Grade 7 Mathematics Unit 5 Operations with · PDF fileGrade 7 Mathematics Unit 5 Operations...

Grade 7 Mathematics Curriculum Outcomes 159

Outcomes with Achievement Indicators

Unit 5

Grade 7 Mathematics

Unit 5

Operations with Fractions

Estimated Time: 24 Hours

[C] Communication [PS] Problem Solving [CN] Connections [R] Reasoning

[ME] Mental Mathematics [T] Technology

and Estimation [V] Visualization



Unit 5

Unit 5: Operations with Fractions

Grade 7 Math Curriculum Guide 161

Unit 5 Overview

Introduction

Students will focus on developing skills and understanding the addition and subtraction of fractions. The

big ideas in this unit are:

• Equivalent fractions represent the same quantities.

• The concept of equivalent fractions is very useful when comparing, ordering, simplifying, and

operating with fractions.

• The use of manipulatives such as fraction strips and fraction circles, number lines, and pattern

blocks is an effective way to model the addition and subtraction of fractions. It creates a concrete base for a traditionally difficult concept.

• Addition and subtraction of fractions requires common denominators.

• Estimation strategies for these two operations are based on using benchmarks like 0,

4

3,

2

1,

4

1etc.

Context The students will model, using manipulatives, the addition and subtraction of fractions. They will be

encouraged to informally generalize rules for these operations that are based on their investigations.

Through the use of these investigations, and guidance from the teacher, the students will discover the need

to use common denominators when adding, subtracting, comparing and ordering fractions.

They will discover the algorithm for adding and subtracting fractions. Once again estimation will play an

important role in helping students to decide if their answers are “sensible.” The students will then apply

these algorithms to adding and subtracting mixed numbers.

Why are these concepts important?

Developing a good understanding of adding and subtracting fractions will permit students to:

• Understand real-life situations that require fractions such as;

The clock ("a quarter 'till").

Electricians (gauge/length of wires).

Plumbing (thickness of pipe, diameter of pipe, length of pipe).

Carpenters (thickness/length/width of wood).

Engineers (just math equations). Metal fabrication (length/width/gauge of metal).

Taxes/budgeting (obvious math involved).

Cooking (measurements like HALF a cup...).

In your car (km PER hour, km PER liter).

Paying for things in general (1 penny is 1/100 of a dollar, writing out checks.)

• Be ready to learn and understand future topics in math such as algebra and proportions.

“It isn't that they can't see the solution. It is that they can’t see the problem.”

G. K. Chesterton (1874 – 1936)

Strand: Number



Unit 5

General Outcome: Develop Number Sense

Specific Outcome

It is expected that students will: 7N5. Demonstrate an

understanding of adding

and subtracting positive

fractions and mixed

numbers, with like and

unlike denominators,

concretely, pictorially and

symbolically (limited to

positive sums and

differences).

[C, CN, ME, PS, R, V]

Achievement Indicators

Elaborations: Suggested Learning and Teaching Strategies

Lesson 5.1 in the student text briefly models like fractions

using pattern blocks, clocks and fraction circles. It primarily

demonstrates like denominators, but includes some examples

in which one of the denominators is a simple multiple of the

other. Teachers will need to model several more examples

using these manipulatives in order to ensure student

understanding. Students should also have the opportunity to

model using the manipulatives since they are hands-on

experiences.

Lesson 5.2 uses fraction strips and number lines to support the

same indicators. Students should be able to use the models to

understand fractional equivalents and how they can be useful

when adding fractions and changing them to their simplest

form.

Using the fractions strips and number line masters in the

ProGuide pp. 64–67, students will combine both the fraction

strips and number lines to model sums and to illustrate the

concept of common denominators.

7N5.2 Determine the sum

of two given positive

fractions with like

denominators.

7N5.1 Model addition of

positive fractions, using

concrete representations,

and record symbolically.

7N5.3 Determine a

common denominator for

a given set of positive

fractions.

7N5.4 Simplify a given

positive fraction by

identifying the common

factor between the

numerator and

denominator.

Strand: Number



Unit 5


Suggested Assessment Strategies

Pencil and Paper

Write an addition sentence to represent the total fraction of each

hexagon that is shaded. Use an addition sentence to find the total

value of the shaded hexagons in each case.

A.

B.

C.

Informal Observation

An alternative, but similar activity would be to create cards with

addition sentences and their equivalents in pattern blocks as used in

the Pencil and Paper exercise above. Each student would receive a

card with either the addition sentence, or the pattern block

representation. They mix-up and match-up within the class to find

their partner. Each group must then explain to another group, or to

their class, why they belong together.

