GAP CLOSING
Intermediate / Senior
Student Book
Diagnostic...........................................................................................3
Comparing Fractions .......................................................................6
Adding Fractions ............................................................................13
Subtracting Fractions ....................................................................21
Multiplying Fractions .....................................................................28
Dividing Fractions...........................................................................34
Relating Situations to Fraction Operations ..............................40
Template
Fraction Tower ..........................................................................46
Diagnostic
1. List three fractions equivalent (equal) to each fraction.
2
a) –
3
b)
8
––
10
c)
24
––
100
2. Use a greater than (>) or less than (<) sign to make these statements true.
3
–
8
1
a) –
2
5
c) –
6
8
–
9
5
1
1–
4
e) 2–
3
3
b) –
8
3
––
10
9
d) –
5
5
–
9
f)
3
1
1–
8
1––
10
3. Order these values from least to greatest:
4
––
10
1
–
3
2
–
7
6
––
10
1
1–
3
7
2 ––
10
4. Draw a picture to show why each statement is true:
2 1 13
b) –
+ – = ––
5 4 20
2 2 4
a) –
+–=–
5 5 5
5. Add each pair of numbers.
4 2
a) –
+–
9 9
2 1
b) –
+–
3 5
3 5
c) –
+–
8 6
9 7
d) –
+–
4 4
8
1
e) –
+ 2–
3
2
2
5
f) 3–
+ 4–
3
8
2
1
6. Write a story problem that you could solve by adding –
and 1–
.
3
2
3
September 2011
© Marian Small, 2011
Fractions (IS)
Diagnostic
(Continued)
7. Draw a picture to show why each statement is true:
7 2 5
a) –
––=–
8 8 8
9
5
2
b) ––
– – = ––
10 5 10
8. Subtract:
7 3
a) –
––
8 8
2 1
b) –
––
3 5
5 1
c) –
––
6 4
8 2
d) –
––
5 3
2
e) 4 – 1–
3
f)
1
3
4–
– 2–
5
3
1
1
from 3 –
.
9. Write a story problem that you could solve by subtracting 1–
3
4
10. Draw a picture to show why each statement is true:
2 3 2
a) –
×–=–
3 5 5
5 10
2
b) –
×–
= ––
3
8 24
11. Multiply each pair of numbers.
3 5
a) –
×–
5 6
4 2
b) –
×–
5 3
9 2
c) –
×–
4 3
d) 2 –
× 2–
3
4
1
1
2
5
12. Describe a situation where you might multiply –
× –.
3 6
4
September 2011
© Marian Small, 2011
Fractions (IS)
Diagnostic
(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
September 2011
© Marian Small, 2011
Fractions (IS)
Comparing Fractions
Learning Goal
‡ VHOHFWLQJDVWUDWHJ\WRFRPSDUHIUDFWLRQVEDVHGRQWKHLUQXPHUDWRUVDQG
denominators.
Open Question
‡ &KRRVHWZRSDLUVRIQXPEHUVIURPWRXVHDVQXPHUDWRUV
10
4
and denominators of two fractions. For example, you could use –
and ––
.
6
20
–
Make sure that some of your fractions are improper and some are proper.
–
Make sure that some of your fractions use the same numerator and some
do not.
Tell which of your fractions is greater and how you know.
‡ 0DNHPRUHIUDFWLRQVIROORZLQJWKHUXOHVDERYHDQGFRPSDUHDWOHDVWVL[SDLUVRI
them. Tell how you know which fraction is greater each time.
6
September 2011
© Marian Small, 2011
Fractions (IS)
Comparing Fractions
(Continued)
Think Sheet
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
5
‡ 7RFRPSDUH–
5 and –
, we can use a picture that shows the two fractions lined
7
up, so we can see which extends farther.
3
_
5
3
_ < 5
_
5 7
5
_
7
3
_
5
5
_
7
‡ :HFDQXVHSDUWVRIVHWV)RUH[DPSOHXVHDQXPEHURIFRXQWHUVVXFKDV
WKDWLVHDV\WRGLYLGHLQWRERWKÀIWKVDQGVHYHQWKV
3
1
– of 35 counters is 7 counters, so – of 35 counters is 21 counters.
5
5
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
September 2011
© Marian Small, 2011
Fractions (IS)
Comparing Fractions
(Continued)
Renaming Fractions
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.
