Gap Closing
Fractions
Intermediate / Senior
Facilitator Guide
Contents
Introduction
Fractions
Decimals
Integers
Proportional Reasoning
Powers and Roots
Algebraic Expressions and Equations
Solving Equations
2-D Measurement
Volume and Surface Area
Permission is given to reproduce these materials for non-commercial educational purposes. Any references in this document to particular
commercial resources, learning materials, equipment, or technology reflect only the opinions of the writers of this resource, and do not reflect
any official endorsement by the Ministry of Education.
© Marian Small, 2011
INTRODUCTION
The Gap Closing package is designed to help Grade 9 teachers provide precisely targeted remediation for students who they identify as being significantly behind in mathematics.
The goal is to close gaps in Number Sense, Algebra, and Measurement so that the students can be successful
in learning grade-appropriate mathematics. However, the materials may also be useful to Grades 7 and 8 teachers in providing alternative approaches they had not considered to some of the content or to Grade 10 applied
math or Grade 11 or 12 teachers, particularly teachers of Mathematics for Workplace courses.
For each topic, there is a diagnostic and a set of intervention materials that teachers can use to help students be
more successful in their learning.
The diagnostics are designed to uncover the typical problems students have with a specific topic. Each diagnostic should be used as instructional decisions are being made for the struggling student.
Each set of intervention materials includes a single-task Open Question approach and a multiple-question Think
Sheet approach. These approaches both address the same learning goals, and represent different ways of engaging and interacting with students.
Suggestions are provided for how best to facilitate learning before, during, and after using your choice of approaches. This three-part structure consists of:
• Questions to ask before using the approach
• Using the approach
• Consolidating and reflecting on the approach
The style and content of the consolidating questions may be useful to teachers in their
regular course preparation as well.
Getting Started
• Identify students who are underperforming in Number Sense, Algebra, and Measurement or in
mathematics in general – students who are not where you think they should be.
• Select the Number, Algebra, and Measurement topics to focus on.
• Administer the diagnostic, allowing 20-40 minutes or as much additional time as seems reasonable for
a particular student.
For students with accommodations, e.g., a scribe, supports should remain in place; they may require a longer
time to complete the diagnostic.
• Refer to the Facilitator’s Guide as you administer and mark the diagnostic.
• Use the chart, Evaluating Diagnostic, Results to identify which of the intervention materials aligns with the
student’s learning needs.
This is the stage where you personalize remediation based on the precisely identified area of concern. It may
be appropriate to use more than one of the intervention materials with the student.
• Select either the Open Question or the Think Sheet approach:
The Open Question provides the student with a variety of possible responses and approaches.
The Think Sheet offers a more guided and structured approach.
Teachers should consider student preferences and readiness when deciding which of these two approaches
to use.
• Use the suggestions for the three-part structure included for each topic to provide remediation.
Allow approximately 45 minutes to complete each topic, recognizing that time may vary for different students.
Considerations
Materials required for each intervention approach are indicated in the Facilitator’s Guide.
There are templates included in the student book should the concrete materials not be available.
Sample answers to questions are included to give teachers language that they might use in further discussions with students. These samples sometimes may be more sophisticated than the responses that students
would give. Teachers should probe further with individual students in their questioning, as needed.
Some students might need more practice with the same topic than other students.
Teachers should use their professional judgment when working with individual students to decide how much
practice they need. For additional practice, teachers can create similar questions to those provided using alternate numbers.
Access the Gap Closing materials at www.edugains.ca
Fractions
Diagnostic..........................................................................................5
Administer the diagnostic.......................................................................5
Using diagnostic results to personalize interventions.............................5
Solutions.................................................................................................5
Using Intervention Materials.......................................................8
Comparing Fractions.................................................................................9
Adding Fractions.....................................................................................14
Subtracting Fractions..............................................................................21
Multiplying Fractions...............................................................................27
Dividing Fractions...................................................................................33
Relating Situations to Fraction Operations................................................39
Fractions
Relevant Expectations for Grade 9
MPM1D
Number Sense and Algebra
• substitute into and evaluate algebraic expressions involving exponents (i.e., evaluate
expressions involving natural-number exponents with rational-number bases)
• simplify numerical expressions involving integers and rational numbers, with and without
the use of technology
• solve problems requiring the manipulation of expressions arising from applications of
percent, ratio, rate, and proportion
• solve first-degree equations, including equations with fractional coefficients, using a variety
of tools and strategies
Linear Relations
• identify, through investigation, some properties of linear relations (i.e., numerically, the first
difference is a constant, which represents a constant rate of change; graphically, a straight
line represents the relation), and apply these properties to determine whether a relation is
linear or non-linear
Analytic Geometry
• determine, through investigation, various formulas for the slope of a line segment or a line
and use the formula to determine the slope of a line segment or a line
MPM1P
Number Sense and Algebra
• illustrate equivalent ratios, using a variety of tools
• represent, using equivalent ratios and proportions, directly proportional relationships arising
from realistic situations
• solve for the unknown value in a proportion, using a variety of methods
• make comparisons using unit rates
• solve problems involving ratios, rates, and directly proportional relationships in various
contexts, using a variety of methods
• solve problems requiring the expression of percents, fractions, and decimals in their
equivalent forms
• simplify numerical expressions involving integers and rational numbers, with and without
the use of technology
• substitute into algebraic equations and solve for one variable in the first degree
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September 2011
© Marian Small, 2011 Fractions (IS)
Linear Relations
• identify, through investigation, some properties of linear relations (i.e., numerically, the first
difference is a constant, which represents a constant rate of change; graphically, a straight
line represents the relation), and apply these properties to determine whether a relation is
linear or non-linear
• determine, through investigation, that the rate of change of a linear relation can be found by
choosing any two points on the line that represents the relation, finding the vertical change
between the points (i.e., the rise) and the horizontal change between the points (i.e., the
rise
run) and writing the ratio ​ _
​
run  
Possible reasons why a student might struggle when
working with fractions
Students may struggle with fraction comparisons and operations as well as when to apply the
various operations.
Some of the problems include:
• misunderstandings about how to create equivalent fractions, e.g., adding the same amount
to both numerator and denominator
• misunderstandings about how to compare fractions when the denominators are not the
same, e.g., assuming that if the numerator and denominator are closer, the fraction is
9
7 _
1
4
_
_
greater although this is not always true; even though ​ _
10  ​is greater than ​ 9 ​, ​  3 ​is not greater than ​ 7 ​
• adding (or subtracting) fractions by adding the numerators and adding (or subtracting)
the denominators
• not recognizing that fractions can only be added or subtracted if they are parts of the
same whole
• lack of understanding that ​ _ab ​ × _​  dc ​is a portion of ​ _dc ​ if _​  ab ​is a proper fraction
• misinterpreting a remainder when dividing fractions, e.g., thinking that ​ _12 ​ ÷ ​ _13 ​= 1​ _16 ​
(since _​ 12 ​ – _​  13 ​ = _​  16 ​) instead of 1​ _12 ​
• inverting the wrong fraction when dividing fractions using an invert-and-multiply strategy
• inability to extend knowledge of comparisons and operation rules with proper fractions to
improper fractions and/or mixed numbers
• multiplying mixed numbers by separately multiplying whole number parts and fraction parts
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© Marian Small, 2011 Fractions (IS)
Diagnostic
Administer the diagnostic
Materials:
Using the diagnostic results to personalize
interventions
• a grid and counters or
fraction strips/circles
(optional)
Intervention materials are included on each of these topics:
• comparing fractions
• adding fractions
• subtracting fractions
• multiplying fractions
• dividing fractions
• relating situations to fraction operations
You may use all or only part of these sets of materials, based on student performance
with the diagnostic. If students need help in understanding the intent of a question in the
diagnostic, you are encouraged to clarify that intent.
Evaluating Diagnostic Results
Suggested Intervention Materials
If students struggle with Questions 1-3
use Comparing Fractions
If students struggle with Questions 4-6
use Adding Fractions
If students struggle with Questions 7-9
use Subtracting Fractions
If students struggle with Questions 10-12
use Multiplying Fractions
If students struggle with Questions 13-15
use Dividing Fractions
If students struggle with Questions 6, 9,
12, 15, and 16
use Relating Situations to Fraction
Operations
Solutions
12
1.a)e.g., _​  46 ​, _​  69 ​ and ​ _
18 ​
16
32
4 _
_
b) e.g., ​  5 ​, ​  20 ​ and ​ _
40 ​
6 _
18
12 _
_
c) e.g., ​  25  ​, ​  50 ​, ​  75 ​
2.a)>
b) >
c) <
d) >
e) >
f) <
3.
2
_
7
1
_
3
4
_
10
6
1
7
_
_
10
1_
3 2 10
4.a)e.g.,
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September 2011
© Marian Small, 2011 Fractions (IS)
b) e.g.,
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
5.a)​ _69 ​
29
c)​ _
24 ​
e) 5​ _16 ​
b) _
​  13
15 ​ Note: Equivalent forms are acceptable.
_
d) ​  16
4  ​
7
f) 8​ _
24  ​
6. e.g., There were 1​ _12 ​Hawaiian pizzas and ​ _23 ​of a vegetarian pizza left. How many pizzas
were left altogether?
7.a)e.g.,
8.a)​ _48 ​
7
c)​ _
12  ​
e) 2​ _13 ​
b) e.g.,
X
X
X
X
X
X
X
X
X
7
b) _
​  15
  ​
d) _
​  14
15 ​
11
f) 1​ _
15 ​ 
9. e.g., I had 3​ _14 ​cartons of eggs and used 1​ _13 ​of them for some baking. How many cartons of
eggs are left?
10.a) e.g.,
11.a)
b)
c)
d)
b) e.g.,
​ _12 ​
8
​ _
15  ​
_
​  32 ​
_
​  63
12 ​
12. e.g., If _​ 56 ​of a project had been completed, and one student had done ​ _23 ​of it, then _​ 23 ​ × _​  56 ​ tells
what part of the whole project she had completed.
13.a) e.g.,
14.a)
b)
c)
d)
e)
f)
b)
3
2​ _12 ​
6
16
​ _
5  ​
9
_
​  25
  ​
21
_
​  26 ​
1
  ​of a room
15. _
​  10
16.a) 1 – _​ 58 ​= ?
b) ​ _13 ​ × _​  58 ​= ?
c) _​  58 ​ ÷ ​ _15 ​= ?
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September 2011
© Marian Small, 2011 Fractions (IS)
Diagnostic
Diagnostic
1. List three fractions equivalent (equal) to each fraction.
7. Draw a picture to show why each statement is true:
2
a) –
3
b)
8
––
10
c)
(Continued)
7 2 5
a) –
––=–
8 8 8
24
––
100
9
5
2
b) ––
– – = ––
10 5 10
2. Use a greater than (>) or less than (<) sign to make these statements true.
3
–
8
1
a) –
2
c)
5
–
6
3
b) –
8
8
–
9
5
1
e) 2–
3
1–
4
3
––
10
d)
9
–
5
f)
1–
8
5
–
9
8. Subtract:
3
1
1––
10
3. Order these values from least to greatest:
4
––
10
1
–
3
6
––
10
2
–
7
1
1–
3
2 1
b) –
––
3 5
5 1
–––
6 4
8 2
d) –
––
5 3
c)
7
2 ––
10
2
f)
e) 4 – 1–
3
1
3
4–
– 2–
5
3
1
4. Draw a picture to show why each statement is true:
1
from 3 –
.
9. Write a story problem that you could solve by subtracting 1–
3
4
2 1 13
b) –
+ – = ––
5 4 20
2 2 4
a) –
+–=–
5 5 5
7 3
a) –
––
8 8
10. Draw a picture to show why each statement is true:
2 3 2
×–=–
a) –
3 5 5
5. Add each pair of numbers.
4 2
a) –
+–
9 9
2 1
b) –
+–
3 5
c)
3 5
–+–
8 6
9 7
d) –
+–
4 4
e)
8
1
– + 2–
3
2
2
5 10
2
b) –
×–
= ––
3
8 24
11. Multiply each pair of numbers.
3 5
×–
a) –
5 6
5
f) 3–
+ 4–
3
8
c)
2
4 2
b) –
×–
5 3
9 2
–×–
4 3
1
1
d) 2 –
× 2–
3
4
1
6. Write a story problem that you could solve by adding –
and 1–
.
3
2
2
5
12. Describe a situation where you might multiply –
× –.
3 6
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September 2011
© Marian Small, 2011
Diagnostic
Fractions (IS)
4
September 2011
© Marian Small, 2011
Fractions (IS)
(Continued)
13. Draw a picture to show why each statement is true:
8
2
a) ––
÷ –– = 4
10 10
8
3
2
÷ –– = 2–
b) ––
10 10
3
14. Divide:
6 2
÷–
a) –
9 9
5 2
b) –
÷–
8 8
9 3
c) –
÷–
4 8
8 5
d) –
÷–
3 6
3
5
e) ––
÷–
10 6
f)
1
1
1
3–
÷ 4–
2
3
1
15. A painter uses 2–
cans of paint to paint –
of a room. How much of a room could
2
4
he paint with 1 can of paint?
16. Write an equation involving fractions and an operation sign that you would
complete to solve the problem.
5
a) Mia read –
of her book. How much of her book does she have left to read?
8
5
1
b) Mia read –
of her book. She read –
of that amount on Monday. What fraction
8
3
of the whole book did she read on Monday?
5
1
c) Mia read –
of a book. If she read –
of the book each hour, how many hours
5
8
was she reading?
5
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September 2011
© Marian Small, 2011
September 2011
Fractions (IS)
© Marian Small, 2011 Fractions (IS)
USING INTERVENTION MATERIALS
The purpose of the suggested work is to help students build a foundation for working with
rational expressions, proportions, and slope and solving equations where fractions are
involved.
Each set of intervention materials includes a single-task Open Question approach and
a multiple-question Think Sheet approach. These approaches both address the same
learning goals, and represent different ways of engaging and interacting with learners.
You could assign just one of these approaches, or sequence the Open Question
approach before, or after the Think Sheet approach.
Suggestions are provided for how best to facilitate learning before, during, and after
using your choice of approaches. This three-part structure consists of:
• Questions to ask before using the approach
• Using the approach
• Consolidating and reflecting on the approach
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September 2011
© Marian Small, 2011 Fractions (IS)
Comparing Fractions
Learning Goal
• selecting a strategy to compare fractions based on their numerators and denominators.
