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2012-13 and 2013-2014 Transitional Comprehensive Curriculum

Grade 5Mathematics

Unit 8: Multiplying and Dividing Fractions, Exponents

Time Frame: Approximately three weeks

Unit Description

The focus of the unit is on multiplying and dividing fractions. There is also an emphasis on writing word problems, looking for patterns, and creating line plots.

Student Understandings

Students develop, understand, and use conceptual knowledge of fractions Students create visual models to aid in multiplying and dividing fractions. They will use the models to assist them in creating algorithms to perform operations with fractions. Students will also use exponents to show powers of 10.

Guiding Questions

1. Can students use previous understandings of multiplication to multiply a whole number by a fraction?

2. Can students multiply whole numbers, fractions, and mixed numbers?3. Can students divide a unit fraction by a whole number and a whole number by

a unit fraction? 4. Can students apply knowledge of fractions to real-world problems?5. Can students create a line plot to display a data set of fractional measures?6. Can students explain patterns in the number of zeros when multiplying by

powers of 10?7. Can students use whole number exponents to denote powers of 10?

Unit 8 Grade Level Expectations (GLEs) and Common Core State Standards (CCSS)CCSS for Mathematical Content

CCSS# CCSS TextNumber and Operations in Base Ten5.NBT.2 Explain patterns in the number of zeros of the product when

multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

Number and Operations – Fractions5.NF.2 Solve word problems involving addition and subtraction of fractions

referring to the same whole, including cases of unlike denominators, (e.g., by using visual fraction models or equations to represent the problem.) Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.

Grade 5 Mathematics Unit 8 Multiplying and Dividing Fractions, Exponents 8-1

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5.NF.3 Interpret fractions as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, (e.g., by using visual fraction models or equations to represent the problem.)

5.NF.4 Apply and extend previous understanding of multiplication to multiply a fraction or whole number by a fraction.

a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b.

b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

5.NF.6 Solve real-world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.

b. Interpret division of a whole number by a unit fraction, and compute such quotients.

c. Solve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem.

Measurement and Data5.MD.2 Make a line plot to display a data set of measurements in fractions of a

unit (1/2, ¼, 1/8). Use operations on fractions for this grade to solve problems involving information present in line plots.

ELA CCSSCCSS# CCSS TextWriting StandardsW.5.2a Write informative/explanatory texts to examine a topic and convey

ideas and information clearly.a. Introduce a topic clearly, provide a general observation and

focus, and group-related information logically; include formatting (e.g., headings), illustrations, and multimedia when useful to aiding comprehension.



Sample Activities

Activity 1: Fraction as Division (CCSS: 5.NF.3, W.5.2a)

Materials List: paper, pencils

Display the fraction 3/5. Ask the students to read the fraction and interpret its meaning. The students may say 3 parts of 5, 3 out of 5, or 3 divided by 5. Tell students that fractions can be interpreted as a division of the numerator by the denominator (a/b = a ÷ b). Therefore, 3/5 can be interpreted as 3 ÷ 5. Have students use a visual fraction model to show three being divided by five. Ask the student to think of 3 candy bars to be shared among five people and ask them how could this be done? Have them draw three bars as shown to represent 3 candy bars.

Ask them to divide each bar into fifths to represent each person getting an equal share of each bar. Each section of each rectangle is equal to 1/5. Each person would get 3/5 of 3 candy bars as shown below.

Have the students consider the following: Your mother gives you 7 packs of gum to share among yourself and three friends. How much of the gum does each person get? Have the students create a visual model to solve the problem. Their model may look like the one shown.



Each friend will receive ¼ of each pack of gum or 7/4 of the total of 7 packs. Students should understand that 7/4 is interpreted as 7 ÷ 4. Each person would get 1 whole pack and ¾ of a second pack (1 ¾).

Have students answer the following problems by creating a visual fraction model or equation to represent the problem.1. Ms. Johnson is buying pizza for a pizza party. With Toby’s Typical Pizzeria, she can

buy 3 pizzas for every 5 students. For the same price, with Peter’s Peppy Pizzeria she can buy 3 pizzas for every 4 students. She is trying to find the best bargain. How much pizza would each student get to eat from each pizzeria? (Toby’s Typical Pizzeria- 3/5 of a pizza; Peter’s Peppy Pizzeria-3/4 of a pizza) Which pizzeria should she buy from? (Peter’s Peppy Pizzeria)

2. The seven fifth grade classrooms have been given a total of 37 boxes of pencils. How many boxes will each classroom receive? (37/7 or 5 2/7)

In their math learning logs (view literacy strategy descriptions), have students explain how to write a fraction as division by using visual models.

