Module 2
Comparing Fractions
Diagnostic ......................................................................................... 4
Administer the diagnostic ......................................................................4
Using diagnostic results to personalize interventions ............................4
Solutions ................................................................................................5
Using Intervention Materials ...................................................... 6
Comparing Fractions Using Pictures ..........................................................7
Comparing Fractions with the Same Denominator .................................... 11
Comparing Fractions with the Same Numerator........................................13
Equivalent Fractions ...............................................................................15
Comparing Fractions to _12 and to 1 ...........................................................18
COMPARING FRACTIONS
Relevant Learning Expectation for Grade 6
… compare, and order fractional amounts with unlike denominators, including proper and improper
fractions and mixed numbers, using a variety of tools (e.g., fraction circles, Cuisenaire® rods, drawings,
number lines, calculators) and using standard fractional notation
Possible reasons a student might struggle in comparing
fractions or deciding if they are equal
Many students struggle with comparing fractions and/or determining their equivalence. Some of the
problems include:
• not realizing that you may have to reorient one diagram to match another to determine which fraction
is greater (e.g., thinking that _23 < _35 since the blue is farther to the right with _35 )
• not realizing that you can only compare visually if the fractions refer to the same whole
• not recognizing that numerators are all that matter when comparing fractions with the same
denominator
• assuming that if fractions have the same numerator, the greater one has the greater denominator
• showing more discomfort when comparing improper fractions than when comparing proper fractions
• thinking that if both numerator and denominator are larger, the fraction is automatically larger
7
4
_
(e.g., thinking that _
10 > 5 since 7 and 10 are more than 4 and 5)
• not realizing that fractions are equal only if numerator and denominator are multiplied or divided by
the same amount, not if the same amount is added or subtracted
• thinking that _46 > _23 since a picture of 4 out of 6 items shows more items than a picture of 2 out of
3 items
• not realizing that comparisons to the benchmarks of _12 and 1 are often an easy way to compare
fractions and that this can be done simply by comparing the numerator to the denominator
Part-of-whole models are used to close the gap in student understanding about fractional comparisons
and equivalence rather than part-of-set models because the latter requires more sophistication.
Although students could figure out why, for example, _23 of 12 is greater than _12 of 12 by determining
the count for each, it would be much more difficult to see why _23 of 11 is greater than _12 of 11. For this
reason, parts of wholes are used; parts of measures could also be used.
Additional consideration
Before beginning the diagnostic or using the intervention materials, students need to know that >
means “greater than” and < means “less than.” Post a reminder as reference.
Comparing Fractions
© Marian Small, 2010
3
DIAGNOSTIC
Administer the diagnostic
Materials
If students need help in understanding the directions of the diagnostic, clarify an item’s
intent.
If available, students can use concrete fraction pieces instead of the pictorial models
provided in the templates. However, after Question 1, the purpose is to see how students
work symbolically. If they are dependent on the materials for many questions, assume
that they need the relevant intervention materials.
• Fraction Circles and
Rectangles (1) template
Diagnostic
1. Circle the greater shaded amount. If the fractions are equal, circle both.
a)
b)
c)
d)
2. Circle the greater fraction. Tell why it is greater.
Using the diagnostic results to personalize
interventions
a) –– ––
7
12
5
12
because
b) –
3
4
3
–
2
because
c) –
5
6
5
–
8
because
4
6
1
–
3
because
d) –
2
.
3. Circle all of the fractions below that are greater than –
5
4
1
b) –
5
a) –
5
5
2
3
c) –
5
8
3
2
d) –
e) –
7
Comparing Fractions
20
6
f) –
7
8
g) ––
h) –
© Marian Small, 2010
3
Provide Template 1 for students to use if needed.
Intervention materials are included for each of these topics:
• comparing fractions using pictures
• comparing fractions with the same denominator
• comparing fractions with the same numerator
• equivalent fractions
• comparing fractions to _12 and 1
Diagnostic
(Continued)
1
2
4. Circle all of the fractions below that are between 0 and –.
5
6
2
b) –
a) –
5
4
9
2
3
c) –
1
10
d) –
4
e) ––
f) –
7
4
6
5. Circle the fractions below that are equivalent (equal) to –.
3
2
3
5
b) –
5
a) –
20
30
c) –
7
32
48
d) ––
e) ––
6. Complete the statement: The best way to create an equivalent fraction
3
is to
to –
5
You can use all or only part of these sets of materials, based on student performance
with the diagnostic.
4
Evaluating Diagnostic Results
Suggested Intervention Materials
If students struggle with Question 1
Use Comparing Fractions using Pictures
If students struggle with Questions 2a,
3a, b
Use Comparing Fractions with the Same
Denominator
© Marian Small, 2010
Fraction Circles and Rectangles (1)
1
–
3
If students struggle with Questions 2b, c,
3c, d, 4e
If students struggle with Questions 1c,
2d, 5, 6
If students struggle with Questions 2d,
3f–h, and 4
4
Use Equivalent Fractions
© Marian Small, 2010
1
–
3
1
–
3
1
–
3
1
–
4
1
–
4
1
–
4
1
–
4
1
–
4
1
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4
1
–
4
1
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4
1
–
6
1
–
6
1
–
6
1
–
6
1
–
6
1
–
8
1
–
8
Use Comparing Fractions to and 1
1
–
3
1
–
3
Use Comparing Fractions with the Same
Numerator
1
_
2
Comparing Fractions
1
–
6
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1 1
1 –– –– 1
–– 12 12 ––
12 1
1 12
––
––
12
12
1
1
––
––
12 1
1 12
–– 1 1 ––
12 –– –– 12
12 12
20
1
–
6
1
–
6
1
–
6
1
–
6
1
–
6
1
–
6
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1 1
1 –– –– 1
–– 12 12 ––
12 1
1 12
––
––
12
12
1
1
––
––
12 1
1 12
–– 1 1 ––
12 –– –– 12
12 12
© Marian Small, 2010
1
–
3
1
–
3
1
–
3
1
–
3
1
–
3
1
–
3
1
–
4
1
–
4
1
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4
1
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4
1
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4
1
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4
1
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4
1
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4
1
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6
1
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6
1
–
6
1
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6
1
–
6
1
–
6
1
–
6
1
–
6
1
–
6
1
–
6
1
–
6
1
–
6
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1 1 1 1 1 1 1 1 1 1 1 1
–– –– –– –– –– –– –– –– –– –– –– ––
12 12 12 12 12 12 12 12 12 12 12 12
1 1 1 1 1 1 1 1 1 1 1 1
–– –– –– –– –– –– –– –– –– –– –– ––
12 12 12 12 12 12 12 12 12 12 12 12
Comparing Fractions
Comparing Fractions
Solutions
1. a)
b)
c)
d)
2. a)
7
_
12
e.g., because the pieces are the same size but there are more of them; the
7
numerator is greater; they draw pictures and show that _
12 uses more space, but
the wholes must be the same size
b) _32 e.g., because _32 is more than 1 and _34 is less than 1
c) _56 e.g., because there are 5 pieces each time but sixths are bigger than eighths
d) _46 e.g., because _46 is more than _12 and _13 is less than _12 ; I drew pictures and _13 took
up less space.