Open Question
Questions to Ask Before Using the Open Question
Materials
• fraction strips or other
fraction materials
◊ How can you decide whether one fraction is greater than another? (e.g., You can
draw them and see which takes up more space.)
◊ What if the fractions were really close and you could not tell from the drawing?
(e.g., You could use equivalent fractions with the same denominator.) Then what?
(e.g., The one with the greater numerator is greater.)
◊ How could you tell that ​ _43 ​ > _​  35 ​without using common denominators or a drawing?
(e.g., _​ 43 ​is more than 1 whole and ​ _35 ​is not, so _​ 43 ​is greater.)
Using the Open Question
Make sure students understand that they must use four numbers for each
comparison, but they can use the same number twice. (This is required for fraction
pairs with the same numerator.) Also ensure that they explain their thinking to justify
their decisions.
By viewing or listening to student responses, note:
• how or whether they use benchmarks to compare fractions;
• whether they always use a common denominator to compare fractions;
• how they compare fractions with a common numerator;
• whether they are comfortable with improper fractions.
Consolidating and Reflecting on the Open Question
3
◊ What fractions did you use with the same numerator? (e.g., ​ _38 ​ and ​ _
12  ​) How did
3
1
_
_
you know which was greater? (e.g., I knew that ​ 12  ​is only ​ 4 ​and that is _​ 28 ​and so ​ _38 ​
3
2
_
is more.) Could you have known without changing ​ _
12  ​ to ​  8 ​? (e.g., Yes, I know that
3
eighths are bigger than twelfths, so ​ _38 ​has to be more than ​ _
12  ​.)
◊ Which of your fractions did you compare by using common denominators?
(e.g., When I compared _​ 38 ​ and ​ _49 ​, I got a common denominator of 72 and then
compared them.)
◊ Did you compare any fractions by deciding how they compared to ​ _12 ​? (Yes, e.g., I
9
9
1
_
_
knew that _​ 38 ​was less than ​ _12 ​and that _
​ 10
  ​was more than ​ 2 ​, so ​  10  ​was greater.)
Solutions
e.g.,
10
3
3
1
1 _
4
_
_
_
_
_
• _​  46 ​ > _
​  10
20 ​ I know that ​ 20 ​ is ​  2 ​and that ​ 6 ​is also ​ 2 ​. ​  6 ​is more sixths than ​ 6 ​, so it is greater.
4
• ​ _49 ​ > _
​  12
  ​ I know that ninths are bigger than twelfths since if you cut a whole into nine parts,
each part will be bigger than if you cut the whole into 12 parts. So four big parts is more
than four little parts.
6
6
12
_
_
• ​ _
I know that _
​ 12
6  ​ > ​  16  ​ 6  ​is 2 but ​ 16  ​is not even 1.
20
12
12
_
_
_
• _
​  20
4  ​ > ​  6  ​ I know that ​ 4  ​is 5, but ​ 6  ​is only 2.
3
3
_
• ​ _
20  ​ < ​  8 ​ I know that twentieths are smaller than eighths since, if you cut a whole into
20 parts, each part will be smaller than if you cut the whole into eight parts. So three
big parts is more than three little parts.
9
8
9
8
1
1
1
_
_
_
_
_
_
• ​ _
10  ​ > ​  9 ​ I know that ​ 10  ​is only ​ 10  ​away from 1 but ​ 9 ​ is ​  9 ​away. I know that ​ 10  ​away is closer
1
1
since _
​  10  ​ < _​  9 ​. I know that because, if you cut a whole into 10 pieces, each piece is smaller
than if you cut it into nine pieces.
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September 2011
© Marian Small, 2011 Fractions (IS)
Think Sheet
Materials
• fraction strips or other
fraction materials
Questions to Ask Before Assigning the Think Sheet
◊ How can you decide whether one fraction is greater than another? (e.g., You can
draw them and see which takes up more space.)
◊ Why is it important that you use the same size whole when you are drawing?
(e.g., You can make even a little fraction seem big if it is part of a big whole.)
◊ Do you think that ​ _35 ​ or _​  45 ​is more? ​( _​  45 ​  )​Do you need a drawing to tell? (e.g., No, since
I know that _​ 45 ​is more fifths.)
Using the Think Sheet
Read through the introductory box with the students; respond to any questions they
might have.
Make sure they realize that different comparisons suit different strategies.
Assign the tasks.
By viewing or listening to student responses, note:
• how or whether they use benchmarks to compare fractions;
• whether they always get a common denominator to compare fractions;
• how they compare fractions with a common numerator;
• whether they are comfortable with improper fractions;
• whether they gravitate toward using decimals to compare fractions.
Consolidating and Reflecting: Questions to Ask After Using the Think Sheet
6
◊ How can ​ _25 ​ = _
​  15
  ​even though 2 and 5 are only 3 apart, but 6 and 15 are 9 apart?
(e.g., It does not matter how far apart the numerator and denominator are; what
matters is what fraction of the denominator each numerator is.
6
3
6
_
_
◊ Which fraction would you rename to compare ​ _38 ​ and ​ _
17  ​? (e.g., I would rename ​ 8 ​ to ​  16  ​
6
6
_
and I know ​ _
16  ​ > ​  17  ​.)
◊ Think of a different way to compare each of these pairs of fractions: ​ _35 ​ and ​ _37 ​; _​  38 ​
9 _
2
4
_
and ​ _
10  ​; ​  5 ​ and ​  15  ​. Which way would you choose for each? (e.g., I would know that
fifths are more than sevenths and just know that ​ _35 ​ > _​  37 ​. I would know that ​ _38 ​is less
9
3
9
6
2
2
_
_
_
_
_
than ​ _12 ​ and ​ _
10  ​is more to compare ​ 8 ​ and ​  10  ​. I would change ​ 5 ​ to ​  15  ​to compare ​ 5 ​
4
and ​ _
15  ​).
Solutions
1.a)​ _25 ​ < _​  47 ​
b) _​  78 ​ > _​  56 ​
c) _​  57 ​ > _​  35 ​
d) _​  56 ​ < _​  89 ​
2.a)e.g.,
b) e.g.,
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September 2011
© Marian Small, 2011 Fractions (IS)
3. e.g., You can look at the models and see.
4.a)​ _69 ​since there are more ninths
b) _​  59 ​since each ninth is bigger than each eleventh and there are the same number
of ninths and elevenths.
8
5.a)e.g., _​  45 ​ as _
​  10
  ​since 8 tenths is more than 6 tenths
8
b) e.g., ​ _45 ​ as _
​  10
  ​since 8 tenths is more than 8 fifteenths (fifteenths are smaller)
28
9
27
14
_
_
_
c) ​  8 ​ as ​  24 ​ and ​ _
12 ​ as ​  24 ​(since 27 twenty-fourths is less than 28 of them)
d) ​ _83 ​ as _
​  16
6  ​(since 16 sixths is less than 17 sixths)
6.a)e.g., Division means sharing. If 9 people share 4 things, they each get ​ _19 ​of each of
the things; that is a total of _​ 49 ​for each person
b) _​  37 ​= 0.428571428571…. and ​ _49 ​= 0.4444….
4
4
_
c) Since ​ _37 ​is not even 43 hundredths and _
9 is more than 44 hundredths, ​ 9 ​is greater.
7.a)e.g., I know that _​ 83 ​is more than 1 and ​ _25 ​is less than 1, so _​ 83 ​is greater.
17
_
b) e.g., _​  73 ​is a bit more than 2, but _
​ 17
4  ​is more than 4, so ​ 4  ​is greater.
c) e.g., 3​ _15 ​is more since it is more than 3, but 2​ _13 ​is not even 3.
1
1
7
1
_
_
_
d) e.g., _
​  10
  ​is less since it is less than ​ 2 ​ and ​  9 ​is more than ​ 2 ​.
9
1
_
e) e.g., _​  78 ​is less since it is ​ _18 ​away from 1, but _
​ 10
  ​is only ​ 10  ​away from 1, which is closer
8
2
_
_
f) e.g., ​  9 ​is less since it is less than 1, but 1​ 3 ​is more than 1.
8.a)e.g., If you were deciding which of two groups was closer to reaching their
fund-raising goal, when one is ​ _35 ​of the way and the other is ​ _12 ​the way
b) e.g., If you were deciding which of two turkeys needed more cooking time if the
number of hours for one was 2​ _12 ​and for the other was 2​ _23 ​.
9. e.g., _​  23 ​ < _​  55 ​ < _​  84 ​
12
_
_
​  43 ​ < _
​  10
5  ​ < ​  4  ​
​ _13 ​ < _​  35 ​ < _​  34 ​
10.a) _​  73 ​, _​  83 ​
11
_
b) _​  94 ,​ _
​  10
4  ​, ​  4  ​
c) e.g., No since there will be fractions with each denominator and there are all the
possible counting numbers that could be denominators
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September 2011
© Marian Small, 2011 Fractions (IS)
Open Question
Comparing Fractions
Learning Goal
• selectingastrategytocomparefractionsbasedontheirnumeratorsand
denominators.
Open Question
• Choosetwopairsofnumbersfrom3,4,6,8,9,10,12,16,20touseasnumerators
10
4
and––
20.
anddenominatorsoftwofractions.Forexample,youcoulduse–
6
– Makesurethatsomeofyourfractionsareimproperandsomeareproper.
– Makesurethatsomeofyourfractionsusethesamenumeratorandsome
donot.
Tellwhichofyourfractionsisgreaterandhowyouknow.
• Makemorefractionsfollowingtherulesaboveandcompareatleastsixpairsof
them.Tellhowyouknowwhichfractionisgreatereachtime.
6
September 2011
© Marian Small, 2011
Fractions (IS)
Think Sheet
Comparing Fractions
(Continued)
Comparing Fractions
Think Sheet
(Continued)
Renaming Fractions
Pairs of fractions are either equal or one fraction is greater than the other fraction.
We can decide which statement is true by using a model, by renaming one or
both fractions, or by using benchmarks.
Using a Model
3
To rename a fraction, we can think about how to express the fraction as an
equivalent fraction or equivalent decimal.
• Equivalent Fractions
Two fractions are equivalent, or equal, if they take up the same part of a
whole or wholes.
5
• Tocompare–
5 and –
,wecanuseapicturethatshowsthetwofractionslined
7
up,sowecanseewhichextendsfarther.
3
_
5
3
3
6
Each section of the –
model is split into two sections in the ––
model. So, there
5
10
are twice as many sections in the second model, and twice as many are
shaded. The numerator and denominator have both doubled.
3
_ < 5
_
5 7
5
_
7
3
_
5
6
_
10
6
= ––.
For example, –
5 10
3
_
5
We can multiply the numerator and denominator by any amount (except 0)
and the same thing happens as in the example above. There are more sections
shaded and more sections unshaded, but the amount shaded does not change.
5
_
7
3
3×3
9
For example, if we multiply by 3: –
= ––– = ––.
5 3 × 5 15
• Wecanusepartsofsets.Forexample,useanumberofcounters,suchas35,
thatiseasytodivideintobothfifthsandsevenths.
or
3
1
– of 35 counters is 7 counters, so – of 35 counters is 21 counters.
5
5
3
10 × 3
30
= ––– = ––.
If we multiply by 10: –
5 10 × 5 50
• Common Denominators and Common Numerators
Renaming fractions to get common denominators or common numerators
is helpful when comparing fractions. If we compare fractions with the same
denominator, the one with the greater numerator is greater. If we compare
fractions with the same numerator, the one with the lesser denominator is greater.
5
2
To determine if –
or –
is more, we might use common denominators:
8
3
5
1
– of 35 counters is 5 counters, so – of 35 counters is 25 counters.
7
7
5
3
25 is more than 21, so –
> –.
7 5
7
12
September 2011
© Marian Small, 2011
September 2011
Fractions (IS)
8
5 10 15
– = –– = ––
8 16 24
8
10 12 14 16
2 4 6
– = – = – = –– = –– = –– = –– = ––
3 6 9 12 15 18 21 24
5 15
2 16
Since –
= –– and –
= ––, 24 is a common denominator.
8 24
3 24
5 2
Since 15 < 16, then –
< –.
8 3
5 10
– = ––
8 16
10
8
2 4 6
– = – = – = –– = ––
3 6 9 12 15
5 10
2 10
Since –
= –– and –
= ––, 10 is a common numerator.
8 16
3 15
10 10
2 5
Sincefifteenthsarebiggerthansixteenths,––
15 > ––
, and –
> –.
16
3 8
September 2011
© Marian Small, 2011 © Marian Small, 2011
Fractions (IS)
Fractions (IS)
Comparing Fractions
(Continued)
• Decimals
Comparing Fractions
(Continued)
1. What fraction comparison is being shown?
a)
Anotherwaytorenameafractionistorepresentitasadecimal.
Forexample,
2
–
=2÷3.Usingacalculator,weseethat2÷3=0.6666…
3
b)
Thereason–
b =a÷bisbecausewecanimagineaobjects,eachbeingshared
bybpeople.Eachpersongetsonesectionofeachobject,soaltogethereach
1
a
.
getsasectionsofsize–
;thatis–
b
b
c)
Forexample,iffourpeoplesharethreebarseachonegetsonefourthfrom
a
3
eachofthe3wholebarsandthatis–
.
4
d)
2
4
Wecanusedecimalrenamingtocomparefractions,suchas–
to–.
5
3
2
4
2
4
Since–
=0.6666…and– =0.8,weseethat–
3 <–
5.
5
3
2. Draw a picture to show why this statement is true.
Using Benchmarks
3
6
a) –
= ––
8 16
Wecanusebenchmarkstocomparefractions.
• Sometimeswecantellthatonefractionismorethananotherbycomparing
17
3
17
3
4 islessthan1,so––
12>–
4.
themto1.Forexample,––
12ismorethan1,and–
5 10
b) –
= ––
4
8
• Sometimeswecantellthatonefractionismorethananothersinceoneis
greaterthanonehalfandoneisnot.Wecanmentallyrenameonehalfto
comparetheotherfractionstoit.
5
3
3 5
4
3. Tell or show why –
is not equivalent to –
.
8
9
Forthefraction–
,sinceonehalfis–,–
ismorethanonehalf.
6
6 6
3
5 3
,––
islessthanonehalf.