Resources/Notes

The national library of

virtual manipulatives

provides an interesting

activity on adding using

common denominators

with various models at

http://nlvm.usu.edu/en/na

v/frames_asid_106_g_3_

t_1.html?from=category_

g_3_t_1.html

Math Makes Sense 7

Lesson 5.1

Lesson 5.2

Unit 5: Operations with

Fractions

TR: ProGuide, pp. 4–6 &

pp. 7–11

Master 5.13, 5.18, 5.27

Master 5.10, 5.11, 5.14,

5.15, 5.16, 5.17, 5.19,

5.28

PM 28, PM 25

CD-ROM Unit 5 Masters

ST: pp. 178–180

ST: pp. 181–185

Practice and HW Book

pp. 106–108

pp. 109–111

Strand: Number



Unit 5


Specific Outcome




fractions and mixed





positive sums and

differences).


(Cont’d)



In the previous lessons, students used models to add using like

denominators. They also modelled unlike denominators when

one denominator was a multiple of the other.

Lesson 5.3 develops the addition algorithm for fractions.

The addition of fractions with unlike denominators that are not

simple multiples of each other will require students to multiply

the numerator and denominator of each fraction by the same

number. Example:

Ideally, students should use the Least Common Multiple

(LCM) of the unlike denominators.

Through the use of benchmarks (close to 1

0, ,12

) developed in

Unit 3, students will estimate the solution and use their

estimate to verify the reasonableness of the answer obtained

using the algorithm.

(This elaboration is continued on the next two page spread…)


positive fractions with


using concrete

representations, and

record symbolically.



fractions with unlike

denominators.

Strand: Number



Unit 5



Pencil and Paper

1. Create three addition sentences that give the same sum as

6 3

12 12+ . You cannot use like denominators in the sentences

you create.

2. Magic square. The sum of each row, column and diagonal in

this magic square must equal 1. Find the missing values.

Magic Square Solution

5

12

7

12

1

3

1

4

1

6

5

12

5

12

7

12

1

3

1

12

1

4

1

4

1

2

3. A tangram is a square puzzle that is divided into seven shapes.

A. Suppose piece A is1

4. What are the values of pieces B, C,

D, E, F and G?

B. What is the sum of A and B?

C. If you subtract D from the whole puzzle, what value

remains?

D. Which two tangram pieces add up to the value of C?

E. Invent a problem on your own and solve it.

Resources/Notes

Math Makes Sense 7

Lesson 5.3


Fractions

TR: ProGuide, pp. 12–15

Master 5.14, 5.15, 5.16,

5.17, 5.20, 5.29

PM 27


ST: pp. 186–189


pp. 112–114

Strand: Number



Unit 5


Specific Outcome




fractions and mixed





positive sums and

differences).


(Cont’d)



Here is another example of adding fractions with unlike

denominators.

Find the sum of the fractions:

6

1

4

3+

Students should think 4

3is a little bit more than a half and

6

1 is

less than a half so the answer should be close to 1. Then they

can use the previous algorithm to calculate:

12

11

12

2

12

9

2

2

6

1

3

3

4

3

6

1

4

3

=

+=

×+×=

+

Finally, they should look at their answer and ask themselves if

12

11is reasonable based on their estimate of 1.

Note: When a common denominator must be found, the

common denominator that is chosen should be the lowest

common denominator. Simply multiplying the denominators

of the fractions being adding or subtracted will not guarantee a

lowest common denominator. The lowest common

denominator for 6

1

4

3+ is 12, not 24.




using concrete

representations, and

record symbolically.

(continued)



fractions with unlike

denominators.

(continued)

Strand: Number



Unit 5



Performance

Use pattern blocks to create a design on triangular grid paper

(Program Master 27). Then use fraction addition to name the

design. Consider the flower design illustrated in Appendix 5-A. It is

possible to use several different addition sentences to name the

same design.

Journal

1. If a problem required you to add fourths and thirds, is it

possible for the sum to be sixths? Why or why not? You may

use an example or a diagram to help you explain your answer.

2. If a problem required you to add fourths and thirds, is it

possible for the sum to be sevenths? Why or why not? You may

use an example or a diagram to help you explain your answer.

Interview

A classmate missed yesterday’s class. When solving a problem

today he suggested that5 5 10

6 8 14+ = . How would you convince him

that this is not a reasonable solution?

Game/Activity

Refer to Appendix 5-B for the Connect Three game.

Resources/Notes

Math Makes Sense 7

Lesson 5.3

(continued)

Strand: Number



Unit 5


Specific Outcome




fractions and mixed





positive sums and

differences).