3
6
3
_
5
6
_
10
= ––.
For example, –
5 10
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.
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.
3
3×3
9
For example, if we multiply by 3: –
= ––– = ––.
5 3 × 5 15
or
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
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
–– and – = ––, 10 is a common numerator.
Since –
=
8 16
3 15
10 10
2 5
6LQFHÀIWHHQWKVDUHELJJHUWKDQVL[WHHQWKV––
> ––, and –
> –.
15 16
3 8
September 2011
© Marian Small, 2011
Fractions (IS)
Comparing Fractions
(Continued)
‡ Decimals
Another way to rename a fraction is to represent it as a decimal.
For example,
2
– = 2 ÷ 3. Using a calculator, we see that 2 ÷ 3 = 0.6666…
3
a
= a ÷ b is because we can imagine a objects, each being shared
The reason –
b
by b people. Each person gets one section of each object, so altogether each
1
a
gets a sections of size –
; that is –
.
b
b
For example, if four people share three bars each one gets one fourth from
3
each of the 3 whole bars and that is –
.
4
2
4
We can use decimal renaming to compare fractions, such as –
to –
.
5
3
2
4
2
4
Since –
= 0.6666… and –
= 0.8, we see that –
< –.
5
3
3 5
Using Benchmarks
We can use benchmarks to compare fractions.
‡ 6RPHWLPHVZHFDQWHOOWKDWRQHIUDFWLRQLVPRUHWKDQDQRWKHUE\FRPSDULQJ
17
3
17
3
them to 1. For example, ––
is more than 1, and –
is less than 1, so ––
> –.
12
4
12 4
‡ 6RPHWLPHVZHFDQWHOOWKDWRQHIUDFWLRQLVPRUHWKDQDQRWKHUVLQFHRQHLV
greater than one half and one is not. We can mentally rename one half to
compare the other fractions to it.
5
3 5
3
5
For the fraction –
, since one half is –
, – is more than one half.
6
6 6
3
For the fraction ––
, since one half is ––
, –– is less than one half.
10
10 10
5
3
> ––.
So, –
6 10
‡ 6RPHWLPHVZHFDQWHOORQHIUDFWLRQLVPRUHWKDQDQRWKHUE\FRPSDULQJ
them to whole numbers other than 1.
8 11
11
For the fraction ––
, since 2 is –
, –– is more than 2.
4
4 4
5
6 5
For the fraction –
, since 2 is –
, – is less than 2.
3
3 3
11
5
So, ––
is more than –
.
4
3
9
September 2011
© Marian Small, 2011
Fractions (IS)
Comparing Fractions
(Continued)
1. What fraction comparison is being shown?
a)
b)
c)
d)
2. Draw a picture to show why this statement is true.
3
6
a) –
= ––
8 16
5 10
b) –
= ––
4
8
3
4
3. Tell or show why –
is not equivalent to –
.
8
9
4. &LUFOHWKHJUHDWHUIUDFWLRQ([SODLQZK\
5
6
–
a) –
or
9
9
5
5
b) –
or ––
11
9
10
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.
6
4
a) –
and ––
5
10
8
4
b) –
and ––
5
15
9
14
c) –
and ––
8
12
8
17
––
d) –
and
3
6
4
6. a) Why does it make sense that –
= 4 ÷ 9?
9
3
4
b) What are the decimal equivalents of –
and –
?
7
9
3
4
c) How could you use decimal equivalents to compare –
and –
?
7
9
11
September 2011
© Marian Small, 2011
Fractions (IS)
Comparing Fractions
(Continued)
7. How can each of these fraction pairs be compared without renaming them as
other fractions or as decimals?
8
2
a) –
and –
5
3
1
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
1
2
b) 2– to 2–
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.
–– < –– < ––
5
3
4
–– < –– < ––
5
3
4
–– < –– < ––
5
3
4
10. a) List all the fractions with a denominator of 3 between 2 and 3.
b) List all the fractions with a denominator of 4 between 2 and 3.
c) Is it possible to list all the fractions between 2 and 3? Explain.
12
September 2011
© Marian Small, 2011
Fractions (IS)
Adding Fractions
Learning Goal
‡ VHOHFWLQJDQDSSURSULDWHXQLWDQGDQDSSURSULDWHVWUDWHJ\WRDGGWZRIUDFWLRQV
Open Question
‡ &KRRVHWZRGLIIHUHQWQRQHTXLYDOHQWIUDFWLRQVWRDGGWRPHHWWKHVHFRQGLWLRQV
– Their sum is a little more than 1.
