Gap Closing
Comparing Fractions
Junior / Intermediate
Student Book
Module 2
Comparing Fractions
Diagnostic..........................................................................................3
Comparing Fractions Using Pictures.......................................5
Comparing Fractions with the Same Denominator............. 8
Comparing Fractions with the Same Numerator............... 11
Equivalent Fractions....................................................................14
Comparing Fractions to _​ 12 ​and to 1..........................................17
Templates
Fraction Circles and Rectangles (1)................................20
Fraction Circles and Rectangles (2)................................21
Pairs of Fractions...................................................................22
Fraction Tower (1)..................................................................23
Fraction Tower (2)..................................................................24
Fraction Tower (3)..................................................................25
Fraction Tower (4)..................................................................26
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.
7
12
5
12
because
3
4
3
2
because
5
6
5
8
because
4
6
1
3
because
––  ​
a)​ ––  ​​
b)​ – ​​ – ​
c)​ – ​​ – ​
d)​ – ​​ – ​
2
3. Circle all of the fractions below that are greater than –
​ 5 ​.
4
​ a)​ –
5
4
1
b) ​ –
​ 5
5
c) ​ –
​ 5
2
3
d) ​ – ​ 2
e) ​ –
​ 7
© Marian Small, 2010
8
3
f) ​ – ​ 20
6
g) ​ ––  ​ 7
8
h) ​ – ​
Comparing Fractions
Diagnostic(Continued)
4
6
4. Circle the fractions below that are equivalent (equal) to –
​   ​.
2
3
a)​ – ​
3
b) ​ –
​ 5
5
c) ​ –
​ 7
20
30
d) ​ ––  ​ 32
48
e) ​ ––  ​
5. Complete the statement: The best way to create an equivalent fraction
3
to –
​  5 ​is to
1
2
6. Circle all of the fractions below that are between 0 and –
​   ​.
2
​ a)​ –
5
5
5
6
b) ​ – ​ 4
9
c) ​ – ​ 2
3
d) ​ – ​ 1
10
e) ​ ––  ​
© Marian Small, 2010
4
f) ​ –
7
Comparing Fractions
Comparing Fractions Using Pictures
Learning Goal
• reasoning about how two fractions shown pictorially can be compared.
Open Question
1. Choose 4 of the 6 fraction circles to put in order from least to greatest
depending on how much is shaded.
Describe your strategies.
C
B
A
E
D
F
2. Choose 4 of the 6 fraction rectangles to put in order from least to
greatest depending on how much is shaded.
Describe your strategies.
B
A
C
D
E
6
F
© Marian Small, 2010
Comparing Fractions
Comparing Fractions Using Pictures
(Continued)
Think Sheet
Think of comparing two different fractions.
Each is a part of the same whole.
2
3
1
4
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.
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
5
2
3
3
2
3
For example, –
​ 5 ​ > –
​   ​since we can move the shaded area from one
3
2
3
fraction ( –
)  into the unshaded area of the other fraction ( – ) and see
5
3
2
3
that –
takes up less area than –.
5
We can only use pictures to compare fractions if the fractions are parts
of the same whole.
7
© Marian Small, 2010
Comparing Fractions
Comparing Fractions Using Pictures
(Continued)
Circle the fraction that is greater.
2.
1.
3
1
4
–
​   ​​ –
​​ – ​
5
3
4
4
​ –
​ 5
4.
3.
5
6
3
​ – ​​ –
​
5
5.
​ – ​
6.
2
1
1
​ –
​ ​ – ​
5 3
2
​   ​
​ –
​ –
5 8
5
8
1
3
3
​ – ​
8
7. Shade in –
​   ​of a rectangle. Use these pictures to show at least six other
5
8
fractions greater than –
​   ​.
8
© Marian Small, 2010
Comparing Fractions
Comparing Fractions with the Same Denominator
Learning Goal
• reasoning about how two fractions with the same denominator can be
compared.
Open Question
3
5
9
9
2
1
4
11
––  ​ ​ ––  ​
​ – ​ –
​   ​ ​ – ​ –
​   ​ ​ – ​ ​ ––  ​ ​
3
3
8
8
8
10
10
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.