3. a, c, d, f, g, h
4. a, c, e
5. a, d, e
6. e.g., multiply the numerator and denominator by the same amount; draw a picture of
3 out of 5 and keep making more rows, etc.
Comparing Fractions
© Marian Small, 2010
5
USING INTERVENTION MATERIALS
The purpose of the suggested student work is to help them build a foundation with
comparing fractions that they can use with ratios, percents, and decimals as they move
into Grade 7.
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
6
© Marian Small, 2010
Comparing Fractions
Comparing Fractions Using Pictures
Materials
• Fraction Circles and
Rectangles (1) (2)
templates
• pattern blocks
Open Question
Questions to Ask Before Using the Open Question
Comparing Fractions Using Pictures
Open Question
1. Choose 4 of the 6 fraction circles to put in order from least to greatest.
Describe your strategies.
Place a pattern block rhombus on top of a hexagon and beside it a pattern block
triangle on top of a hexagon. Ask:
◊ What fraction of the hexagon is each shape? (_13 and _16 )
◊ Which fraction is greater? How can you be sure? (_13 since it takes up more space; I
can be sure by putting the triangle right on top of it to show it’s smaller)
Place the rhombus in different positions on the hexagon.
◊ Does where I put the fraction change its size? (no)
◊ Suppose I had a picture of two fractions. How could I compare them to see which
took up more space? (e.g., cut one out and put it on top of the other)
◊ Why do the wholes have to be the same size if you’re comparing the fractions?
(e.g., even a little part of something really big is a lot more than a big part of
something really little)
2. Choose 4 of the 6 fraction rectangles to put in order from least to
greatest.
Describe your strategies.
Comparing Fractions
1
–
3
1
–
3
1
–
4
1
–
4
1
–
4
1
–
4
1
–
4
1
–
4
1
–
4
1
–
4
1
–
6
1
–
6
1
–
6
1
–
6
1
–
6
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
6
1
–
6
1
–
6
1
–
6
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1 1
1 –– –– 1
–– 12 12 ––
12 1
1 12
––
––
12
12
1
1
––
––
12 1
1 12
–– 1 1 ––
12 –– –– 12
12 12
20
1
–
3
1
–
3
1
–
3
1
–
3
1
–
3
1
–
3
1
–
4
1
–
4
1
–
4
1
–
4
1
–
4
1
–
4
1
–
4
1
–
4
1
–
6
1
–
6
1
–
6
1
–
6
1
–
6
1
–
6
1
–
6
1
–
6
1
–
6
1
–
6
1
–
6
1
–
6
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1 1 1 1 1 1 1 1 1 1 1 1
–– –– –– –– –– –– –– –– –– –– –– ––
12 12 12 12 12 12 12 12 12 12 12 12
1 1 1 1 1 1 1 1 1 1 1 1
–– –– –– –– –– –– –– –– –– –– –– ––
12 12 12 12 12 12 12 12 12 12 12 12
© Marian Small, 2010
Comparing Fractions
Fraction Circles and Rectangles (2)
1
–
6
1
–
6
1
–
6
1
–
6
1
–
6
1
–
5
1
–
5
Consolidating and Reflecting on the Open Question
2.
1
–
8
1 1
1 –– –– 1
–– 10 10 ––
10
10
1
1
––
––
10
10
1
1
–– 1 1 ––
10 –– –– 10
10 10
1
–
6
1
–
5
© Marian Small, 2010
1
–
6
1
–
6
1 1
1 –– –– 1
–– 12 12 ––
12 1
1 12
––
––
12
12
1
1
––
––
12 1
1 12
–– 1 1 ––
12 –– –– 12
12 12
1.
Comparing Fractions
1
–
3
1
–
6
Depending on student responses, use your professional judgement to guide specific
follow-up.
Observe the comparisons students made and the strategies they used. Ask:
1
◊ Why was it easy to see that _
10 was the smallest? (e.g., It took up hardly any space)
◊ When did you turn the circles to line things up to compare the fractions? Why?
9
(e.g., for _56 and _
10 since they didn’t start at the same place)
◊ Which fractions were really close in size? Was it hard to compare them even when
7
7
_
you moved the fraction? (e.g., _23 and _
10 - I think 10 was more but it was so close, I’m
not sure)
◊ When you are ordering a whole set of fractions, what’s a good strategy for starting?
(e.g., start with really tiny ones at one end and really big ones at the other end and
then compare two at a time in the middle)
1
–
3
1
–
3
1
–
3
Provide the templates of circle and rectangle models for fractions.
By viewing or listening to student responses, note if they:
• use visual estimates when appropriate (when fractions are easily compared
visually)
• appropriately match to compare when the comparison is not as obvious
5
Fraction Circles and Rectangles (1)
Using the Open Question
Explain that they are to use only circles for Question 1 and only rectangles for
Question 2.
© Marian Small, 2010
1
–
8
1
–
5
1
–
5
1
–
8
1
–
8
1
–
8
1
–
8
1
–
8
1
–
3
1
–
3
1
–
3
1
–
3
1
–
3
1
–
3
Comparing Fractions
1
–
4
1
–
4
1
–
4
1
–
4
1
–
8
1
–
4
1
–
4
1
–
4
1
–
4
1
––
1 10
––
10
1
––
10 1
––
10
1
–– 1
10 ––
10 1
––
10
1
––
1 10
1 ––
–– 10
10
1 1 1 1 1 1 1 1 1 1
–– –– –– –– –– –– –– –– –– ––
10 10 10 10 10 10 10 10 10 10
1
–
5
1
–
5
1
–
5
1
–
5
1
–
5
1 1 1 1 1 1 1 1 1 1
–– –– –– –– –– –– –– –– –– ––
10 10 10 10 10 10 10 10 10 10
© Marian Small, 2010
21
7
Solutions
Circles: Students show only 4 of these circle fractions:
1
___
1
__
2
__
3
__
5
__
9
___
10
4
5
4
6
10
Sample strategies:
1
_
10
is very tiny so I knew it was least.
I overlapped _56 and _25 and _25 was less.
9
I could see that _
10 was the most since it was almost the whole thing.
Rectangles: Students show only 4 of these rectangle fractions:
1
___
10
1
__
3
3
__
5
5
__
8
2
__
3
7
___
10
Sample strategies:
1
_
10
is very tiny so I knew it was least.
7
2
_
I overlapped _23 and _
10 and 3 was less
I could just see that _13 < _23 .
8
© Marian Small, 2010
Comparing Fractions
Think Sheet
Materials
Questions to Ask Before Assigning the Think Sheet
Place a pattern block rhombus on top of a hexagon and beside it a pattern block
triangle on top of a hexagon. Ask: What fraction of the hexagon is each shape?
(_13 and _16 )
◊ Which fraction is greater? How can you be sure? (e.g., _13 since it takes up more
space; I can be sure by putting the triangle right on top of it to show it’s smaller)
• Pairs of Fractions
template
• pattern blocks
• coloured pencils
Comparing Fractions Using Pictures
(Continued)
Think Sheet
Think of comparing two different fractions.
Each is a part of the same whole.
Sometimes, it’s just easy to see which is greater.
2 1
2
1
– > – since – is more than half the circle and – is less and it’s easy
3 4
3
4
to see.
