Operations with Fractions Suggested Time: 4 · PDF file82 GRADE 8 MATHEMATICS DRAFT CURRICULUM...

Operations with Fractions

Suggested Time: 4 Weeks

GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE82

OPERATIONS WITH FRACTIONS

Unit Overview

Focus and Context

Math Connects

In this unit, students will apply prior knowledge of fractions and whole

number operations to multiply and divide positive fractions and mixed

numbers concretely, pictorially and symbolically. Work with equivalent

fractions was developed in Grade 5, relating improper fractions and

mixed numbers was developed in Grade 6, and fraction addition,

subtraction and comparison was developed in Grade 7. This will now

be extended to include multiplication and division.

Multiplying fractions by whole numbers will fi rst be presented as

repeated addition. Building from this, along with extensive work with

concrete representations such as fraction strips, pattern blocks, number

lines and area models, students will generalize a rule for multiplying

fractions. Following this, the concept of grouping, modelling on a

number line, and the idea of inverse operations will allow students to

generalize a rule for dividing fractions. Estimating with benchmarks of

zero, one-half, and one whole is encouraged throughout the unit to help

students determine the reasonableness of answers. Finally, students will

consolidate the four operations with fractions by applying the order of

operations.

Work with fractions allows students to build a greater facility for

working with numbers. A solid understanding of fractions is essential

for future work with rational expressions. Fractions are also a necessary

component of the foundation of algebra and trigonometry.

Fractions are used everyday by doctors, nurses, mechanics and stock

brokers, to name just a few. The knowledge of multiplying and dividing

fractions will often be used in daily life. Whether purchasing fl oor

covering or material for four bridesmaids’ dresses, modifying recipes,

or fi guring out the amount and size of lumber for a particular project,

there are many activities that require multiplication and division of

fractions.

GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE 83


Process Standards

Key

Curriculum

Outcomes

STRAND OUTCOME

PROCESS STANDARDS

Number

Demonstrate an understanding of multiplying and dividing positive fractions and mixed numbers, concretely, pictorially and symbolically. [8N6]

C, CN, ME, PS

[C] Communication [PS] Problem Solving [CN] Connections [R] Reasoning [ME] Mental Mathematics [T] Technology and Estimation [V] Visualization

84 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE

Outcomes


Elaborations—Strategies for Learning and Teaching

Students will be expected to

Strand: Number

8N6 Demonstrate an

understanding of multiplying

and dividing positive fractions

and mixed numbers, concretely, pictorially and symbolically.

[C, CN, ME, PS]

Achievement Indicator:

8N6.1 Model multiplication of a positive fraction by a whole number concretely or pictorially and record the process.

Multiplication and division of fractions is similar to multiplication

and division of whole numbers, even though the algorithms differ. It is

important for students to realize that the meaning of the operation has

not changed just because they are now working with fractions.

In Grade 7, students used models and an algorithm to add and subtract

positive fractions. Benchmarks were used for estimation, and extensive

work was done on equivalency, ordering and reducing to simplifi ed

form.

Research indicates that the teaching of fractions through memorizing

rules has signifi cant dangers; the rules do not help students think in any

way about the meanings of the operations or why they work and the

mastery observed in the short term is often quickly lost (Van de Walle

2001, p.228).

Exploring operations with fractions through the use of models such as

number lines, the area model, counters, fraction circles and strips helps

solidify understanding of such concepts.

When multiplying a fraction by a whole number, a common

misconception is that both the numerator and denominator must be

multiplied by the whole number. The use of a concrete model should

help address this. A model reinforces that a denominator indicates the

number of equal parts that make up the whole and this does not change

when multiplying by a whole number. Samples of concrete models

illustrating 13

6× are provided here.

It is important that the student be exposed to a concrete model,

followed by representing the concrete model pictorially, which leads to

an understanding of multiplying fractions symbolically.

85GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE

Suggested Assessment Strategies Resources/Notes


General Outcome: Develop Number Sense

Paper and Pencil

• Draw a number line to show why each of the following is true:

(i) 13

3 1× =

(ii) 13

3 1× =(iii) Use a different model to verify the above.

(8N6.1)

Problem Solving

• Wayne fi lled 5 glasses with 7

8 of a litre of soda in each glass.

(i) Estimate how much soda Wayne used.(ii) Use a model to determine how much soda Wayne used. (8N6.1, 8N6.4)

Math Makes Sense 8

Lesson 3.1: Using Models to

Multiply Fractions and Whole

Numbers

ProGuide: pp. 4-9, Master 3.16

CD-ROM: Master 3.27

SB: pp. 104-109

Practice and HW Book: pp. 50-51


Outcomes




Strand: Number

8N6 Continued

Achievement Indicators:

8N6.2 Model multiplication of a positive fraction by a positive fraction concretely or pictorially using an area model and record the process.

8N6.3 Provide a context that requires the multiplication of two given positive fractions.

A variety of models can demonstrate the meaning of fraction

multiplication. The area model for multiplying two fractions is

emphasized in this achievement indicator.

To model 2 23 5× , create the rectangle based on the factors of the

multiplication. The denominators determine the dimensions of the

rectangle and the numerators indicate the required shading.

First, divide the rectangle vertically into fi fths and shade two-fi fths.

Next, to determine two-thirds of the shaded two-fi fths, divide the

rectangle into thirds along the horizontal dimension.

Finally, shade two-thirds horizontally. The product will be the area that

is double shaded (four pieces out of fi fteen).

Therefore, 2 2 43 5 15× = .

Relating multiplication of fractions to real-life situations helps solidify

student understanding. When asked to provide a context that requires

the multiplication of two given positive fractions, some students may

use original contexts for their problem and others may adopt the

wording of earlier problems. Encourage students to share their problems

so that they are exposed to some that show originality.

It should be shown that “of” means multiplication. This may be done

by comparing results in examples such as 12

of 6 and 12

6× .





Journal

• Lisa has 34

of a large candy bar. She gave 13

of what she had to Shannon.

(i) Demonstrate that Shannon got less than 13

of what would have been a whole bar.

(ii) What fraction of the whole bar does Shannon receive?(iii) What fraction of the whole bar does Lisa have left?

(8N6.2)

• Refer students to problems like the assessment item above as a starting point for creating their own problems.

(8N6.1, 8N6.2, 8N6.3)

• Explain how you could use a diagram to fi nd 3 24 5× . (8N6.2)

Math Makes Sense 8


Multiply Fractions


CD-ROM: Master 3.28

SB: pp. 110-114



Multiply Fractions

Lesson 3.3: Multiplying Fractions

Lesson 3.4: Multiplying Mixed

Numbers

ProGuide: pp. 14, 18-19, 25-26,

Master 3.16

SB: pp. 114,119,126


Outcomes




Strand: Number

8N6 Continued


8N6.4 Estimate the product of two given positive proper fractions to determine if the

product will be closer to 0, 12

or

1.

Estimation keeps the focus on the meaning of the numbers and the

operations, encourages refl ective thinking and helps build number sense

with fractions.

To estimate products close to 0, 12

or 1, consider the following

properties:0 0,where is any number

1 ,where is any number

1 1 1

n n

n n n

× =

× =

× =

Applying these properties and using benchmarks of 0, 12

, and 1 for

given factors, students can estimate a product.

To estimate the product of 19

and 89

, encourage students to think about

19

being close to 0. Since 89

0 0× = , 819 9× would be close to 0. Similarly,

the following products can be estimated using benchmarks.

Determine Benchmarks

Multiply using Benchmarks

Estimate Product

8 49 9× 8 4 1

9 9 21, B B 1 1

2 21× = 8 4 1

9 9 2× B

8 89 9× 8

91B 1 1 1× = 8 8

9 91× B

Estimation helps fraction computations make sense. It should play a

signifi cant role in the development of multiplication strategies.