Forthefraction––
10,sinceonehalfis––
10 10
5
3
>––
.
So,–
6 10
4. Circle the greater fraction. Explain why.
• Sometimeswecantellonefractionismorethananotherbycomparing
themtowholenumbersotherthan1.
5
6
or –
a) –
9
9
8 11
11
Forthefraction––
4,since2is–
4,––
4ismorethan2.
5
6 5
–
–
Forthefraction3,since2is3,–
3 islessthan2.
11
5
5
b) –
or ––
11
9
5
3.
So,––
4ismorethan–
9
September 2011
© Marian Small, 2011
Fractions (IS)
Comparing Fractions
(Continued)
5. How might you rename one or both fractions as other fractions to make it easier
to compare them? Tell how it helps.
10
September 2011
Comparing Fractions
8
2
a) –
and –
5
3
1
8
4
and ––
b) –
5
15
(Continued)
b)
7
17
– and ––
3
4
7
9
1
10
1
c) 2–
and 3–
5
3
d) –– and –
7
9
e) –
and ––
8
10
f)
8
2
– and 1–
9
3
8. Describe a real-life situation when you might make each comparison:
2
3
3
1
a) –
to –
5
2
d)
Fractions (IS)
7. How can each of these fraction pairs be compared without renaming them as
other fractions or as decimals?
6
4
a) –
and ––
5
10
9
14
and ––
c) –
8
12
© Marian Small, 2011
1
2
b) 2– to 2–
8
17
– and ––
3
6
9. A fraction with a denominator of 5 is between one fraction with a denominator of 3
and one fraction with a denominator of 4. Fill in all 9 blanks to show three ways this
could be true.
4
= 4 ÷ 9?
6. a) Why does it make sense that –
9
–– < –– < ––
5
3
4
3
–– < –– < ––
5
3
4
–– < –– < ––
5
3
4
10. a) List all the fractions with a denominator of 3 between 2 and 3.
4
and –
?
b) What are the decimal equivalents of –
7
9
b) List all the fractions with a denominator of 4 between 2 and 3.
3
4
and –
?
c) How could you use decimal equivalents to compare –
7
9
c) Is it possible to list all the fractions between 2 and 3? Explain.
11
13
September 2011
© Marian Small, 2011
September 2011
Fractions (IS)
12
September 2011
© Marian Small, 2011 © Marian Small, 2011
Fractions (IS)
Fractions (IS)
Adding Fractions
Learning Goal
• selecting an appropriate unit and an appropriate strategy to add two fractions.
Open Question
Materials
• fraction strips or other
fraction materials
Questions to Ask Before Using the Open Question
◊ Why is it easy to add _​ 25 ​ + _​  25 ​? (e.g., It is just 4 sections that are fifths, so it is _​ 45 ​.)
◊ How would you add ​ _34 ​ and ​ _23 ​? (e.g., I would use a model, such as fraction strips.)
How would you do that? (e.g., I would look for equivalent strips for ​ _34 ​ and ​ _23 ​with the
same denominator. Then I would count how many sections were shaded.)
◊ Which way would you add ​ _23 ​ and ​ _23 ​? (e.g., The same way I did _​ 25 ​ + _​  25 ​.)
◊ How do you know that ​ _13 ​ + _​  14 ​is more than ​ _12 ​? (e.g., I know that ​ _14 ​ + _​  14 ​ is _​  12 ​ and ​ _13 ​is more
than ​ _14 ​, so the answer has to be more.)
Using the Open Question
Make sure students understand that they must create at least four fraction sums,
each time predicting why the sum would only be a bit more than 1, how they
calculated the sum, and a verification that the sum meets the required conditions.
Some students might choose to use models.
By viewing or listening to student responses, note if they:
• can estimate fraction sums;
• realize that if both fractions are slightly greater than ​ _12 ​, the sum will be slightly
greater than 1;
• realize that if one fraction is ​ _12 ​and the other slightly greater than half, the sum will
be greater than 1;
• realize that they can take any fraction, figure out how much less than 1 it is and add
a bit more;
• realize that they can start with a fraction greater than, but not a lot greater than, 1
and add a very small amount.
Consolidating and Reflecting on the Open Question
◊ Suppose your first fraction had been ​ _12 ​. How would you have picked the second
one? (e.g., I would pick something just a little more than ​ _12 ​.) What might it have
been and how would you have added? (e.g., It had to have an odd denominator,
so maybe ​ _35 ​. To add _​  35 ​ to _​  12 ​, I would write them both as tenths and count how many
tenths.)
◊ Could one of your fractions have been greater than 1? (Yes) How would you pick
your other one? (e.g., I would make it really small.)
◊ What strategy did you use most often to add your fractions? (e.g., I used equivalent
fractions with the same denominator.)
Solutions
e.g.,
Pair 1: _​  47 ​ + _​  12 ​ =
I knew that _​ 47 ​was just a little more than half since _​ 48 ​is half. Sevenths are bigger than eighths,
so ​ _47 ​is more than ​ _48 ​. I knew that if I added exactly _​ 12 ​, I would be just a little more than 1.
15
8
1
7
_
_
_
The sum is ​ _
14 ​which equals 1​ 14  ​. I got the sum by using equivalent fractions: ​ 14  ​ and ​  14  ​.
1
_
​  14  ​is not much, so it is just a little more than 1.
14
September 2011
© Marian Small, 2011 Fractions (IS)
Pair 2: _​  35 ​ + _
​  12
25 ​
3
12
1
1
_
_
_
I knew that ​ _
25 ​was just a little under ​ 2 ​and thought that ​ 5 ​was enough more than ​ 2 ​that the sum
would be more than 1.
3
15
12
27
2
_
_
_
_
_
_
​  15
25 ​ + ​  25 ​ = ​  25 ​= 1​ 25  ​. I got the sum by writing ​ 5 ​as the equivalent fraction ​ 25 ​.
2
_
​  25
  ​is not much, so the sum is just a little more than 1.
Pair 3: _​  56 ​ and ​ _15 ​.
I knew that _​ 56 ​+ _​  16 ​would be exactly 1 and I knew that ​ _15 ​was just a bit more than ​ _16 ​so the sum
would be just a bit more than 1.
5
25
1
1
_
_
_
_
_
​  56 ​ + _​  15 ​ = _
​  31
30 ​= 1​ 30  ​. I got the sum by writing ​ 6 ​as the equivalent fraction ​ 30 ​ and ​  5 ​as the equivalent
6
31
_
_
fraction ​  30  ​. The sum is ​ 30 ​.
1
_
Pair 4: _
​  10
   ​. 
9  ​ + ​  100
10
1
_
I knew that ​ 9  ​is already just a little more than 1, but that ​ _
   ​ is so small that adding it keeps the
100
answer close to 1.
109
1
1
1
_
_
_
The sum is 1​ _
​. I wrote _
​ 10
    
​using the common denominator of 900.
900  
9  ​as 1 + ​ 9 ​. I added ​ 9 ​ and ​  100
100
9
109
_
_
_
The equivalent fractions were ​ 900  ​ + ​  900    ​ = ​  900  ​. But there was still the 1 whole.
109
109
​ _
​is close to _
​ 100
​which is ​ _19 ​and that is not much more than 0, so 1​ _
​is not that much more
900  
900  
900  
than 1.
15
September 2011
© Marian Small, 2011 Fractions (IS)
Think Sheet
Materials
• fraction strips or other
fraction materials
• grids and counters
Questions to Ask Before Assigning the Think Sheet
◊ Why is it easy to add _​ 25 ​ + _​  25 ​? (e.g., It is just 4 sections that are fifths, so it is _​ 45 ​.)
◊ How would you add ​ _34 ​ and ​ _23 ​? (e.g., I would use a model, such as fraction strips.)
How would you do that? (e.g., I would look for equivalent strips for ​ _34 ​ and ​ _23 ​with the
same denominator. Then I would count how many sections were shaded.)
◊ Which way would you add ​ _23 ​ and ​ _23 ​? (e.g., The same way I did _​ 25 ​ + _​  25 ​.)
◊ How do you know that ​ _13 ​ + _​  14 ​is more than ​ _12 ​? (e.g., I know that ​ _14 ​ + _​  14 ​ is _​  12 ​ and ​ _13 ​is more
than ​ _14 ​, so the answer has to be more.)
Using the Think Sheet
Read through the introduction box with students and make sure students understand
the material.
Assign the tasks.
By viewing or listening to student responses, note if they:
• relate appropriate fraction models to fraction addition;
• recognize how to add fractions with the same denominator as well as those with
different denominators;
• can estimate fraction sums;
• add mixed numbers by adding the whole number parts and fraction parts
separately;
• relate situations to fraction additions;
• recognize why they cannot just add numerators and denominators.
Consolidating and Reflecting: Questions to Ask After Using the Think Sheet
◊ What different strategies could you use to add ​ _16 ​ to _​  78 ​? (e.g., I could use fraction
strips, or grids or equivalent fractions.)
◊ How do you know the result would be a little more than 1? (e.g., If you add _​ 18 ​ to _​  78 ​, it
would be 1, but ​ _16 ​is a little more than ​ _18 ​.)
◊ What size grid would you use? (e.g., 6 by 8) How would you use it? (e.g., I would
fill 7 of the 8 columns. Then I would fill 1 of the 6 rows. I would move any doubledup counters. I think I will need a second grid, though.)
◊ If you used equivalent fractions, what denominator would you use? (e.g., I would
use 24, since you can write eighths and sixths as twenty-fourths.)
Solutions
1.a)e.g., A whole has 3 rows and 5 columns. Each row is ​ _13 ​and each column is ​ _15 ​. _​  43 ​is four
rows and _​ 15 ​is one column.
8
_
_
b) ​  23
15 ​or 1​ 15  ​
2.a)​ _25 ​ + _​  25 ​
b) _​  38 ​ + _​  28 ​
c) _​  23 ​ + _​  18 ​
d) _​  38 ​ + _​  13 ​
e) _​  95 ​ + _​  43 ​
3.a)MORE than 1
b) LESS than 1
c) MORE than 1
d) MORE than 1
16
September 2011
© Marian Small, 2011 Fractions (IS)
4.a)​ _98 ​
b) _​  75 ​
c) _
​  31
40 ​
7
_
_
d) ​  22
15 ​or 1​ 15  ​
49
4
_
_
e) ​  15 ​ or 3​  15
  ​
4
_
f) 4​ 15  ​
Model for part c:
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Model for part e:
5.a)e.g., 3 and 9 since you could have changed the thirds to be ninths to combine them
with the other ninths
b) e.g., 9 and 9
6.a)e.g., _​  48 ​ + _​  14 ​
b) e.g., _​  66 ​ + _​  12 ​
7.a)e.g., because 1​ _12 ​+ 1​ _12 ​is 3 and ​ _23 ​ and ​ _34 ​are both more than ​ _12 ​
5
b) 3​ _
12  ​ cups
8. e.g., A baker used ​ _23 ​cup of sugar for a cake and ​ _34 ​cup of sugar for some cookies. How
much sugar was used?
9.a)​ _37 ​ + _​  59 ​ = _
​  62
63 ​
b) _​  93 ​ + _​  75 ​ = _
​  66
15 ​
10.a) yes
b) e.g., You can always get equivalent fractions with the same denominator if you
multiply the numerator and denominator of each fraction by the other denominator
and that is what Kyle did. Once the denominators are the same, you just combine
numerators.
11. e.g., I would use an example, such as ​ _23 ​ and ​ _34 ​. If you add numerators and denominators,
you get _​ 57 ​which is less than 1. But I know that _​ 23 ​ and ​ _34 ​are each almost 1, so the sum will
be more than 1.Open
17
Question
September 2011
© Marian Small, 2011 Fractions (IS)
Adding Fractions
Learning Goal
• selectinganappropriateunitandanappropriatestrategytoaddtwofractions.
Open Question
• Choosetwodifferent,nonequivalent,fractionstoaddtomeettheseconditions.
– Theirsumisalittlemorethan1.
– Theirdenominatorsaredifferent.
– Atleastonedenominatorisodd.
Tellhowyoupredictedthesumwouldbealittlemorethan1.
Calculatethesumandexplainyourprocess.
Verifythatthesumisjustalittlemorethan1.
• Repeatthestepsatleastthreemoretimeswithotherfractions.
13
September 2011
© Marian Small, 2011
Fractions (IS)
Think Sheet
Adding Fractions
(Continued)
Adding Fractions
Think Sheet
(Continued)
If the fractions had the same denominator, we could count sections.
1
That denominator has to be a multiple of both 3 and 4, so 12 is a possibility
because 3 × 4 = 12.
Same Denominators
3
1
4
1
– = –– and – = ––, so we add 4 twelfths + 3 twelfths to get 7 twelfths. In this
3 12
4 12
• Tocombinefractionswiththesamedenominatorwecancount.
3
5
5
8
picture, there are 7 sections shaded and each is a twelfth section.
+ –is3fourths+5fourths.Ifwecountthefourths,weget8
For example, –
4 4
3
8
fourths,so–
4 + –
= –. Another name for –
is 2.
4 4
4
3
5
8
1
_
4
1
1
3
_
12
7
1
So, –
+ – = ––.
4 3 12
Usingfractionpieces,wecanseethat–
+ – = –.Ifwecombinethelast–
4 part of
4 4 4
3
1
_
3
4
_
12
Noticeweaddthenumerators,notthedenominators,becauseweare
combiningfourths.Ifwecombinedthetwodenominators,wewouldget
eighthsnotfourths.
5
1
and –
that use the same denominator.
We can create equivalent fractions for –
3
4
Adding fractions means combining them.
8
the –
withthe–
,weseethat2wholesareshaded.Thatshowsthat–
4 is 2.
4
4
Use a Grid
• Anotherwaytocombinefractionsistouseagrid.
2
1
and –
, we could create a 3-by-5 grid to show thirds and
For example, to add –
5
3
3
_
4
2
1
– is two rows and – is one column.
5
3
fifths
5
_
4
2
_
3
Different Denominators
• Tocombinefractionswithdifferentdenominatorsrequiresmorethinking.For
1
1
_
5
1
example,lookatthispictureof–
+ –.
3 4
2
2
4 .
Thetotallengthisnotasmuchas–
3 butitismorethan–
1
.
show –
5
2
_
3
1
_
4
2
Usetheletterxtofilltworowstoshow–
.Usetheletterotofillacolumnto
3
Move counters to empty sections of the grid so each section holds only one
counter.
