5th Grade Mathematics - Investigationselementarymath.dmschools.org/uploads/1/3/2/2/... · Web viewDeveloping fluency with addition and subtraction of fractions, and developing understanding

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5th Grade Mathematics - Investigations

Unit 5: Multiplication and Division of FractionsTeacher Resource Guide

2012-2013

In Grade 5, instructional time should focus on four critical areas:1. Developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of

division of fractions (limited to unit fractions divided by whole numbers and whole numbers divided by unit fractions);Students apply their understanding of fractions and fraction models (set model, area model, linear model) to represent addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators (¼ + 2/8 = ¼ + ¼). They develop fluency in calculating sums and differences of fractions, and make reasonable estimates of them. Students also use the meaning of fractions and the relationship between multiplication and division to explain why the procedures for multiplying and dividing fractions make sense.

2. Extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations;

Students develop understanding of why division procedures work based on place value and properties of operations. They are fluent with multi-digit addition, subtraction, multiplication, and division. They apply their understandings of models for decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths. Students use the relationship between decimals and fractions, and the relationship between decimals and whole numbers (i.e., a decimal multiplied by anpower of 10 is a whole number) to understand and explain why procedures for multiplying and dividing decimals make sense. They compute products and quotients of decimals to hundredths efficiently and accurately.

3. Developing understanding of volume.Students understand that volume is an attribute of three-dimensional space and can be measured by finding the total number of same-size units of volume required to fill the space without gaps or overlaps. They understand that a 1-unit by 1-unit by 1-unit cube is the standard unit for measuring volume. They decompose three-dimensional shapes and find volumes of right rectangular prisms by viewing them as decomposed into layers of arrays of cubes.

Unit 5: Multiplication and Division of Fractions March 25- May 30 (9 weeks)

5th Grade Mathematics 2012-2013

Unit Time Frame Testing Window

TRIM

ESTE

R 1 1: Multi-Digit Multiplication and Division 7 weeks 8/22 – 10/12 October 12

2: Measurement/Geometry4 weeks 10/15 – 11/9 November 9

TRIM

ESTE

R 2 3: Addition and Subtraction of Fractions

8 weeks 11/12 – 1/18 January 18

4: Decimals 8 weeks 1/22 – 3/14 March 14

TRIM

ESTE

R 3

5: Multiplication and Division of Fractions9 weeks 3/25 – 5/30 May 30

DMPS Wiki: http://dmps-mathematics.wikispaces.com/

5th Grade 2012-2013 Page 2

http://dmps-mathematics.wikispaces.com/




Big Ideas Essential QuestionsThe size of a product depends on the factors. If both factors are greater than one whole, the product will be greater than both factors. If both factors are less than one whole, the product will be less than both factors. If only one factor is greater than one whole, the product will be less than that factor.

When does a multiplication problem have a product greater than the factors or less than the factors?

The size of a quotient depends on the divisor. When a divisor is greater than 1, the quotient is less than the dividend. When a divisor is less than 1, the quotient is greater than the dividend.

When does a division problem have a quotient greater than the dividend or less than the dividend?

Identifier Standards Mathematical Practices

STAN

DARD

S

5.NF.3 Interpret a fraction 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. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

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.

5.NF.6

5.NF.4

5.NF.5

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

Apply and extend previous understandings 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. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (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.

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.

5.NF.7c

5.NF.7

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 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.1

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.

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

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.



Identifier Standards Bloom’s Skills Concepts5.NF.3 Interpret a fraction 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. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

Understand (2)

Apply (3)

Interpret (a fraction as division)

Solve (word problems involving division of whole numbers w/ quotients as fractions and mixed numbers)

fractionmixed numberdivisionnumeratordenominator

5.NF.6

5.NF.4

5.NF.5

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

Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

c. 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. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)

d. 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.

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

Apply (3) Solve (problems involving multiplication of fractions and mixed numbers)

multiplicationequations



STANDARDS

5.NF.7c

5.NF.7a, b

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 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.2

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.

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

Apply (3) Solve (problems involving division of unit fractions by whole numbers & whole numbers by unit fractions)

divisionunit fractionswhole numbers

2 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.