– Their denominators are different.
– At least one denominator is odd.
Tell how you predicted the sum would be a little more than 1.
&DOFXODWHWKHVXPDQGH[SODLQ\RXUSURFHVV
Verify that the sum is just a little more than 1.
‡ 5HSHDWWKHVWHSVDWOHDVWWKUHHPRUHWLPHVZLWKRWKHUIUDFWLRQV
13
September 2011
© Marian Small, 2011
Fractions (IS)
Adding Fractions
(Continued)
Think Sheet
Adding fractions means combining them.
Same Denominators
‡ 7RFRPELQHIUDFWLRQVZLWKWKHVDPHGHQRPLQDWRUZHFDQFRXQW
3
5
5
8
+ – is 3 fourths + 5 fourths. If we count the fourths, we get 8
For example, –
4 4
3
8
fourths, so –
+ – = –. Another name for –
is 2.
4 4 4
4
Notice we add the numerators, not the denominators, because we are
combining fourths. If we combined the two denominators, we would get
eighths not fourths.
3
5
8
1
Using fraction pieces, we can see that –
+ – = –. If we combine the last –
part of
4 4 4
4
5
3
8
the –
with the –
, we see that 2 wholes are shaded. That shows that –
is 2.
4
4
4
3
_
4
5
_
4
Different Denominators
‡ 7RFRPELQHIUDFWLRQVZLWKGLIIHUHQWGHQRPLQDWRUVUHTXLUHVPRUHWKLQNLQJ)RU
1
1
example, look at this picture of –
+ –.
3 4
2
2
but it is more than –
.
The total length is not as much as –
3
4
2
_
3
1
_
4
1
_
3
2
_
4
14
September 2011
© Marian Small, 2011
Fractions (IS)
Adding Fractions
(Continued)
If the fractions had the same denominator, we could count sections.
1
1
and –
that use the same denominator.
We can create equivalent fractions for –
3
4
That denominator has to be a multiple of both 3 and 4, so 12 is a possibility
because 3 × 4 = 12.
3
1
4
1
– = –– and – = ––, so we add 4 twelfths + 3 twelfths to get 7 twelfths. In this
3 12
4 12
picture, there are 7 sections shaded and each is a twelfth section.
1
_
3
4
_
12
1
_
4
1
3
_
12
7
1
So, –
+ – = ––.
4 3 12
Use a Grid
‡ $QRWKHUZD\WRFRPELQHIUDFWLRQVLVWRXVHDJULG
2
1
and –
, we could create a 3-by-5 grid to show thirds and
For example, to add –
5
3
2
1
– is two rows and – is one column.
5
3
ÀIWKV
2
_
3
1
_
5
2
8VHWKHOHWWHU[WRÀOOWZRURZVWRVKRZ–
8VHWKHOHWWHURWRÀOODFROXPQWR
3
1
show –
.
5
Move counters to empty sections of the grid so each section holds only one
counter.
13
2
1
13
Since ––
of the grid is covered, –
+ – = ––.
15
3 5 15
2
10
1
3
Notice that –
= –– and –
= ––, so this makes sense.
5 15
3 15
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
o
Fractions (IS)
Adding Fractions
(Continued)
‡ 6RPHWLPHVWKHVXPRIWZRIUDFWLRQVLVJUHDWHUWKDQ
To model this with fraction strips, we might need more than one whole strip. For
3 3
example, to show –
+ – we can use 20 as the common denominator.
5 4
3
_
_ 12
=
5 20
3
_
_ 15
=
4 20
8
7
Move ––
XSWRWKHÀUVWVWULSWRÀOOWKHZKROH7KHQ––
is left in the second strip.
20
20
3 3 27
7
– + – = –– or 1––
5 4 20
20
We can model the sum of two fractions that is greater than 1 with a grid. When
you move the counters so there is only one counter in each section, the grid is
ÀOOHGZLWKFRXQWHUVH[WUD
3 12
– = –– (marked with o’s)
5 20
3 15
– = –– (marked with x’s)
4 20
x
x
x
x
x
x
x
x
x
x
xo xo xo
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
o
3 3
7
– + – = 1––
5 4
20
16
September 2011
© Marian Small, 2011
Fractions (IS)
Adding Fractions
(Continued)
Improper Fractions
To add improper fractions, we can use models or we can use equivalent fractions
with the same denominators and count.