Activity 2: Multiplying a Whole Number by a Unit Fraction (CCSS: 5.NF.4a, W.5.2a)

Materials List: counters, paper, pencils

Have students work in pairs. Give each pair eight counters. Ask them to divide the counters into 4 equal sets. How many counters are in each set? (2) Their grouping could look like the one shown.

Ask students to look at the 8 counters again. This time, ask them to find ¼ of the counters. Ask how many counters would there be in each ¼ (2). Help the students


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understand that they divided the counters into four equal shares and each equal share is 2 counters. Tell the students that whether multiplying by1/4 or dividing by 4, the counters are divided into equal shares. In both instances, the answer is 2; therefore, ¼ of 8 = ¼ × 8 = 8 ÷ 4 = (8/4) = 2.

Have students work with their partners to answer the following questions. Circulate through the class to listen to discussions and help with understanding. Have students use the counters or draw pictures to help in understanding.

1. What is 1/3 × 18? (6) What is 18 ÷ 3? (6) What is 18/3? (6)2. Six students each worked for 1/3 hour. For how many total hours did the 6

students work? (Students should draw 6 rectangles each divided into thirds with 1/3 shaded. This model should show that the total is equal to 6/3 or 2 whole hours.)

3. Derrick has worked ¼ of his work shift. If he works 16 hours, how many hours has he worked? (4 hours)

4. ¾ of the students in Mr. Clark’s class are attending the school play. If Mr. Clark has 28 students, how many students are attending the play? (21 students)

When students have finished, have them use drawings and words to explain why 1/5 of 10 is the same as 10 ÷ 5 in their math their learning logs (view literacy strategy descriptions).

Activity 3: Multiplying Fractions Strategies (CCSS: 5.NF.4a)


This activity will focus on strategies for multiplying a fraction by a whole number.

Strategy 1 : Find the unit fraction then multiply.Display 2/3 × 18. Question students how they would solve the problem. Ask students to consider how many thirds are in 2/3 (2). Tell them to draw a number line representing 18 and show thirds by dividing the line into three equal parts like the figure shown.





Ask students if 1/3 of 18 + 1/3 of 18 is the same as 2/3 of 18. Ask them to find 1/3 of 18 and then multiply by 2. Tell them that this is one strategy that can be used to multiply a fraction by a whole number. In this method, students multiply the whole number by a unit fraction and then multiply that result by the numerator of the original fraction. Ask students to explain in words what was done in the problem using the strategy. (They found 1/3 of 18 and then multiplied it by 2.)

Ask students to solve the expression 2/9 of 18 by finding 1/9 of 18 and multiplying by 2. Students may draw a number line like the one shown to help with solving the problem.

Strategy 2: Multiply, then divide.Display 2/3 × 18 again. Ask the students to think about multiplying 2 × 18 and then dividing by 3. Ask them what answer they will get if they do this (12). Inform students that this is another method that can be used to multiply a fraction by a whole number. In this method, students multiply the numerator of the fraction times the whole number and then divide that result by the denominator of the fraction. Have them solve the following problem using this method 2/5 × 35 = (2 × 35)/ 5 = 14).

Strategy 3: Simplify, then multiply.Students may find it challenging to find equivalent fractions. Review ways to find equivalent fractions prior to introducing this method. Tell students that it is possible to simplify before multiplying. Remind students that in the first strategy they multiplied 18 by 1/3 to get 6. Then they multiplied 6 by 2 to get 12. Display 2/3 × 18 again.

Instruct student to simplify 18/3 to 6/1 by dividing the numerator and denominator by 3.

Have the students multiply = = 12. In this method, students simplify the expression before multiplying. Have students solve ¾ × 12 by simplifying first (9).

Discuss the three strategies used. Ask them if simplifying first before multiplying makes solving the problem easier and if so, to explain their thoughts. Students should relate that the numbers to multiply were smaller which made it easier to multiply.

Have students solve the following problems using the strategy of their choice. Have them simplify their answers. 2/3 × 15 = ? (10); 5/8 × 40 = ? (25); ½ × 42 = ? (21); 1/9 × 81 = ? (9)


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Activity 4: Area Model of Multiplication with Fractions (CCSS: 5.NF.4b, W.5.2a)

Materials List: Area Model of Multiplication with Fractions BLM, pencils

Remind the students of the area model of multiplication used with whole numbers. Have students find 3 × 5 by drawing a rectangle to find the area. The students should see that the side lengths are multiplied to find the area of the rectangle (15). This method was used in Unit 2.