Observation

• Spinner Game: Use a 4 section spinner. Label each section with

fractions such as:

9 51 119 10 12 11

, , , .

Spin twice and estimate the product.

Score 0 points if the closest benchmark is zero, 1 point if the closest

benchmark is 12

, and 2 points if the closest benchmark is 1. The

student who scores 20 points fi rst wins the game.

(8N6.4)

Math Makes Sense 8


ProGuide: pp. 15-20

CD-ROM: Master 3.29

Student Book (SB): pp. 115-120


Outcomes




Strand: Number

8N6 Continued


8N6.5 Generalize and apply rules for multiplying positive proper fractions, including mixed numbers.

Patterning provides a benefi cial transition from the concrete to the

symbolic. When multiplying a whole number by a fraction, the

following pattern could be used:

8 81 12 1 2 2

8 81 14 1 4 4

8 81 18 1 8 8

8 4 32

8 2 16

8 1 8

8 4 4

8 2 2

8 1 1

× =

× =

× =

× = → × = =

× = → × = =

× = → × = =

This pattern can be extended to include other fraction products and

ultimately to a generalization about multiplying fractions.

After working with models, students should observe that when you

multiply two fractions, the numerator is the product of the numerators,

and the denominator is the product of the denominators. For example,

2 2 2 2 43 5 3 5 15

××× = = .

Estimation is valuable once students have moved to the symbolic level.

To check the reasonableness of the solution, think about 23

as a little less

than 1 and 25

as a little less than 12

. Since 1 12 2

1× = , and each fraction

is slightly less than these factors, the product should be less than 12

. The

product, 4

15 , is less than 12 , so it is a reasonable answer.

A common misconception is that multiplying always makes things

bigger. When one of the factors is between zero and one, this is not the

case. The use of models, as well as estimation, should help overcome

this.

Modelling multiplication of mixed numbers should be done prior to

multiplying the equivalent improper fractions. No reference to improper

fractions is necessary when using the models.

Continued





Paper and Pencil

• The last time that Ms. Martinez ordered pizza, there was 23

of a 12

slice pizza left. Bobby came in and ate 12 of what was left. The other

students were mad that Bobby ate 12

of it. Bobby said “I only ate 2 pieces.” Was he right? How many pieces did he eat? What fraction of the whole pizza did he eat?

(8N6.5)

• 35

1 m of fabric is needed to sew one blouse. How many metres of

fabric are needed to sew 12 such blouses? (8N6.5)

Math Makes Sense 8


ProGuide: pp. 15-20

CD-ROM: Master 3.29

SB: pp. 115-120


Math Makes Sense 8

Lesson 3.4: Multiplying Mixed Numbers


CD-ROM: Master 3.30

SB: pp. 121-126


Journal

• Jared calculated 3 25 5× as follows: 3 62

5 5 5× = .

(i) What mistake did Jared make?(ii) How could you use estimation to show Jared that he made a

mistake?(iii) What is the correct procedure? (8N6.1, 8N6.5)

Problem Solving

• In your job as a gardener, you must decide how to use your garden.

You mark 12

of the garden for potatoes. You use 14

of the remaining

area for corn. Then you plant cucumbers in 13

of what is left. The rest of your garden is used for carrots. What fraction of your garden is used for carrots? (8N6.5)


Outcomes




Strand: Number

An area model to multiply 15

1 by 15

2 is shown below.

( ) ( ) ( ) ( )1 1 1 13 5 5 3

1 2 13 5 15

5 6 115 15 15

1215

45

1 2 1 2

2

2

2

2

× + × + × + ×

= + + +

= + + +

=

=Students may eventually be able to do these calculations without having

to draw the area model.

A common error when fi nding the product of mixed numbers is

to multiply the whole numbers together and multiply the fractions

together. Use of the area model clearly demonstrates why this is

incorrect.