1
_
3
13
2
2
2
_
4
14
18
September 2011
1
13
Since ––
of the grid is covered, –
+ – = ––.
15
3 5 15
10
1
3
= –– and –
= ––, so this makes sense.
Notice that –
5 15
3 15
© Marian Small, 2011
September 2011
Fractions (IS)
15
x
x
x
x xo
x
x
x
x
o
x
x
x
x xo
x
x
x
x
o
o
x
x
September 2011
© Marian Small, 2011 © Marian Small, 2011
o
Fractions (IS)
Fractions (IS)
Adding Fractions
(Continued)
Adding Fractions
Improper Fractions
• Sometimesthesumoftwofractionsisgreaterthan1.
Tomodelthiswithfraction strips,wemightneedmorethanonewholestrip.For
(Continued)
To add improper fractions, we can use models or we can use equivalent fractions
with the same denominators and count.
3 3
example,toshow–
+–
wecanuse20asthecommondenominator.
5 4
9
3
_
_ 12
=
5 20
5
27
10
37
1
1
2
+ – = ––
+ ––
= ––
= 6–
.
For example, –
2 3
6
6
6
6
+ 1–
is close to 4 + 2 = 6.
We can check by estimating. 4–
2
3
Mixed Numbers
3
_
_ 15
=
4 20
8
Move––
20uptothefirststriptofillthewhole.Then,––
20isleftinthesecondstrip.
–
+–
=––
or1––
20
5 4 20
Wecanmodelthesumoftwofractionsthatisgreaterthan1withagrid. When
youmovethecounterssothereisonlyonecounterineachsection,thegridis
filled,with7countersextra.
3
3
27
3
13
1
1
+ 3–
= 5 ––
= 6 ––
2–
3
4
12
12
7
4
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
xo xo xo
x
x
o
o
o
o
o
xo xo xo
x
x
o
o
o
o
o
o
o
o
o
x
x
x
x
x
x
x
x
x
x
o
x
x
x
x
x
o
x
x
x
x
xo
x
xo xo xo
o
1
1. a) How does this model show –
+ –?
3 5
3 12
–
5 =––
20(markedwitho’s)
3 15
–
4 =––
20(markedwithx’s)
3
1
To add mixed numbers, such as 2–
+ 3–
, we could add the whole number parts
3
4
and fraction parts separately.
7
b) What is the sum?
2. What addition is the model showing?
a)
3 3
7
–
4 =1––
20
5 +–
b)
16
September 2011
© Marian Small, 2011
Fractions (IS)
Adding Fractions
(Continued)
17
September 2011
© Marian Small, 2011
Fractions (IS)
Adding Fractions
(Continued)
14
c)
5. The sum of two fractions is ––
.
9
a) What might their denominators have been? Explain.
d)
e)
x
x
x
xo xo xo xo xo
xo xo xo xo xo
xo xo xo xo xo
x
x
xo xo xo
x
x
x
x
x
x
x
x
o
o
o
o
o
o
o
o
o
o
o
o o
xo xo xo xo x
o o o o
o o o o
o
b) List another possible pair of denominators.
o
x
x
x
x
x
x
x x
x x
x x
x
x
x
x x x x
o o o o o
o o o o o
x
o o o o o
o o o o o
o o o o o
6. Choose values for the blanks to make each true.
o
o
3
a) ––
+ –– = –
8
4
3. Estimate to decide if the sum will be more or less than 1. Circle MORE than 1 or
LESS than 1.
5 2
a) –
+–
6 3
MORE than 1
LESS than 1
2 2
+–
b) –
7 5
MORE than 1
LESS than 1
c)
d)
3 3
–+–
8 4
7
3
–– + –
10 5
MORE than 1
LESS than 1
MORE than 1
LESS than 1
b)
3
–– + –– = –
6
2
3
2
7. Lisa used 1–
​cups​of​flour​to​make​cookies​and​another​1​–
​4 ​cups​of​flour​to​
3
bake a cake.
a)​ How​do​you​know​that​she​used​more​than​3​cups​of​flour​for​her​baking?
b)​ How​much​flour​did​she​use?
4. Add each pair of fractions or mixed numbers. Draw models for parts c) and e).
2 7
a) –
+–
8 8
4 3
b) –
+–
5 5
2
3
+ – to solve it. Solve your
8. Write a story problem that would require you to add –
3 4
problem.
18
19
2 3
+–
c) –
5 8
4 2
d) –
+–
5 3
8 3
+–
e) –
3 5
f)
September 2011
3
2
2–
+ 1–
5
3
© Marian Small, 2011
September 2011
Fractions (IS)
19
September 2011
© Marian Small, 2011 © Marian Small, 2011
Fractions (IS)
Fractions (IS)
Adding Fractions
(Continued)
9. a) Use the digits 3, 5, 7 and 9 in the boxes to create the least sum possible.
–– + ––
b) Use the digits again to create the greatest sum possible.
–– + ––
10. Kyle noticed that if you add two fractions, you get the sum’s denominator by
multiplying the denominators and you get the sum’s numerator by multiplying each
3
7
+ –,
numerator by the other fraction’s denominator and adding. For example, for –
5 8
the denominator is 5 × 8 and the numerator is 3 × 8 + 7 × 5.
a) Do you agree?
b) Explain why or why not.
11. You have to explain why it does not work to add numerators and denominators
to add two fractions. What explanation would you use?
20
20
September 2011
© Marian Small, 2011
September 2011
Fractions (IS)
© Marian Small, 2011 Fractions (IS)
Subtracting Fractions
Learning Goal
• selecting an appropriate unit and an appropriate strategy to subtract two fractions
Open Question
Materials
• fraction strips or other
fraction materials
Questions to Ask Before Using the Open Question
◊ How would you subtract ​ _26 ​ from _​  56 ​? (e.g., I would start with _​ 26 ​and count how many
more sixths to get to ​ _56 ​.)
◊ How would you subtract ​ _23 ​ from _​  34 ​? (e.g., I would use a model, such as fraction
strips.) How would you do that? (e.g., I would look for equivalent strips for ​ _34 ​ and ​ _23 ​
with the same denominator. Then I would count how many more sections were
shaded in the ​ _34 ​than in the ​ _23 ​.)
◊ Which way would you subtract ​ _23 ​ from _​  53 ​? (e.g., More as I did _​ 56 ​ – _​  26 ​.)
◊ How do you know that ​ _78 ​ – _​  39 ​is close to _​ 12 ​? (e.g., I know that _​ 78 ​ – _​  38 ​ is _​  12 ​ and ​ _39 ​is close
to _​  38 ​, so it should be close to ​ _12 ​.)
Using the Open Question
Make sure students understand that they must create at least four fraction
differences, each time predicting why the difference would be close to 1, explaining
how they calculated the difference and verifying the fact that the difference meets the
criteria required. Some students might choose to use models.
By viewing or listening to student responses, note if they:
• can estimate fraction differences;
• realize that if one fraction is slightly greater than 1, they should not subtract too much;
• realize that if one fraction is close to, but less than 1, the other fraction should be
fairly small;
• realize that the original fraction cannot be too much less than 1;
• realize that they can use improper fractions whose mixed number equivalents have
wholes about 1 apart.
Consolidating and Reflecting on the Open Question
◊ Suppose your first fraction had been ​ _54 ​. How would you have picked the second
one? (e.g., I would pick something just a little more or less than ​ _14 ​, since _​  54 ​ is _​  14 ​
over 1.) What might it have been and how would you have subtracted? (e.g., It
had to have an odd denominator, so maybe _​ 15 ​. To subtract _​ 15 ​ from _​  54 ​, I would use
equivalent decimals.)
◊ Could one of your fractions have been greater than 3? (Yes) How would you pick
your other one? (e.g., I would choose a fraction near 2.)
◊ What strategy did you use most often to subtract your fractions? (e.g., I used
equivalent fractions with the same denominator.)
Solutions
e.g.,
Pair 1: _​  75 ​ – _​  12 ​
I knew that _​ 75 ​was _​ 25 ​more than 1, so I wanted to subtract a little more than ​ _25 ​. _​  12 ​works since it is 2​ _12 ​fifths.
9
5
14
_
_
_
​  75 ​ – _​  12 ​ = _
​  10
  ​. I used equivalent fractions that were tenths (​ ​  10 ​ and ​  10  ​  )​.
9
10
_
_
​  10
  ​is almost ​ 10 ​so it is just a little less than 1.
21
September 2011
© Marian Small, 2011 Fractions (IS)
1
_
Pair 2: _
​  24
   ​ 
25 ​ – ​  100
24
1
_
I knew that ​ 25 ​is already close to 1 and ​ _
   ​ is not very much, so if you subtract it, the answer will still be close to 1.
100
95
24
1
24
_
_
_
​  25 ​ – ​  100    ​ = ​  100  ​,  I used a hundredths grid and shaded in nine columns and 6 more squares (96 squares) for ​ _
25 ​ and
then I removed the shading on one square, so there were 95 squares left shaded.
Pair 3: _​  32 ​ – _​  35 ​
I knew that _​ 32 ​ is _​  12 ​more than 1, so I wanted to take away more than ​ _12 ​, but not a lot more. That
is why I chose ​ _35 ​.
9
_
​  32 ​ – _​  35 ​ = _
​  10
  ​
2
3
_
I figured it out using fraction strips. It takes 2 strips to represent _
2 . I saw that there were ​ 5 ​ left
5
9
1
2
1
4
_
_
_
_
_
_
on the first strip and then another ​ 2 ​on the second strip. I know that ​ 5 ​ + ​  2 ​ = ​  10  ​ + ​  10  ​ = ​  10  ​
9
_
10  
​  ​is almost a whole, so it is close to 1.
Pair 4: 2​ _13 ​– 1​ _14 ​.
I know that 2​ _13 ​is just a bit more than 2 and 1​ _14 ​is a bit more than 1, so the difference should be
close to 1.
1
2​ _13 ​– 1​ _14 ​= 1​ _
12  ​
I subtracted the whole number parts and then the fraction parts. I used a grid for the fraction
parts.
1
1
_
1​ _
12  ​is close to 1 since ​ 12  ​is quite small.
22
September 2011
© Marian Small, 2011 Fractions (IS)
Think Sheet
Materials
• fraction strips or other
fraction materials
• grids and counters
Questions to Ask Before Assigning the Think Sheet
◊ How would you subtract ​ _26 ​ from _​  56 ​? (e.g., I would start with _​ 26 ​and count how many
more sixths to get to ​ _56 ​.)
◊ How would you subtract ​ _23 ​ from _​  34 ​? (e.g., I would use a model, such as fraction
strips.) How would you do that? (e.g., I would look for equivalent strips for ​ _34 ​ and ​ _23 ​
with the same denominator. Then I would count how many sections were shaded in
the _​  34 ​than the _​ 23 ​.)
◊ Which way would you subtract ​ _23 ​ from _​  53 ​? (e.g., More as I did _​ 56 ​ – _​  26 ​.)
◊ How do you know that ​ _78 ​ – _​  39 ​is close to _​ 12 ​? (e.g., I know that _​ 78 ​ – _​  38 ​ is _​  12 ​ and ​ _39 ​is close
to _​  38 ​, so it should be close to ​ _12 ​.)
Using the Think Sheet
Read through the introductory box with students and make sure students understand
the material explained in the instructional box.
Assign the tasks.
By viewing or listening to student responses, note if they:
• relate appropriate fraction models to fraction subtraction;
• recognize how to subtract fractions with the same denominator as well as those
with different denominators;
• can estimate fraction differences;
• subtract mixed numbers by using the whole number parts and fraction parts separately;
• relate situations to fraction subtractions;
• recognize why they cannot just subtract numerators and denominators.
Consolidating and Reflecting: Questions to Ask After Using the Think Sheet
◊ What different strategies could you use to subtract ​ _16 ​ from _​  78 ​? (e.g., I could use
fraction strips, grids, or equivalent fractions.)
◊ How do you know the result would be a little less than ​ _34 ​? (e.g., If you subtract _​ 18 ​, it
would be ​ _68 ​, which is ​ _34 ​, but _​ 16 ​is a little more than ​ _18 ​to take away.)
◊ What size grid would you use? (e.g., 6 by 8) How would you use it? (e.g., I would
fill 7 of the 8 columns. Then I would move one counter to fill a row and remove the
counters in one row.)
◊ If you used equivalent fractions, what denominator would you use? (e.g., I would
use 24, since you can write eighths and sixths as twenty-fourths)
Solutions
1.a)e.g., each row is ​ _13 ​of a grid and 4 rows are covered and one column is removed;
each column is ​ _15 ​of a grid.
2
b)
1​ _
15  ​
2.a)​ _56 ​ – _​  16 ​
5
7
_
b) _
​  12
  ​ – ​  12  ​
2
4
c) ​ _5 ​ – _​  3 ​
d) _​  56 ​ – _​  25 ​
e) _​  83 ​ – _​  35 ​
3.a)closer to _​ 12 ​
b) closer to 1
c) closer to ​ _12 ​
d) closer to ​ _12 ​
23
September 2011
© Marian Small, 2011 Fractions (IS)
4.a)​ _65 ​
5
b) _
​  24
  ​
6
c) _
​  35
  ​
5
17
_
d) ​  12 ​= 1​ _
12  ​
e) 1​ _16 ​
13
f) 2​ _
40 ​
Example 4 c)
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Example 4 f)
1
1
6
0
1
2
3
3 56
4
5
6
5.a)e.g., 4 and 3 since fourths and thirds can both be written as twelfths
b) e.g., 12 and 2
6.a)e.g., ​ _43 ​ – _​  12 ​
1
_
b) e.g., _
​  17
12 ​ – ​  12  ​
7.a)e.g., about 1 cup
b) 1​ _13 ​ cups
8. e.g., I had 3​ _13 ​cups of flour and used 1​ _12 ​cups for a cake. How much flour do I have left?
9.a)e.g., 3​ _79 ​– 2​ _56 ​
b) e.g., 9​ _67 ​ – 3​ _25 ​
10.a) yes
b) e.g., When you do that, you are just getting equivalent fractions for the original
fractions since you are multiplying the numerator and denominator by the same
thing — the other denominator.