Place the rhombus in different positions on the hexagon. Ask:
◊ Does where I put the fraction change its size? (no)
◊ Suppose I had a picture of two fractions. How could I compare them to see which
took up more space? (e.g., cut one out and put it on top of the other)
◊ Why do the wholes have to be the same size if you’re comparing the fractions?
(e.g., even a little part of something really big is a lot more than a big part of
something really little)
Using the Think Sheet
Sometimes we have to work a little harder. One fraction is greater than
another if its picture has “extra” when we overlap them as much as
possible.
3
2
3
other and see that – has a little extra.
Then
We can only use pictures to compare fractions if the fractions are parts
of the same whole.
6
© Marian Small, 2010
Comparing Fractions
Comparing Fractions Using Pictures
(Continued)
Circle the fraction that is greater.
2.
1.
Read the shaded introductory box with the students.
4
–
5
Make sure they understand that sometimes you can just “see” which fraction is more
and at other times you have to cut out and rearrange.
Assign the tasks.
1
–
3
3
–
4
4
–
5
3.
4.
5
–
6
3
–
5
5.
3
–
8
1
–
3
1
–
5
2
–
8
6.
1
–
3
2
–
5
5
8
7. Shade in – of a rectangle. How could you use pictures to show at least
5
six other fractions greater than –?
8
By viewing or listening to student responses, note if they:
• can visually compare quickly, when appropriate
• know how to rearrange visuals to compare, when that is needed
• recognize that when only part of a whole is coloured, then it is easy to get greater
fractions by colouring in more parts
Depending on student responses, use your professional judgement to guide further
follow-up.
Comparing Fractions
1.
© Marian Small, 2010
7
5.
1
–
6
1
–
6
1
–
6
1
–
6
1
–
5
1
–
5
1
–
6
1
–
5
1
–
3
1
–
5
1
–
3
1
–
5
1
–
5
1
–
5
1
–
4
1
–
4
1
–
4
1
–
4
– – – – – – – –
8 8 8 8 8 8 8 8
1
–
5
1
–
3
1
–
3
6.
1
–
5
1
–
3
1
–
5
1
–
5
4. 1 1 1 1 1 1 1 1
1
–
5
1
–
5
1
–
5
1
–
5
1
–
3
1
–
3
1
–
5
1
–
5
1
–
6
2.
1
–
5
1
–
5
1
–
5
Consolidating and Reflecting: Questions to Ask After Using the Think Sheet
◊ Why was it easy to answer Question 1? (e.g., You can just look to see _13 is not
even _12 and _45 is almost the whole thing.)
◊ Why was Question 2 easier than Question 3? (e.g., The fractions were shaded
starting at the same place at the top so it was easy to overlap them.)
◊ Does your strategy change if the shape was a rectangle and not a circle? (e.g., no;
I still had to turn things sometimes or fold them to compare them.)
◊ Why is it easy to see that _78 > _58 if you colour in _58 of a shape? (just colour in two more
sections - if you colour more, the fraction is greater)
© Marian Small, 2010
Pairs of Fractions
3.
Comparing Fractions
2
3
For example, –
> – since we can move one fraction to overlap the
5
1
–
5
1
–
5
1
–
3
1
–
5
1
–
5
1
–
5
1 1 1 1 1 1 1 1
– – – – – – – –
8 8 8 8 8 8 8 8
1
–
3
7.
1
–
8
1
–
8
1
–
8
1
–
4
1
–
5
1
–
6
1
–
8
1
–
8
1
–
4
1
–
5
1
–
6
1
–
8
1
–
5
1
–
6
1
–
8
1
–
4
1
–
5
1
–
6
1
–
8
1
–
4
1
–
5
1
–
6
1
–
6
1 1 1 1 1 1 1 1 1 1
–– –– –– –– –– –– –– –– –– ––
10 10 10 10 10 10 10 10 10 10
22
1
–
8
1
–
8
1
–
8
1
–
4
1
–
5
1
–
6
1
–
8
1
–
8
1
–
4
1
–
5
1
–
6
1
–
8
1
–
5
1
–
6
1
–
8
1
–
4
1
–
5
1
–
6
1
–
8
1
–
6
1
–
8
1
–
8
1
–
4
1
–
5
1
–
6
1 1 1 1 1 1 1 1 1 1
–– –– –– –– –– –– –– –– –– ––
10 10 10 10 10 10 10 10 10 10
© Marian Small, 2010
1
–
8
1
–
4
1
–
5
1
–
6
1
–
8
1
–
8
1
–
4
1
–
5
1
–
6
1
–
8
1
–
5
1
–
6
1
–
8
1
–
4
1
–
5
1
–
6
1
–
8
1
–
4
1
–
6
1
–
5
1
–
6
1 1 1 1 1 1 1 1 1 1
–– –– –– –– –– –– –– –– –– ––
10 10 10 10 10 10 10 10 10 10
Comparing Fractions
9
Solutions
1.
2.
3.
4.
5.
6.
4
_
5
4
_
5
5
_
6
3
_
8
2
_
5
2
_
8
7. Sample responses:
6
8
I could see that _8 , _78 and _8 were bigger by using the same rectangle since more of it
would be used.
I could use the rectangle with 4 sections and colour 3 or 4 of them and line them up to
3
see that _4 and _44 are also bigger.
5
I lined up the rectangle in 6 sections underneath _8 and saw that I would have to go to _46
to get another bigger fraction.
10
© Marian Small, 2010
Comparing Fractions
Comparing Fractions with the Same Denominator
Materials
• Fraction Tower (1)
template
• pattern blocks
Open Question
Questions to Ask Before Using the Open Question
1
_
3
2
_
3
Ask students to use pattern blocks to show and then . Ask:
◊ How do your models show why _23 > _13 ? (e.g., they cover more of the hexagon)
◊ Did you find that surprising? (no, 2 of something is more than 1 of it)
◊ How could you use the fraction tower to show the same thing? (e.g., I go to the _13
row to show both fractions. _23 is two _13 s and that goes farther to the right than _13 does)
Comparing Fractions with the Same Denominator
Open Question
2
–
3
3
–
8
1
–
3
5
–
8
9
–
8
4
––
10
11
––
10
9
––
10
Choose two fractions with the same denominator.
How do you know that the two fractions are not the same size?
How could you decide which is greater without using a picture?
Show your work in the box.
Repeat with two other pairs of fractions.
Using the Open Question
Provide the Fraction Tower (1) template for students to use, if needed.
8
© Marian Small, 2010
1
_
3
1
_
8
1
1
_
_
10 10
1
_
8
1
_
10
Fraction Tower (1)
1
_
10
1
_
8
1
_
10
1
_
8
1
_
10
1
1
_
8
1
_
10
1
_
3
1
1
_
_
8
8
1
1
1
_
_
_
10 10 10
◊ What two fractions did you compare first? (e.g., _38 and _58 )
◊ How did you decide which was greater? (e.g., I knew _58 was greater since it was
more eighths.)
3
4
4
_
_
◊ Would it be easy to compare _
20 to 20 without a picture? How? (e.g., yes, 20 is more
since there are more twentieths)
1
_
8
Depending on student responses, use your professional judgement to guide specific
follow-up.
