Grade 5 - Home Page - Newark Public Schools · Develop fluency in calculating sums and differences...

$: Grade 5 - Home Page - Newark Public Schools · Develop fluency in calculating sums and differences of fractions, and make reasonable estimates of them. Also, use the meaning of ...$
Grade 5 Number and Operations - Fractions

5.NF.4

2012 COMMON CORE STATE STANDARDS ALIGNED MODULES

THE NEWARK PUBLIC SCHOOLS THE OFFICE OF MATHEMATICS

THE NEWARK PUBLIC SCHOOLS THE OFFICE OF MATHEMATICS

2012 COMMON CORE STATE STANDARDS ALIGNED MODULES

Page 2 of 35

Essential Questions:

What happens to a fraction when you multiply it

by a whole number?

What do the values of the numerator and

denominator tell you about the value of a

fraction?

Prerequisites: Whole Numbers

Fractions

Fraction models

Addition

Subtraction

Factors

Multiplication

Division

Goal:

Students will 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. Also,

students will 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.

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Number and Operation – Fractions 5.NF.4

Apply and extend previous understandings of multiplication to

multiply a fraction or whole number by a fraction.

Lesson 1 5.NF.4.a Multiplying whole numbers by fractions

Lesson 2 5.NF.4.a Multiplying fractions by

fractions

Lesson 3 5.NF.4.b Tiling to find fractional area

Lesson 4 5.NF.4.b Multiplying to find

fractional area

Lesson 5 5.NF.4.a-b Golden Problem

Lesson Structure: Introductory Task Prerequisite Skills Focus Questions Guided Practice

Homework Journal Question

Embedded Mathematical Practices MP.1 Make sense of problems and persevere in solving

them

MP.2 Reason abstractly and quantitatively

MP.3 Construct viable arguments and critique the

reasoning of others

MP.4 Model with mathematics

MP.5 Use appropriate tools strategically

MP.6 Attend to precision

MP.7 Look for and make use of structure

MP.8 Look for and express regularity in

repeated reasoning.

Page 3 of 35

Understand Multiplication

Distributive Property

The Distributive Property states that when you multiply the sum of two or more addends by a factor, the product is the

same as if you multiplied each addend by the factor and then added the partial products. The Distributive Property is

illustrated below graphically, arithmetically, and algebraically. At this time, students do not need to know the algebraic

explanation of the Distributive Property.

To find the total number of squares, you can multiply 3 × (2 + 6) or you can add (3 × 2) and (3 × 6).

arithmetically

3 × (2 + 6) = (3 × 2) + (3 × 6)

3 × 8 = 6 +18

24 = 24

algebraically

a × (b + c) = (a × b) + (a × c)

Multiplication by a One-Digit Number

One way to multiply a number by a one-digit number is to multiply the value of each digit by that one-digit number and

then find the sum of the partial products. The traditional multiplication algorithm for multiplying by a one-digit number has

the product written in place with necessary regrouping recorded above the number being multiplied.

Examples:

The above example shows how the Distributive Property applies to the multiplication algorithm. 546 × 7 = (500 × 7) + (40

× 7) + (6 × 7) Emphasize that the traditional algorithm starts by multiplying in the ones place and that makes recording

and regrouping of the product easier. Students sometimes find it difficult to multiply numbers with internal zeros, such as

302 or 10,809, and may need extra practice with such examples.

To multiply money amounts by a one-digit number, students multiply as if the numbers were whole numbers. They place

the decimal point so the answer is given in dollars and cents.

Multiplication Patterns

Patterns can be used when multiplying multiples of 10.

8 × 7 = 56

80 × 7 = 560

Page 4 of 35

800 × 7 = 5,600

8,000 × 7 = 56,000

80 × 70 = 5,600

800 × 700 = 560,000

Think: (8 × 7) × 10

Think: (8 × 7) × 100

Think: (8 × 7) × 1,000

Think: (8 × 7) × (10 × 10)

Think: (8 × 7) × (100 × 100)

In each example, the number of zeros in the product is the same as the sum of the number of zeros in each factor.

Multiplication by a Two-Digit Number

To solve problems such as 392 × 50, students can use patterns and what they know about multiplication by 1-digit

numbers to find (392 × 5) × 10.