9
© Marian Small, 2010
Comparing Fractions
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
8
8
2
1
​ – ​is 2 copies of –
​   ​
8
8
​ – ​is 3 copies of –
​   ​
3 copies > 2 copies
3
8
2
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
one
9
1
whole
​is 9 copies of –
​ 5 ​
​ –
5
12 copies > 9 copies
12
9
​ ––
​ > –
​  5 ​
5
1. Circle the greater fraction in each pair.
3
1
–
​  5 ​
b) ​ –
​ 5
2
9
9
–
​   ​
9
d) ​ – ​  ​ a)​ –
5
c)​ – ​ 10
3
8
–
​  5 ​
4
3
2
–
​   ​
3
© Marian Small, 2010
Comparing Fractions
Comparing Fractions with the Same Denominator
5
12
(Continued)
7
12
2. How can you explain to your friend why ––
​    ​ < ––
​    ​?
3. Choose different values for the two boxes to make each statement true.
2
3
2
3
3
5
8
8
– ​ < ––
a)​ – ​ < ––
​     ​​
​     ​
3
5
8
– ​ > ––
b) ​ – ​ > ––
​     ​​
​     ​
8
7
12
7
12
––  ​ < ––
c) ​ ––  ​ < ––
​    ​​
​    ​
12
12
4. Fill in the boxes. Use each of the numbers: 0, 1, 2, 3, 4, and 5.
​ ––
​ < ––
​  5   ​
5
2
8
​   ​
​ ––   ​ > –
8
​ < ––
​    ​
​ ––
5
11
© Marian Small, 2010
Comparing Fractions
Comparing Fractions with the Same Numerator
Learning Goal
• reasoning about how two fractions with the same numerator can be
compared.
Open Question
5
5
5
6
6
6
4
4
4
– ​ ​ – ​ ​ – ​ ​ ––  ​ ​
– ​ ​ – ​ ​ – ​
​ – ​ ​ ––  ​ ​
8
10
3
8
2
10
9
8
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.
Repeat with two other pairs of fractions.
12
© Marian Small, 2010
Comparing Fractions
Comparing Fractions with the Same Numerator
(Continued)
Think Sheet
If two fractions have the same numerator, they are easy to compare.
3
8
3
For example, –
​   ​ < –
​  5 ​
3
1
8
8
3
1
​ –
​is 3 copies of –
​
​
5
5
​ – ​is 3 copies of –
​   ​
1
8
1
3
8
3
​ – ​ < –
​  5 ​since the whole is shared into more pieces, so
the pieces are smaller.
​ – ​ < –
​  5 ​
The same is true if the fractions are greater than 1.
8
8
3
​ –
​ < –
​   ​
5
8
1
3
3
8
1
​ –
​is 8 copies of –
​
​
5
5
1 1
​ –
​ < –
​   ​
5 3
8 8
​ –
​ < –
​   ​
5 3
​ – ​is 8 copies of –
​   ​
1. Circle the greater fraction in each pair.
3
8
3
––
​     ​
10
b) ​ ––  ​  2
9
2
–
​   ​
3
d) ​ – ​ a)​ – ​ c)​ – ​ 13
4
12
5
3
© Marian Small, 2010
4
–
​  5 ​
5
–
​   ​
2
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.
3. Choose different values for the two boxes to make each statement true.
2
3
2
2
3
2
5
8
5
5
8
5
– ​ > ––
​    ​​
​    ​
a)​ – ​ > ––
– ​ > ––
b) ​ – ​ > ––
​    ​​
​    ​
3
3
3
3
c) ​ ––  ​ > –
​  5 ​​ ––  ​ > –
​  5 ​
4. Fill in the boxes. Use each of the numbers 1, 2, 3, 4, and 5.
4
10
​ ––  ​ > ––
​    ​ 12
3
​ –
​ > ––
​     ​
5
8
1
2
1
​ – ​ > ––
​    ​
3
3
​ ––  ​ > ––
​    ​
14
© Marian Small, 2010
Comparing Fractions
Equivalent Fractions
Learning Goal
• representing the same fraction in different ways.
Open Question
Look at the Fraction Tower.
