jeremy spent on his homework and the least?

1 Number of Hours x x x x x x

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69 What is the total number of hours that Jeremy spent on his homework?

1 Number of Hours x x x x x x

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70 Jeremy said he will need to spend twice the number of hours on his homework this week as he did last week. How many hours will he spend on his homework?

1 Number of Hours x x x x x x

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71 If Jeremy spends the same amount of time every week

• n his homework for the next 8 weeks, how many total

hours will he have spent doing homework?

1 Number of Hours x x x x x x