9
5
27
10
37
1
1
2
For example, –
+ – = ––
+ ––
= ––
= 6–
.
2 3
6
6
6
6
+ 1–
is close to 4 + 2 = 6.
We can check by estimating. 4–
2
3
Mixed Numbers
3
1
To add mixed numbers, such as 2–
+ 3–
, we could add the whole number parts
3
4
and fraction parts separately.
1
3
13
1
2–
+ 3–
= 5 ––
= 6 ––
3
4
12
12
4
1
1. a) How does this model show –
+ –?
3 5
x
x
x
x
x
x
x
x
x
x
o
x
x
x
x
x
o
x
x
x
x
xo
b) What is the sum?
2. What addition is the model showing?
a)
b)
17
September 2011
© Marian Small, 2011
Fractions (IS)
Adding Fractions
(Continued)
c)
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
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
o
o
3. (VWLPDWHWRGHFLGHLIWKHVXPZLOOEHPRUHRUOHVVWKDQ&LUFOH025(WKDQRU
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
3 3
c) –
+–
8 4
MORE than 1
LESS than 1
7
3
–
d) ––
+
10 5
MORE than 1
LESS than 1
4. Add each pair of fractions or mixed numbers. Draw models for parts c) and e).
18
2 7
a) –
+–
8 8
4 3
b) –
+–
5 5
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
Fractions (IS)
Adding Fractions
(Continued)
14
5. The sum of two fractions is ––
.
9
a) What might their denominators have been? Explain.
b) List another possible pair of denominators.
6. &KRRVHYDOXHVIRUWKHEODQNVWRPDNHHDFKWUXH
3
a) ––
+ –– = –
8
4
b)
3
–– + –– = –
6
2
3
2
7. Lisa used 1–
FXSVRIÁRXUWRPDNHFRRNLHVDQGDQRWKHU–
4 FXSVRIÁRXUWR
3
bake a cake.
a) +RZGR\RXNQRZWKDWVKHXVHGPRUHWKDQFXSVRIÁRXUIRUKHUEDNLQJ"
b) +RZPXFKÁRXUGLGVKHXVH"
2
3
8. Write a story problem that would require you to add –
+ – to solve it. Solve your
3 4
problem.
19
September 2011
© Marian Small, 2011
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
September 2011
© Marian Small, 2011
Fractions (IS)
Subtracting Fractions
Learning Goal
‡ VHOHFWLQJDQDSSURSULDWHXQLWDQGDQDSSURSULDWHVWUDWHJ\WRVXEWUDFWWZR
fractions.
Open Question
‡ &KRRVHWZRIUDFWLRQVWRVXEWUDFWWRPHHWWKHVHFRQGLWLRQV
– The difference is close to 1, but not exactly 1.
– Their denominators are different.
– One denominator is odd.
Tell how you predicted the difference would be close to 1.
&DOFXODWHWKHGLIIHUHQFHDQGH[SODLQ\RXUSURFHVV
Verify that the difference is close to 1.
‡ 5HSHDWWKHVWHSVDWOHDVWWKUHHPRUHWLPHVZLWKRWKHUIUDFWLRQV
21
September 2011
© Marian Small, 2011
Fractions (IS)
Subtracting Fractions
(Continued)
Think Sheet
Subtracting can involve take away, comparing, or looking for a number to add.
‡ 7RVXEWUDFWfractions with the same denominator we count.
5
3
– – is 5 fourths – 3 fourths. Since the denominators are fourths, if
For example, –
4 4
5
3
2
1
we take 3 from 5, there are 2 left, so –
– – = – or –
.
4 4 4
2
We subtracted the numerators because the fourths were there just to tell the
size of the pieces. Nothing was done to the 4s.
5 3 2 1
We can see that –
– – = – – using fraction pieces.
4 4 4 2
( )
‡ 7RVXEWUDFWfractions with different denominators requires a number of steps. For
2
1
2
– –. The difference is — the part of the –
that
example, look at the picture of –
5 4
5
1
extends beyond the –
.
4
It is hard to tell how exactly long the dark piece is by looking at the picture.