1
1
_
2
Two fractions are equivalent, or equal, if
they take up the same amount of area. For
1
example, use a ruler at the end of the –
​   ​ and
3
2
1 2
–
​   ​sections to see why –
​   ​ = –
​   ​.
6
3 6
1
_
2
1
_
3
1
_
3
1
_
4
1
_
4
1
_
5
1
_
4
1
_
5
1
_
6
1
_
5
1
_
6
1
_
8
1
_
9
1
_
10
1
_
10
1
_
12
1
_
8
1
_
10
1
_
12
1
_
9
1
_
10
1
_
12
1
_
12
1
_
5
1
_
6
1
_
8
1
_
9
1
_
4
1
_
5
1
_
6
1
_
8
1
_
9
1
_
12
1
_
3
1
_
6
1
_
8
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
_
10
1
_
12
1
_
12
1
_
8
1
_
9
1
_
9
1
_
10
1
_
10
1
_
12
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?
15
© Marian Small, 2010
Comparing Fractions
Equivalent Fractions
(Continued)
Think Sheet
Two fractions are equivalent if they describe exactly the same
amount.
2
3
4
6
​   ​ The fractions are equivalent since:
For example, –
​   ​ = –
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
​
​ = ––
​  15  ​
For example, –
​ 5 ​ = –––
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?
12
––
​    ​
18
b) ​ –
​ 5
5
8
10
––
​    ​
16
d) ​ ––  ​  c)​ – ​ 16
3
2
3
a)​ – ​ 3
10
© Marian Small, 2010
8
––
​    ​
10
12
––
​    ​
40
Comparing Fractions
Equivalent Fractions
(Continued)
5
12
2. a) Name three fractions equivalent to ––
​    ​.
b) Choose one of your fractions from part a). How would you convince
5
12
someone who wasn’t sure why it is equivalent to ––
​    ​?
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
​     ​  What can the values of
5.​ ––  ​ = ––
3
=
=
17
and
© Marian Small, 2010
be?
Comparing Fractions
Comparing Fractions to _​ 12 ​and to 1
Learning Goal
• reasoning about how two fractions can be compared using
1
2
benchmarks of –
​   ​and 1.
Open Question
Choose values for the missing parts of the fractions so that they are less
1
2
than –
​   ​. Do not use any number twice. Tell how you chose the values.
3
5
8
––   ​ ​
––  ​ ​
––  ​ ​
––  ​ ​
––  ​
​ ––   ​ ​
5
8
10
Now choose values so that the fractions are greater than 1. Tell how you
chose the values.
3
5
8
––   ​ ​
––  ​ ​
––  ​ ​
––  ​ ​
––  ​
​ ––   ​ ​
5
8
18
10
© Marian Small, 2010
Comparing Fractions
Comparing Fractions to _​ 12 ​and to 1
(Continued)
Think Sheet
1
2
Comparing to –
​   ​
1
2
3 1
​   ​since 6 = 2 × 3.
For example, ​ – ​ = –
6 2
1
A fraction is more than –
​   ​if the numerator is more than half the denominator.
2
5 1
3 1
5 3
​   ​since half of 6 is 3. So –
​   ​ = –
​   ​ and –
​   ​ > –
​   ​.
For example, ​ – ​ > –
6 2
6 2
6 6
A fraction equals –
​   ​if the denominator is 2 times the numerator.
1
_
2
1
2
A fraction is less than –
​   ​if the numerator is less than half of the denominator.
6
6
4
1
1
4
​ ––  ​ < –
​   ​since half of 12 is 6. So –
​   ​ = ––
​    ​ and ––
​    ​ < ––
​    ​.
12 2
2 12
12 12
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, –
​ 5 ​> 1 since 1 = ​ –
​ and –
​  5 ​ > –
​  5 ​.
5
1
A fraction is less than 1 if the numerator is less than the denominator.
5
4
​ 5 ​.
For example, –
​ 5 ​< 1 since it is only 4 out of 5 parts, not the whole –
1
2
1. Circle the fractions that are less than –
​   ​.
3
8
4
10
2
2
9
2
3
– ​​ – ​​ – ​
​ – ​​ ––  ​​
5
1
2
2. Circle the fractions that are between –
​   ​and 1.
