operations with fractions suggested time: 4 · pdf file82 grade 8 mathematics draft curriculum...
Post on 04-Feb-2018
217 Views
Preview:
TRANSCRIPT
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
OPERATIONS WITH FRACTIONS
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
OPERATIONS WITH FRACTIONS
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
OPERATIONS WITH FRACTIONS
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
86 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Outcomes
OPERATIONS WITH FRACTIONS
Elaborations—Strategies for Learning and Teaching
Students will be expected to
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× .
87GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Suggested Assessment Strategies Resources/Notes
OPERATIONS WITH FRACTIONS
General Outcome: Develop Number Sense
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
Lesson 3.2: Using Models to
Multiply Fractions
ProGuide: pp. 10-14, Master 3.17
CD-ROM: Master 3.28
SB: pp. 110-114
Practice and HW Book: pp. 52-53
Lesson 3.2: Using Models to
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
88 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Outcomes
OPERATIONS WITH FRACTIONS
Elaborations—Strategies for Learning and Teaching
Students will be expected to
Strand: Number
8N6 Continued
Achievement Indicator:
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.
89GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Suggested Assessment Strategies Resources/Notes
OPERATIONS WITH FRACTIONS
General Outcome: Develop Number Sense
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
Lesson 3.3: Multiplying Fractions
ProGuide: pp. 15-20
CD-ROM: Master 3.29
Student Book (SB): pp. 115-120
90 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Outcomes
OPERATIONS WITH FRACTIONS
Elaborations—Strategies for Learning and Teaching
Students will be expected to
Strand: Number
8N6 Continued
Achievement Indicator:
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
91GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Suggested Assessment Strategies Resources/Notes
OPERATIONS WITH FRACTIONS
General Outcome: Develop Number Sense
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
Lesson 3.3: Multiplying Fractions
ProGuide: pp. 15-20
CD-ROM: Master 3.29
SB: pp. 115-120
Practice and HW Book: pp. 54-55
Math Makes Sense 8
Lesson 3.4: Multiplying Mixed Numbers
ProGuide: pp. 21-26, Master 3.19
CD-ROM: Master 3.30
SB: pp. 121-126
Practice and HW Book: pp. 56-57
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)
92 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Outcomes
OPERATIONS WITH FRACTIONS
Elaborations—Strategies for Learning and Teaching
Students will be expected to
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
Achievement Indicator:
8N6.5 Continued
93GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Suggested Assessment Strategies Resources/Notes
OPERATIONS WITH FRACTIONS
General Outcome: Develop Number Sense
Practice and HW Book: pp. 48-49
“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)
94 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Outcomes
OPERATIONS WITH FRACTIONS
Elaborations—Strategies for Learning and Teaching
Students will be expected to
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
Achievement Indicator:
8N6.6 Model division of a whole number and a positive proper fraction, concretely or pictorially and record the process.
95GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Suggested Assessment Strategies Resources/Notes
OPERATIONS WITH FRACTIONS
General Outcome: Develop Number Sense
Math Makes Sense 8
Lesson 3.5: Dividing Whole
Numbers and Fractions
ProGuide: pp. 29-34, Master 3.20
CD-ROM: Master 3.31
SB: pp. 129-134
Practice and HW Book: pp. 58-59
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)
96 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Outcomes
OPERATIONS WITH FRACTIONS
Elaborations—Strategies for Learning and Teaching
Students will be expected to
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
Achievement Indicator:
8N6.6 Continued
97GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Suggested Assessment Strategies Resources/Notes
OPERATIONS WITH FRACTIONS
General Outcome: Develop Number Sense
Math Makes Sense 8
Lesson 3.5: Dividing Whole
Numbers and Fractions
ProGuide: pp. 29-34, Master 3.20
CD-ROM: Master 3.31
SB: pp.129-134
Practice and HW Book: pp. 58-59
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)
98 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Outcomes
OPERATIONS WITH FRACTIONS
Elaborations—Strategies for Learning and Teaching
Students will be expected to
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
Achievement Indicator:
8N6.6 Continued
99GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Suggested Assessment Strategies Resources/Notes
OPERATIONS WITH FRACTIONS
General Outcome: Develop Number Sense
Math Makes Sense 8
Lesson 3.5: Dividing Whole
Numbers and Fractions
ProGuide: pp. 29-34, Master 3.20
CD-ROM: Master 3.31
SB: pp. 129-134
Practice and HW Book: pp. 58-59
Pencil and Paper
• Use a diagram to calculate the following.
