Grade 5 Unit 4 Extensions and Applications of Multiplication and Division of Fractions (5 Weeks)

Stage 1 – Desired Results

Established Goals Unit Description Students will relate different fractional units to a common fractional unit: 1 third + 1 fourth = 4 twelfths + 3 twelfths = 7 twelfths. Relating different fractional units together back to the whole unit requires extensive work with area and number line models, fluency, and bar diagrams used in word problems. The Mathematical Practices should be evident throughout instruction and connected to the content addressed in this unit. Students should engage in mathematical tasks that provide an opportunity to connect content and practices. Common Core Learning Standards Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 5NF4: Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. a. Interpret the product (a/b) x q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a x q ÷ b. For example, use a visual fraction model to show (2/3) x 4 = 8/3, and create a story context for this equation. Do the same with (2/3) x (4/5) = 8/15. [In general, (a/b) x (c/d) = ac/bd.]

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

5NF5: Interpret multiplication as scaling (resizing), by: a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without

performing the indicated multiplication. b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given

number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1.

5NF6: Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. 5NF7: 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. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 =1/3. 1 Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.

b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.

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. For example, how much chocolate will each person get if 3 people share ½ lb of chocolate equally? How many 1/3-cup servings are 2 cups of raisins?

Common Core Standards of Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

ESL Language Standards Standard 1: 1.5: Formulate, ask, and respond to various question forma to obtain, clarify and extend information and meaning 1.7: Present information clearly in a variety of oral and written forms for different audiences and purpose related to all academic content areas. 1.13: Engage in collaborative activities through a variety of student groupings to read, gather, share, discuss interpret, analyze and present information. Standard 3: 3.4: Evaluate students’ own and other’s work, individually and collaboratively, on the basis of a variety of criteria. 3.7: Engage in collaborative activities through a variety of groupings to discuss, share, reflect on, develop, and express opinion and evaluations about a variety of experiences, ideas, and information. 4.4: Listen attentively, take turns speaking, and build on others’ ideas when engaged in pair Bridge Guidance Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5. (1/4), recording the conclusion by the equation 5/4 = 5. (1/4). b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3. (2/5) as 6. (1/5), recognizing this product as 6/5. (In general, n. (a/b) = (n . a)/b.) c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

Big Ideas Operations Meanings & Relationships: The same number sentence (e.g. 12-4=8) can be associated with different concrete or real world situations, AND different number sentences can be associated with the same concrete or real-world situations.

Essential Questions a. How can I use what I know about operations with

whole numbers and apply it to operations with

rational numbers?

b. If I multiply a whole number by a fraction greater than

1 and a fraction less than 1, what happens to the

product and why?

c. How can I reason about the size of a product based

one of its factors?

d. How do solve and represent real world problems

involving fractions and mixed numbers?

Content (Students will know….) A. Visual fraction models can represent multiplication

and division with whole numbers, fractions, and mixed numbers (5.NF4, 5.NF.5, 5.NF.6, 5.NF.7)

B. Multiplication of a fraction or whole number by a fraction (5.NF.4a)

C. Area of rectangles with fractional side lengths (5.NF.4b)

D. Multiplication can interpreted as scaling or resizing

(5.NF.5a)

E. Products change according to the size of the fraction being multiplied (5.NF.5b)

F. Real world problems involving multiplication of fractions and mixed numbers (5.NF.6)

G. Division of unit fraction by whole numbers and vice versa (5.NF.7)

Skills (Students will be able to…) A1. Visual fraction models include:

Number lines

Fraction Bars

Area models

Fraction circles

Set models

Partition models B1. Multiply fraction by a whole number B2. Interpret multiplication of fraction by a whole number in two ways such that (a/b) x q = (a x q) ÷ b AND a x (q/b) B3. Create a story context for each interpretation of the problem B4. Represent each interpretation of the problem using a visual fraction model (tape model, number line model or area model). C1. Find area of rectangles with fractional side lengths C2. Represent multiplication of fractions using area models D1. Compare the size of a product to the size of one factor without performing the actual operation. For example, know that the product of 30 x 3/5 will be twice the product of 15 x 3/5. D2. Represent or explain the above multiplication with a model and an equation. E1. Compare the products when multiplying by a fraction greater then one (product will be greater then original factor) and less than one (product will be less than the original factor) E2. Represent and explain thinking using a visual model F1. Use various strategies to solve real world problems including fraction with a fraction, fraction with a mixed number, or a mixed number by mixed number F2. Represent real world problems using visual fraction models and equations G1. Represent division of unit fraction by whole number using a visual fraction model and an equation G2. Represent division of a whole number by a unit fraction using a visual fraction model and an equation G3. Solve real world problems using visual fraction models and equations G4. Create story context for division of unit fraction by whole and whole by unit fraction