Remind students of multiplying decimals using a hundreds grid in Unit 2. Each column and row was 1/10 of the whole grid. One of the small squares represented 0.1 times 0.1 or 0.01. Students used the area model to multiply decimals.

The students can use this knowledge to help them understand multiplying fractions using the area model of multiplication. The area model of multiplication is an efficient way to multiply and helps students visualize the fractional factors as side lengths. Through this alternative way to multiply, students discover that the area is the same as multiplying the side lengths. A process guide (view literacy strategy descriptions) accompanies this activity and will be used to teach this method. Provide the students with the Area Model of Multiplication with Fractions BLM and have them review the guide. Group the students into small groups. Explain the guide’s features (explore, explain, understand, apply, and reason) and tell them that they will use the process guide to work through the area model of multiplication. Remind them of their work with area model of multiplication when multiplying whole numbers and decimals in Unit 2. Tell them that this process guide will help them understand multiplication problems that involve fractions.




Guide the students through the process by helping them see how the area model of multiplication is used to solve 4/5 × 2/3. Allow for discussion and listen as they explain how to break the 4/5 × 2/3 rectangle into smaller rectangular sections. Guide students in understanding this idea. They may need assistance in understanding that the model is one square unit (5/5 × 3/3) and each small rectangle has a length of 1/5 and width of 1/3.

Facilitate their completion of the guide, providing feedback and additional explanations as needed. The following are ideas for students to understand through the process guide:

With the area model of multiplication with fractions, students should represent the fractions as unit fractions for both the length and width.

Students should shade in the rectangles represented by the fractional sides. This shading represents the area of the rectangle which is the answer to the multiplication problem.

Multiplying the numerator by numerator and denominator by denominator will give the same answer as the area model.

Students should be encouraged to use the area model of multiplication and to draw their own rectangles even though their drawings may not be drawn to scale.

Activity 5: Making Models (CCSS: 5.NF.4b, 5.NF.6)

Materials List: paper, pencils, Grid Paper BLM

Have students work in groups of four. Tell students that they will work together to create word problems with multiplication of fractions and represent the problems with an area model. Give each student the Grid Paper BLM to use to create their models. Have groups work to create word problems and area models for the following equations. Remind students to find the simplest form of the fraction answers. Answers are shown. Word problems will vary. Sample word problems are shown.

1/3 of the class are girls. 5/6 of the girls are wearing ponytails. What fraction of the class are girls with ponytails?1/3 5/6 = 5/18

1/3 of the cereal was left in the box. Keisha ate 4/9 of what was left. What fraction of the cereal did Keisha eat?1/3 4/9 = 4/27

Susan had 2/3 feet of wrapping paper. She used 2/7 of the paper to wrap presents. How much does she have left?2/3 2/7 = 4/21



5/8 of the class eats breakfast. 4/5 of those students eat breakfast at school. What fraction of the class eats breakfast at school?

5/9 of Jonathan’s newspaper customers paid for their subscription. 3/5 of those who paid did so with a credit card. What fraction of customers paid with a credit card?

3/7 of the boys in band class play in the horn section. 2/5 of those boys play trombone. What fraction of the boys play trombone?

5/8 4/5 = 20/40 = 1/2 5/9 3/5 = 15/45 = 3/9 3/7 2/5 = 6/35

Have groups share their word problems with other groups in the class and allow them to solve their problems.

Activity 6: Area Model of Multiplication with Fractions, Whole Numbers and Mixed Numbers (CCSS: 5.NF.4b, W.5.2a)

Materials List: More Area Model of Multiplication BLM, pencils

Remind the students of the area model of multiplication used with fractions in the previous activities. Provide the students with the More Area Model of Multiplication BLM. Group the students into small groups. Explain to the students that they will explore the examples of multiplying a whole number by a fraction and a mixed number by a mixed number.

Tell the students to look at the first example, Problem 1. Have a student read the paragraph. Ask students how they know that the product will be less than 2. Students need to understand that the rectangle will be divided into 2 equal rectangle horizontally and 3 equal sections vertically. Help them see that when they shade the area of the rectangles needed, not all of each rectangle will be shaded. Two sections of each rectangle will be shaded as shown on the BLM. Each section is 1/3 of each rectangle, so 2/3 of the first rectangle is shaded and 2/3 of the second rectangle is shaded for a total of 4/3.