1
11. e.g., I would use an example: I know that ​ _13 ​ – _​  14 ​ = _
​  12
  ​, but if I subtracted numerators and
0
_
denominators, I would get -1 , which, I know, is wrong.
24
September 2011
© Marian Small, 2011 Fractions (IS)
Open Question
Subtracting Fractions
Learning Goal
• selectinganappropriateunitandanappropriatestrategytosubtracttwo
fractions.
Open Question
• Choosetwofractionstosubtracttomeettheseconditions.
– Thedifferenceiscloseto1,butnotexactly1.
– Theirdenominatorsaredifferent.
– Onedenominatorisodd.
Tellhowyoupredictedthedifferencewouldbecloseto1.
Calculatethedifferenceandexplainyourprocess.
Verifythatthedifferenceiscloseto1.
• Repeatthestepsatleastthreemoretimeswithotherfractions.
21
September 2011
© Marian Small, 2011
Fractions (IS)
Think Sheet
Subtracting Fractions
(Continued)
Subtracting Fractions
Think Sheet
1
5
wecouldcreatea4-by-6gridtoshowbothfourthsandsixths.–
6 isfiveofsix
5
1
columnsand–
4 isoneoffourrows.Usingcounterstomodel–
6,werearrange
• Tosubtractfractions with the same denominator we count.
1
themsothat–
ofthecounters(onefullrow)canberemoved.Since14sections
4
5 3
– –is5fourths–3fourths.Sincethedenominatorsarefourths,if
For example, –
4 4
5 3 2
1
wetake3from5,thereare2left,so–
– – = – or –
.
4 4 4
2
14
remainfull,thedifferenceis––
24.
Wesubtractedthenumeratorsbecausethefourthsweretherejusttotellthe
sizeofthepieces.Nothingwasdonetothe4s.
( )
5 3 2 1
Wecanseethat–
4 – –
= – – usingfractionpieces.
4 4 2
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
5
_
_ 20
=
6 24
2
1
5
_
3
2
1
.
extendsbeyondthe–
4
Itishardtotellhowexactlylongthedarkpieceisbylookingatthepicture.
Sowecancreateequivalentfractionsfor–
5 and –
thatusethesame
4
2
25
25
12
8
2
_ _
=
5 20
5
bycovering5rows.
Usingtwo3-by-5gridstosolvethesamequestion,cover–
3
September 2011
13
4
Removecountersfrom4columnstoremove–
5 ofagrid.––
15ofagridareleft
covered.
5
1
_ _
=
4 20
© Marian Small, 2011
12
13
September 2011
4
––
–––
=––
15 15 15
5
_
3
3
2 1
– – – = ––
5 4 20
22
25
Renamingthefractionsasfifteenthscanhelp.–
=––
–
5 =––
15
3 15
5twentiethsfrom8twentieths,
leaving3twentieths.
Inthediagramthereare3extra
sectionsinshadedontopand
eachisatwentiethsection.
5
1
denominator.Thatdenominatorhastobeamultipleofboth5and4,so
20(5×4)isapossibility.
x
Ifoneofthefractionsisgreaterthan1,morethanonestriporgridmayneed
5 4
5:
tobeused.Forexample,toshow–
3 ––
.Thedifferenceis—thepartofthe–
5 that
example,lookatthepictureof–
5 – –
4
8
5
2
1
– = –– and – = ––,sowesubtract
5 20
4 20
x
20
6
14
_ – _
_
=
24
24 24
• Tosubtractfractions with different denominatorsrequiresanumberofsteps.For
5
• Tosubtractfractions,wecanuseagrid.Forexample,tosubtract–
4 from–
,
6
Subtracting can involve take away, comparing, or looking for a number to add.
(Continued)
Fractions (IS)
23
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
September 2011
© Marian Small, 2011 © Marian Small, 2011
Fractions (IS)
Fractions (IS)
Subtracting Fractions
(Continued)
3
1
• Tosubtractmixed numbers,suchas4–
3 –1–
4,wecouldaddup.
3
1
Subtracting Fractions
(Continued)
2. What subtraction is the model showing?
1
a)
1
to1–
4,wegetto2.Ifweaddanother2–
3 wegetto4–
3,
Forexample,ifweadd–
4
3 1
7
1
1
–1–
4 =–
4 +2–
3 =2––
12.
so4–
3
1
2_
3
1
_
4
0
3 2
1_
4
1
b)
3
4
_
41
3
1
c)
3
Wecouldsubtractthewholenumberpartsandthefractionparts.Since–
3 >–
4,it
1
4
as3+–
3 first.
mightmakesensetorename4–
3
9
1
4
4 3 16
3 –
3 ––
4 =––
12–––
12
4–
3 =3–
4
–1–
4 =1–
3
d)
3
7
2––
12
4 1
1. a) Howdoesthismodelshow–
3 ––
5?
e)
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
4
1
x
x
x
x
x
1
3. Estimate to decide if the difference will be closer to –
or 1. Circle your choice.
2
b) Howmuchis–
3 ––
5?
24
September 2011
© Marian Small, 2011
Fractions (IS)
Subtracting Fractions
(Continued)
4. Subtract each pair of fractions or mixed numbers. Draw models for parts c) and f).
9 3
a) –
––
5 5
7 2
b) –
––
8 3
5 1
a) –
––
6 4
closer to –
2
7 7
b) –
––
3 5
closer to –
2
closer to 1
c)
3
1
– – ––
5 10
1
closer to –
2
closer to 1
d)
8 7
–––
5 6
1
closer to –
2
closer to 1
25
c)
d)
1
7
f)
© Marian Small, 2011
Subtracting Fractions
closer to 1
Fractions (IS)
(Continued)
9. a) Use the digits 2, 3, 5, 6, 7 and 9 in the boxes to create a difference that is
less than 1.
11 9
–– – –
3
4
––
b) Use the digits again to create a difference greater than 5.
–– –
e) 4 –
– 1–
5
8
1
September 2011
–– –
3 3
–––
5 7
1
––
10. Kyla noticed that if you subtract two fractions, you get the denominator of
the answer by multiplying the denominators and you get the numerator of the
answer by multiplying each numerator by the other fraction’s denominator and
8 4
subtracting. For example, for –
– –, the denominator is 5 × 3 and the numerator is
5 3
8 × 3 – 4 × 5.
5
5 – 3–
6
11
.
5. The difference of two fractions is ––
12
a) What do you think their denominators might have been? Explain.
a) Do you agree?
b) Explain why or why not.
b) List another possible pair of denominators.
6. Choose values for the blanks to make each equation true.
5
a) ––
– –– = –
3
6
11. You have to explain why it does not work to subtract numerators and denominators
to subtract two fractions. What explanation would you use?
1
b) ––
– –– = 1–
12
3
2
7. Sakurastartedwith3cupsofflour.Sheused1–
cups for one recipe.
3
a) Abouthowmuchflourdidshehaveleft?
b) Howmuchflourdidshehaveleft?
1
1
from 3–
to solve it.
8. Write a story problem that would require you to subtract 1–
2
3
Solve the problem.
26
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September 2011
© Marian Small, 2011
September 2011
Fractions (IS)
27
September 2011
© Marian Small, 2011 © Marian Small, 2011
Fractions (IS)
Fractions (IS)
Multiplying Fractions
Learning Goal
• representing multiplication of fractions as repeated addition or determining area.
Open Question
Materials
• grid paper
Questions to Ask Before Using the Open Question
◊ What is _​ 13 ​ of _​  35 ​? (​ _​  15 ​  )​Why do you think you might write that as ​ _13 ​ × _​  35 ​? (e.g., 4 × _​ 35 ​
means 4 of them, so _​ 13 ​ × _​  35 ​should be ​ _13 ​of them.)
◊ How would you multiply ​ _35 ​ and ​ _43 ​? (e.g., I would think of _​ 43 ​as 1 + _​ 13 ​and multiply 1
× _​  35 ​to get _​ 35 ​and add it to ​ _13 ​ × _​  35 ​. I know that _​ 13 ​ × _​  35 ​ means ​ _13 ​ of _​  35 ​, so it is _​ 15 ​. The answer,
then is ​ _35 ​ + _​  15 ​ = _​  45 ​.)
◊ Would you multiply ​ _23 ​ × _​  23 ​the same way? (e.g., No, I would draw a rectangle and
divide it into equal sections and figure out what part of the whole a rectangle ​ _23 ​ × _​  23 ​is.)
◊ How do you know that ​ _43 ​ × _​  35 ​is more than ​ _35 ​, but not a lot more? (e.g., I know that
1 × ​ _35 ​ is _​  35 ​. Since _​ 43 ​is more than 1, then ​ _43 ​ × _​  35 ​is more _​ 35 ​s. But ​ _43 ​is not a lot more than
1, so ​ _43 ​ × _​  35 ​is not a lot more than ​ _35 ​.)
Using the Open Question
Make sure students understand that they must create at least four fraction products,
each time predicting why the product would be a bit more than 1, how they calculated
the product and a verification that the product is the right size. Some students might
choose to use models.
By viewing or listening to student responses, note if they:
• can estimate fraction products;
• realize that if one fraction is slightly greater than 1, they should multiply by
something close to 1;
• realize that if one fraction is close to, but less than 1, the other fraction should be a
bit more than 1;
• realize that they can use any mixed number if they multiply by a unit fraction with a
denominator equal to the whole number part of the mixed number (e.g., 4​ _13 ​ × _​  14 ​)
Consolidating and Reflecting on the Open Question
◊ Suppose your first fraction had been ​ _54 ​. How would you have picked the second
one? (e.g., I would pick something really close to 1, maybe a bit less.) What might
9
it have been and how would you have multiplied? (e.g., I might have picked ​ _
10  ​ and
then just multiplied numerators and denominators.)
◊ Could one of your fractions have been greater than 3? (Yes) How would you pick
your other one? (e.g., I would choose a fraction that is about ​ _13 ​.)
◊ How could you multiply 3​ _13 ​ × _​  25 ​? (e.g., I know that you can add _​ 25 ​× 3 to _​ 25 ​ × _​  13 ​, so I
would do that.)
Solutions
e.g.,
Pair 1: _​  85 ​ × _​  23 ​
I started with _​ 85 ​. I realized that if I multiplied by ​ _12 ​, it would be ​ _45 ​and that was too low, so I
picked a fraction more than ​ _12 ​but not more than 1.
​ _85 ​ × _​  23 ​ = _
​  16
15 ​.
16
1
_
_
​  15 ​= 1​ 15  ​which is just a little more than 1.
27
September 2011
© Marian Small, 2011 Fractions (IS)
Pair 2: _​  25 ​ × 2​ _35 ​
2
I started with _​ 25 ​. I knew that double was only _​ 45 ​, so I needed more than _
5 x 2. I thought about
1
2
_
_
2​ 2 ​times but it would be exactly 1, so I went higher than ​ 5 ​for the fraction part.
26
1
_
_
_
​  25 ​ × 2​ _35 ​ = _​  25 ​ × _
​  13
5  ​ = ​  25 ​= 1​ 25  ​.
1
1
_
​  25  ​is not very much, so 1​ _
25  ​is just a little more than 1.
Pair 3: 3​ _13 ​ × _​  13 ​
I started with 3​ _13 ​which is more than 3. I need _​ 13 ​of it to be close to 1.
3​ _13 ​ × _​  13 ​= 1​ _19 ​. I took _​ 13 ​of 3, which I know is 1, and ​ _13 ​ of _​  13 ​ is _​  91 ​since there are three 3s in 9.
1​ _19 ​just a little more than 1, since ​ _19 ​is not much.
Pair 4: _​  89 ​ × _
​  10
8  ​
I knew that ​ _89 ​ × _​  98 ​is 1 (since ​ _89 ​ × _​  98 ​ = _
​  72
72 ​= 1). So I needed the second fraction to be a bit more
9
10
_
_
than ​  8 ​. That means I could use ​ 8  ​.
10
8
8
1
1
1
_
_
_
_
_
_
_
​  89 ​ × _
​  10
8  ​ = ​  9  ​ I know that ​ 8 ​ of ​  9 ​ = ​  9 ​since there are 8 parts in ​ 9 ​and 1 out of 8 of them is ​ 9 ​. That
10
10
_
_
means that ​ 8  ​is 10 times as much; that is ​ 9  ​.
10
1
1
_
_
I know that ​ _
9  ​= 1​ 9 ​ and ​  9 ​is not that much more than 1.
28
September 2011
© Marian Small, 2011 Fractions (IS)
Think Sheet
Materials
• grid paper
Questions to Ask Before Assigning the Think Sheet
◊ How much is ​ _23 ​ of _​  35 ​? (​ _​  25 ​  )​Why does that make sense? (e.g., _​ 13 ​ of _​  35 ​means 1 of the 3
sections in ​ _35 ​ and ​ _23 ​means 2 of the sections. The sections are fifths.)
◊ Why might you write ​ _23 ​ × _​  35 ​ for _​  23 ​ of _​  35 ​? (e.g., 4 × _​ 35 ​means 4 of them, so _​ 23 ​ × _​  35 ​ should
be ​ _23 ​of them.)
◊ About how much do you think ​ _23 ​ × _​  78 ​might be? (e.g., I think just a little less than ​ _23 ​
since ​ _23 ​× 1 is _​ 23 ​ and ​ _78 ​is less than 1.)
◊ How do you know that ​ _43 ​ × _​  35 ​is more than ​ _35 ​, but not a lot more? (e.g., I know that 1
× ​ _35 ​ is _​  35 ​. Since _​ 43 ​is more than 1, then ​ _43 ​ × _​  35 ​is more _​ 35 ​s. But ​ _43 ​is not a lot more than 1,
so ​ _43 ​ × _​  35 ​is not a lot more than ​ _35 ​.)
Using the Think Sheet
Read through the introductory box with students and make sure students understand
the material.
Assign the tasks.
By viewing or listening to student responses, note if they:
• relate appropriate fraction models to fraction multiplication;
• recognize how to multiply both improper and proper fractions;
• can estimate fraction products;
• relate situations to fraction multiplications;
• recognize why they can multiply numerators and denominators.
Consolidating and Reflecting: Questions to Ask After Using the Think Sheet
◊ What model could you draw to show what ​ _34 ​ × _​  45 ​ means? (e.g., I could use a ​ _45 ​
fraction strip and just use 3 out of the 4 shaded sections; I could also draw a
rectangle that has length ​ _45 ​and width ​ _34 ​. I would do that inside a 1 × 1 rectangle.)