Terms/ Vocabulary Fraction, numerator, denominator, operations, multiplication/multiply, division/divide, mixed numbers, product, quotient, partition, equal parts, equivalent, factor, unit fraction, area, side lengths, fractional side lengths, scaling, comparing, whole numbers, dividend, divisor, equation, inverse operations

Stage 2 – Assessment Evidence

Initial Task: Brian’s Birthday Party Final Performance Task: School Bake Sale

Other Evidence Teacher observation, conferencing, teacher designed assessment pieces, student work, exit slips, journal entries

Stage 3 – Learning Plan

Everyday Mathematics/Impact Mathematics Lessons – The following lessons may support some of the CCLS & essential questions outlined in this unit map:

5.NBT.7: 2-2, 2-3, 2-4, 2-5, 2-6, 2-7, 2-8, 2-9, 4-5, 4-6, 5-11, 6-5, 6-7, 7-10, 9-8, 9-10, 10-6, 12-12 5.NF.4: 1-4, 8-8, 9-1 5.NF.5: 5-3, 10-2 5.NF.6: 8-6, 8-8 Additional Resources: Impact Mathematics Course I 5.NBT.7: 3-5, 4-2, 4-3, 4-4 5.NF4, 5.NF.5, 5.NF.6: 7-2, 7-4 K-5 Math Teaching Resources Click on any activity listed under standards 5.NBT.7, 5.NF.4, 5.NF.5 or 5NF.6 http://www.k-5mathteachingresources.com/5th-grade-number-activities.html Georgia Grade 5 Unit 4 Framework This Georgia unit addresses all fraction operations including addition and subtraction. For the purposes of this unit, use any activities or lessons that align to multiplication and division of fractions (5.NF.4, 5.NF.5, 5.NF.6) https://www.georgiastandards.org/Common-Core/Pages/Math-K-5.aspx Fractionbars.com Scroll down to find lesson plans for 5.NF4, 5.NF.5 and 5.NF.6 http://fractionbars.com/CommonCore/Grade5.html 5.NF.4 Lessons: Multiplying Fractions Times Fractions Areas of Rectangles with Fractional Side Lengths 5.NF5 Lessons: Comparing Sizes of Products and Estimating Products Obtaining and Recognizing Equal Fractions 5.NF.6 Lessons: Solving Problems Involving Multiplication of Fractions and Mixed Numbers North Carolina Unpacked Standards for Grade 5 Outlines what each standard requires a student to know and understand as well as gives example problems and many visual models for fraction operations http://www.ncpublicschools.org/acre/standards/common-core-tools/#unmath Utah Education Network – standards breakdown in terms of concepts, skills and representations. Plus, web resources, lesson plans and sample tasks 5.NF.4 http://www.uen.org/core/displayLinks.do?courseNumber=5150&standardId=71122&objectiveId=71126 5.NF.5 http://www.uen.org/core/displayLinks.do?courseNumber=5150&standardId=71122&objectiveId=71129 5.NF.6

http://www.uen.org/core/displayLinks.do?courseNumber=5150&standardId=71122&objectiveId=71132 5.NF.7 http://www.uen.org/core/displayLinks.do?courseNumber=5150&standardId=71122&objectiveId=71133

Name: _________________________________ Date: _______________

Grade 5 Unit 4 Initial Task: Brian’s Birthday Party

1. A diagram of Brian’s birthday cake is shown below. The shaded area represents the amount

of the whole cake that was eaten during the party.

a) Write a multiplication equation that represents the amount of cake that was eaten and solve

(assume each piece is equal in size).