Answer any questions the student may have about creating the model.

Work the first “Try It” problem, Problem 2, with the students to assist in understanding. Examples of the visual model are found on the BLM. Have the groups work together to complete Problem 3.

Have students look at the second example, Problem 4. This problem will involve a visual model for multiplying two mixed numbers. Have a student read the paragraph while the class reads silently. Ask students how they know that the product will be greater than 6. Have a discussion with the students about the explanation used to visually model 2 ½ groups of 3 ½. Answer any question the students may have about creating the model.

Work the “Try It” problem, Problem 5, with the students to assist in understanding. Have the group work together to complete Problem 6.

If time permits, have groups find the products of the following problems 4/7 × ¾ = ? (3/7); 3 ¼ × 2 1/3= ? (7 7/12); 4 × 3/5 = ? (12/20). Have students display and discuss their visual models with the class.



Activity 7: Multiplication with Fractions Word Problems (CCSS: 5.NF.6)


Have students work in groups. Have them solve the following word problems using the strategies discussed previously.

1. There were 3 ½ pounds of mixed nuts under the counter to be sold. 3/5 of the nuts are peanuts. How many pounds of the nuts are peanuts? (21/10 = 2 1/10 pounds)

2. Eva bought 12 roses for her mother. 2/3 of them were red. How many red roses were there? (8)

3. Twelve of the 18 workers ate crawfish at the company picnic. Of those who ate crawfish, ¾ ate crabs as well. What fraction of the workers ate crawfish and crabs? (1/2)

After groups have successfully answered the word problems, have them create word problems of their own and have other groups solve the problems.

Activity 8: Division of a Unit Fraction by a Whole Number (CCSS: 5.NF.7a, W.5.2a)


Review division of whole numbers. Ask the students this question: If there were 20 cookies to share among five friends, how many cookies would each person get? Elicit answers directed toward division 20 ÷ 5 or multiplication where each friend would receive 1/5 of the 20 cookies (1/5 of 20).

Display the following problem: Rachel has 1/2 yard of ribbon. She wants to make 6 wreaths. How much of the ribbon will she use for each wreath?

Tell students that they will use visual models to represent their understanding of the problem. Ask the students what the problem is asking them to find (the amount of ribbon per wreath to be used). Ask them what division expression would be used to find the amount of ribbon used (½ ÷ 6). Some students may be able to solve the problem mentally; however, encourage them to support their answer with explanations and visual models (i.e., number line, area, or symbols). Examples are shown:



or ½ ÷ 6 = ½ ÷ 6/1 = 1/2 × 1/6 = 1/12

Have a volunteer explain how they solved the problem. Students should exhibit understanding that the amount of ribbon (1/2 yard) will be divided into 6 equal shares. In order to do so, they must show that ½ is less than one whole and that ½ is divided into 6 parts (i.e. ½ 6 or 1/6 of ½). Each wreath would have 1/12 of the original ribbon. To justify their answer, the student may add each 1/12 together to get 6/12 which is ½ or use the inverse operation (multiplication) in their symbol representation to get the original amount of ribbon (1/12 × 6 = 6/12 = ½ yard)

Have students find ¼ ÷ 2 = ? (1/8); 1/3 ÷ 3 = ? (1/9); 1/8 ÷ 5 = ? (1/40) by creating visual models.

Place students in groups of four. Ask them to demonstrate their understanding of fractional division by completing a RAFT writing (view literacy strategy descriptions) assignment. This form of writing gives students the freedom to project themselves into unique roles and to look at content from unique perspectives. From these roles and perspectives, RAFT writing can be used to describe a point of view, envision a potential job or assignment, or solve a problem. It is the kind of writing that when crafted appropriately, should be creative and informative.R – Role (role of the writer – advice column writer)A – Audience (to whom the RAFT is being written – a person seeking advice)F – Form (the form the writing will take – a letter)T – Topic (the subject of the focused writing – to answer a question posed by Perplexed

Patty. See below.)

Have students answer the following letter sent to them for advice:Dear Ask the Expert:I told my friend Rachel that dividing a unit fraction by 3 is the same as dividing by 1/3. I gave her the example 1/3 ÷ 3 is the same as 1/3 ÷ 1/3. I said both answers would be 1/9 but Rachel doesn’t think so. Did I do something wrong? What do you think?