Consolidating and Reflecting on the Open Question
Comparing Fractions
1
_
3
By viewing or listening to student responses, note if they require the support of the
visuals or recognize that if fractions have the same denominator, the fraction with the
greater numerator is the greater fraction.
Comparing Fractions
© Marian Small, 2010
23
Solutions
Sample responses
2
_
3
and _13 are not the same size since _23 is two sets of _13 , so that’s more. It makes sense
since two of anything is more than one of that thing.
3
_
8
and _58 are not the same size since _38 is 3 sections that are each _18 long, but _58 is 5 of
those sections and 5 is more than 3. I don’t need a picture to see that.
4
_
10
11
4
11
_
_
and _
10 are not the same size since 10 is less than a whole and 10 is more than a
10
_
whole and 10 is a whole. Something more than a whole is greater than something less
than a whole.
Comparing Fractions
© Marian Small, 2010
11
Think Sheet
Materials
Questions to Ask Before Assigning the Think Sheet
Students to use pattern blocks to show _13 and then _23 . Ask:
◊ How do your models show why _23 > _13 ? (e.g., They cover more of the hexagon.)
◊ Did you find that surprising? (no, 2 of something is more than 1 of it)
◊ How could you use the fraction tower to show the same thing? (I go to the _13 row to
show both fractions. _23 is two _13 s and that goes farther to the right than _13 does)
• Fraction Tower (2)
template
• pattern blocks
Comparing Fractions with the Same Denominator
(Continued)
Think Sheet
If two fractions have the same denominator, they are easy
to compare.
3
8
2
8
For example, – > –
3
1
– is 3 copies of –
8
8
2
1
– is 2 copies of –
8
8
3 copies > 2 copies
3 2
–>–
8 8
The same is true even if the fractions are greater than 1.
12 9
–– > –
5
5
12
1
–– is 12 copies of –
5
5
9
1
– is 9 copies of –
5
5
one
whole
Using the Think Sheet
12 copies > 9 copies
12 9
–– > –
5
5
Read the shaded introductory box with the students.
1. Circle the greater fraction in each pair.
Make sure they understand that the numerator tells how many sections and the
denominator indirectly indicates the size of the section.
a) –
5
3
1
–
5
b) –
5
3
8
–
5
2
9
9
–
9
d) –
4
3
2
–
3
c) –
Comparing Fractions
© Marian Small, 2010
9
Assign the tasks.
Indicate that students are free to use the Fraction Tower (2) template, if needed.
By viewing or listening to student responses, note if they observe that when fractions
have the same denominator, the fraction with the greater numerator is the greater
fraction and why.
Depending on student responses, use your professional judgement to guide further
follow-up.
Comparing Fractions with the Same Denominator
5
12
(Continued)
7
12
2. How can you explain to your friend why –– < ––?
3. Choose two different values for each box to make the statement true.
2
3
3
5
8
8
2
– < ––
3
3
a) – < ––
b) – > ––
5
– > ––
8
8
7
c) –– < ––
12
12
7
–– < ––
12
12
4. Fill in the boxes. Use each of the numbers: 0, 1, 2, 3, 4, and 5.
–– < ––
5
5
2
–– > –
8
8
Consolidating and Reflecting: Questions to Ask After Using the Think Sheet
–– < ––
5
10
© Marian Small, 2010
Comparing Fractions
1
1
_
_
10 10
1 _
1
_
12 12
1
_
9
1
_
8
1
_
9
1
_
10
1
_
12
1
_
5
1
_
5
1
1
_
_
8
8
1
1
1
_
_
_
9
9
9
1
1
1
_
_
_
10 10 10
1 _
1 _
1 _
1
_
12 12 12 12
1
_
8
1
_
3
1
_
5
1
1
_
8
1
_
3
1
1
_
_
5
5
1
1
_
_
8
8
1
1
1
_
_
_
9
9
9
1
1
1
_
_
_
10 10 10
1 _
1 _
1 _
1
_
12 12 12 12
1
_
8
1
_
9
1
_
10
1
_
12
Fraction Tower (2)
1
_
3
◊ Is there more than one way to figure out, for Question 1b, which fraction is greater?
(e.g., Yes, since _85 is more fifths so that makes it greater, but it’s also more than
1 and _35 is less than 1, so that’s another reason it’s greater.)
◊ For Question 3, how did you know that 1 could not be an answer for 3a?
(e.g., I know that the number has to be more than 2.)
◊ How did you start Question 4? Why did you start that way? (e.g., I knew that the
missing number for the second one couldn’t be 0, 1, or 2. I know 0 of anything is
small, so it had to be the last numerator or the first numerator.)
◊ What rule could you give for comparing fractions with the same denominator and
why does it work? (e.g., The fraction with the greater numerator is greater since
there are more pieces and all the pieces are the same for both fractions.)
24
© Marian Small, 2010
Comparing Fractions
Solutions
1. a)
3
_
5
b)
8
_
5
c)
9
_
9
d)
4
_
3
5
1
7
1
_
_
_
2. e.g., _
12 is 5 sections of 12 and 12 is 7 sections of 12 so there are two extra sections;
that makes it more.
3. a) e.g., 3 and 4
4.
12
1
_
5
< _25
3
_
8
> _28
b) e.g., 3 and 4
0
_
5
c) e.g., 3 and 4
< _45
© Marian Small, 2010
Comparing Fractions
Comparing Fractions with the Same Numerator
Materials
• Fraction Tower (3)
template
Open Question
Comparing Fractions with the Same Numerator
Questions to Ask Before Using the Open Question
3
_
3
3
_
4
3
_
4
3
_
3
Open Question
4
–
8
3
_
4
◊ Which do you think is more or ? (e.g., since is a whole and is less)
1
_
3
4
––
10
4
–
3
5
–
8
5
–
2
5
––
10
6
–
9
6
–
8
6
–
4
Choose two fractions from the list with the same numerator.
How do you know that the two fractions are not the same size?
How could you decide which is greater without using a picture?
Show your work in the boxes.
1
_
8
Have the student look at the and the on the tower. Ask:
◊ Which fraction is greater? Why? (_13 since it is a bigger piece)
◊ Why is it bigger? (since the whole is only 3 sections instead of 8 so there is more in
each section)
◊ What about _23 and _28 which is more? Why? (_23 , if there are two bigger sections and
two littler sections, the two bigger ones are worth more)
Repeat with two other pairs of fractions.
Comparing Fractions
© Marian Small, 2010
11
Using the Open Question
1
_
8
1
_
9
1
_
10
1
_
8
1
_
4
1
_
8
1
_
4
1
_
8
1
1
1
1
_
_
_
_
9
9
9
9
1
1
1
1
1
_
_
_
_
_
10 10 10 10 10
1
_
3
1
_
2
1
_
8
1
_
9
1
_
10
Consolidating and Reflecting on the Open Question
1
1
_
_
8
8
1
1
1
_
_
_
9
9
9
1
1
1
_
_
_
10 10 10
1
_
4
1
_
3
1
_
2
Depending on student responses, use your professional judgement to guide specific
follow-up.