Equation: ____________________________________________ Solve:

b) Explain how your equation connects to the diagram:

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

c) Brian’s mom is splitting the leftover cake (non- shaded in the diagram) among 6 guests to take home. What fraction of the cake will each of the 6 guests take home? Write and equation AND show your thinking with a diagram:

2. Brian’s mom is deciding between two party rooms at the Party Palace. Room A is 20 feet long and 15-1/2 feet wide. Room B is half as long but the same width. How do the areas of the two rooms compare? Draw a diagram to prove your answer.

3. Brian needs four tables for his party. A diagram of one of the tables is shown below:

L= 5-1/3 feet

W= 3-1/4 feet

a) What is the area of each table? Show your thinking:

b) What is the area of all four tables? Show your thinking:

4. Brian’s dad is deciding between two different “Happy Birthday!” Banners.

Banner #1 is 5 feet long and 6/5 feet wide.

Banner #2 is 5 feet long and 5/6 feet wide.

Without calculating, which banner will have an area greater than 5 square feet? Explain: ___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

5. Brian buys three 2- liter bottles of fruit punch for his party.

a) How many 1/5 liter servings does he have? Explain or show your thinking with a visual

fraction model or an equation

b) If Brian wants each person to have 2 servings of fruit punch, what is the greatest number of

people he can have at his party? Explain or show your thinking

Grade 5 Unit 4 Initial Task Scoring Guide

Brian’s Birthday Party Scoring Guide Points Section

Points

1. 5.NF.4, 5.NF.7

a) Student writes correct equation: ¾ x 5/6 = (any variable)

Student solves correctly with answer of 15/24 or 5/8

b) Student makes a connection between the equation and the diagram.

For example: The cake was cut into 24 equal pieces. Only 3 out of the 4

rows were eaten and only 5 out of the 6 columns were eaten. I found

that 15 pieces out of the 24 were eaten”

c) Student finds fraction of cake leftover = 9/24 or 3/8. Student sets up and

correctly solves 3/8 ÷ 6 = 1/16 AND shows a diagram that represents the

above equation such as:

1/16 1/16 1/16

1/8

1/8

1/8

1/8

1/8 1/16 1/16 1/16

2

2

3

7

2. 5.NF.5a, 5.NF.6

Student correctly states that Room B will be half the area of Room A OR

that Room A will be twice the area of Room B.

Student shows a correct diagram proving the above such as:

20 ft.

15.5 ft

Room A

10 ft

15.5 ft

Room B

10 ft.

1

2

3

3. 5.NF.6

a) Student finds the area of one table by multiplying 5-1/3 x 3-1/4 = 17-1/3

square feet OR shows an appropriate area model with correct

calculations such as:

5 x 3 = 15

¼ x 4 = 1

1/3 x 3 = 1

¼ + 1/12 = 1/3

And: 15 + 1 + 1 + 1/3 = 17-1/3 square feet

2

4

5 1/3

1/3

1/3

1/3

1/4 1/4 1/4 1/4 1/4 1/12

b) Student finds area of four tables 17-1/3 x 4 = 69-1/3 square feet

by multiplying or using other viable method such as adding, partial

products, or drawing a visual model

2

4. 5.NF.5b

Student correctly states that Banner #1 will have an area greater than 5

square feet.

Student correctly explains that the reason why Banner #1 will have an area

greater than 5 is because if you multiply a given number with a fraction

greater than 1, the product will be greater than the given number.

[Conversely, if you multiply a given number with a fraction less than 1, the

product will be less than the given number (as is the case with banner #2)]

1

2

3

5. 5.NF.4, 5.NF.7

a) Student correctly answers there are 30 servings and shows their thinking

with an equation (6 ÷ 1/5 = 30) OR a visual model such as:

1 2 3 4 5 6 (liters)

1/5 liter 1/5 liter 1/5 liter 1/5 liter 1/5 liter 1/5 liter

1/5 liter 1/5 liter 1/5 liter 1/5 liter 1/5 liter 1/5 liter

1/5 liter 1/5 liter 1/5 liter 1/5 liter 1/5 liter 1/5 liter

1/5 liter 1/5 liter 1/5 liter 1/5 liter 1/5 liter 1/5 liter

1/5 liter 1/5 liter 1/5 liter 1/5 liter 1/5 liter 1/5 liter

b) Student correctly proves that if there are 30 servings and each person

will receive 2 servings then, the greatest number of people he can have

at his party will be 30 ÷ 2 = 15 people.