◊ How do you know the result would be less than ​ _45 ​? (e.g., I am only taking a part of
the ​ _45 ​, so it has to be less.)
◊ How would you multiply 2​ _12 ​ × 2​ _12 ​? (e.g., I would change to ​ _52 ​ × _​  52 ​and just multiply the
numerators and denominators.) Why can you not just multiply the whole number
parts and the fraction parts separately? (e.g., If you draw the rectangle that is 2​ _12 ​by
2​ _12 ​, you would only be using two of the four sections in the rectangle.)
Solutions
20
4
_
1. e.g., I would multiply 5 × 4 and remember it is ninths, so it is ​ _
9  ​. If you added ​ 9 ​five times,
that is also what you would get.
2.a)e.g., The rectangle has a length of ​ _56 ​and a width of ​ _34 ​, so the area is the product.
b) _
​  15
24 ​
3.a)5 × _​ 23 ​
b) _​  23 ​ × _​  58 ​
c) _​  43 ​ × _​  85 ​
d) 1​ _23 ​× 2​ _25 ​
4.a)closer to _​ 12 ​
b) closer to 1
c) closer to ​ _12 ​
d) closer to 1
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September 2011
© Marian Small, 2011 Fractions (IS)
12
5.a)​ _
45 ​
6
_
b) ​  35
  ​
20
_
c) ​  9  ​
d) _
​  42
20 ​ ​
e) _
​  24
10 ​
1
f) 9​ _
15  ​
NOTE: Answers may be written as equivalent fractions. ​
Model for part b:
Model for part e:
1
1
3
_
5
1
_
2
20
6.a)e.g., _​  43 ​ × _​  52 ​since, if you multiply numerators and denominators, you get ​ _
6  ​.
2
10
_
_
b)
e.g., ​  3 ​ × ​  2  ​
7. e.g., _​  53 ​ × _​  54 ​
8.a)e.g., about 3 cups
28
1
_
b) ​ _
9  ​cups or 3​ 9 ​ cups
9. e.g., Jane read ​ _34 ​of her book. She read ​ _23 ​of that amount on Monday. What fraction of the
whole book did she read on Monday?
10.a) e.g., 5​ _34 ​ × 6​ _12 ​= 37​ _38 ​
1
b) e.g., 1​ _56 ​× 2​ _34 ​= 5​ _
24  ​
11. Yes, e.g., if you multiply _​ 23 ​ × _​  32 ​, you get 1.
12. e.g., I would say that if you are multiplying by ​ _23 ​, you are taking ​ _23 ​of something and that is
less than the whole thing; it is only part of it.
30
September 2011
© Marian Small, 2011 Fractions (IS)
Open Question
Multiplying Fractions
Learning Goal
• representingmultiplicationoffractionsasrepeatedadditionordeterminingarea.
Open Question
• Choosetwofractionsormixednumberstomultiplytomeettheseconditions.
– Theproductisalittlemorethan1.
– Ifamixednumberisused,thewholenumberpartisnot1.
Tellhowyoupredictedtheproductwouldbealittlemorethan1.
Calculatetheproductandexplainyourprocess.
Verifythattheproductiscloseto1.
• Repeatthestepsatleastthreemoretimeswithotherfractions.
28
September 2011
© Marian Small, 2011
Fractions (IS)
Think Sheet
Multiplying Fractions
(Continued)
Multiplying Fractions
Think Sheet
• Tomultiplytwomixednumbers,wecaneitherrewriteeachmixednumberas
animproperfractionandmultiply,orwecanbuildarectangle.Forexample,
2
• Wehavelearnedthat4x3means4groupsof3.Soitmakessensethat4x–
3
means4groupsoftwo-thirds:
2 2 2 2 2 8
4×–
=–
+–
+–
+–
=–
3 3 3 3 3 3
or
(Continued)
1
1
for3–
2 ×2–
3.
1
2_
3
2thirds+2thirds+2thirds+2thirds=8thirds
Wemultiply4×2togetthenumerator8.Thedenominatorhastobe3since
wearecombiningthirds.
Legend:
6
whole
3
_
3
• Tomultiplytwoproper fractions,wecanthinkoftheareaofarectanglewith
thefractionsasthelengthsandwidthsjustaswithwholenumbers.
Forexample,4×3isthenumberofsquareunitsina
rectanglewithlength4andwidth3.Thatisbecause
thereare4equalgroupsof3.
2
3
1
_
6
4 × 3 = 12
4
Tomultiply–
×–
,wethinkoftheareaofarectanglethat
3 5
2
2
_
2
1
3_
2
4
4
5 long.
is–
3 wideand–
2×4sectionsoutofthetotalnumberof3×5sectionsareinsidetherectangle.
–––
5
3 ×–
5 =–––
3×5
.
Thatmeanstheareais
3×5andso–
2×4
4
_
2
4
Theproductofthenumeratorstellsthenumberof
sectionsintheshadedrectangleandtheproduct
ofthedenominatorstellshowmanysectionsmakea
whole.
2
_
3
5
_
• Wecanmultiplyimproper
fractions
4
thesamewayasproperfractions.
1
_
4
1
5
1
whole
3
_
2
1
1
3
2
1
1
3–
×2–
3 =6+–
+–
+–
=8–
6
2
3 2 6
×2– isthesameas–
2 ×–
3 =––
6=8–
6.
3–
2
3
1
7
1
7
49
1
4
3
Forexample,–
×–
istheareaofarectangle
4 2
1
thatis1–
unitswideand1–
2 unitslong.There
4
1
2×4
are15sectionsintheshadedrectangleand
1. Howwouldyoucalculate5×–
?Whydoesyourstrategymakesense?
9
3
5
6?
2. a) Howdoesthismodelshow–
4 ×–
1
eachis–
8 of
5
3
5×3
2 =–––
4×2
.
1whole,sotheareais–
4 ×–
1
_
2
Theshadedpartofthediagramlooks
likeitmightbecloseto2wholessothe
answerof––
makessense.Eachshaded
8
b) Whatistheproduct?
15
squareis1whole.
29
31
September 2011
© Marian Small, 2011
September 2011
Fractions (IS)
30
September 2011
© Marian Small, 2011 © Marian Small, 2011
Fractions (IS)
Fractions (IS)
Multiplying Fractions
(Continued)
3. What multiplication is being modelled?
Multiplying Fractions
(Continued)
5. Multiply each pair of fractions or mixed numbers. Draw models for parts b) and e).
You may use grid paper.
a)
b)
3 4
a) –
×–
5 9
2 3
b) –
×–
7 5
4 5
×–
c) –
3 3
6 7
d) –
×–
5 4
1
3
f)
× 1–
e) 1–
5
2
c)
2
2
2–
× 3–
5
3
1
1
20
6. The product of two fractions is ––
.
6
a) What fractions might they have been? Explain.
1
d)
b) List another possible pair of fractions.
1
7. Choose values for the blanks to make this equation true.
1
–– × –– = 2––
3
12
1
4. Estimate to decide if the product will be closer to –
or 1. Circle your choice.
2
3 7
a) –
×–
8 8
closer to –
2
5 5
–×–
4 6
1
closer to –
2
b)
1
9
3
×–
c) ––
10 8
2
1
closer to –
2
1
1
×–
d) 2–
3 3
31
closer to –
2
September 2011
closer to 1
2
​cups​of​flour.
8. A recipe to serve 9 people requires 4–
3
a)​ About​how​much​flour​is​needed​to​make​the​recipe​for​6​people?
closer to 1
closer to 1
b)​ How​much​flour​is​actually​needed?
closer to 1
© Marian Small, 2011
Fractions (IS)
Multiplying Fractions
32
September 2011
© Marian Small, 2011
Fractions (IS)
(Continued)
2
3
9. Write a story problem that would require you to multiply –
× – to solve it.
3 4
Then solve the problem.
10. a) Use the digits 1, 2, 3, 4, 5, and 6 in the boxes to create a fairly high product.
–– ×
––
b) Use the digits again to create a fairly low product.
–– ×
––
11. Is it possible to multiply two fractions and get a whole number product? Explain.
2
12. You have to explain why multiplying a number by –
results in less than you started
3
with. What explanation would you use?
33
32
September 2011
© Marian Small, 2011
September 2011
Fractions (IS)
© Marian Small, 2011 Fractions (IS)
Dividing Fractions
Learning Goal
• representing division of fractions as counting groups, sharing or determining a unit rate.
Open Question
Materials
• fraction strips or Fraction
Tower template
Questions to Ask Before Using the Open Question
Note: Provide fraction strips as you work through these questions.
◊ How many times does ​ _13 ​ fit into ​ _12 ​? (1​ _12 ​times) Why do you think you might write that
as ​ _12 ​ ÷ ​ _13 ​ = 1​ _12 ​? (e.g., because you divide when you are trying to figure out how many
of one thing fits into another.)
◊ How would you figure out ​ _43 ​ ÷ ​ _23 ​? (e.g., I know that there are 2 groups of _​ 23 ​ in _​  43 ​, so
the answer is 2.)
◊ Why would ​ _45 ​ ÷ ​ _25 ​have the same answer? (e.g., It is still how many groups of 2 of
something are in 4 of something.)
◊ What about _​ 56 ​ ÷ ​ _26 ​? (e.g., It would be 2​ _12 ​ since you can get to ​ _46 ​with two groups of ​ _26 ​
but you need another half of ​ _26 ​.) When you write 2​ _12 ​as an improper fraction, what do
you notice? (e.g., It is _​ 52 ​.) Where do you see the ​ _52 ​in the original problem? (the two
numerators)
Using the Open Question
Make sure students understand that they must create at least four fraction quotients,
each time predicting why the quotient would be about ​ _32 ​, explaining how they
calculated the quotient and verifying that the quotient is the right size. Some students
might choose to use models.
By viewing or listening to student responses, note if they:
• can estimate fraction quotients;
• realize that the divisor should fit into the dividend between 1 and 2 times;
• relate division to multiplication;
• use either a common-denominator or invert-and-multiply strategy to divide.
Consolidating and Reflecting on the Open Question
◊ Suppose your first fraction had been ​ _54 ​. How would you have picked the second
one? (e.g., I would pick something between ​ _58 ​, which fits into _​ 54 ​twice and ​ _54 ​.) What
fraction might you have chosen? (e.g.,​ _56 ​)
◊ Could your first fraction be any size? (Yes) Why? (e.g., You just get another one
that is smaller and fits into it about one and a half times.)
◊ Once you chose your fractions, what strategies did you use to divide?
(e.g., Sometimes I used the fraction tower and looked for a fraction that fit into
another one about one and a half times. If it was exactly one and a half, then I just
used a close fraction for one of them. Sometimes I wrote the division as a missing
multiplication. I had to use an equivalent fraction for ​ _32 ​, though, to figure out what to
multiply by. Other times I wrote the fractions as decimals and divided that way.)
33
September 2011
© Marian Small, 2011 Fractions (IS)
Solutions
e.g.,
9
Pair 1: _​  32 ​ ÷ ​ _
10  ​
I knew that ​ _32 ​÷ 1 = _​ 32 ​, so I just chose a fraction close to 1, but not 1.
9
2
_
_
​  32 ​ ÷ ​ _
10  ​= 1​ 3 ​
15
6
_
I wrote 3.2 as a decimal (1.5) and divided by 0.9. I realized 1.5 ÷ 0.9 = ​ _
9  ​and that is 1​ 9 ​which
is 1​ _23 ​.
1​ _23 ​is pretty close to _​ 32 ​, which is 1​ _12 ​.
Pair 2: _​  35 ​ ÷ ​ _38 ​
I knew that _​ 35 ​ ÷ ​ _25 ​ is _​  32 ​since there are 1​ _12 ​groups of ​ _25 ​ in _​  35 ​. But the answer was not supposed to
be exactly ​ _32 ​, so instead of _​ 25 ​, I used a close fraction. I used a fraction tower to help me see
what was close and I picked ​ _38 ​.
_
​  35 ​ ÷ ​ _38 ​ = _​  85 ​= 1​ _35 ​
[]
To figure the answer out, I thought about multiplication. ​ _38 ​ × _​  [] ​ = _​  35 ​. I changed ​ _35 ​ to _
​  24
40 ​, so that I
8
_
could multiply 8 by something to get the new denominator. The answer was ​ 5 ​.
_
​  35 ​is close to _​ 12 ​, so 1​ _35 ​is close to _​ 32 ​which is 1​ _12 ​.
Pair 3: ​ _45 ​ ÷ ​ _12 ​
I started with _​ 34 ​ ÷ ​ _12 ​, since I know that _​ 12 ​fits into _​ 34 ​one and a half times. But I did not want an
exact answer, so I changed ​ _34 ​to a close-by fraction. I chose ​ _45 ​.
_
​  45 ​ ÷ ​ _12 ​= 0.8 ÷ 0.5 = _​ 85 ​= 1​ _35 ​.
I wrote each fraction as a decimal and divided. I realized that 8 tenths ÷ 5 tenths = 8 ÷ 5.
_
​  35 ​is fairly close to ​ _12 ​, so 1​ _35 ​is fairly close to 1​ _12 ​, which is ​ _32 ​.
Pair 4: ​ _69 ​ ÷ ​ _25 ​
One way you can divide two fractions is to invert and multiply. So I needed two fractions that
multiplied to something close to ​ _32 ​and then I would just flip the second one. Since ​ _32 ​ = _
​  30
20 ​, I
30
_
picked ​  18 ​to be close.
30
6
2
_
_
Since ​ _69 ​ × _​  52 ​multiply to be ​ _
18 ​, I used ​ 9 ​ and ​  5 ​.
12
1
_
_
_
​  69 ​ ÷ ​ _25 ​ = _
​  30
18 ​= 1​ 18 ​, which is close to 1​ 2 ​.
I checked by looking at fraction strips. I see that ​ _25 ​fits into _​ 23 ​(the same as _​ 69 ​) more than
1 time, but less than two times, so that makes sense.
60
6
_
_
I also multiplied: ​ _25 ​ × _
​  30
18 ​ = ​  90 ​ = ​  9 ​
34
September 2011
© Marian Small, 2011 Fractions (IS)
Think Sheet
Materials
• grid paper
• fraction strips
Questions to Ask Before Assigning the Think Sheet
Note: Provide fraction strips as you work through these questions.