1
_
4
1
_
8
1
_
3
By viewing or listening to student responses, note if they:
• require the support of the visuals
• recognize that if fractions have the same numerator, the fraction with the greater
denominator is the lesser fraction
Fraction Tower (3)
1
Students can use the Fraction Tower (3) template, if needed.
Comparing Fractions
© Marian Small, 2010
25
4
◊ What two fractions did you compare first? (e.g., _48 and _
10 )
4
4
1
_
_
◊ How did you decide which was greater? (e.g., I knew 8 is _12 and _
10 is not even 2
5
1
_
_
since 2 is 10 )
4
◊ How could you use the section sizes to help figure out that _48 is more than _
10 ? (e.g.,
I know that eighths are bigger than tenths and so 4 eighths is more than 4 tenths.)
◊ Why is it difficult to tell whether _35 or _46 is greater but easier to tell if _35 or _36 is greater?
(e.g., I know fifths are bigger than sixths so if I have the same number of each, I
know _35 is more. But even though sixths are smaller, if I have more of them (like with
4 sixths compared to 3 fifths), I don’t know which is more in total.)
◊ How could you advise someone to compare two fractions like _4 and _4 if you don’t
know exactly what the denominators are? (e.g., I would tell them that whichever is
the bigger denominator is the smaller fraction since the pieces are bigger and 4 big
pieces is always more than 4 little ones.)
Solutions
Sample responses:
4
_
8
4
and _
10 are not the same size since if the whole is divided into 10, the pieces are
smaller than when it is divided into 8. So I am comparing 4 smaller pieces to 4
4
bigger ones. That means that _48 > _
10 . I don’t need a picture to tell.
5
_
2
> _58 since _52 is more than one whole (which is only _22 ) and _58 is less than a whole,
since it’s not _88
6
_
9
< _68 since ninths are smaller than eighths so six small pieces are less than six
bigger ones
Comparing Fractions
© Marian Small, 2010
13
Think Sheet
Materials
• Pairs of Fractions
template
Questions to Ask Before Assigning the Think Sheet
Comparing Fractions with the Same Numerator
(Continued)
Think Sheet
If two fractions have the same numerator, they are easy to compare.
3
8
Have the student look at the _13 and the _18 on the tower. Ask:
◊ Which fraction is greater? Why? (e.g., _13 since it is a bigger piece)
◊ Why is it bigger? (e.g., since the whole is only 3 sections instead of 8 so there is
more in each section)
◊ Which is more _23 or _28 ? Why? (_23 , e.g., if there are two bigger sections and two littler
sections, the two bigger ones are worth more)
Using the Think Sheet
3
For example, – < –
5
◊ Which do you think is more _33 or _34 ? (e.g., _34 since _33 is a whole and _34 is less)
3
1
– is 3 copies of –
8
8
3
1
– is 3 copies of –
5
5
1 1
– < – since the whole is shared into more pieces,
8 5
which means the pieces are smaller.
3 3
–<–
8 5
The same is true if the fractions are greater than 1.
8
–
5
8
–
3
8
–
5
1
–
5
8
–
5
1
3
1
is 8 copies of –
is 8 copies of –
5
1
3
8
<–
3
<–
1. Circle the greater fraction in each pair.
3
8
3
––
10
b) ––
2
9
2
–
3
d) –
a) –
c) –
4
12
4
–
5
5
3
12
Read the shaded introductory box with the students.
8
3
<–
5
–
2
© Marian Small, 2010
Comparing Fractions
Comparing Fractions with the Same Numerator
5
12
(Continued)
5
8
2. How can you explain to your friend why –– < –? It doesn’t make sense
to her since 12 > 8. What could you say to help?
Make sure they understand that the numerator tells how many sections and the
denominator indirectly indicates the size of the section.
3. Choose two different values for each part to make the statement true.
2
3
Assign the tasks.
a) – > ––
2
2
2
– > ––
3
5
5
b) – > ––
8
5
5
– > ––
8
3
3
3
–– > –
5
3
c) –– > –
5
Indicate that students can use the Pairs of Fractions template, if needed.
4. Fill in the boxes. Use each of the numbers 1, 2, 3, 4, and 5.
4
–– > ––
10
12
By viewing or listening to student responses, note if they observe that when fractions
have the same numerator, the fraction with the greater denominator is the lesser
fraction and why.
3
– > ––
5
8
1
1
– > ––
2
3
3
–– > ––
Depending on student responses, use your professional judgement to guide further
follow-up.
Consolidating and Reflecting: Questions to Ask After Using the Think Sheet
◊ Is there more than one way to figure out, for Question 1c, which fraction is greater?
(e.g., yes, since _29 is way less than half and _23 is more, I know _23 is bigger but I also
know that 2 thirds is more since thirds are bigger than ninths, so _23 > _29 )
◊ For Question 3, how did you know that 1 could not be an answer for 3a?
(e.g., I know that the number has to be more than 3.)
◊ How did you start Question 4? (e.g., I know it’s easiest to compare if the
denominators or numerators are the same, so I used 4 for the first fraction and 3 for
the second one.)
Comparing Fractions
© Marian Small, 2010
13
Pairs of Fractions
1.
1
–
6
5.
1
–
6
1
–
6
1
–
6
1
–
6
1
–
5
1
–
5
1
–
5
1
–
5
1
–
5
1
–
3
1
–
5
1
–
3
1
–
6
1
–
5
1
–
4
1
–
4
1
–
4
1
–
4
– – – – – – – –
8 8 8 8 8 8 8 8
1
–
5
1
–
3
1
–
3
6.
1
–
5
1
–
3
1
–
5
1
–
5
4. 1 1 1 1 1 1 1 1
1
–
5
1
–
5
1
–
5
1
–
5
1
–
3
1
–
3
1
–
5
1
–
5
3.
2.
1
–
5
1
–
5
1
–
5
1
–
5
1
–
5
1
–
3
1
–
5
1
–
5
1
–
5
1 1 1 1 1 1 1 1
– – – – – – – –
8 8 8 8 8 8 8 8
1
–
3
7.
1
–
8
1
–
8
1
–
8
1
–
4
1
–
6
1
–
8
1
–
8
1
–
4
1
–
5
1
–
5
1
–
6
1
–
8
1
–
6
1
–
8
1
–
4
1
–
5
1
–
6
1
–
8
1
–
4
1
–
5
1
–
5
1
–
6
1
–
6
1 1 1 1 1 1 1 1 1 1
–– –– –– –– –– –– –– –– –– ––
10 10 10 10 10 10 10 10 10 10
22
1
–
8
1
–
8
1
–
8
1
–
4
1
–
6
1
–
8
1
–
8
1
–
4
1
–
5
1
–
5
1
–
6
1
–
8
1
–
6
1
–
8
1
–
4
1
–
5
1
–
6
1
–
8
1
–
6
1
–
8
1
–
8
1
–
4
1
–
5
1
–
6
1 1 1 1 1 1 1 1 1 1
–– –– –– –– –– –– –– –– –– ––
10 10 10 10 10 10 10 10 10 10
© Marian Small, 2010
1
–
8
1
–
4
1
–
5
1
–
6
1
–
8
1
–
8
1
–
4
1
–
5
1
–
5
1
–
6
1
–
8
1
–
6
1
–
8
1
–
4
1
–
5
1
–
6
1
–
8
1
–
4
1
–
5
1
–
6
1
–
5
1
–
6
1 1 1 1 1 1 1 1 1 1
–– –– –– –– –– –– –– –– –– ––
10 10 10 10 10 10 10 10 10 10
Comparing Fractions
Solutions
1. a)
3
_
8
b)
4
_
5
c)
2
_
3
d)
5
_
2
2. e.g., If something is divided into 12 pieces, each piece is smaller since there are
5
5
_
more pieces. That means twelfths are smaller than eighths, so _
12 is less than 8 .