1

1

1

3

Total Points 20 20

Novice Apprentice Practitioner Expert

0 - 7 8 - 12 13 - 17 18 - 20

3

1/4

Grade 5 Unit 4 Final Task: School Bake Sale

1. Jennifer plans to make sugar cookies for the bake sale. She wants to make 36 cookies but is not

sure how to adjust the recipe.

Her grandma says, “Just multiply each measurement by 3” Using Jennifer’s grandma’s method, adjust the recipe for 36 cookies. Show all of your math thinking.

Sugar Cookies

Makes 12 Makes 36

1-1/3 cup flour

1/4 teaspoon baking soda

1/8 teaspoon baking powder

1/3 cup butter, softened

3/4 cup white sugar

1 small egg

1/2 teaspoon vanilla extract

2. Mr. Ross’s class is selling homemade chocolate pudding. If each serving is ¼ cup, how many

servings will they be able to sell if they have 8 cups of pudding? Use a visual fraction model to prove your answer

3. The bake sale will use tables in the school cafeteria. The blue table measures 4 feet by 5-1/4

feet. The green table measures 5-1/4 feet by 8 feet. Without multiplying, what is the relationship between the areas of the blue and green tables? Prove your answer with a diagram or a written explanation.

4. Ms. Carl’s class is selling brownies. The pan of brownies is rectangular and measures 8 inches by 10 inches. a) In a diagram, show how they can cut the pan of brownies into 2 inch by 2 inch squares. How

many brownie squares will they cut?

b) At the end of the bake sale, there are 4 brownies leftover. What fraction of the total brownies did they sell? Prove your thinking with an equation or a visual fraction model.

5. Mrs. Torres’ class is deciding between two different signs to advertise the bake sale.

Sign #1 measures 2 feet by 5/4 feet. Sign #2 measures 2 feet by 4/5 feet. Without multiplying, which sign has the greater area? Explain: ______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

Grade 5 Unit 4 Final Task Scoring Guide

School Bake Sale Scoring Guide Points Section

Points

1. 5.NF.4, 5.NF.6

Student receives 1 point per correct answer and 1 point if they used

multiplication (except the egg). Because these standards specify

multiplication (as opposed to repeated addition), assign ½ credit for an

equation other than multiplication

Makes 12 Makes 36

1-1/3 cup flour 4/3 x 3 = 4 cups

1/4 teaspoon baking soda ¼ x 3 = ¾ teaspoon

1/8 teaspoon baking powder 1/8 x 3 = 3/8 teaspoon

1/3 cup butter, softened 1/3 x 3 = 1 cup

3/4 cup white sugar ¾ x 3 = 9/4 or 2-1/4 cup

1 small egg 1 x 3 = 3 – do not assign/deduct credit here

1/2 teaspoon vanilla extract 1/2 x 3 = 3/2 or 1-1/2 teaspoon

2

2

2

2

2

2

14

2. 5.NF.7

Student correctly answers that there are 32 servings and uses any fraction

model to prove their thinking (can include circle, bar, number line or pictorial

representation of 8 cups with 4 servings in each)

2

2

3. 5.NF.5a

Student correctly explains that the green table’s area is twice that of the blue

table or that the blue table’s area is half the area of the green table based on

the size of the factors. The 5-1/4 remains the same but the length of the

green table is twice the length of the blue table and therefore, the green

table’s area is two times greater OR shows a correct diagram

1

2

3

4. 5.NF.4b

a) Student shows correct diagram and answers correctly that they will cut

20 brownies (each measuring 2 inches x 2 inches) from the 8 inch x 10

inch pan.

1

3

b) Student correctly shows with an equation or a model that the leftover

brownies: 4/20 = 1/5 and therefore the fraction sold: 16/20 = 4/5.

Student may shade in amount sold, etc

2

5. 5.NF.5b

Student correctly states and explains why sign #1 has the larger area. The

width is the same for both signs however, sign #2’s width is less than 1 (4/5)

and sign#1’s width is greater than 1 (5/4), so sign #1 will have a greater area

(greater than 2 square feet)

2

2

Total Points

Novice Apprentice Practitioner Expert

0 - 9 10 - 15 16- 20 21-24