Since the product is the area of the entire rectangle, multiplying only

the whole numbers together and the fractions together misses the two

unshaded pieces.

After using the area model, students can move to rewriting the mixed

numbers as improper fractions before fi nding the product. This

conversion to the equivalent improper fraction was an outcome in grade

6, and was revisited in grade 7. As with multiplying proper fractions, it

is essential that students check the reasonableness of their answer using

estimation.

Students worked with equivalent fractions in Grade 7. As with adding

and subtracting, they should be encouraged to reduce fractions to

simplest form when multiplying as well.

8N6 Continued


8N6.5 Continued






“Activating Prior Knowledge”

provides review on relating mixed

numbers and improper fractions.

CD-ROM: Master 3.37b

“Activating Prior Knowledge”

http://www.visualfractions.com/

MultStrict.html

Paper and Pencil

• Joanne gave the following answer on her homework assignment.

31 1

3 4 122 1 3× =

(i) Use an area model to show why this answer is incorrect.(ii) What mistake did Joanne make?(iii) What is the correct answer?

(8N6.5)

Journal

• Jane multiplied 1 13 2

2 2× as follows: 7 51 13 2 3 2

15146 6

21036

356

56

2 2

5

× = ×

= ×

=

=

=

(i) Was Jane’s fi nal answer correct?(ii) How did Jane make the calculations longer than necessary?

(8N6.5)

Interview

• Ask students to estimate each of the following and to explain their thinking.

(i) 16

5 8×

(ii) 38

4 8× (8N6.5)


Outcomes




Strand: Number

Work with concrete and pictorial models is necessary when students

are fi rst introduced to dividing fractions. It is not enough for students’

knowledge of the division of fractions to be limited to the traditional

invert-and-multiply algorithm. To develop students’ conceptual

understanding of division of fractions, teachers must carefully consider

what students need to learn beyond this algorithmic procedure.

Students were introduced to division of whole numbers in two ways:

sharing and grouping. This idea can be extended to division of fractions.

It is appropriate to think of dividing a fraction by a whole number as

equal sharing.

Consider the following example: You have 23

of a pizza to divide evenly

among 3 people. How much pizza would each person receive?

The diagram shows 23

of a pizza.

To divide it evenly among 3 people, cut each piece into thirds and share

the 6 pieces evenly.

Each person will receive 29

of a pizza.

Alternatively, 23

3÷ can mean 2 thirds is shared by 3 people. Creating

an equivalent fraction with a numerator divisible by 3 gives 69

3÷ . This

means 6 ninths is shared by 3 people, so each person would get 2 ninths.

Continued

8N6 Continued


8N6.6 Model division of a whole number and a positive proper fraction, concretely or pictorially and record the process.





Math Makes Sense 8

Lesson 3.5: Dividing Whole

Numbers and Fractions


CD-ROM: Master 3.31

SB: pp. 129-134


Journal

• Explain the difference between “six divided by one half ” and “six divided in half ”. Write a division statement for each phrase and fi nd each quotient. (8N6.6)

• Explain how the following diagram can be used to calculate 14

3÷ .

Are there other manipulatives or diagrams you could use? Explain. (8N6.6)

Problem Solving

• You have 34

of a pizza to divide equally between 2 people. Use a model to determine how much pizza each person would receive. (8N6.6)


Outcomes




Strand: Number

A number line can also provide a useful model for division.

To model 23

3÷ :

� • Mark off 23

.

� • Divide 23

into 3 equal parts.

23

divided into 3 equal parts gives equal pieces of 29

. Therefore,

2 23 9

3÷ = .

When dividing a whole number by a fraction, ask “How many equal

groups can be made?” You have 3 pizzas. Each person eats 13

of a pizza

and all pizzas are completely eaten. How many people eat the pizzas?

Start with 3 pizzas and divide them into thirds. How many groups of 13

are there?