Sincerely,Perplexed Patty




Students should answer Perplexed Patty in an informal letter where they explain that dividing a unit fraction by the whole number 3 is not the same as dividing the same unit fraction by 1/3. Some possible explanations are the following:

When dividing 1/3 by 3, the answer is 1/9. When dividing 1/3 by 1/3 the answer is 1.

When dividing 1/3 by 3, you are taking the fraction 1/3 and dividing it into 3 equal shares. Each share is 1/9 of the third.

When dividing the fraction 1/3 by 1/3, you are taking 1/3 and dividing it by 1/3. Since 1/3 is a third, the answer is one, because any number divided by itself is equal to 1.

Students should be encouraged to draw visual models to help explain the error. After writing their letters, have students share with partners and the class to check for accuracy and logic.

Activity 9: Fractions on Cards (CCSS: 5.NF.7b)

Materials List: paper, pencils, index cards

Have students work in pairs. Have students create cards with the unit fractions 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, and 1/10 written on them. On another set of cards, have them create ones with the numbers 1-20. Shuffle the unit fractions together and place them in a pile face down. Shuffle the whole number cards together and place them in a separate pile face down. Allow one partner to select one card from each pile and create division where the whole number is divided by the unit fraction, W ÷ . Have the other partner divide the problem and each partner check the answer. Have the students return the cards to the appropriate decks, and repeat the activity taking turns as time permits.

Activity 10: Division with Fractions Word Problems (CCSS: 5.NF.7c)


Give the following problem to students. You have 15 cookies. You want to give 5 cookies to some friends. To how many friends can you give cookies? (3) One way to look at this question is to think “How many 5’s are in 15?”



Tell students that when dividing a whole number by a unit fraction, such as 1/2, it will help them to think, “How many ½’s are in the whole number?” For example, ask students to think of a problem for the equation, 4 ½. Student might say “How many ½ dollars are in $4? (8) If I cut 4 apples into 2 pieces, how many ½ apples would I have? If I have a rope that is 4 ft long, how many ½ ft pieces could I cut from it?” A visual model for this type of problem could look like this:

Have students work in groups. Have them write an equation to model the problem and solve the following word problems. Encourage students to create visual models.

1. Tabitha made 10 sandwiches and cut each one into fourths. How many sandwich fourths does she have? (10 ÷ ¼ = 40. Tabitha has 40 sandwich fourths. )

2. Caleb has ½ of a crate of Louisiana strawberries. He and his 4 friends will share the crate equally. How much of the strawberry crate will each of them get? (1/2 ÷ 5 = 1/10. Each person will get 1/10 of the crate.)

3. Kayla has 1/4 of a watermelon she wants to share with 2 friends. Kayla does not want any of the watermelon. How much of the watermelon will she give to each person? (1/4 ÷ 2 =1/8. 1/8 of the watermelon will be given to each person)

4. Carly wants to share ½ cup of ice cream with her sister. How much of the ice cream will each person get? (1/2 ÷ 2 = ¼. Each sister will get 1/4 cup of ice cream.)

After groups have successfully answered the word problems, have them create word problems of their own and allow each other to solve.



Activity 11: Ask the Experts (CCSS: 5.NF.7a, 5.NF.7b, 5.NF.7c, W.5.2a)

Materials List: paper, pencils, chart paper, markers, dry erase board, dry erase markers

Have students work in groups of four. Allow them time to review dividing whole numbers by unit fractions and dividing unit fractions by whole numbers. To demonstrate that students have learned the content, use professor-know-it-all (view literacy strategy descriptions). Tell students that one group will be called on randomly to come to the front of the class to be a team of “professors-know-it-all.” Have groups generate 3-5 questions, including word problems, about the content they might anticipate being asked and that they can ask other experts.

Inform the students that at least one of their questions must be a word problem where the “professors” will either divide a whole number by a unit fraction or divide a unit fraction by a whole number. The “professors” will demonstrate how to solve the problem as a team. They will be given a moment of time to “consort” in order to solve the problem by creating visual models, checking computations, establishing strategy use, and finding patterns in problem solving. Evidence of this knowledge could be shown in their ability to solve the student-created word problems.

Listed are some sample questions, including sample word problems.1. How is dividing a whole number by a unit fraction the same as multiplying the whole

number by the denominator of the unit fraction? (When dividing a whole number by a unit fraction, you are finding the number of the unit fractions in the whole number.) Students should be encouraged to use visual models to explain.