Instructional Strategies for ALL StudentsReal-world context – For students to reach the level of rigor intended for rational number operations in the Iowa Core, they must develop understanding of multiplying and dividing fractions within real-world contexts. In this unit, students make sense of how multiplying and dividing fractions is similar to and different from multiplying and dividing whole numbers.

Area model and Number Line model – For multiplication and division of fractions, the Area model and the Number Line model help students visualize problems and make sense of the multiplication and division process. These models are also helpful in assessing the reasonableness of answers. It is critical for students to work with these models as opposed to only watching teacher demonstrations. Activities that utilize these instructional strategies to support students’ conceptual understanding for multiplication and division of fractions are listed on p. 7-8 of this guide. The activities are also provided on the Wiki.

The following are two examples of using an area model to show multiplying a fraction by a fraction.

Tony tried the chocolate fudge cake first. He liked it so much that he ate 12

of a pan that was 23

full.

Shade 23 of the square. Then shade

12 of the

23 . Tony ate

12 of a pan that was

23 full or

26 of a full pan. (The second drawing shows

13 of the full pan.

13=26 )

a) b)

Next are two examples of using a number line model to show multiplying a fraction by a fraction.

Use the number line to show 13

of 23

. By marking the remainder of the number line into equal partitions, the first number line shows 16

of the whole shaded. The

second number line show 212

of the whole shaded. Both fractions are equivalent.

14

of the entire 23

is marked 14

of each 13

is marked



0 23

1 0 13

23

1

Here is an example of an area model for dividing a whole number by a unit fraction.

Mike is planning to sell small cheese pizzas at the Iowa State Fair. He discovered that it is less expensive to buy large bars of cheese and shred them. He bought five bars

of cheese. How many pizzas can he make if each pizza needs 13

of a bar of cheese?

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

What multiplication number sentence can we use to solve 5 ÷ 13

? (5 x 3 = 15 )

The ratio table also helps explain the pattern.

Finally, here is an example of an area model and a number line model for dividing a unit fraction by a whole number.

All the models show 12 . To find

12 ÷ 4, the

12 is split into four equal-sized parts. Each part

is 14

of the 12

. One part is shaded to show the amount a student gets if 12

is shared

with four people. 12

÷ 4 is like finding 12of 14

. They both equal 18

.

12


Bars of cheese 1 2 3 4 5Number of pizzas 3 6 9 12 15


14

of12

12

12

14

of 12

14

of 12

0 12

1

Routines/Meaningful Distributed Practice



Distributed Practice that is Meaningful and PurposefulPractice is essential to learn mathematics. However, to be effective in improving student achievement, practice must be meaningful, purposeful, and distributed.

Meaningful: Builds on and extends understanding Purposeful: Links to curriculum goals and targets an identified need based on multiple data sources Distributed: Consists of short periods of systematic practice distributed over a long period of time

Routines are an excellent way to achieve the mandate of Meaningful Distributed Practice outlined in the Iowa Core Curriculum.. The skills presented during routines do not necessarily reinforce the lesson concept for that day. Routines may be used to address a need for small increments of exposure to a skill or review of skills already taught. Routine activities may be repeated several days in a row, allowing for a build-up of conceptual understanding, or can be visited and re-visited over a period of time. Routines can be inserted as the schedule allows; in short intervals throughout the day or as a lesson opener or closer. Selection of the routine should be made based on informal teacher observation and formative assessments.

Concepts taught through Meaningful Distributed Practice during Unit 5:

Skill Standard ResourceMulti-digit multiplication 5.NBT.5Whole-number division 5.NBT.6Interpreting remainders 4.OA.3Read, write and compare decimals 4.NBT.3Add, subtract, multiply and divide decimals 4.NBT.7Other skills students need to develop based on teacher observations and formative assessments.

Note: This unit will begin with continuing work fraction equivalence, mixed/improper fractions, and addition/subtraction of fractions. This will strengthen students understanding of these critical concepts and support the work they will do multiplication and division of fractions.



Investigations Resources for Unit 5 – 5th grade multiplication/division fractionsAll pages referenced are found on the WIKI unless otherwise noted.