The pizzas can be divided into 9 equal groups of 13

.

Continued

8N6 Continued


8N6.6 Continued





Math Makes Sense 8




CD-ROM: Master 3.31

SB: pp.129-134


Problem Solving

• You pay $3 for 34

kg of nuts. Use a model to determine how much

1 kg of these nuts would cost. (8N6.6)

Performance

• Demonstrate, by drawing diagrams, and explain why each of the following is true:

(i) 14

2 8÷ =

(ii) 1 12 4

2÷ = (8N6.6)


Outcomes




Strand: Number

Using a number line to model 13

3 ÷ ,

� • mark off 3

� • starting at zero, mark off the number of groups of 13

Therefore, 13

3 9÷ = .

Students may experience more diffi culty using number lines when the

quotient is not a whole number. Teachers should spend more time on

questions such as the following.

Model 23

3 ÷ on a number line.

� • mark off 3

� • starting at zero, mark off the number of groups of 23

4 wholes can be made using groups of 23

, leaving part of a whole

remaining.

Continued

8N6 Continued


8N6.6 Continued





Math Makes Sense 8




CD-ROM: Master 3.31

SB: pp. 129-134


Pencil and Paper

• Use a diagram to calculate the following.

(i) 13

4 ÷

(ii) 12

3 ÷

(iii) 15

2 ÷ (8N6.6)


Outcomes




Strand: Number

� • fi nd the remaining amount

Two thirds make up one whole. One third remains. Thus, 1 piece out of

2 remains, or 12

. Therefore, 2 13 2

3 4÷ = .

A common misconception about division is that it always makes things

smaller. Students should see here that this is not the case.

8N6 Continued


8N6.6 Continued

8N6.7 Model division of a positive proper fraction by a positive proper fraction pictorially and record the process.

A good understanding of modelling division of a fraction and a whole

number should provide a smooth transition to dividing positive proper

fractions. An example of division of a fraction by a fraction using

fraction strips follows. When dividing 45

by 13

, students should use

diagrams to determine how many groups of 13

are in 45

. The diagram

below shows that the number of groups of 13

in 45

is between 2 and 3.

It is diffi cult to determine precisely how many groups there are. A

common denominator for 5 and 3 is 15. Using a rectangle divided into

fi fteenths will help students determine the exact number of groups.

In 1215

there are 2 whole groups of 515

, plus 25

of another group.

Therefore, 4 1 25 3 5

2÷ = .

Continued





Math Makes Sense 8

Lesson 3.6: Dividing Fractions

ProGuide: pp. 35-40

CD-ROM: Master 3.32

SB: pp. 135-139


Problem Solving

• You have 56

of a litre of ice cream.

(i) About how many 12

litre cartons could you fi ll with the ice cream?

(ii) Calculate how many 12

litre cartons you could fi ll with this ice cream. Include a diagram. (8N6.7)


Outcomes




Strand: Number

Modeling division of a fraction by a fraction using a number line

follows the same pattern as the fraction strip model. Using a common

denominator to mark the intervals on the number line is benefi cial. To

model 4 15 3÷ :

• use a number line with 15ths and mark ( )4 4 125 5 15

= , the fi rst fraction in the operation.

• Starting at zero, mark off groups of ( )5 5115 3 15

= until no more whole

groups of 515

can be made.

Two wholes groups of 515

are formed.

• Five fi fteenths make up one whole and two fi fteenths remain. In

other words, 2 pieces out of 5, or 25

, remain.


2÷ = ; the same result we saw with the fraction strips.

8N6 Continued


8N6.7 Continued





Math Makes Sense 8


ProGuide: pp. 35-40

CD-ROM: Master 3.32

SB: pp. 135-139


Pencil and Paper

• What division expression does this picture represent? (8N6.7)

• Draw a fraction strip model to show 7 18 4

.÷ (8N6.7)


Outcomes




Strand: Number

8N6.9 Generalize and apply rules for dividing positive proper fractions.