2. When you divide a unit fraction by a whole number, will your quotient be a unit fraction? Explain. (Yes. When dividing a unit fraction by a whole number, you are finding one of the equal shares. Students should be encouraged to use visual models to explain.)

3. Sandra had 1/3 of an apple pie left. She shared the pie with two friends. How much of the pie did each person get? (1/9 of the pie)

4. Derrick’s mom made 20 cakes for the bake sale. She divided each cake into fourths. How many pieces of cakes made by Derrick’s mom will be on sale? (80 pieces of cake)

Select one group to come to the front of the class. The group will be asked questions by one of the other groups. Every student in each team should be ready to explain the answers to all questions asked and should be able to explain the visual models they create. Students should listen for accuracy and logic in the “professors’” answers to their questions. After appropriate time, ask a new group to become the “professors” and continue having the groups ask questions of them. If time permit, allow all groups an opportunity to go before the class.





Activity 12: The Plot Thickens, Creating Line Plots (CCSS: 5.NF.2, 5.MD.2)

Materials List: The Plot Thickens, Creating Line Plots BLM, paper, pencil

Have students review types of graphs, such as bar, double bar, pictograph, line, and coordinate graphs used in Unit 3. Inform students that this activity will help them understand line plots and see their usefulness in understanding data. Remind the student that a line plot shows data on a number line with an “x” or another mark to show frequency.

Have students work in pairs and give a The Plot Thickens, Creating Line Plots BLM to each student. Read the stories to the class and have them plot the data. When they complete the line plot correctly, read the end of the story. The indented section is the same information they have on their BLM.

First Story: Mr. Johnston owns a small apartment complex. He is considering building a recreational area and wants to know if a playground area or an arcade would be better to build. He gathered the ages of the children and now needs help in displaying them. The approximate ages in years of the 20 children in the building are 13, 1, 5, 8, 7, 1, 0, 12, 10, 9, 1, 5, 5, 13, 6, 8, 16, 2, 3, 0. He wants to understand the results he collected, but doesn’t want to spend time creating bar or circle graphs. He is sure a line graph would not work. You suggest that he create a line plot and help him do so.

Have students start by analyzing the data and putting it in numerical order (0, 0, 1, 1, 1, 2, 3, 5, 5, 5, 6, 7, 8, 8, 9, 10, 12, 13, 13, and 16). Tell the students that zero represents children under the age of 1 year. Have students group the data items that are the same. (Example 2 are less than 1 year old, 3 are about 1 year old, etc.) Have students determine the range of their line plot (0-16 or 16 years). Have them plot their data by putting an x to represent the frequency of each age. Have them title their line plot. Their line plot should look similar to the following plot:

Once they are finished, continue with the story. (This portion of the story is not included on their BLM.) Mr. Johnston said, “Thanks. This line plot really makes sense. I can see that more than half of the children in my complex are under the age of 10. I’ll build a playground so they’ll have a place in the complex to play this summer.” The first part of the activity is just a way to review making a line plot. Additional questions could be asked such as the following:

What is the age range of the children? (16 years)



Which age (s) occurred most often? (1 year and 5 years) How many children were 13 years old? (2)

Second Story: Cullen, the citrus seller, heard from Mr. Johnston that you are very good with creating line plots. He had created two line plots showing his inventory in the store after his latest Bag of Fruit sale. He is not sure which one is correct. He asks you to analyze his data, look at the graphs, and tell him which one is correct. His data showed three ¼ lb. bags, four ½ lb. bags, and five 1/8 lb. bags.

Have students analyze the data and figure which line plot matches the data. (Bag of Fruit A) After the students find the correct line plot, continue with the story.

Cullen attempts to give you money for your help but you refuse and say that you are just glad to help. “You are terrific when it comes to creating line plots. There are just a few more things that I need help with,” Cullen says.

1. “I started with 10 bags of each weight before the sale. I need your help to find how many bags I sold altogether,” he exclaimed. (18 bags)

2. “I need to know the total amount of fruit in pounds for the 1/8 pound bags that are left.” (5/8 pound)

3. “Are there at least 2 pounds of fruit left in the ½ pound bags of fruit?” (Yes, there are 2 pounds of fruit.)

4. “If there is less than 1 pound left in the ¼ pound bags, you can take them home.” (There is ¾ of a pound left, so you get to take those bags home.)