The algorithm for multiplying by a two-digit number simply extends the algorithm for multiplying by a one-digit number.

The Distributive Property can be used to show the multiplication.

Estimation can be used to check that the answer to a multiplication problem is reasonable. Students round each factor to

a multiple of 10 that has only one nonzero digit. Then they use mental math to recall the basic fact product and patterns to

determine the correct number of zeros in the estimate.

Page 5 of 35

Understand Fractions

Fractions

Fractions are numbers that are needed to solve certain kinds of division problems. Much as the subtraction problem

3 − 5 = −2

creates a need for numbers that are not positive, certain division problems create a need for numbers that are not

integers. For example, fractions allow the solution to 17 ÷ 3 to be written as

17 ÷ 3 = .

When a and b are integers and b ≠ 0, then the solution to the division problem a ÷ b can be expressed as a fraction, .

At this grade level, students should learn to identify fractions with models that convey their properties. Proper fractions

can be modeled in terms of a part of a whole. The whole may be a group consisting of n objects where part of the group

consists of k objects and k < n. The fraction can be modeled as follows.

Equivalently, the whole may consist of a region that is divided into n congruent parts, k of which belong to a subregion.

For example, the fraction can be identified as the shaded part of the region below.

A unit fraction is a fraction with a numerator of 1 (for example, , , , ). The definition of a unit fraction, , is to take one

unit and divide it into n equal parts. One of these smaller parts is the amount represented by the unit fraction. On the

number line, the unit fraction represents the length of a segment when a unit interval on the number line is divided into n

equal segments. The point located to the right of 0 on the number line at a distance from 0 will be .

The fraction can represent the quotient of m and n, or m ÷ n. If the fraction is defined in terms of the unit fraction ,

the fraction means m unit fractions . In terms of distance along the number line, the fraction means the length of m

abutting segments each of length . The point is located to the right of 0 at a distance m × from 0. The numerator of

the fraction tells how many segments. The denominator tells the size of each segment.

Page 6 of 35

A straightforward way to show that fractions represent a solution to a division problem is by using equivalent fractions.

What is 35 ÷ 7? It is 5 because 35 equals 7 × 5. What is 5 ÷ 7? This is more difficult because 5 is not a multiple of 7.

However, 5 = = 5 × × 7 = , and equals 35 unit fractions of . Just as 35 divided by 7 is 5, 35 unit fractions of

divided by 7 is 5 unit fractions of . So 5 ÷ 7 = .

Finding a Fractional Part of a Number

The word of is often used to pose problems involving the multiplication of a whole number by a fraction. At this level,

students have not yet learned to multiply fractions. The problem of finding of 6 can be modeled in terms of a group of 6

objects that has been separated into 3 smaller groups, each of which has 2 objects.

Equivalent Fractions

Two fractions and are equivalent if there exists a number m such that m × × b = . For example, the fact that 2 ×

× 4 = implies that is equivalent to .

Geometrically, this concept can be conveyed in terms of a picture in which there are two ways of representing the same

part of the whole. The fact that is equivalent to can be shown as follows.

Because equivalent fractions represent the same number, they are referred to as equal.

A fraction is in simplest form if the numerators and denominators are as small as possible. A more formal way of stating

this is to say that in a simplest form fraction, the numerator and denominator have no common factors other than 1.

Page 7 of 35

Teaching Tips

Fractions TeachingTip1

Focus on the understanding of fractions and fraction models to represent the addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators. Develop fluency in calculating sums and differences of fractions, and make reasonable estimates of them. Also, use the meaning of fractions, of multiplication and division, and the relationship between multiplication and division to understand and explain why the procedures for multiplying and dividing fractions make sense. (Note: this is limited to the case of dividing unit fractions by whole numbers and whole numbers by unit fractions.)

Division TeachingTip2

Develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of operations. Students use the relationship between decimals and fractions, as well as the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number), to understand and explain why the procedures for multiplying and dividing finite decimals make sense. They compute products and quotients of decimals to hundredths efficiently and accurately.

What do I Focus On? TeachingTip3

Instructional time should focus on three critical areas: (1) developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions); (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; and (3) developing understanding of volume.

Page 8 of 35

Tim really needs to start exercising. So, he plans to run 3/5 of a mile every weekday. If he keeps this schedule,

how far will Tim run in one week? Explain your answer with words, pictures, or mathematical calculations.