Students should always be encouraged to think about the reasonableness

of their answers and to use estimation to check them. When estimating

quotients close to whole numbers, consider the following:

1

0 0,where is any number and 0

1 ,where is any number

1 ,where is any number and 0

1,where is any number and 0

n

n n n

n n n

n n n

n n n n

÷ = ≠

÷ =

÷ = ≠

÷ = ≠

Students are not responsible for notation involving n and they should be

reminded that division by zero is undefi ned. Applying these properties

and using whole number benchmarks, students can estimate a quotient.

Determine Benchmarks

Divide Using Benchmarks

Estimate Quotient

819 9÷ 81

9 90, 1B B 0 1 0÷ = 81

9 90÷ B

4 15 3

2÷ 4 15 3

1, 2 2B B 12

1 2÷ = 4 1 15 3 2

2÷ B

815 9

4 1÷ 815 9

4 4, 1 2B B 4 2 2÷ = 815 9

4 1 2÷ B

8 19 10

2 3÷ 8 19 10

2 3, 3 3B B 3 3 1÷ = 8 19 10

2 3 1÷ B

One approach to generalizing rules for dividing fractions involves

using models to show the connection between division and the related

multiplication.

Problems Modeled Previously Using

Number Line

Related Multiplication

Problem

Therefore (∴)

32 23 1 9÷ = 2 1 2

3 3 9× = 32 2 1

3 1 3 3∴ ÷ = ×

13

3 9÷ = 3 3 91 1 1

9× = = 3 313 1 1

3∴ ÷ = ×

3 92 11 3 2 2

4÷ = = 3 3 9 11 2 2 2

4× = = 3 3 321 3 1 2

∴ ÷ = ×

4 1 2 125 3 5 5

2÷ = = 34 12 25 1 5 5

2× = = 34 1 45 3 5 1

∴ ÷ = ×

Continued

8N6 Continued


8N6.8 Estimate the quotient of two given positive fractions and compare the estimate to whole number benchmarks.





Math Makes Sense 8


Lesson 3.7: Dividing Mixed

Numbers

ProGuide: pp. 35-40, 41-46

SB: pp.135-140, 141-146

Interview

• Estimate each of the following and explain your thinking.

(i) 14

24 4÷

(ii) 34

32 7÷ (8N6.8)



CD-ROM: Master 3.32

SB: pp. 135-140



Outcomes




Strand: Number

Using patterns is also a good way to help increase the understanding of

the division of fractions. When dividing a whole number by a fraction,

consider the following pattern:

81 22 1 1

81 44 1 1

8 818 1 1

8 4 2

8 2 4

8 1 8

8 16 16

8 32 32

8 64 64

÷ =

÷ =

÷ =

÷ = → × =

÷ = → × =

÷ = → × =

This pattern can be extended to include other fraction quotients and

ultimately to a generalization about dividing fractions.

One algorithm for dividing fractions is the common denominator

algorithm. This involves fi nding a common denominator and dividing

the numerators. For example, 54 1 12 12 25 3 15 15 5 5

12 5 2÷ = ÷ = ÷ = = . This

uses the measurement, or equal grouping, interpretation of division that

was referenced earlier.

Continued

8N6 Continued


8N6.9 Continued





Math Makes Sense 8



CD-ROM: Master 3.32

SB: pp. 135-140


Journal

• Sarah carried out the division 3 24 3÷ as follows:

3 2 4 24 3 3 3

89

÷ = ×

=

Do you agree with Sarah’s method and answer? Explain. (8N6.9)

• Explain why 15 516 8÷

is half of 15 5

16 16.÷ (8N6.9)


Outcomes




Strand: Number

The traditional invert-and-multiply algorithm introduces students to

the concept of reciprocal. Reciprocals are two numbers whose product

is 1. For example, 34

and 43

are reciprocals because 3 4 124 3 12

1× = = . It

is important to reinforce that any whole number can be written in

fractional form with a denominator of 1.