2
1
So we can create equivalent fractions for –
and –
that use the same
5
4
denominator. That denominator has to be a multiple of both 5 and 4, so
20 (5 × 4) is a possibility.
8
5
2
1
– = –– and – = ––, so we subtract
5 20
4 20
8
2
_ _
=
5 20
5 twentieths from 8 twentieths,
leaving 3 twentieths.
In the diagram there are 3 extra
sections in shaded on top and
each is a twentieth section.
5
1
_ _
=
4 20
3
2 1
– – – = ––
5 4 20
22
September 2011
© Marian Small, 2011
Fractions (IS)
Subtracting Fractions
(Continued)
1
5
‡ 7RVXEWUDFWIUDFWLRQVZHFDQXVHDJULG)RUH[DPSOHWRVXEWUDFW–
from –
,
4
6
5
we could create a 4-by-6 grid to show both fourths and sixths. –
LVÀYHRIVL[
6
5
1
columns and –
is one of four rows. Using counters to model –
, we rearrange
4
6
1
them so that –
of the counters (one full row) can be removed. Since 14 sections
4
14
remain full, the difference is ––
.
24
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
x
x
20
6
14
_ – _
_
=
24
24 24
If one of the fractions is greater than 1, more than one strip or grid may need
5 4
– –:
to be used. For example, to show –
3 5
5
_
3
5
25
4 12
– = ––
5 15
5HQDPLQJWKHIUDFWLRQVDVÀIWHHQWKVFDQKHOS–
= ––
3 15
25 12 13
–– – –– = ––
15 15 15
5
_
3
5
Using two 3-by-5 grids to solve the same question, cover –
by covering 5 rows.
3
13
4
Remove counters from 4 columns to remove –
of a grid. ––
of a grid are left
5
15
covered.
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
Fractions (IS)
Subtracting Fractions
(Continued)
3
1
‡ 7RVXEWUDFWmixed numbers, such as 4–
– 1–
, we could add up.
3
4
3
1
1
1
to 1–
, we get to 2. If we add another 2–
we get to 4–
,
For example, if we add –
4
4
3
3
3
1
1
7
1
so 4–
– 1–
= – + 2–
= 2––
.
3
4 4
3
12
1
2_
3
1
_
4
0
_ 2
13
4
1
3
4
_
41
3
1
3
We could subtract the whole number parts and the fraction parts. Since –
> –, it
3 4
1
4
might make sense to rename 4–
as 3 + –
ÀUVW
3
3
1
4
3
3
9
4 3 16
– – – = –– – ––
3 4 12 12
4–
= 3–
3
3
–1–
= 1–
4
4
7
2––
12
4
1
1. a) How does this model show –
– –?
3 5
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
4
1
x
x
x
x
x
b) How much is –
– –?
3 5
24
September 2011
© Marian Small, 2011
Fractions (IS)
Subtracting Fractions
(Continued)
2. What subtraction is the model showing?
a)
b)
c)
d)
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
1
3. Estimate to decide if the difference will be closer to –
RU&LUFOH\RXUFKRLFH
2
25
5 1
a) –
––
6 4
closer to –
2
7 7
b) –
––
3 5
closer to –
2
3
1
c) –
– ––
5 10
closer to –
2
8 7
d) –
––
5 6
closer to –
2
1
1
1
1
September 2011
© Marian Small, 2011
closer to 1
closer to 1
closer to 1
closer to 1
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
3 3
c) –
––
5 7
11 9
d) ––
––
3
4
1
7
e) 4 –
– 1–
5
8
f)
5
5 – 3–
6
11
5. The difference of two fractions is ––
.
12
a) What do you think their denominators might have been? Explain.
b) List another possible pair of denominators.
6. &KRRVHYDOXHVIRUWKHEODQNVWRPDNHHDFKHTXDWLRQWUXH
5
–– = –
a) ––
–
3
6
1
–– = 1–
b) ––
–
12
3
2
7. 6DNXUDVWDUWHGZLWKFXSVRIÁRXU6KHXVHG–
cups for one recipe.
3
a) $ERXWKRZPXFKÁRXUGLGVKHKDYHOHIW"
b) +RZPXFKÁRXUGLGVKHKDYHOHIW"
1
1
8. Write a story problem that would require you to subtract 1–
from 3–
to solve it.
2
3
Solve the problem.
26
September 2011
© Marian Small, 2011
Fractions (IS)
Subtracting Fractions
(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.