7
8
6
4
2
10
3
5
3
– ​​ – ​
​ – ​​ – ​​ ––  ​​
5
19
© Marian Small, 2010
Comparing Fractions
Comparing Fractions to _​ 12 ​and to 1
(Continued)
1
3. What possible values could the numerator have if ––
​     ​ < –
​   ​? Why would
9
2
the numerator have just these values?
4. Replace the missing values with counting numbers (1, 2, 3,…).
Why are there more solutions to ––
​     ​ > 1 than to ​ ––   ​ < 1?
6
6
5. Fill in the boxes with 2, 4, 6, 8, 10. Use each number once to make these
statements true.
1
2
4
8
6
1
1
2
––  ​ > ––
––   ​
––  ​ > –
​ – ​ < ––
​    ​ ​
​    ​ ​
> 1 ​
​   ​
20
2
© Marian Small, 2010
Comparing Fractions
Fraction Circles and Rectangles (1)
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
8
1
–
​   ​
8
​ – ​
1
6
​ – ​
​ – ​
1
–
​   ​
8
1
–
​   ​
8
1
–
​   ​
8
1
–
​   ​
6
1
–
​   ​
6
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
21
1
–
​   ​
6
1
–
​   ​
6
1
–
​   ​
8
1
6
​ – ​
1
–
​   ​
8
1
​ – ​
8
1
–
​   ​
8
1
–
​   ​
8
1
–
​   ​
6
1
–
​   ​
8
1
–
​   ​
8
1
–
​   ​
8
1
–
​   ​
8
1 1
1 ––
1
​    ​ ​ ––  ​ ––
––
12
12 ​    ​
​    ​
12
12
1
1
​ ––  ​
​ ––  ​
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
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
Comparing Fractions
Fraction Circles and Rectangles (2)
1.
1
–
​   ​
6
1
–
​   ​
6
1
–
​   ​
6
1
–
​   ​
6
1
–
​  5 ​
22
1
–
​   ​
8
1
–
​   ​
6
1
​
​ –
5
1
–
​  5 ​
2.
1 1
​    ​
1 ––
​    ​  ––
––
​    ​  10 10
10
1
​ ––  ​
10
1
––
​    ​  1 1
10 ––
​    ​  ​ ––  ​
10 10
1
​ – ​
6
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
1
10
​ ––  ​
1
––
​    ​
10
1
––
​    ​
10
1
–
​   ​
4
1
–
​   ​
4
1
–
​   ​
4
1
–
​   ​
4
1
8
​ – ​
© Marian Small, 2010
1
–
​   ​
4
1
–
​   ​
4
1
–
​   ​
4
1
–
​   ​
4
1
1 ––
1
​    ​  ––
––
​    ​  10 ​    ​
10 1
1 10
​ ––  ​
​ ––  ​
10
10
1
1
​ ––  ​
​ ––  ​
10 1
10
1
1 ​ ––
​ ––  ​  ​ ––
​
10 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
​ ––  ​  ​ ––  ​  ​ ––  ​  ​ ––  ​  ​ ––  ​  ​ ––  ​  ​ ––  ​  ​ ––  ​  ​ ––  ​  ​ ––  ​
Comparing Fractions
Pairs of Fractions
1.
1
​
​ –
5
1
–
​  5 ​
1
–
​  5 ​
1
​
​ –
5
3.
1
​
​ –
5
1
–
​   ​
6
1
6
1
​ – ​
6
5. 1
​ –
​
5
1
–
​   ​
6
1
​
​ –
5
1
–
​   ​
3
1
1
​ –
​
5
1
–
​  5 ​
1
​
​ –
5
1
3
​
​ –
5
1
–
​  5 ​
1
​
​ –
5
​ – ​
1
4
​ – ​
1
4
1
–
​   ​
4
1
–
​   ​
4
8 8 8 8 8 8 8 8
1
–
​  5 ​
1
​
​ –
5
1
–
​   ​
3
6.
1
1
–
​   ​
3
1
–
​  5 ​
​ –
​
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
​ – ​
1
–
​  5 ​
4.