This algorithm is probably one of the most poorly understood

procedures in intermediate mathematics. For the benefi t of teachers, a

mathematical justifi cation for this approach is provided here.

Why Multiplying by the Reciprocal Works (Example)

Explanation of Steps

2 43 5÷ =n

23

45

=n Division in fractional form.

( ) ( )23 4 4

5 545

=

n

Multiply each side of the equation

by the denominator: 45

( ) ( )( )

( )2 43 5 4

545

=

n

2 43 5= ×n

Simplify.

5 52 43 4 5 4× = × ×n Isolaten by multiplying both

sides of the equation by 54

, the

reciprocal of 45

.

( )5 52 43 4 5 4× = × ×n

( )523 4

1× = ×n 1 is the product of the reciprocals.

523 4× =n

52 4 23 5 3 4

∴ ÷ = × This is true because 2 43 5÷ =n

and 523 4× =n .

8N6 Continued


8N6.9 Continued





Math Makes Sense 8



CD-ROM: Master 3.32

SB: pp. 135-140



Outcomes




Strand: Number

8N6 Continued


8N6.10 Model, generalize and apply rules for dividing fractions with mixed numbers.

Work with mixed numbers is a natural extension of the modelling

and rules applied to dividing proper fractions. To model 3 24 3

3 1÷ on a

number line:

• write equivalent improper fractions with a common denominator.

3 24 3

9 812 12

45 2012 12

3 1

3 1

÷

= ÷

= ÷

• Using a number line with 12ths, mark ( )45 45 1512 12 4

= , the fi rst fraction in the operation.

• Starting at zero, mark off groups of ( )520 2012 12 3

= until no more whole

groups of 2012

can be made.

Two whole groups of 2012

are formed.

• 20 twelfths make up one whole and 5 twelfths remain. Thus, 5 pieces

out of 20, or 5 120 4= , remain.


3 1 2÷ = .

Continued





Portfolio

• Caitlin decides to make muffi ns for the school picnic. Her recipe

requires 14

2 cups of fl our to make 12 muffi ns. Caitlin found there was exactly 18 cups of fl our in the canister, so she decided to use all of it.

(i) How many muffi ns can Caitlin expect to get?(ii) The principal of the school liked Caitlin’s muffi ns and asked her

to cater the school picnic next year, providing enough muffi ns for all 400 students. How many cups of fl our will Caitlin require? (8N6.10)

Math Makes Sense 8


Numbers


CD-ROM: Master 3.33

SB: pp. 141-146



Outcomes




Strand: Number

8N6 Continued


8N6.10 Continued

Students should always be encouraged to check the reasonableness of

their answers. Since 34

3 4B and 23

1 2B , and 4 2 2÷ = , the answer 14

2

is reasonable because 14

2 2B .

Both the invert-and-multiply algorithm and the common-denominator

algorithm can be applied here. Mixed numbers must be rewritten as

equivalent improper fractions fi rst.

8N6.11 Provide a context that requires the dividing of two given positive fractions.

In addition to giving students problems involving division of fractions

and asking them to explain solution methods, they should also be

required to write word problems that fi t a given fraction division. This

requires a depth of understanding that simple calculations do not and

student responses will provide the teacher with a more well-rounded

view of student thinking. It may be benefi cial to model such a process

with the class. Encourage students to continue to share the problems

they create so that there is exposure to unique situations requiring

division of fractions.

Determining the operation required to solve a mathematical word

problem can be a challenge for students at the intermediate level. Prior

to solving problems, it is important to read through various problems

with students and identify key words that determine the operation(s)

required to solve the problem. The focus here is not to have students

solve the problems. Development of a table including words or concepts

such as the following could be benefi cial.

Addition Subtraction Multiplication Division Sum Difference Product Quotient Total Exceed Multiply Equal Shares

Altogether Subtract Times Equal Groups How much

greater than? Divide

How much less than?