–– –
––
b) Use the digits again to create a difference greater than 5.
–– –
––
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.
a) Do you agree?
b) Explain why or why not.
11. You have to explain why it does not work to subtract numerators and denominators
to subtract two fractions. What explanation would you use?
27
September 2011
© Marian Small, 2011
Fractions (IS)
Multiplying Fractions
Learning Goal
‡ UHSUHVHQWLQJPXOWLSOLFDWLRQRIIUDFWLRQVDVUHSHDWHGDGGLWLRQRUGHWHUPLQLQJDUHD
Open Question
‡ &KRRVHWZRIUDFWLRQVRUPL[HGQXPEHUVWRPXOWLSO\WRPHHWWKHVHFRQGLWLRQV
– The product is a little more than 1.
– If a mixed number is used, the whole number part is not 1.
Tell how you predicted the product would be a little more than 1.
&DOFXODWHWKHSURGXFWDQGH[SODLQ\RXUSURFHVV
Verify that the product is close to 1.
‡ 5HSHDWWKHVWHSVDWOHDVWWKUHHPRUHWLPHVZLWKRWKHUIUDFWLRQV
28
September 2011
© Marian Small, 2011
Fractions (IS)
Multiplying Fractions
(Continued)
Think Sheet
2
‡ :HKDYHOHDUQHGWKDW[PHDQVJURXSVRI6RLWPDNHVVHQVHWKDW[–
3
means 4 groups of two-thirds:
2
2
2
2
2
8
4×–
=–+–+–+–=–
3 3 3 3 3 3
or
2 thirds + 2 thirds + 2 thirds + 2 thirds = 8 thirds
We multiply 4 × 2 to get the numerator 8. The denominator has to be 3 since
we are combining thirds.
‡ 7RPXOWLSO\WZRproper fractions, we can think of the area of a rectangle with
the fractions as the lengths and widths just as with whole numbers.
For example, 4 × 3 is the number of square units in a
rectangle with length 4 and width 3. That is because
there are 4 equal groups of 3.
2
3
4 × 3 = 12
4
To multiply –
× –, we think of the area of a rectangle that
3 5
2
4
4
wide and –
long.
is –
5
3
2 × 4 sections out of the total number of 3 × 5 sections are inside the rectangle.
2×4
2
4
2×4
and so –
× – = –––.
That means the area is –––
3×5
3 5 3×5
4
_
5
The product of the numerators tells the number of
sections in the shaded rectangle and the product
of the denominators tells how many sections make a
whole.
2
_
3
‡ We can multiply improper fractions the same way as proper fractions.
5
5
_
4
1
_
4
1
3
For example, –
× – is the area of a rectangle
4 2
1
1
units wide and 1–
units long. There
that is 1–
4
2
are 15 sections in the shaded rectangle and
1
1
whole
3
_
2
each is –
of
8
5
3
5×3
1 whole, so the area is –
× – = –––.
4 2 4×2
The shaded part of the diagram looks
like it might be close to 2 wholes so the
1
_
2
15
answer of ––
makes sense. Each shaded
8
square is 1 whole.
29
September 2011
© Marian Small, 2011
Fractions (IS)
Multiplying Fractions
(Continued)
‡ 7RPXOWLSO\WZRPL[HGQXPEHUVZHFDQHLWKHUUHZULWHHDFKPL[HGQXPEHUDV
an improper fraction and multiply, or we can build a rectangle. For example,
1
1
for 3–
× 2–
.
2
3
1
2_
3
Legend:
6
whole
3
_
3
2
_
2
1
_
6
1
3_
2
1
1
3
2
1
1
3–
× 2–
=6+–
+ – + – = 8–
2
3
3 2 6
6
1
1
7
7
49
1
× 2–
is the same as –
× – = ––
= 8–
.
3–
2
3
2 3
6
6
4
1. How would you calculate 5 × –
? Why does your strategy make sense?
9
3
5
× –?
2. a) How does this model show –
4 6
b) What is the product?