Give students time to work with their partners to calculate the answers. Students can use addition or multiplication to find the answers. After students calculate the correct answers, finish the story.

Cullen is happy to give you fruit for your hard work. You enjoy the tasty treat as you make your way home. News of your ability to represent data correctly quickly spreads across the town. Now when people need help with creating line plots, they will contact you. The End.

Celebrate success of the activity with the students and encourage them to create word stories with line plots of their own.



Activity 13: Creating Line Plots (CCSS: 5.MD.2)

Materials List: paper, pencils,

Have students work in pairs. Allow them to use the following scenarios to create line plots. Students should check each other’s line plots for accuracy.

1. Mr. Smith’s class measured the lengths of their pencils.

four 1/6 in. four ½ in. five ¼ in. four 1/12 in. five 1/3 in.

2. The amount of ounces of glue left after a craft project was measured.

six ¼ oz. eight ½ oz. two 1/8 oz.3. The diameter of small circles drawn was measured in inches.

eight ¼ in. ten ½ in. six 1/5 in. nine 1/10 in. 4. The size of the seashell collected at the beach was recorded.

seven 1/6 cm. ten 1/3 cm. eight ½ cm. nine 1/12 cm. 5. The amount of sugar per tablespoon in snacks at school was discovered.

ten 1/4 tbsp. seven ½ tbsp. eight 1/8 tbsp.

Have students write 2 questions that could be answered from the information in each line plot. Have students share their questions with the rest of the class.

Activity 14: Patterns in Zero (Whole Numbers) (CCSS: 5.NBT.2, W.5.2a)

Materials List: overhead calculator, paper, pencils

Have students work in pairs. Have students multiply 10 × 1; 10 × 10; 10 × 10 × 10; 10 × 10 × 10 × 10. Ask them to discuss any patterns they see with the zeros. They may notice that every time they multiply by 10, a zero is added to the end of the number 10. This is because the value of 10 becomes ten times larger every time they multiply by 10. Inform the students that they can use exponents as a shorter way to write powers of ten. Demonstrate that 10 × 10 can be written as 102 where 2 is the exponent and 10 is the base. Have students create the exponential form of 10 × 10 × 10 (103) and 10 × 10 × 10 × 10 (104). Ask students if they can see the relationship between the exponent and the number of zeros in the product. They may see that in the exponential expression 104 written in standard form, 10,000, there are 4 zeroes in the number which is the same number of the exponent.

Display 4 × 102. Ask the students to state what is being multiplied. (4 × 10 × 10 or 4 × 100) Help students see that 400 has two zeros represented by the exponent in 102. Display 6 × 103. Ask the students to state what is being multiplied. (6 × 10 × 10 × 10 or 6 × 1000) Help students see that 6,000 has 3 zeros as does the exponent in 103. Explain that the answer 6,000 is the standard notation for 6 × 103. Display 105. Explain that the



standard notation of 105 can be found by writing the expression as repeated factors 10 × 10 × 10 × 10 × 10. Students should see the number of zeros in the product 100,000 is the same number of the exponent in 105.

Tell pairs that they will work together to find exponential expressions. Work one problem as an example. Have the students consider 7 × 102. Lead students to see that 7 × 10 =70 and 70 × 10 = 700 or 7 × 100 = 700. Have them note that 700 has 2 zeros which is the same number as the exponent. Have students write the following in standard notation: 2 × 102 (200); 56 × 103 (56,000); 104 × 12 (120,000); 5 × 103 (5,000). As they work to find the answer, have them discuss the patterns they see.

Allow students to create problems of their own and have their partners solve the problems. When appropriate time is given to the activity, have several volunteers explain to the class the relationship of the number of zeros and the exponent.

Activity 15: Patterns in Zero (Decimals) (CCSS: 5.NBT.2; W.5.2a)

Materials List: overhead calculator, paper, pencils

Have students work in pairs. Ask students to simplify 25 × 10 (250). Ask students to simplify 25 × 102 (2500). Have students review relationships discovered in the previous activity. (The number of zeros in 2500 is the same number as the exponent in 102.)