(Cont’d)



Lesson 5.4 of the student text begins with subtraction

involving unlike denominators using pattern blocks. Students

will learn that addition and subtraction of fractions with unlike

denominators uses the same algorithm. Teachers may wish to

model several examples using fraction circles or fraction

strips.

For example: 4 1

5 5−

In this case, students must understand that they are simply

removing one part of a set of equivalent quantities. This can be

demonstrated by modelling 4

5 using fraction strips or fraction

circles and removing one portion representing1

5. The answer

is the remaining portion of3

5.

The subtraction of fractions with unlike denominators that are

not simple multiples of each other will require students to

multiply the numerator and denominator of each fraction by

the same number. This is identical to the algorithm used for

addition.

Ideally, students should use the Least Common Multiple

(LCM) of the unlike denominators.

Through the use of benchmarks (close to 1

0, ,12

) developed in

Unit 3, students will estimate the solution and use their

estimate to verify the reasonableness of the answer obtained

using the algorithm.


7N5.7 Model subtraction

of positive fractions, using



7N5.8 Determine the

difference of two given

positive fractions with like

denominators.

7N5.9 Determine the



unlike denominators.

Strand: Number



Unit 5



Observation

Ask students to use concrete materials or diagrams to show why the

following is an incorrect procedure.

3 1 3 1 2 1

8 4 8 4 4 2

−− = = =

−

Informal Observation

Students can play the game Tic-Tac-Toe Fractions. A really useful

game for adding and subtracting fractions. See ProGuide (Page V)

and Master 5.8a, 5.8b and 5.8c.

Resources/Notes

Math Makes Sense 7

Lesson 5.4

Lesson 5.5


Fractions


& pp. 21–24

Master 5.12, 5.14, 5.15,

5.16, 5.17, 5.21, 5.30

Master 5.14, 5.15, 5.16,

5.17, 5.22, 5.31


ST: pp. 191–194

ST: pp. 195–198


pp. 115–117

pp. 118–120

Strand: Number



Unit 5


Specific Outcome




fractions and mixed





positive sums and

differences).


(Cont’d)



Find the difference of the fractions:

3

1

9

4−

Students should think 9

4is a little bit less than a half and

3

1 is

a little less than a half. The difference between them should

therefore be almost 0 or just a little bit more than 0.

9

1

9

34

9

3

9

4

3

3

3

1

9

4

3

1

9

4

=

−=

−=

×−=

−

Finally, they should look at their answer and ask themselves if

9

1 is reasonable based on their estimate of something a little

bit more than 0.

7N5.7 Model subtraction

of positive fractions, using



(continued)

7N5.8 Determine the


positive fractions with like

denominators.

(continued)

7N5.9 Determine the




(continued)

Strand: Number



Unit 5



Resources/Notes

Math Makes Sense 7

Lesson 5.4

Lesson 5.5

(continued)

Strand: Number



Unit 5


Specific Outcome

It is expected that students will:

7N5. Demonstrate an



fractions and mixed





positive sums and

differences).


(Cont’d)



Now that the models for addition and subtraction have been

studied separately by the students, the same models and skills

can now be used in the study of mixed fractions.

Lessons 5.6 and 5.7 explore the subtraction of mixed numbers

using fraction circles, number lines and fraction strips. Lesson

5.7 also introduces Cuisenaire rods as a model for subtracting

mixed fractions. Teachers may consult the link for use of this

model in the resource section of this guide.

When adding and subtracting mixed fractions students may

approach the problem in different ways. They may choose to

keep the mixed fraction form or, they may change the mixed

fractions to improper fractions.

For addition:

Mixed Fraction Form Improper Fraction Form

18

13

18

112

18

192

18

151

18

41

3

3

6

51

2

2

9

21

6

51

9

21

=

=

=

+=

×+×=

+

and

18

13

...54,36,1818

55

18

33

18

22

3

3

6

11

2

2

9

11

6

11

9

11

6

51

9

21

=

=

+=

×+×=

+=

+


7N5.11 Determine the

sum or difference of two

mixed numbers with like

denominators.

7N5.10 Model addition

and subtraction of mixed

numbers with like

denominators, using





numbers with unlike

denominators, using




sum and difference of two

mixed numbers with


Strand: Number



Unit 5



Interview

Consider the following two problems: 3

14

− and 3

110

−

Without calculating, explain how you could determine which

answer would be greater.

Journal

Describe at least two ways you can calculate1 5

4 22 6

− .