3. a) e.g., 4 or 5
4.
14
4
_
10
4
>_
12
3
_
5
3
> _8
b) e.g., 9 or 10
1
_
2
> _15
3
_
1
c) e.g., 3 or 4
> _32
© Marian Small, 2010
Comparing Fractions
Equivalent Fractions
Materials
• Fraction Tower (4)
template
• ruler (or any object with
straight edge)
Open Question
Questions to Ask Before Using the Open Question
◊ Suppose a pie was cut into 8 equal pieces. How many pieces would be half of it?
Why? (4 pieces since there would be two equal shares of 4)
◊ How does that tell you that _12 = _48 ? (_48 is 4 pieces if there are 8 pieces in the pie and
it’s half the pie)
◊ We call fractions that are equal equivalent fractions. Can you think of another
5
fraction equivalent to _12 ? (e.g., _
10 )
Help the student see why _12 = _24 on the tower by showing how a vertical line down the
right edge of _12 passes through the right edge of _24 .
Using the open Question
Equivalent Fractions
Open Question
Look at the Fraction Tower.
2
1 2
– sections to see why – = –.
6
3 6
1
_
2
1
_
3
1
_
3
1
_
4
1
3
example, use a ruler at the end of the – and
1
_
4
1
_
5
1
_
6
1
_
9
1
_
9
1
_
10
1
_
10
1
_
12
1
_
8
1
_
10
1
_
12
1
_
12
1
_
6
1
_
8
1
_
9
1
_
10
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
_
12
1
_
3
1
_
4
1
_
5
1
_
9
1
_
8
1
_
9
1
_
10
1
_
10
1
_
12
1
_
12
1
_
6
1
_
8
1
_
9
1
_
10
1
_
12
1
_
8
1
_
9
1
_
10
1
_
12
1
_
9
1
_
10
1
_
12
1
_
12
1
_
10
1
_
12
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1 _
1
1 _
1 _
_
15 15 15 15 15 15 15 15 15 15 15 15 15 15 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
Find as many sets of equivalent fractions on the Fraction Tower as
you can.
What do you notice about the numerators and denominators of equivalent
fractions?
14
© Marian Small, 2010
Comparing Fractions
Fraction Tower (4)
Encourage students to list as many possible equivalent fractions as they can, using
the Fraction Tower (4) template.
1
1_
2
© Marian Small, 2010
1_
4
1_
5
1_
6
1_
9
1_
9
1
_
10
1
_
10
1
_
12
1
_
15
1_
8
1
_
10
1
_
15
1_
9
1
_
10
1
_
12
1
_
15
1
_
15
1_
9
1
_
12
1
_
15
1
_
15
1_
6
1_
8
1_
9
1
_
10
1
_
12
1
_
15
1_
6
1_
8
1
_
10
1
_
12
1_
5
1_
6
1_
8
1_
9
1
_
12
1
_
15
1_
5
1_
6
1_
8
1_
4
1_
5
1_
6
1_
8
1
_
12
1_
3
1_
4
1_
5
1
_
15
◊ Which fraction did you find the most equivalent fractions for? Why do you think that
happened? (e.g., 1 had the most equivalent fractions since each line was another
name for 1)
◊ Once you knew that _13 = _26 , how did that help you predict other equivalent fractions?
(e.g., I just doubled the numerators to get _23 = _46 and then I also doubled the
4
numerator and denominator of _26 to get _
12 .)
1
_
◊ Look at all the equivalent fractions for 3 . What do you notice about the relationship
between the numerator and denominator? (The denominator is always 3 times the
numerator.)
◊ How could that help you get more equivalent fractions? (e.g., I would just write any
numerator I wanted and multiply by 3 to get the denominator.)
◊ What rule could you give someone for getting equivalent fractions? (e.g., multiply
the numerator and denominator by the same thing.)
1_
3
1_
4
Depending on student responses, use your professional judgement to guide specific
follow-up.
Consolidating and Reflecting on the Open Question
1_
2
1_
3
Encourage them to look at the values of the equivalent fractions to see what
they notice.
Comparing Fractions
1
1
_
2
Two fractions are equivalent, or equal, if
they take up the same amount of area. For
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
_
15
1
_
12
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
26
© Marian Small, 2010
Comparing Fractions
15
Solutions
Any or all of:
10
15
18
20
12
_
_
_
_
1 = _22 = _33 = _44 = _55 = _66 = _88 = _99 = _
10 = 12 = 15 = 18 = 20
1
_
2
5
6
9
10
_
_
_
= _24 = _36 = _48 = _
10 = 12 = 18 = 20
1
_
3
5
6
4
_
_
= _26 = _39 = _
12 = 15 = 18
2
_
3
8
10
12
_
_
= _46 = _69 = _
12 = 15 = 18
1
_
4
3
5
_
= _28 = _
12 = 20
3
_
4
9
15
_
= _68 = _
12 = 20
1
_
5
3
2
4
_
_
=_
10 = 15 = 20
2
_
5
6
8
4
_
_
=_
10 = 15 = 20
3
_
5
6
9
12
_
_
=_
10 = 15 = 20
4
_
5
8
16
12
_
_
=_
10 = 15 = 20
1
_
6
3
2
_
=_
12 = 18
5
_
6
10
15
_
=_
12 = 18
1
_
9
2
=_
18
4
_
9
8
=_
18
5
_
9
10
=_
18
7
_
9
14
=_
18
8
_
9
16
=_
18
1
_
10
2
=_
20
3
_
10
6
=_
20
7
_
10
14
=_
20
9
_
10
18
=_
20
I notice that every time you need to multiply the numerator and denominator of one
fraction by the same amount to get the other fraction.
16
© Marian Small, 2010
Comparing Fractions
Think Sheet
Materials
• Pairs of Fractions
template
Questions to Ask Before Assigning the Think Sheet
Equivalent Fractions
(Continued)
Think Sheet
Two fractions are equivalent if one is another name for the same
amount.
2
3
◊ Suppose a pie was cut into 8 equal pieces. How many pieces would be half of it?
Why? (4 pieces since there would be two equal shares of 4)
◊ How does that tell you that _12 = _48 ? (_48 is 4 pieces if there are 8 pieces in the pie and
it’s half the pie)
◊ We call fractions that are equal “equivalent fractions”. Can you think of another
fraction equivalent to _12 ? (e.g., 5 out of 10)
4
6
For example, – = –
The fractions are equivalent since:
2
• We can split the sections of – equally and it is the same part of the
3
whole.
2 out of every 3 is the same as 4 out of every 6.