1 1 1 1 1 1 1 1
–
​   ​ –
​   ​ –
​   ​ –
​   ​ –
​   ​ –
​   ​ –
​   ​ –
​   ​
1
–
​  5 ​
1
​
​ –
5
1
​
​ –
5
1
–
​   ​
3
1
–
​   ​
3
1
​ – ​
6
​ – ​
1
–
​   ​
6
1
​ – ​
3
2.
1
–
​  5 ​
​ – ​
7.
1 1 1 1 1 1 1 1
–
​   ​ ​ – ​ ​ – ​ ​ – ​ ​ – ​ ​ – ​ ​ – ​ ​ – ​
8 8 8 8 8 8 8 8
1
4
1
–
​   ​
4
​ – ​
1
–
​  5 ​
1
6
​ – ​
1
–
​  5 ​
1
6
​ – ​
1
–
​   ​
4
1
–
​  5 ​
1
6
​ – ​
1
–
​   ​
4
1
​ –
​
5
1
–
​   ​
6
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
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
​ – ​ ​ – ​ ​ – ​ ​ – ​ ​ – ​ ​ – ​ ​ – ​ ​ – ​
1
4
1
4
​ – ​
1
​ –
​
5
1
–
​   ​
6
1
4
​ – ​
1
1
​ –
​
5
1
–
​   ​
6
​ – ​
1
​ –
​
5
1
–
​   ​
6
1
4
​ – ​
​ –
​
5
1
–
​   ​
6
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
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
​ – ​ ​ – ​ ​ – ​ ​ – ​ ​ – ​ ​ – ​ ​ – ​ ​ – ​
1
–
​   ​
4
1
–
​  5 ​
1
–
​   ​
6
1
–
​   ​
4
1
–
​  5 ​
1
–
​   ​
6
1
–
​   ​
4
1
–
​  5 ​
1
–
​   ​
6
1
–
​   ​
4
1
–
​  5 ​
1
–
​   ​
6
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
​ ––  ​ ​ ––  ​ ​ ––  ​ ––
​    ​ ––
​    ​ ––
​    ​ ––
​    ​ ​ ––  ​ ––
​    ​ ​ ––  ​  ​ ––  ​ ​ ––  ​ ​ ––  ​ ​ ––  ​ ​ ––  ​ ​ ––  ​ ​ ––  ​ ​ ––  ​ ​ ––  ​ ​ ––  ​  ––
​    ​ ––
​    ​ ––
​    ​ ––
​    ​ ––
​    ​ ––
​    ​ ––
​    ​ ––
​    ​ ––
​    ​ ––
​    ​
23
© Marian Small, 2010
Comparing Fractions
1
_
8
1
_
10
1
_
3
1
1
_
_
8
8
1
1
1
_
_
_
10 10 10
24
1
_
8
1
_
10
1
_
3
1
1
_
8
1
_
10
1
_
3
1
1
_
_
8
8
1
1
1
_
_
_
10 10 10
1
_
8
1
_
10
Fraction Tower (1)
© Marian Small, 2010
Comparing Fractions
25
1
_
8
1
_
9
1
_
10
1
_
12
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
_
5
1
_
3
© Marian Small, 2010
1
_
9
1
_
8
1
1
_
_
10 10
1 _
1
_
12 12
1
_
8
1
_
3
1
_
5
1
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
_
3
1
_
8
1
_
9
1
_
10
1
_
12
Fraction Tower (2)
Comparing Fractions
1
_
8
1
_
9
1
_
10
26
1
_
8
1
1
_
_
9
9
1
1
_
_
10 10
1
_
4
1
_
3
1
_
2
1
_
8
1
_
8
1
_
8
1
_
4
1
_
8
1
_
2
1
_
3
1
_
4
1
_
8
1
1
1
1
1
_
_
_
_
_
9
9
9
9
9
1
1
1
1
1
1
_
_
_
_
_
_
10 10 10 10 10 10
1
_
4
1
_
3
1
1
_
8
1
_
9
1
_
10
Fraction Tower (3)
© Marian Small, 2010
Comparing Fractions
Fraction Tower (4)
1
1_
2
1_
2
1_
3
1_
3
1_
4
1_
4
1_
5
1_
9
1_
9
1
_
10
1
_
10
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
_
15
1_
4
1_
5
1_
6
1
_
12
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
_
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
27
© Marian Small, 2010
Comparing Fractions