(i) 13
4 ÷
(ii) 12
3 ÷
(iii) 15
2 ÷ (8N6.6)
100 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Outcomes
OPERATIONS WITH FRACTIONS
Elaborations—Strategies for Learning and Teaching
Students will be expected to
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
Achievement Indicators:
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
101GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Suggested Assessment Strategies Resources/Notes
OPERATIONS WITH FRACTIONS
General Outcome: Develop Number Sense
Math Makes Sense 8
Lesson 3.6: Dividing Fractions
ProGuide: pp. 35-40
CD-ROM: Master 3.32
SB: pp. 135-139
Practice and HW Book: pp. 60-61
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)
102 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Outcomes
OPERATIONS WITH FRACTIONS
Elaborations—Strategies for Learning and Teaching
Students will be expected to
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.
Therefore, 4 1 25 3 5
2÷ = ; the same result we saw with the fraction strips.
8N6 Continued
Achievement Indicator:
8N6.7 Continued
103GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Suggested Assessment Strategies Resources/Notes
OPERATIONS WITH FRACTIONS
General Outcome: Develop Number Sense
Math Makes Sense 8
Lesson 3.6: Dividing Fractions
ProGuide: pp. 35-40
CD-ROM: Master 3.32
SB: pp. 135-139
Practice and HW Book: pp. 60-61
Pencil and Paper
• What division expression does this picture represent? (8N6.7)
• Draw a fraction strip model to show 7 18 4
.÷ (8N6.7)
104 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Outcomes
OPERATIONS WITH FRACTIONS
Elaborations—Strategies for Learning and Teaching
Students will be expected to
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
Achievement Indicators:
8N6.8 Estimate the quotient of two given positive fractions and compare the estimate to whole number benchmarks.
105GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Suggested Assessment Strategies Resources/Notes
OPERATIONS WITH FRACTIONS
General Outcome: Develop Number Sense
Math Makes Sense 8
Lesson 3.6: Dividing Fractions
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)
Lesson 3.6: Dividing Fractions
ProGuide: pp. 35-40, Master 3.21
CD-ROM: Master 3.32
SB: pp. 135-140
Practice and HW Book: pp. 60-61
106 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Outcomes
OPERATIONS WITH FRACTIONS
Elaborations—Strategies for Learning and Teaching
Students will be expected to
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
Achievement Indicator:
8N6.9 Continued
107GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Suggested Assessment Strategies Resources/Notes
OPERATIONS WITH FRACTIONS
General Outcome: Develop Number Sense
Math Makes Sense 8
Lesson 3.6: Dividing Fractions
ProGuide: pp. 35-40, Master 3.21
CD-ROM: Master 3.32
SB: pp. 135-140
Practice and HW Book: pp. 60-61
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)
108 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Outcomes
OPERATIONS WITH FRACTIONS
Elaborations—Strategies for Learning and Teaching
Students will be expected to
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
Achievement Indicator:
8N6.9 Continued
109GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Suggested Assessment Strategies Resources/Notes
OPERATIONS WITH FRACTIONS
General Outcome: Develop Number Sense
Math Makes Sense 8
Lesson 3.6: Dividing Fractions
ProGuide: pp. 35-40, Master 3.21
CD-ROM: Master 3.32
SB: pp. 135-140
Practice and HW Book: pp. 60-61
110 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Outcomes
OPERATIONS WITH FRACTIONS
Elaborations—Strategies for Learning and Teaching
Students will be expected to
Strand: Number
8N6 Continued
Achievement Indicator:
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.
Therefore, 3 2 14 3 4
3 1 2÷ = .
Continued
111GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Suggested Assessment Strategies Resources/Notes
OPERATIONS WITH FRACTIONS
General Outcome: Develop Number Sense
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
Lesson 3.7: Dividing Mixed
Numbers
ProGuide: pp. 41-46, Master 3.22
CD-ROM: Master 3.33
SB: pp. 141-146
Practice and HW Book: pp. 62-63
112 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Outcomes
OPERATIONS WITH FRACTIONS
Elaborations—Strategies for Learning and Teaching
Students will be expected to
Strand: Number
8N6 Continued
Achievement Indicators:
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.
113GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Suggested Assessment Strategies Resources/Notes
OPERATIONS WITH FRACTIONS
General Outcome: Develop Number Sense
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
Lesson 3.6: Dividing Fractions
Lesson 3.7: Dividing Mixed
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
114 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Outcomes
OPERATIONS WITH FRACTIONS
Elaborations—Strategies for Learning and Teaching
Students will be expected to
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?
Achievement Indicators:
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).
115GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Suggested Assessment Strategies Resources/Notes
OPERATIONS WITH FRACTIONS
General Outcome: Develop Number Sense
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
OPERATIONS WITH FRACTIONS
top related