• Notice that if we multiply both the numerator and denominator by
the same amount (but not 0), we get an equivalent fraction. If we
have 2 times or 3 times or 4 times as many pieces, we will have 2
times or 3 times or 4 times as many pieces of the type we want.
3
3×3
5×3
9
For example, –
= ––– = ––
5
15
We can either split all the fifths into 3 equal pieces or we can make
3 rows of 3 out of 5.
1. Which of these pairs of fractions are equivalent?
Using the Think Sheet
3
8
––
10
a) –
2
3
12
––
18
b) –
5
5
8
10
––
16
d) ––
c) –
3
10
Comparing Fractions
12
––
40
© Marian Small, 2010
15
Read the shaded introductory box with the students.
Make sure they understand that an equivalent fraction either uses the same space or
represents the same ratio, i.e., ____ out of every ____.
Equivalent Fractions
(Continued)
5
12
2. a) Name three fractions equivalent to ––.
b) Choose one of your fractions. How would you convince someone
5
12
who wasn’t sure why it is equivalent to ––?
Assign the tasks.
By viewing or listening to student responses, note if students:
• can identify equivalent fractions
• can create equivalent fractions either by multiplying both terms or dividing both
terms by the same amount
• recognize patterns in equivalent fractions
• recognize the difference in the effect of adding the same amount to both terms of a
fraction as compared to multiplying both terms by the same amount
Depending on student responses, use your professional judgement to guide further
follow-up.
2
3
3. Jane listed equivalent fractions for –. She noticed that the denominators
were always at least 3 apart. Do you agree? Explain.
4. How do you know that you cannot add 2 to the numerator and
3
4
denominator of – and end up with an equivalent fraction?
6
5. –– = ––
2
_
3
◊ Why are and equal? (e.g., divide each third into 6 equal sections and you
12
have _
18 )
8
◊ Why are _35 and _
10 not equal even though 8 is 5 more than 3 and 10 is 5 more than
3
8
_
5? (e.g., 5 is just a little bit over half but _
10 is almost the whole thing)
3
_
◊ List some equivalent fractions for 4 . How far apart are the denominators? Why?
9 _
12
(e.g., _68 , _
12 , 16 - the denominators are 4 apart - if you split the fourths into smaller
pieces, you get 4 extra pieces each time)
◊ For the last question, how did you know that if is big, then is little? (e.g., if is
big, then the fraction _6 is little so you need a little numerator for 3.)
be?
© Marian Small, 2010
Comparing Fractions
Pairs of Fractions
1.
5.
1
–
6
1
–
6
1
–
6
1
–
6
1
–
5
1
–
5
1
–
5
1
–
3
1
–
5
1
–
3
1
–
5
1
–
6
1
–
5
1
–
5
1
–
4
1
–
4
1
–
4
1
–
4
– – – – – – – –
8 8 8 8 8 8 8 8
1
–
5
1
–
3
1
–
3
6.
1
–
5
1
–
3
1
–
5
1
–
5
4. 1 1 1 1 1 1 1 1
1
–
5
1
–
5
1
–
5
1
–
5
1
–
3
1
–
3
1
–
5
1
–
5
3.
2.
1
–
5
1
–
5
1
–
6
12
_
18
and
=
16
1
–
5
Consolidating and Reflecting: Questions to Ask After Using the Think Sheet
What can the values of
3
=
1
–
5
1
–
5
1
–
3
1
–
5
1
–
5
1
–
5
1 1 1 1 1 1 1 1
– – – – – – – –
8 8 8 8 8 8 8 8
1
–
3
7.
1
–
8
1
–
8
1
–
8
1
–
4
1
–
5
1
–
6
1
–
8
1
–
8
1
–
4
1
–
5
1
–
6
1
–
8
1
–
5
1
–
6
1
–
8
1
–
4
1
–
5
1
–
6
1
–
8
1
–
4
1
–
5
1
–
6
1
–
6
1 1 1 1 1 1 1 1 1 1
–– –– –– –– –– –– –– –– –– ––
10 10 10 10 10 10 10 10 10 10
22
1
–
8
1
–
8
1
–
8
1
–
4
1
–
5
1
–
6
1
–
8
1
–
8
1
–
4
1
–
5
1
–
6
1
–
8
1
–
5
1
–
6
1
–
8
1
–
4
1
–
5
1
–
6
1
–
8
1
–
6
1
–
8
1
–
8
1
–
4
1
–
5
1
–
6
1 1 1 1 1 1 1 1 1 1
–– –– –– –– –– –– –– –– –– ––
10 10 10 10 10 10 10 10 10 10
© Marian Small, 2010
1
–
8
1
–
4
1
–
5
1
–
6
1
–
8
1
–
8
1
–
4
1
–
5
1
–
6
1
–
8
1
–
5
1
–
6
1
–
8
1
–
4
1
–
5
1
–
6
1
–
8
1
–
4
1
–
6
1
–
5
1
–
6
1 1 1 1 1 1 1 1 1 1
–– –– –– –– –– –– –– –– –– ––
10 10 10 10 10 10 10 10 10 10
Comparing Fractions
Solutions
1. a, c, d
10 _
15 _
20
2. a) e.g., _
24 , 36 , 48
5
b) e.g., I would draw _
12 and break up each twelfth into 2 equal parts. Then there
10
would be 24 equal parts and 10 parts would be shaded, so that’s _
24 .
3. yes e.g., The equivalent fractions come from multiplying 2 and 3 by the same
amount. When you multiply by 3, there is an extra group of 3 for each extra amount
in the number, so there is always 3 or 6 or 9, …
4. e.g., I know that _56 is _16 away from 1, but _34 is _14 away and since _16 is less, it has to be
closer to 1.
18
6
9
6
6
6
3
_
_
_
_
_
_
5. e.g., _61 = _
3 or 2 = 3 or 3 = 3 or 6 = 3
Comparing Fractions
© Marian Small, 2010
17
Comparing Fractions to _12 and 1
Materials
• Fraction Tower (4)
template
Open Question
Questions to Ask Before Using the Open Question
Comparing Fractions to _21 and to 1
Open Question
Choose values for the missing parts of the fractions so that they are less
1
2
than –. Use the same number only once. Tell how you chose the values.
Examine the fraction tower with the students. Ask:
◊ How many thirds do you need to be past _12 ? (_23 )
◊ How many fourths? (_34 )
◊ How many fifths? (_35 )
◊ What do you notice? (e.g., There were two lines in a row that needed the same
number of sections to get past _12 and then it changed.)
◊ What if you used sixths and sevenths? (It would be 4 both times.)
5
7
1
4
1
_
_
_
_
◊ How could you predict that _
10 is more than 2 but 10 is less? (e.g., I know that 2 = 10
1
_
so I have to go higher than 5 in the numerator to be more than 2 .)
Using the Open Question
––
8
––
5
5
––
3
––
––
10
8
––
Now choose values so that the fractions are greater than 1. Tell how you
chose the values.
––
8
––
5
5
––
3
––
––
10
Comparing Fractions
8
––
© Marian Small, 2010
17
Fraction Tower (4)
1
Students can use the Fraction Tower (4) template, if needed.
1_
2
Encourage students to come up with several values for each numerator or
denominator.