Introductory Task Guided Practice Homework Assessment

Focus Questions

Journal Question

Why does a whole number get

smaller when multiplied by a

fraction? Draw a picture to

explain your answer.

Question 1: What happens to a fraction when it is multiplied?

Question 2: How can we compare fractions?

Students will 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. MP: Make sense of problems

and persevere in solving them. Reason abstractly and

quantitatively. Model with mathematics. Use appropriate tools

strategically. Attend to precision. Look for and make use of

structure

Numbers and Operation – Fractions 5.NF.3, 5.MP.1, 5.OMP.2, 5.MP.3,

5.MP.4, 5.MP.5, 5.MP.6, 5.MP.7

Introductory Task

Lesson 1

Page 9 of 35

1. Isabel had 6 feet of wrapping paper. She used 3/5 of the paper to wrap some presents. How much does

she have left?

2. Kelly had 12 pencils in her pencil case. She lent out 2/3 of them to her friends. How many pencils did

she let her friends borrow?



5.MP.4, 5.MP.5, 5.MP.6, 5.MP.7

Lesson 1: Guided Practice

Teachers model with students.







structure

Page 10 of 35

3. Manny’s mom gave him $20 to spend on lunch for the week. He used 1/8 of it on Monday. How much

did he have left for the rest of the week?

4. Sarah is reading a 150 page book. She read 4/5 of the books already? How many pages did she read so

far?

5. Nicole baked 24 cupcakes. Her friends came over and ate 5/6 of them. How many does she have left?

Page 11 of 35

6. 3/4 x 2 =

7. 7/8 x 9 =

8. 1/2 x 5 =

9. 4/5 x 3 =

10. 7/9 x 3 =

11. 2/3 x 5 =

Page 12 of 35

1. Isabel had 12 feet of wrapping paper. She used 2/3 of the paper to wrap some presents. How much does

she have left?

2. Kelly had 48 pencils in her pencil case. She lent out 2/3 of them to her friends. How many pencils did

she let her friends borrow?


Name____________________________ Date_____________________


5.MP.4, 5.MP.5, 5.MP.6, 5.MP.7

Lesson 1: Homework

Students practice skills

at home.







structure.

Page 13 of 35

3. Manny’s mom gave him $20 to spend on lunch for the week. He used 1/3 of it on Monday. How

much did he have left for the rest of the week?

4. Sarah is reading a 100 page book. She read 3/5 of the books already? How many pages did she read

so far?

5. Nicole baked 20 cupcakes. Her friends came over and ate 3/4 of them. How many does she have left?

Page 14 of 35

6. 3/4 x 6 =

7. 7/8 x 12 =

8. 1/2 x 8 =

9. 4/5 x 9 =

10. 7/9 x 4 =

11. 2/3 x 10 =

Page 15 of 35

Mr. Oliveira posed a Math question to his class. It stated, “Three-fourths of the class are boys. Two-thirds of

the boys are wearing tennis shoes. What fraction of the class are boys with tennis shoes?” Solve the problem

and provide an explanation that would help a classmate understand your thinking.



5.MP.4, 5.MP.5, 5.MP.6, 5.MP.7

Lesson 2

Introductory Task

Students will 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. MP: Make sense of problems and persevere

in solving them. Reason abstractly and quantitatively. Model with

mathematics. Use appropriate tools strategically. Attend to

precision. Look for and make use of structure.

Journal Question

In your own words, describe

what happens when you multiply

a fraction by a fraction. Use

pictures to illustrate your point.

Focus Questions

Question 1: Why is the result of a fraction multiplied by

a fraction smaller than the original numbers?

Question 2: What does ½ of ¾ represent?

Page 16 of 35

1. 1/2 of the monkeys in the zoo are female. 1/3 of them are pregnant. What fraction of the monkeys in

the zoo are pregnant?

2. Manny has a board that is 7/8 of a ft. long. He cut it in half. What is the length of the two sections?



5.MP.4, 5.MP.5, 5.MP.6, 5.MP.7


Teachers model with students.




in solving them. Reason abstractly and quantitatively. Model

with mathematics. Use appropriate tools strategically. Attend to


Page 17 of 35

3. Kelly has ¼ of a gallon of milk left in her refrigerator. She drank ¼ of it. What fraction of a gallon of

milk is left in the refrigerator?