This table can be added to at any time.

Continued

8N6.12 Identify the operation required to solve a given problem involving positive fractions.





Paper and Pencil

• Shelley’s salsa recipe is very popular. This ingredient list makes enough salsa for 6 people.

14

2 cups of diced tomatoes

12

cup of onions

34

teaspoon of salt

18

teaspoon of sugar

23

cup of green pepper

(i) Shelly is having a party and will have 17 guests. How should she change the ingredient list to ensure she has enough salsa for her party? Write out the new ingredient list.

(ii) If Shelly is having a movie night and will only have 2 people to share her salsa, how much smaller will the batch be? Write out the

new ingredient list. (8N6.4 and 8N6.11 )

• Write a real world problem for the following operations using fractions:

(i) 3 divided by 14

(ii) 23

1 divided by 16

(8N6.11)

• Create a problem you might solve by dividing 34

by 3. Solve your problem. (8N6.11)

• Give students a number of problems. Ask them to identify the operation involved in solving the problem and explain why they know this. Do not have students solve the problem. Any alternate math text may be used as a source of problems for discussion.

(8N6.12)

Math Makes Sense 8



Numbers

Lesson 3.8: Solving Problems

with Fractions

ProGuide: pp. 39, 45,52, Master

3.9a, 3.9b

SB: pp. 140, 146, 152

http://mathforum.org/paths/

fractions/frac.recipe.html


Outcomes




Strand: Number

8N6 Continued

To emphasize the importance of reading problems carefully, have

students compare the solutions to two problems, which differ in only

one word, such as the following.

o Jack can usually drive home at an average speed of 50 km/h. One day, a winter storm reduced his speed by three-fi fths of his usual speed. What was his average speed on his drive home that day?

o Jack can usually drive home at an average speed of 50 km/h. One day, a winter storm reduced his speed to three-fi fths of his usual speed. What was his average speed on his drive home that day?


8N6.12 Continued

Students have already been exposed to the order of operations from

their work with whole numbers and decimals in previous grades. Work

with integers in this course also applies order of operations. To extend

the order of operations to work with fractions, a review of addition and

subtraction of fractions may be necessary.

Students could use a mnemonic, such as BEDMAS, to remember the

order. However, exponents are not included as part of this outcome. If

students are relying on this memory device, it is important to reiterate

that division and multiplication are completed in the order they appear

from left to right, as are addition and subtraction.

At this level students are working with positive fractions only and

questions must be limited to those that have positive solutions.

Concrete and pictorial representations continue to be helpful if students

are still having diffi culties with the basic operations of addition,

subtraction, multiplication and division of fractions while solving

problems that involve the order of operations.

8N6.13 Solve a given problem involving positive fractions taking into consideration order of operations (limited to problems with positive solutions).





Journal

• Margie is entering a competition to win a cell phone. She must answer the following skill-testing question.

What is the value of 12

10 2− × ?

(i) How could Margie determine a possible answer of 4?(ii) How could Margie determine a possible answer of 9?(iii)What is the correct answer? Explain. (8N6.13)

• How does knowing the order of operations help ensure that you get

the same answer to 3 514 4 12+ × as other students in the class?

(8N6.13)

Pencil and Paper

• Insert one set of brackets to make the following statements true, and justify your answer.

(i) 1 1 2 12 4 3 2+ × =

(ii)

3 51 2 14 5 3 3 12

1× + × = (8N6.13)

Math Makes Sense 8

Lesson 3.8: Solving Problems

with Fractions

Lesson 3.9: Order of Operations

with Fractions

ProGuide: pp. 47-52, 53-55.

master 3.24

CD-ROM: Master 3.34, 3.35

SB: pp. 147-152, 153-155

Practice and HW Book: pp. 64-

66, 67-68

Sample word problems can be found

at http://math.about.com/od/frac-

tionsrounding1/a/freefractions.htm

GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE116


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