Display 2.5 × 102. Show that the expression 2.5 × 102 can be written as 2.5 × 10 × 10 or 2.5 × 100. Remind the students that they are multiplying by a decimal in the tenths place and their answer will be 1/10 smaller than 25 × 100. Have them use either expression (2.5 × 10 × 10 or 2.5 × 100) to find 250. Display 2.5 × 103. Ask them how many zeros the 3 in 103 implies (3). Ask them if there will be 3 zeros in the answer and to explain their thoughts. (The students should state that the answer will be 1/10 smaller than 25 × 103.) Have the students find the answer (2,500). Have them to simplify the expression 2.5 × 104 (25,000). Help students understand that the number 2.5 is 1/10 smaller than 25 and when multiplied by 10,000, the answer will be 25,000 and not 250,000.

Display 0.25 × 102. Show that the expression is equal to 0.25 × 10 × 10 or 0.25 × 100. If 25 × 100 is 2500, and if 2.5 × 100 = 250, ask students what they think 0.25 × 100 will equal (25). Ask them how many zeros the 2 in 102 implies (2). Ask them why are there not 2 zeros in the final answer. (The students should state that it is because the answer will be 1/100 smaller than 25 × 102.) Have the students find the answer (250). Ask them to simplify the expression 0.25 × 104 (2,500). Help students understand that the number 0.25 is 1/100 smaller than 25 and when multiplied by 10,000, the answer will be 2,500 and not 250,000.

Display 0.025 × 103. Ask students what 25 × 103 would equal and have them predict the answer to 0.025 × 103. (25,000. The students should state that the answer would be 1/1000 smaller than 25 × 103 or 1/1000 × 25,000 or 25.) Ask the students if they see a



pattern emerging. (The students may state that with decimal numbers the amount of zeros in the answer is 1 less than the exponent if multiplied by a tenth, 2 less than the exponent if multiplied by a hundredth and 3 less than the exponent if multiplied by a thousandth.)

Have students work with their partners to write the following exponential expressions in standard form:

3.1 × 104 (31,000) 0.45 × 103 (450); 2.068 × 104 (20,680); 7.2× 102 (720); 0.51 × 104 (5,100); 0.031 × 104 (310).

Have the partners discuss any patterns in zeros they see. Allow students to create problems of their own and have their partners solve the problems.

When appropriate time is given to the activity, have students complete a SPAWN writing activity. Through SPAWN writing (view literacy strategy descriptions) prompts, such as “What If?” a thought-provoking activity can be created related to exponents. Ask students to answer this prompt: What if you were asked to explain how to simplify the exponential expression 6.8 × 106, how would you do so? In groups of four, have students brainstorm (view literacy strategy descriptions) how they could discover the answer to the question and write their ideas in their learning logs (view literacy strategy descriptions). Have students share their responses with the class.

Sample Assessments

General Assessments

Portfolio assessments could include the following:o Anecdotal notes made during teacher observationo Any of the learning logs, or one of the explanations from the specific

activitieso Corrections to any of the missed items on the tests

On any teacher-made written tests, include at least one of the following:o One problem that requires the use of manipulatives or drawings such as

this: Using visual models explain why multiplying 1/3 of 12 the same as dividing 12 by 3.

o One problem that requires the student to explain his/her reasoning such as this: Answer the following question and explain his/her reasoning: Explain the patterns in zeros found when simplifying the following expressions: 35 × 103; 3.5 × 103, 0.35 × 103, and 0.035 × 103.

Learning log entries could include the following:







o Give your answer to the question: Explain why dividing a number by a unit fraction is the same as multiplying the number by the denominator of the unit fraction. Use visual models to explain.

o Show how to find the answer to this question: Explain why dividing a number by ½ is the same as multiplying the number by 2.

o Demonstrate your answer to the question: Explain why dividing a whole number by a fraction less than a whole gives an answer greater than the whole number.

Activity-Specific Assessments

Activity 2 : Derrick worked ¼ of his work shift. If he works 16 hours, how many hours does he have left to work? (12 hours)

Activity 12 : The students should complete the following real-life problem: A cookie maker had 10 sacks containing the following amounts of sugar: seven ½ kg, four ¼ kg, four 1/8 kg Plot the measurements on a line plot. Give the line plot a title and label the axis.If the maker redistributed the sugar equally among ten bags, how much sugar would be in each bag? Explain your thinking.

(The bags total would be 5 kg of sugar. To equally distribute the amount of sugar among the 10 bags, each bag would get ½ kg. The student may draw a visual model to show five kg divided by 10 bags.

Activity 15: Write the following exponential expressions in standard form: o 5.1 × 104 (51,000) o 0.32 × 103 (320)o 0.98 × 104 (9,800)o 1.062× 102 (16.2)



o 0.051 × 104 (510)


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