4. Stewart used 3/6 of a can of oil to fix a squeaky hinge. He used ½ of what was left over to oil his bike.

How much oil does he have left in the can?

5. Nicole baked a lot of cupcakes, but her daughter ate half. Her friends came over and ate 3/4 of what

was left. What fraction of cupcakes does she have left?

Page 18 of 35

6. 3/4 x 1/2 =

7. 7/8 x 4/5 =

8. 1/2 x 1/3 =

9. 4/5 x 7/9 =

10. 7/9 x 1/4 =

11. 2/3 x 7/10 =

Page 19 of 35

Name: ____________________________ Date: _______________________

1. 3/4 of the turtles in the aquarium are female. 1/3 of them are pregnant. What fraction of the turtles in

the aquarium are pregnant?

2. Manny has a board that is 11/12 of a ft. long. He cut it in half. What is the length of the two sections?


Numbers and Operation – Fractions 5.NF.3, 5.MP.1, 5.OMP.2, 5.MP.3, 5.MP.4,

5.MP.5, 5.MP.6, 5.MP.7

Lesson 2: Homework


at home.




in solving them. Reason abstractly and quantitatively. Model with

mathematics. Use appropriate tools strategically. Attend to


Page 20 of 35

3. Kelly has 1/2 of a gallon of milk left in her refrigerator. She drank 3/4 of it. What fraction of a gallon

of milk is left in the refrigerator?

4. Stewart has a1/2 a bottle of soda. He drank 1/3 of it. What fraction of the bottle of soda does he have

left?

5. Bill has 5/7 of a candy bar. He gave his friend ½ of it. What fraction of the candy bar does he still

have left?

Page 21 of 35

6. 1/4 x 1/2 =

7. 9/11 x 4/5 =

8. 2/5 x 1/3 =

9. 6/7 x 7/9 =

10. 3/4 x 1/4 =

11. 6/7 x 7/10 =

Page 22 of 35

A gardener is going to plant tomatoes and other vegetables in his backyard. The gardener will only use 2/3 of

his backyard for planting and only 1/4 of that area will be for the tomatoes. What fractional part of the

backyard will be used to plant tomatoes?


Focus Questions

Journal Question

If the area of a rectangle is

equal to base × height, then

what would the area be for the

following rectangle?

b =2/3 ft. h = 7 ft

Question 1: What does ½ of 7 look like?

Question 2: How can you calculate are if a measurement is a

fraction?


5.MP.4, 5.MP.5, 5.MP.6, 5.MP.7

Lesson 3

Introductory Task

Students will 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.MP: Make sense of

problems and persevere in solving them. Reason abstractly and


strategically. Attend to precision. Look for and make use of structure.

Page 23 of 35

1. 2/3 x 3/5 =

2. 4/5 x 3/7 =



5.MP.4, 5.MP.5, 5.MP.6, 5.MP.7


Teachers model with

students.


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.MP: Make sense of




Use the grid to find the solution.

Page 24 of 35

3. 1/2 x 3/4 =

4. 2/6 x 1/4 =

5. 1/5 x 1/2 =

6. 3/4 x 2/5 =

Page 25 of 35

7. 1/2 x 1/4 =

8. 5/6 x 3/4 =

9. 2/5 x 1/2 =

10.. 3/4 x 4/5 =

Page 26 of 35

Name _______________________ Date __________________

1. 2/3 x 4/5 =

2. 3/5 x 2/5 =



5.MP.4, 5.MP.5, 5.MP.6, 5.MP.7

Lesson 3: Homework


at home.









Page 27 of 35

3. 1/2 x 9/10 =

4. 5/6 x 3/4 =

5. 3/5 x 2/3 =

6. 3/4 x 1/5 =

Page 28 of 35

7. 1/2 x 1/4 =

8. 1/6 x 3/4 =

9. 4/5 x 1/2 =

10. 3/4 x 4/9 =

Page 29 of 35

Joel went to the deli counter of the supermarket and ordered the following:

½ lb of ham

¼ lb of swiss cheese

¼ lb of cheddar cheese

½ lb of turkey

He used exactly 1/3 of each of the ingredients to make sandwiches for his brother and himself. How much of

each item was left after he made the sandwiches?