◊ How many times does ​ _13 ​ fit into ​ _12 ​? (1​ _12 ​times) Why do you think you might write that
as ​ _12 ​ ÷ ​ _13 ​ = 1​ _12 ​? (e.g., You divide when you are trying to figure out how many of one
thing fits into another.)
◊ How would you figure out ​ _43 ​ ÷ ​ _23 ​? (e.g., I know that there are 2 groups of _​ 23 ​ in _​  43 ​, so
the answer is 2.)
◊ Why would ​ _45 ​ ÷ ​ _25 ​have the same answer? (e.g., It is still how many groups of 2 of
something are in 4 of something.)
◊ What about _​ 56 ​ ÷ ​ _26 ​? (e.g., It would be 2​ _12 ​since you can get to ​ _46 ​with two groups of ​ _26 ​
but you need another half of ​ _26 ​.) When you write 2​ _12 ​as an improper fraction, what do
you notice? (e.g., It is _​ 52 ​.) Where do you see the ​ _52 ​in the original problem? (the two
numerators)
Using the Think Sheet
Read through the introductory box with students and make sure students understand
the material.
Assign the tasks.
By viewing or listening to student responses, note if they:
• can estimate fraction quotients;
• relate division to multiplication;
• use either a common-denominator or invert-and-multiply strategy to divide.
Consolidating and Reflecting: Questions to Ask After Using the Think Sheet
◊ What model could you draw to show what ​ _23 ​ ÷ ​ _34 ​ means? (e.g., I could use fraction
strips to show how many ​ _34 ​fits into _​ 23 ​.)
◊ How do you know the result would be less than 1? (e.g., ​ _23 ​is not as much as _​ 34 ​.)
◊ How might you calculate ​ _23 ​ ÷ ​ _34 ​if you did not have fraction strips? (e.g., I would use
8
9
_
equivalent fractions with the same denominator. It would be ​ _
12  ​ and ​  12  ​. I know that I
8
9
8
_
_
_
could only fit ​ 9 ​of a ​ 12  ​ into ​  12  ​.)
◊ Why and what would you divide to solve this problem: Kelly went ​ _38 ​of the way to
her grandmother’s home in ​ _25 ​of an hour. How far could she get in a whole hour?
(e.g., It is a rate problem. You would divide ​ _38 ​by 2 to figure out how far in ​ _15 ​of an
hour and multiply by 5 to figure out a whole hour. Since you are doing ​ _38 ​ × _​  52 ​, you are
actually dividing ​ _38 ​by _​ 25 ​.)
Solutions
3
_
1.a)e.g., I renamed _​ 68 ​ as _
​  12
16 ​so I could share into 4 equal groups of ​ 16  ​ each.
4
12
4
_
_
_
b) e.g., I renamed ​ 9 ​ as ​  27 ​so I could share into 3 equal groups of ​ 27
  ​
2.a)e.g., It shows how many times _​ 13 ​fits into _​ 58 ​.
b) e.g., almost 2
3.a)2 ÷ _​ 23 ​
b) _​  74 ​ ÷ ​ _34 ​
4.a)closer to ​ _12 ​
b) closer to 2
c) closer to 1
d) closer to ​ _12 ​
35
September 2011
© Marian Small, 2011 Fractions (IS)
3
5.a)​ _
20  ​
b) 2​ _12 ​
c) 12
d) _
​  70
72 ​
e) _
​  70
6  ​
40
_
f) ​  51 ​
Model for part a:
Model for part c:
NOTE: Answers may be written as equivalent fractions.
6.a)about ​ _34 ​ cup
b) _​  56 ​ (​  or _
​  10
12 ​  )​ cups
8
7.a)​ _
15  ​of the floor
b) division
8. e.g., A small paint can holds ​ _18 ​as much as a large one. If I pour _​ 35 ​of a large can of paint
into small cans, how many full and part small cans can I fill?
16
3
_
9.a)e.g., _​  43 ​ ÷ ​ _
9  ​ = ​  4 ​
b) _​  23 ​ ÷ ​ _89 ​ = _​  34 ​
10.a) e.g., 3​ _56 ​ ÷ ​ _92 ​ = _
​  46
54 ​
7
b) e.g., 6​ _35 ​ ÷ ​ _29 ​= 29​ _
10  ​
11.a) e.g., Since _​ 38 ​is more than ​ _13 ​, _​  13 ​fits into _​ 38 ​more than once, but ​ _38 ​does not fit into _​ 13 ​
even once.
9
8
9
8
9
8
9
8
1
_
_
_
_
_
_
_
_
b) e.g. If you show _​ 38 ​ as _
​  24
  ​ and ​  3 ​ as ​  24  ​, you know that ​ 24  ​ ÷ ​  24  ​ is ​  8 ​, but ​ 24  ​ ÷ ​  24  ​ is ​  9 ​since you
are either fitting 8 things into 9 or else only using 8 parts of the 9.
12. e.g., I would use an example such as 2 ÷ ​ _56 ​. That asks how many times _​ 56 ​fits into 2. I know
that ​ _16 ​fits into 1 six times, so _​ 16 ​fits into 2, 2 × 6 times. But _​ 56 ​is five times as big, so only ​ _15 ​
as many will fit in. That means you have to divide 2 × 6 by 5. That is the same as 2 × ​ _65 ​.
36
September 2011
© Marian Small, 2011 Fractions (IS)
Open Question
Dividing Fractions
Learning Goal
• representingdivisionoffractionsascountinggroups,sharingordeterminingaunit
rate.
Open Question
• Choosetwofractionstodividetomeettheseconditions.
3
– Thequotient(theansweryougetwhenyoudivide)isabout,butnotexactly,–
2.
– Thedenominatorsaredifferent.
3
.
Tellhowyoupredictedthequotientwouldbeabout–
2
Calculatethequotientandexplainyourprocess.
3
.
Verifythatthequotientisabout–
2
• Repeatthestepsatleastthreemoretimeswithotherfractions.
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September 2011
© Marian Small, 2011
Fractions (IS)
Think Sheet
Dividing Fractions
(Continued)
Dividing Fractions
Think Sheet
5
(Continued)
2
0
Sharing
• When we divide by a whole number, we can think about sharing. For
4
5
2
• Tomodel–
÷–,weneedtoseehowmanygroupsof–
6 arein–
.Wefigureout
6 6
6
howmany2sarein5or5÷2.
Just as with whole numbers, dividing fractions can describe the result of sharing,
can tell how many of one size group fits in another, or can describe rates.
1
_
6
2
_
6
3
_
6
4
_
6
5
_
6
5
_ ÷2
_ = 21
_
6 6
2
1
• Iftwofractionshavethesame denominator,allofthesectionsarethesame
size.Togetthequotient,wefigureouthowmanytimesonenumeratorfitsinto
4
example, –
÷ 2 means that 2 people share –
of the whole.
5
5
7
3
,wecalculate7÷3sincethattellshowmany
theother.Forexample,for–
8 ÷–
8
3
6
groupsof3ofsomethingarein7ofthatthing.For–
÷–,wecalculate3÷6,
8 8
6
3
1
whichis –
,sincethattellshowmuchofthe– fitsin–.
2
8
8
We divide the 4 sections into 2 and remember that we are thinking about fifths.
2
Each person gets –
.
5
• Iftwofractionshavedifferent denominators,wecanuseamodeltoseehow
manytimesonefractionfitsinanother,orwecanuseequivalentfractionswith
thesamedenominatoranddividenumerators.
5
• Sometimes the result is not a whole number of sections. For example, –
÷2
3
5
.
means that 2 people share –
3
1
1
1
1
Forexample,for–
2 ÷–
3,thepictureshowsthatthe–
3 sectionfitsintothe–
section
2
1
1–
times.
2
1
2–
2
Each gets ––
.
3
We can either multiply numerator and denominator by 2 to get the equivalent
5
5
fraction –
or we can use an equivalent fraction model for –
.
6
3
1
3
1
3
2
1
Wecouldwrite–
÷– as–
6 ÷–
=–
or1–
.Itisthesameasrepresentingthemodel
2 3
6 2
2
usingequivalentfractions.
The thirds were split in half since there needed to be an even number of
Using a Unit Rate
5
5
sections for 2 people to share them. Each person gets –
. If it had been –
÷ 4,
6
3
• Anotherwaytothinkaboutdivisionisthinkingaboutunitrates.Forexample,
ifwecandrive80kmintwohours,wethinkof80÷2asthedistance
wecantravelinonehour.
the thirds could be split into fourths so that it would be possible to divide the
total amount into four equal sections.
Counting Groups
Sometimes it makes sense to count how many groups. For example, there are 5
1
5
1
1
day,wedivide– ÷– tofigureouthowmuchshecancomplete
aprojectin–
2
3 2
1
0
37
1
1
1
groups of –
in –
.
6
6
35
1
Ifagirlcancomplete–
ofaprojectintwodays,wedivide– ÷2tofigureout
3
3
of
howmuchshecancompleteinoneday.Similarly,ifshecancomplete–
3
September 2011
1
_
6
2
_
6
3
_
6
4
_
6
5
_
6
1
in1day.Sinceyouknowthatwecouldalsomultiply–
3 ×2;itmakessense
5
1
_ ÷_
=5
6 6
© Marian Small, 2011
September 2011
1
1
1
that–
3 ÷–
=–
×2.
2 3
Fractions (IS)
36
September 2011
© Marian Small, 2011 © Marian Small, 2011
Fractions (IS)
Fractions (IS)
Dividing Fractions
Dividing Fractions
(Continued)
(
a)
0
)
2
1
1
1
1
4 ofaspace,then–
3 ofacancovers–
8 ofthespace –
4 ÷2 If–
3 ofacancovers–
3
3
1
1 2 3
,whichis1,cancovers– ofthespace 3×–
.Thatmeans– ÷– =–
.
andthen–
3
8
8
4 3 8
3
8
3
1 2
Sinceweknowthat–
4 ÷–
3 =––
12÷––
=–
,thismakessense.
12 8
(
0
1 2 1
1 3
=–
÷2×3=–
×–
.
So–
4 ÷–
3 4
4 2
Wereversedthedivisorandmultipliedbythereciprocal,thefractionweget
byswitchingthenumeratoranddenominator.
• Anotherwaytothinkaboutthisis:
1
1
1÷–
3 =3sincethreeare3groupsof–
in1whole.
3
1÷–
=–
sincethereareonlyhalfasmanygroupsof–
3 in1wholeasthere
3 2
–
÷– =–
×–
sincethereareonlyonefourthasmanygroupsof–
3 in–
4 4 3 4 2
2
3
2
1
wouldbegroupsof–
3.
1
2
1
3
2
1
2
_
3
4
_
3
2
5
_
3
1
_
4
2
_
4
1
3
_
4
5
_
4
6
_
4
2
7
_
4
9
_
4
1
4. Estimate each quotient as closer to –
or 1 or 2. Circle your choice.
2
3 7
1
–
a) –
÷
closer to –
closer to 1
8 8
2
1
Whatwedidwasdividethe–
by2andmultiplyby3.
4
1
_
3
b)
)
(Continued)
3. What division is being modelled?
2
1
• Supposeweknowthat–
3 ofacanofpaintcancover–
ofaspace.Tofigure
4
1 2
.
outhowmuchofthespaceonefullcanofpaintcancover,wecalculate–
4 ÷–
3
closer to 2
b)
5 2
–÷–
4 3
1
closer to –
2
closer to 1
closer to 2
c)
11 13
–– ÷ ––
3
4
1
closer to –
2
closer to 1
closer to 2
d)
2
1
÷ 4–
2–
3
2
1
closer to –
2
closer to 1
closer to 2
5. Divide each pair of fractions or mixed numbers. Draw models for parts a) and c).
You might use grid paper.
1
3
a) –
÷4
5
2
inawhole.
asgroupsof–
3
• Improper fractionscanbedividedinthesamewayasproperfractions.
5 1
b) –
÷–
6 3
c) 8 ÷ –
3
7
9
d) –
÷ ––
8 10
14 2
÷–
e) ––
5
3
f)
2
• Mixed numbersareusuallywrittenasimproperfractionstodividethem.
1. Howwouldyoucalculateeach?Explainwhyyourstrategymakessense.
6
a) –
8 ÷4
4
b) –
9 ÷3
5
2
2
2–
÷ 3–
5
3
1
​cups​of​flour​to​divide​into​equal​batches​for​four​recipes.
6. You have 3–
3
1
2. a) Howdoesthismodelshow–
÷–?
8 3
a)​ About​how​much​flour​is​available​for​each​batch?
b)​ How​much​flour​is​that​per​batch?
b) Estimatethequotient.
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September 2011
© Marian Small, 2011
Fractions (IS)
Dividing Fractions
38
(Continued)
© Marian Small, 2011
Fractions (IS)
Dividing Fractions
3
2
September 2011
(Continued)
7. You can tile –
​of​a​floor​area​in​​​–
of a day.
5
4
a)​ How​much​of​the​floor​can​you​tile​in​one​full​day?
3
1
9
1
3
8
÷ – is –, but –
÷ – = –.
11. Kevin noticed that –
8 3 8
3 8 9
a) Explainwhyitmakessensethatthefirstquotientisgreaterthan1,butthe
second one is less than 1.
b)​ What​computation​can​you​do​to​describe​this​situation?
3
1
8.​ Write​a​story​problem​that​would​require​you​to​divide​​–
​5 ​by​​–
​8 ​to​solve​it.​Solve​
the​problem.
b) Explainwhyitalsomakessensethatthequotientsarereciprocals.
9. a)​ Choose​values​for​the​blanks​to​make​this​true.
3
–– ÷ –– = –
3
4
b)​ List​another​possible​set​of​values.
3
–– ÷ –– = –
3
4
5
–– ÷
6
12.Youhavetoexplainwhydividingby–
isthesameasmultiplyingby–
. What
5
6
explanation would you use?
10. a)​ Use​the​digits​2,​3,​5,​6,​and​9​in​the​boxes​to​create​a​small​quotient.
––
b)​ Use​the​digits​again​to​create​a​large​quotient.
–– ÷
39
38
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September 2011
© Marian Small, 2011
September 2011
Fractions (IS)
40
September 2011
© Marian Small, 2011 © Marian Small, 2011
Fractions (IS)
Fractions (IS)
Relating Situations to Fraction Operations
Learning Goal
• connecting fraction calculations with real-life situations.