◊ Was it easier to create fractions greater than 1 or less than _12 to make up the
numerator? (e.g., greater than 1 since I just could add 1 to the denominator)
◊ How could you be sure your fraction was less than _12 ? (If I took half of the
denominator and the numerator was lower, I would be right.)
◊ Is it easier to create fractions greater than 1 if you are given the denominator or the
numerator? (e.g., denominator, since I just keep adding 1s to get more and more
numberators, but if I get the numerator, I can subtract, but I don't have as many
possiblities)
10
14
1
1
14
_
_
_
_
◊ Is _
20 more or less than 2 ? How do you know? (e.g., more since 20 is 2 and 20 is more
10
_
than 20 )
1_
3
1_
4
1_
9
1_
9
1
_
10
1
_
10
1
_
12
1
_
15
1_
8
1
_
10
1
_
15
1_
9
1
_
10
1
_
12
1
_
15
1
_
15
1_
9
1
_
12
1
_
15
1
_
15
1_
6
1_
8
1_
9
1
_
10
1
_
12
1
_
15
1_
6
1_
8
1
_
10
1
_
12
1_
5
1_
6
1_
8
1_
9
1
_
12
1
_
15
1_
5
1_
6
1_
8
1_
4
1_
5
1_
6
1_
8
1
_
12
1_
4
1_
5
1_
6
1
_
15
1_
3
1_
4
1_
5
Depending on student responses, use your professional judgement to guide specific
follow-up.
Consolidating and Reflecting on the Open Question
1_
2
1_
3
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
_
15
1
_
12
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
26
© Marian Small, 2010
Comparing Fractions
Solutions
e.g., _18
2
_
5
4
_
10
3
_
10
5
_
30
8
_
20
I had to make sure that the denominator was more than double the numerator.
e.g., _98
6
_
5
11
_
10
3
_
2
5
_
4
8
_
3
I had to make sure that the numerator was more than the denominator.
18
© Marian Small, 2010
Comparing Fractions
Think Sheet
Materials
• Fraction Tower (4)
template
Questions to Ask Before Assigning the Think Sheet
Comparing Fractions to _21 and to 1
(Continued)
Think Sheet
1
2
Comparing to –
1
2
1
2
1
2
5 1
3 1
5 3
For example, – > – since half of 6 is 3. So – = – and – > –.
6 2
6 2
6 6
A fraction equals – if the denominator is 2 times the numerator.
Examine the fraction tower with the students. Ask:
◊ How many thirds do you need to be past _12 ? (_23 )
◊ How many fourths? (_34 )
◊ How many fifths? (_35 )
◊ What do you notice? (e.g., There were two lines in a row that needed the same
number of sections to get past _12 and then it changed.)
◊ What if you used sixths and sevenths? (It would be 4 both times.)
5
7
1
4
1
_
_
_
_
◊ How could you predict that _
10 is more than 2 but 10 is less? (e.g., I know that 2 = 10
1
so I have to go higher than 5 in the numerator to be more than _2 .)
Using the Think Sheet
3
6
For example, – = – since 6 = 2 × 3.
A fraction is more than – if the numerator is more than half the denominator.
1
_
2
1
2
6
6
4
1
1
4
–– < – since half of 12 is 6. So – = –– and –– < ––.
12 2
2 12
12 12
A fraction is less than – if the numerator is less than half of the denominator.
1
_
2
Comparing to 1
A fraction equals 1 if the numerator and denominator are equal.
4 5 6
4
6
For example, –, –
, –.
5
A fraction is more than 1 if the numerator is greater than the denominator.
6
5
6
5
For example, –
> 1 since 1 = –
and –
> –.
5
5
5 5
1
A fraction is less than 1 if the numerator is less than the denominator.
5
4
< 1 since it is only 4 out of 5 parts, not the whole –
.
For example, –
5
5
1
2
1. Circle the fractions that are less than –.
3
–
8
4
––
10
2
–
5
2
–
3
2
–
9
1
2
2. Circle the fractions that are between – and 1.
7
–
8
6
–
4
3
–
5
2
––
10
5
–
3
18
© Marian Small, 2010
Comparing Fractions
Comparing Fractions to _21 and to 1
(Continued)
1
3. What possible values could the numerator have if –– < –? Why would
9
2
the numerator have just these values?
Read the shaded introductory box with the students.
4. Replace the missing values with counting numbers (1, 2, 3,…).
Why are there more solutions to –– > 1 than to –– < 1?
6
Indicate that the students can use the Fraction Tower (4) template, if needed.
6
Assign the tasks.
5. Fill in the boxes with 2, 4, 6, 8, 10. Use each number once to make these
statements true.
By viewing or listening to student responses, note if they realize that:
• to compare to 1, they simply compare numerator and denominator and a greater
numerator means greater than 1
• to compare to _12 , they can either take half the denominator and compare to the
numerator or double the numerator and compare to the denominator
Depending on student responses, use your professional judgement to guide further
follow-up.
8
1
–– > ––
1
4
– < ––
2
Comparing Fractions
© Marian Small, 2010
19
Fraction Tower (4)
1
1_
2
Consolidating and Reflecting: Questions to Ask After Using the Think Sheet
5
4
1
1
4
_
_
_
_
◊ How did you know that _
10 < 2 ? (I know that 10 = 2 and 10 is less.)
2
1
_
_
◊ How did you know that 9 was less than 2 ? (e.g., I know that half of 9 is about 4, so a
little more than _49 = _12 ; _29 is less.)
◊ Is it easy to decide if a fraction is greater than 1? (Yes, the numerator is more than
the denominator.)
◊ What rule could you give for deciding if a fraction is between _12 and 1? (e.g., The
numerator has to be less than the denominator or it would not be less than 1. But
if you take half of the denominator, the numerator must be more than that or it
wouldn’t be more than _12 .)
6
1
–– > –
2
–– > 1
2
1_
2
1_
3
1_
3
1_
4
1_
5
1_
9
1_
9
1
_
10
1
_
10
1
_
12
1
_
15
1_
8
1
_
10
1
_
15
1_
9
1
_
10
1
_
12
1
_
15
1
_
15
1_
9
1
_
12
1
_
15
1
_
15
1_
6
1_
8
1_
9
1
_
10
1
_
12
1
_
15
1_
6
1_
8
1
_
10
1
_
12
1_
5
1_
6
1_
8
1_
9
1
_
12
1
_
15
1_
5
1_
6
1_
8
1_
4
1_
5
1_
6
1_
8
1
_
12
1_
4
1_
5
1_
6
1
_
15
1_
3
1_
4
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
_
15
1
_
12
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
26
© Marian Small, 2010
Comparing Fractions
Solutions
1.
3 _
4 _
2 _
2
_
8 10 5 9
2.
7
_
8
,
, ,
and _35
3. 0, 1, 2, 3, or 4 since half of 9 is 4 _12 , so 5 would be too much
4. e.g., Any number greater than 6 makes _6 >1 so you could go to the millions or
higher but only 0, 1, 2, 3, 4, and 5 work if the fraction is less than 1.
5. Sample responses:
8
4
1
4
1
_
_
_
_
_
2 > 1
10 > 2
2 < 6
Comparing Fractions
6
_
8
> _12
© Marian Small, 2010
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