5.MP.4, 5.MP.5, 5.MP.6, 5.MP.7

Lesson 4

Introductory Task



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.MP:

Make sense of problems and persevere in solving them. Reason

abstractly and quantitatively. Model with mathematics. Use

appropriate tools strategically. Attend to precision. Look for and make

use of structure

Focus Questions

Journal Question

What happens to a fraction that

is multiplied by another

fraction? Explain why.

Question 1: How can multiplication help with division?

Question 2: What does multiplying a fraction really

mean?

Page 30 of 35

Multiply.

1.

1/3 x 2/4 =

2.

2/3 x 5/6 =

3.

2/5 x 3/10 =

4.

6/7 x 1/8 =

5.

2/5 x 2/3 =

6.

6/11 x 3/8 =



Teachers model with

students.


5.MP.4, 5.MP.5, 5.MP.6, 5.MP.7


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.MP: Make sense of




Page 31 of 35

7.

4/5 x 1/6 =

8.

8/9 x 3/10 =

9.

2/3 x 1/3 =

10.

¾ x ¼ =

11.

5/7 x ½ =

12.

1/6 x ¾ =

13.

6/9 x 1/12 =

14.

2/3 x 3/12 =

Page 32 of 35

Name _______________________ Date __________________

1.

1/3 x 2/9 =

2.

2/3 x 5/7 =

3.

2/9 x 3/10 =

4.

6/7 x 1/3 =

5.

4/5 x 2/3 =

6.

6/11 x 3/15 =


Lesson 4: Homework


at home.


5.MP.4, 5.MP.5, 5.MP.6, 5.MP.7









Page 33 of 35

7.

4/5 x 1/16 =

8.

8/9 x 3/4 =

9.

1/3 x 1/3 =

10.

¾ x ¾ =

11.

6/7 x ½ =

12.

5/6 x ¾ =

13.

6/10 x 1/12 =

14.

2/7 x 3/12 =

Page 34 of 35

Parent Volunteers Mrs. Smith and Mrs. Jones both volunteer in their children’s classrooms. Mrs. Smith volunteers every 3rd

school day for 1/3 of a day. Mrs. Jones volunteers every 5th day of school for 1/2 of a day. In a given month,

which parent spends more time volunteering?!



5.MP.4, 5.MP.5, 5.MP.6, 5.MP.7

Lesson 5

Golden Problem

Students will 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.Students will

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.MP:

Make sense of problems and persevere in solving them. Reason abstractly and

quantitatively. Model with mathematics. Use appropriate tools strategically. Attend

to precision. Look for and make use of structure.

Focus Questions

Journal Question

Why do fractions play an important

role in our daily lives? Provide at

least three examples of how you use

math outside of school.

Question 1: What strategies can be used to answer a

multiplication problem involving fractions?

Question 2: How do you know an answer is reasonable?

Question 2: What information do we know?

Page 35 of 35

LESSON 5 RUBRIC

GOLDEN PROBLEM

Score Description

3

M T W Th F

x

x

x

x

This student determines that for a given month (assuming a 4 week

month), that both Mrs. Smith (every 3rd

day) and Mrs. Jones (every 5th

day) both work the same amount of time, which is a total of 2 days each.

All work is shown, and the student’s approach and reasoning is

explained. Math representations are labeled, clear and accurate.

2 The student will achieve a correct solution. However, the student’s

approach and reasoning may be incomplete or contain major errors

leading the scorer to make inferences. Math language will be used

throughout, and math representations will be clear and accurate.

1 The student shows some understanding of the task. The student may be

able to determine the number of days each parent worked but then may

not be able to calculate the time. The student may also achieve an

incorrect solution due to a computation mistake. Some math language

will be used, and representations will be attempted.

0 Does not address task, unresponsive, unrelated or inappropriate.

= Every 3rd day. = Every 5th day. x

6 days occur every 3rd day and 4 days occur

every 5th day, therefore….

and

Grade 5 - Home Page - Newark Public Schools · Develop fluency in calculating sums and differences...

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Transcript of Grade 5 - Home Page - Newark Public Schools · Develop fluency in calculating sums and differences...