Open Question
Questions to Ask Before Using the Open Question
◊ What might make a problem an addition one? (e.g., You have to be combining
things.)
◊ What might make it a subtraction one? (e.g., take-away.) Could there be other
problems that are not take-away that are subtraction problems? (e.g., Yes, it could
be comparing.)
◊ What might make a problem a multiplication one? (e.g., You multiply to get areas or
to count equal groups.)
◊ What might make a problem a division one? (e.g., You divide when you are
sharing.) When else might you divide? (e.g., When you are figuring out how many
of something fits into something else.)
Using the Open Question
Make sure students understand that they must create at least two story problems
for each operation. By suggesting that the problems are different, help students
understand that not only might the scenario change, but even the meaning of the
operation.
By viewing or listening to student responses, note if they:
• relate real-life situations to fraction operations;
• recognize the different meanings operations can hold when applied to fractions.
Consolidating and Reflecting on the Open Question
◊ How were your subtraction questions different? (e.g., One involved comparing and
one involved take-away.)
◊ How were your division questions different? (e.g., One involved a unit rate and one
involved how many of one thing fits into another.)
◊ Why did you not use a sharing problem for division? (e.g., I was dividing by ​ _13 ​ rather
than by a whole number.)
Solutions
e.g.,
3
_
5 
​  ​ + _​  13 ​
I read _​ 35 ​of my book on Monday and ​ _13 ​of it on Tuesday. How much of the book have I read
now? (This involves combining, as addition should.)
I bought ​ _35 ​of my groceries at one store and ​ _13 ​of them at the other store. How much of my
groceries does that account for? (This involves combining, as addition should.)
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September 2011
© Marian Small, 2011 Fractions (IS)
3
_
5 
​  ​ – _​  13 ​
I read _​ 35 ​of my book on Monday and ​ _13 ​of it on Tuesday. How much more of the book did I
read on Monday than Tuesday? (This involves comparing and that is one of the things
subtraction means.)
I finished ​ _35 ​of my project, but decided some of it wasn’t good enough. I decided I should
revise ​ _13 ​of the project. How much of my project is actually ready? (This involves take away
and that is one of the things subtraction means.)
3
_
5 
​  ​ × _​  13 ​
We have to turn in _​ 35 ​of the money for a field trip by next week. My grandparents and parents
and I are each paying ​ _13 ​. How much of the full amount do I have to pay by next week? (This is
multiplication since it is finding ​ _13 ​of a group of something.)
3
_
5 
​  ​ × _​  13 ​
Out classroom is ​ _35 ​as wide and _​ 13 ​as long as the gym. How much of the area of the gym is the
area of our classroom? (This is an area problem so it is about multiplying.)
3
_
5 
​  ​ ÷ ​ _13 ​
You need ​ _35 ​of a cup of flour, but only have a _​ 13 ​cup measure. How many times will you have to
fill it?
3
_
5 
​  ​ ÷ ​ _13 ​
Small cans of paint are ​ _13 ​of the size of large ones. You have ​ _35 ​of a large can of paint. How
many small ones could you fill? (This is division since it is about how many of something fits
in something else.)
40
September 2011
© Marian Small, 2011 Fractions (IS)
Think Sheet
Questions to Ask Before Assigning the Think Sheet
◊ What might make a problem an addition one? (e.g., You have to combine things.)
◊ What might make it a subtraction one? (e.g., take-away) Could there be other
problems that are not take-away that are subtraction problems? (e.g., Yes, it could
be comparing.)
◊ What might make a problem a multiplication one? (e.g., You multiply to get areas or
to count a lot of equal groups.)
◊ What might make a problem a division one? (e.g., You divide when you are
sharing.) When else might you divide? (e.g., When you are figuring how many of
something fits into something else.)
Using the Think Sheet
Read through the introductory box with students and make sure students understand
the material.
Assign the tasks.
By viewing or listening to student responses, note if they:
• relate real-life situations to fraction operations;
• recognize the different meanings operations can hold when applied to fractions.
Consolidating and Reflecting: Questions to Ask After Using the Think Sheet
◊ What made Question 1b a multiplication question? (e.g., It is multiplication because
you are taking a part of a part.)
◊ Why might you solve Question 1e with either addition or subtraction? (e.g., You
could figure out what to add to ​ _14 ​to get to _​ 78 ​or you can take ​ _14 ​ from _​  78 ​.)
◊ What operation would you use to figure out what fraction of Fred’s money his sister
has? (e.g., Division, since it is like imagining Fred’s amount is 1 whole and then
you divide that by 2​ _12 ​.)
◊ Do you think of division problems where you divide by whole numbers the same
way as ones where you divide by fractions? (e.g., No, since I can use sharing when
I divide by whole numbers, but not when I divide by fractions.)
Solutions
1.a)1​ _12 ​ ÷ ​ _25 ​ division
b) _​  14 ​ × _​  34 ​ multiplication
c) _​  15 ​ + _​  13 ​ addition
d) 5 × 1 _​ 23 ​ multiplication
e) _​  78 ​ – _​  14 ​ subtraction
f) _​  23 ​÷ 1​ _12 ​ division
2.a)e.g., You are taking ​ _34 ​of 2​ _12 ​and when you take a part of a part, you are multiplying.
b) e.g., You are finding out how many ​ _13 ​’s are in 5​ _12 ​and finding how many of one thing is
in another is dividing.
c) e.g., It is a take-away problem.
3.a)​ _34 ​× 2​ _12 ​= ?
b) e.g. 5​ _12 ​ ÷ ​ _13 ​= ?
c) e.g., 3​ _12 ​– ? = 1​ _34 ​
4. e.g., To divide 3​ _13 ​by _​ 23 ​to find out how many ​ _23 ​are in 3​ _13 ​, you could figure out what to
multiply ​ _23 ​by to get 3​ _13 ​.
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September 2011
© Marian Small, 2011 Fractions (IS)
5.a)e.g., Anne checks the turkey 3 times an hour. How many times does she check it in
5​ _12 ​hours?
b) e.g., A turkey needs to be in the oven for 5​ _12 ​hours. It has cooked for 2​ _34 ​hours so far.
How much time does it still need?
6. e.g., Alexander has 3​ _12 ​dozen eggs. If he is making omelets that use ​ _14 ​of a carton at a
time, how many omelets can he make?
7. e.g., If it is take-away, it is subtraction.
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September 2011
© Marian Small, 2011 Fractions (IS)
Open Question
Relating Situations to Fraction Operations
Learning Goal
• connectingfractioncalculationswithreal-lifesituations.
Open Question
• Createtwoproblemsthatcouldbesolvedusingeachoftheequationsbelow.
Maketheproblemsasdifferentasyoucan.Youdonothavetosolvethem.
3 1
–
+–
=
5 3
3 1
–
––
=
5 3
3 1
–
×–
=
5 3
3 1
–
÷– =
5 3
• Explainyourthinkingforeach.
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September 2011
© Marian Small, 2011
Fractions (IS)
Think Sheet
Relating Situations to Fraction Operations
(Continued)
Relating Situations to Fraction Operations
Think Sheet
(Continued)
1
• Anexampleofdetermining areas of rectanglesis:Onerectangleis–
aslong
3
2
1
64
2
aswideasa4×8rectangle.Whatisitsarea?Answer:– ×–
×4×8=––
and–
3
3 3
9
It is important to be able to tell what fraction operations make sense to use for
solving fraction problems.
2
• Anexampleofapplying ratesis:Janecanpaintawallin–
3 ofanhour.How
1
1
2
5
longwouldittakehertopaint2–
walls?Answer:2–
2 ×–
3 =–
3 hours
2
Addition
Addition situations always involve combining. For example:
1
Division
1
of my essay done. I did another –
​of​the​essay.​How​much​of​it​is​finished​
I had –
3
2
+ – = – of the essay
now? Answer: –
3 2 6
Divisionsituationscouldinvolvesharing, determining how many groups, or
determining unit rates
Subtraction
• Anexampleofsharingis:–
3 ofaroommustbepainted.Fourfriendsaregoing
1
1
5
2
tosharethejob.Whatfractionoftheroomwilleachpaintiftheyworkatthe
Subtraction situations could involve take away, comparing, or deciding what
to add.
3
2
3
1
5
• Anexampleofdetermining how many groupsis:Youwanttomeasure–
3 ofa
5
1
– – = –– cup
much of a cup of juice is left? Answer: –
4 3 12
cupofflour,butonlyhavea–
2 cupmeasure.Howmanytimesmustyoufillthe
5
1
1
1
–
cupmeasure?Answer:– ÷– =3–
3 times
2
3 2
3
1
=–
project.​How​much​more​did​I​finish?​Answer:​​–
​8 – –
4 8
2
ofanhourtoclean–
5 of
• Anexampleofdetermining unit rateis:Ittakesyou–
2
2
5
1
4
÷– =–
thehouse,howmuchofthehousecouldyoucleanin1hour?Answer:–
5 2 5
ofthehousein1hour.
of my project. How
•​ An​example​of​deciding what to add​is:​I​have​finished​​​–
8
5
5
1
1
​4 of her
•​ An​example​of​comparing​is:​I​finished​​–
​8 ​of​my​project.​Angela​finished​​–
5
2
oftheroom
samerate?Answer:–
3 ÷4=––
12
1
cup of juice. I poured out –
cup. How
•​ An​example​of​take away is: I had –
4
3
3
=–
much more is left? Answer: 1 – –
8 8
More Than One Option
Multiplication
Sometimesthesamesituationcanbeapproachedusingmorethanone
operation.Forexample,ifyoucansolveaproblembydividing,youcanalsosolve
itbymultiplying.
Multiplication situations could involve counting parts of sets, determining areas of
rectangles, or applying rates.
1
of the way.
•​ An​example​of​counting parts of sets​is:​Jeff​filled​8​cups,​each​​​–
3
1
2
Thelastproblemabovecouldbesolvedbythinking:–
2 ×––
=–
5.
If​he​had​filled​the​cups​to​the​top​instead,​how​many​cups​would​he​fill?
1 8
Answer: 8 × –
= –.
3 3
•Everydivisioncanalsobesolvedasamultiplication.
•Everysubtractioncanalsobesolvedasanaddition.
2
of the money they needed to go on a trip.
Another example is: A class raised –
3
2
of the class’s contribution. What fraction of the whole amount
The boys raised –
5
of money needed did the boys raise?
42
43
September 2011
© Marian Small, 2011
September 2011
Fractions (IS)
43
September 2011
© Marian Small, 2011 © Marian Small, 2011
Fractions (IS)
Fractions (IS)
Relating Situations to Fraction Operations
Relating Situations to Fraction Operations
(Continued)
1
(Continued)
2
f)
1. Tell what operation or operations you could use to solve each problem. Write the
equation to represent the problem.
1
of the way to your grandmother’s home in 1–
hours. How
You can travel –
3
2
much of the way can you travel in 1 hour?
2
a) Cynthia has 1–
containers of juice. Each smaller container holds –
as much
5
2
as a large container. How many small containers can she fill?
2. a) What makes this a multiplication problem?
3
1
3
Fred has 2–
times as much money as his sister. Aaron has –
as much money as
2
4
1
Fred. How many times as much money as Fred’s sister does Aaron have?
of the athletes in a school play basketball. About –
of those players are
b) About –
4
4
in Grade 9. What fraction of the students in the school are Grade 9 basketball
players?
b) What makes this a division problem?
1
1
A turkey is in the oven for 5–
hours. You decide to check it every –
of an hour.
2
3
1
How many times will you check the turkey?
1
c) Stacey read –
of her book yesterday and –
of it today. How much of the book
5
3
has she read?
c) What makes this a subtraction problem?
3
1
d)
Ed has 3–
rolls of tape. He uses about 1–
rolls to wrap gifts. How many rolls of
2
4
2
2
hours to drive to work in the morning and 1–
hours
It takes Lea’s mom about 1–
3
3
tape does he have left?
to drive home every afternoon. If it is Wednesday at noon and she went to
work Monday, Tuesday and Wednesday (and is still there), about how many
hours has she spent driving to and from work?
3. Write an equation you could use to solve each problem in Question 2.
a)
7
e) The gas tank in Kyle’s car was –
full when they started a trip. Later in the day,
8
1
full. How much of the tank of gas had been used?
the tank registered –
4
b)
c)
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September 2011
© Marian Small, 2011
Relating Situations to Fraction Operations
Fractions (IS)
45
September 2011
© Marian Small, 2011
Fractions (IS)
Fraction Tower
(Continued)
4. Why might you solve this problem using either multiplication or division? You have
1
2
worked out 3–
hours this week. Each workout was –
of an hour. How many times
3
3
1
did you workout?
1_
2
1_
2
1_
3
5. How would you keep part of the information in Question 2b, but change some
of it so that:
1_
3
1_
4
a) it becomes a multiplication question
1_
4
1_
5
b) it becomes a subtraction question
7. What hints can you suggest to decide if a problem requires subtraction to solve it?
1_
9
1_
9
1
_
10
1
_
10
1
_
12
1
_
15
1
_
12
1
_
15
1_
8
1
_
10
1
_
15
1
_
15
1_
9
1
_
10
1
_
12
1
_
12
1
_
15
1
_
15
1_
5
1_
6
1_
8
1_
9
1
_
12
1_
5
1_
6
1_
8
1_
4
1_
5
1_
6
1_
8
1
Alexander has 3–
dozen eggs.
2
1_
4
1_
5
1_
6
6. Finish this problem so that it is a division problem:
1_
3
1_
8
1_
9
1
_
10
1
_
12
1
_
12
1
_
15
1
_
15
1_
6
1_
8
1_
9
1
_
10
1
_
15
1_
6
1_
8
1_
9
1
_
10
1
_
10
1
_
12
1
_
12
1
_
15
1
_
15
1_
8
1_
9
1_
9
1
_
10
1
_
10
1
_
12
1
_
15
1
_
12
1
_
15
1
_
12
1
_
15
1
_
15
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1
_
18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1
_
20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20
46
44
September 2011
© Marian Small, 2011
September 2011
Fractions (IS)
47
September 2011
© Marian Small, 2011 © Marian Small, 2011
Fractions (IS)
Fractions (IS)