Resources/Notes

An introduction to

Cuisenaire rods and their

use in the study of

fractions can be found at

http://teachertech.rice.ed

u/Participants/silha/Lesso

ns/cuisen2.html

Math Makes Sense 7

Lesson 5.6

Lesson 5.7


Fractions


& pp. 30–34

Master 5.13, 5.14, 5.15,

5.16, 5.17, 5.23, 5.32

Master 5.13, 5.14, 5.15,

5.16, 5.17, 5.24, 5.33

PM 28


ST: pp. 199–203

ST: pp. 204–208


pp. 121–122

pp. 123–124

Strand: Number



Unit 5


Specific Outcome

It is expected that students will:

7N5. Demonstrate an



fractions and mixed





positive sums and

differences).


(Cont’d)


(All Cont’d)


For subtraction:

Mixed Fraction Form Improper Fraction Form

21

141

21

122

7

7

3

21

3

3

7

42

3

21

7

42

−=

×−×=

−

Students will be challenged

by 12–14 and therefore must

think about regrouping.

Students should think:

21

141

21

12

21

211 −andand

which will allow them to

calculate:

21

19

21

141

21

331

=

−

21

19

21

35

21

54

7

7

3

5

3

3

7

18

3

5

7

18

3

21

7

42

=

−=

×−×=

−=

−



mixed numbers with like

denominators.



numbers with like

denominators, using





numbers with unlike

denominators, using




sum and difference of two

mixed numbers with


Strand: Number



Unit 5



Resources/Notes

An introduction to

Cuisenaire rods and their

use in the study of

fractions can be found at

http://teachertech.rice.ed

u/Participants/silha/Lesso

ns/cuisen2.html

Math Makes Sense 7

Lesson 5.6

Lesson 5.7

(continued)

Strand: Number



Unit 5


Specific Outcome




fractions and mixed





positive sums and

differences).


(Cont’d)



Throughout the sections on adding and subtracting fractions, it

is necessary for students to simplify their answers. Simplified

answers may be proper fractions, improper fractions or mixed

numbers in simplest form depending on the context of the

problem.

Example: Kyra is making cookies. She has 1

24

bags of

chocolate chips. She adds 2

13

of these bags to her cookie

dough.

a) What fraction of the total amount of chocolate chips is left?

b) Kyra then decides to add 11

12 bags of butterscotch chips to

the dough as well. How many bags of chips does Kyra use in

total to bake her cookies?

For part a), students should think “1

24

bags” is a little more

than two bags. Kyra then uses 2

13

bags which is a little less

than two bags. Therefore she has two little bits or about half a

bag left over.

Then they calculate:

12

7

12

20

12

27

4

4

3

5

3

3

4

9

3

5

4

9

3

21

4

12

=

−=

×−×=

−=

−

Kyra has 7

12of a bag of

chocolate chips left.

Students must reflect

upon their answer to

determine if it is

reasonable.

In this case, seven

twelfths is very close to a

half.


7N5.14 Simplify the

solution to a given

problem involving the


positive fractions or

mixed numbers.

7N5.15 Solve a given


addition or subtraction of


mixed numbers, and

determine if the solution is

reasonable.

Strand: Number



Unit 5



Journal

Is it possible to find two mixed numbers which add together to form

a whole number?

Explain your answer and, if possible, give an example to support

your explanation.

Pencil and Paper

1. Andrew plays guitar in a rock band. For a song that is 36

measures long he plays for 1

42

measures, rests for 3

88

measures, plays for another 16 measures, rests for 1

24

measures

and plays for the last section. How many measures are in the

last section?

2. This week, Mark practised piano for 1

32

h, played soccer for

16

4 h, and talked on the phone for

14

3 h.

A. How many hours did Mark spend practising piano and

playing soccer?

B. Hour many more hours did Mark spend playing soccer

than talking on the phone?

Resources/Notes

Math Makes Sense 7

Lesson 5.6

Lesson 5.7

(continued)

Strand: Number



Unit 5


Specific Outcome




fractions and mixed





positive sums and

differences).


(Cont’d)



For part b) students should think 1

24

bags is a little more than

two bags and 11

12is almost one full bag, but not quite.

Therefore Kyra uses a little more than 3 bags of chips in total.

Then they calculate:

6

13

6

112

6

72

2

2

12

142

12

142

12

11

12

32

12

11

3

3

4

12

12

11

4

12

=

=

=

÷=

=

+=

+×=

+

and

Kyra used 1

36

bags of chips

in total.

Note that the final answer

must be simplified.

Students must reflect upon

their answer to determine if

it is reasonable.

In this case, the answer is

very close to the estimate.

7N5.14 Simplify the

solution to a given




mixed numbers.

(continued)

7N5.15 Solve a given


addition or subtraction of


mixed numbers, and

determine if the solution is

reasonable.

(continued)

Strand: Number



Unit 5



Resources/Notes

Math Makes Sense 7

Lesson 5.6

Lesson 5.7

(continued)

Strand: Number



Unit 5

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