Lessons Teacher Directions Materials StandardsAddressed

Equivalent Fractions Spend 1 - 2 weeks depending on student needClock Fractions Review Investigations Unit 4

3.1, 3.2, 3.34.NF.1

Fraction Tracks Review Investigations Unit 43.4, 3.5, 3.6

4.NF.1

Missing Number Equivalencies ActivitySlicing Squares Activity

p. 85 – activity 3.15p. 85 – activity 3.16

Additional WorksheetsShips Ahoy! P. 7A Giant Dinosaur p. 8Practice 5-5 p. 39

4.NF.1

Mixed and Improper Fractions Spend up to 1 week depending on student need 4.NF.1Fractional Parts Counting Activity p. 67 4.NF.1Mixed Numbers and Improper Fractions Activity

P. 69 – Activity 3.1Additional WorksheetsHuge, Mysterious Life Form p. 10Practice 5-6 p. 40

4.NF.1

Adding and Subtracting Fractions Spend up to 1 week depending on student needNOTE: Make sure you spend ample time on problems involving addition and subtraction of mixed fractions that require regrouping.

5.NF.15.NF.2

Comparing Fractions to Benchmarks Use this time to extend comparisons of fractions to numbers greater than 1 whole. For example 2 2/3 and 3 7/8. This will help when estimating sums, differences, products and quotients.

2.3 p. 23 - 24 5.NF.15.NF.2

First Estimates This activity requires students to use their knowledge of a benchmark to estimate a sum or difference.

p. 80 – Activity 3.10practice 6-1 p. 45

5.NF.15.NF.2

Addition and Subtraction of Fractions Give students problems in context as much as possible so they have to think about which operation to use.

Additional WorksheetsPractice 6-2 – 6-5Worksheets p. 12 - 20

5.NF.15.NF.2



Investigations Resources for Unit 5 – 5th grade multiplication/division fractionsAll pages referenced are found on the WIKI unless otherwise noted.

Lessons Teacher Directions Materials StandardsAddressed

Multiplication of Fractions This section will take about 2 - 3 weeks. 5.NF.45.MF.6

Wiki: Multiplying a Fraction by a Whole Number Activity (lesson)

PBIT One: Watering a Garden 5.NF.45.MF.6

Wiki: Practice with Multiplying a Fraction by a Whole Number

Some practice problems are provided, but keep practicing this type of problem throughout the rest of the unit.

Multiply in a Fraction by a Whole Number Practice Sheet

5.NF.4, 5.MF.6

Wiki: Multiplying a Fraction by a Fraction Activities (2 lessons)

PBIT Two: Planning A GardenPBIT Three: How Much Cake Was Eaten?

5.NF.45.MF.6

Wiki: Practice with Multiplying a Fraction by a Fraction


Multiplying a Fraction by a Fraction Practice Sheet,Fraction by Fraction and Fraction by Whole Numbers Practice Sheet

5.NF.4, 5.MF.6

Wiki: Multiplying a Whole Number by a Mixed Fraction (lesson)

PBIT Four: Picking Vegetables 5.NF.4, 5.MF.6

Wiki: : Multiplying Mixed Fractions (2 lessons)

PBIT Five: Mixed Fraction ArraysPBIT Six: Flower Gardens

5.NF.45.MF.6

Wiki: Practice with Multiplying a Fraction by a Mixed Fraction Practice Some practice problems are provided, but keep practicing

these types of problems throughout the rest of the unit.

Practice with Multiplying a Fraction by a Mixed Fraction Practice sheet

5.NF.4, 5.MF.6

Wiki: Practice with Multiplying Mixed Fractions

Multiplying Mixed Fractions Practice sheet

5.NF.4, 5.MF.6

Division of Fractions This section will take about 2 weeks. 5.NF.7Wiki: Whole Number Divided by a Unit Fraction (2 lessons)

5.NF.7

Wiki: Practice with Whole Number Divided by Unit Fraction


5.NF.7

Wiki: Dividing a Unit Fraction by a Whole Number

5.NF.7

Wiki: Practice Some practice problems are provided, but keep practicing this type of problem throughout the rest of the unit.

5.NF.7




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5th Grade Mathematics - Investigationselementarymath.dmschools.org/uploads/1/3/2/2/... · Web viewDeveloping fluency with addition and subtraction of fractions, and developing understanding