30
September 2011
© Marian Small, 2011
Fractions (IS)
Multiplying Fractions
(Continued)
3. What multiplication is being modelled?
a)
b)
c)
1
1
1
d)
1
1
4. Estimate to decide if the product will be closer to –
RU&LUFOH\RXUFKRLFH
2
3 7
a) –
×–
8 8
closer to –
2
5 5
b) –
×–
4 6
closer to –
2
9
3
c) ––
×–
10 8
closer to –
2
2
1
1
1
1
1
d) 2–
×–
3 3
31
closer to –
2
September 2011
© Marian Small, 2011
closer to 1
closer to 1
closer to 1
closer to 1
Fractions (IS)
Multiplying Fractions
(Continued)
5. Multiply each pair of fractions or mixed numbers. Draw models for parts b) and e).
You may use grid paper.
3 4
a) –
×–
5 9
2 3
b) –
×–
7 5
4 5
×–
c) –
3 3
6 7
d) –
×–
5 4
1
3
e) 1–
× 1–
5
2
f)
2
2
2–
× 3–
5
3
20
6. The product of two fractions is ––
.
6
a) What fractions might they have been? Explain.
b) List another possible pair of fractions.
7. &KRRVHYDOXHVIRUWKHEODQNVWRPDNHWKLVHTXDWLRQWUXH
1
–– × –– = 2––
3
12
2
8. A recipe to serve 9 people requires 4–
FXSVRIÁRXU
3
a) $ERXWKRZPXFKÁRXULVQHHGHGWRPDNHWKHUHFLSHIRUSHRSOH"
b) +RZPXFKÁRXULVDFWXDOO\QHHGHG"
32
September 2011
© Marian Small, 2011
Fractions (IS)
Multiplying Fractions
(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
September 2011
© Marian Small, 2011
Fractions (IS)
Dividing Fractions
Learning Goal
‡ UHSUHVHQWLQJGLYLVLRQRIIUDFWLRQVDVFRXQWLQJJURXSVVKDULQJRUGHWHUPLQLQJDXQLW
rate.
Open Question
‡ &KRRVHWZRIUDFWLRQVWRGLYLGHWRPHHWWKHVHFRQGLWLRQV
3
– The quotient (the answer you get when you divide) is about, but not exactly, –
.
2
– The denominators are different.
3
Tell how you predicted the quotient would be about –
.
2
&DOFXODWHWKHTXRWLHQWDQGH[SODLQ\RXUSURFHVV
3
Verify that the quotient is about –
.
2
‡ 5HSHDWWKHVWHSVDWOHDVWWKUHHPRUHWLPHVZLWKRWKHUIUDFWLRQV
34
September 2011
© Marian Small, 2011
Fractions (IS)
Dividing Fractions
(Continued)
Think Sheet
Just as with whole numbers, dividing fractions can describe the result of sharing,
FDQWHOOKRZPDQ\RIRQHVL]HJURXSÀWVLQDQRWKHURUFDQGHVFULEHUDWHV
Sharing
‡ :KHQZHGLYLGHE\DZKROHQXPEHUZHFDQWKLQNDERXWVKDULQJ)RU
4
4
example, –
÷ 2 means that 2 people share –
of the whole.
5
5
:HGLYLGHWKHVHFWLRQVLQWRDQGUHPHPEHUWKDWZHDUHWKLQNLQJDERXWÀIWKV
2
Each person gets –
.
5
5
‡ 6RPHWLPHVWKHUHVXOWLVQRWDZKROHQXPEHURIVHFWLRQV)RUH[DPSOH–
÷2
3
5
means that 2 people share –
.
3
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
The thirds were split in half since there needed to be an even number of
5
5
sections for 2 people to share them. Each person gets –
. If it had been –
÷ 4,
6
3
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
groups of –
in –
.
6
6
0
35
September 2011
1
_
6
2
_
6
3
_
6
4
_
6
5
_
6
1
© Marian Small, 2011
5
_ ÷1
_=5
6 6
Fractions (IS)
Dividing Fractions
5
(Continued)
2
5
2
‡ 7Rmodel –
÷ –, we need to see how many groups of –
are in –
:HÀJXUHRXW
6 6
6
6
how many 2s are in 5 or 5 ÷ 2.
0
1
_
6
2
_
6
3
_
6
4
_
6
5
_
6
5
_ ÷2
_ = 21
_
6 6
2
1
‡ ,IWZRIUDFWLRQVKDYHWKHsame denominator, all of the sections are the same
VL]H7RJHWWKHTXRWLHQWZHÀJXUHRXWKRZPDQ\WLPHVRQHQXPHUDWRUÀWVLQWR
7
3
the other. For example, for –
÷ –, we calculate 7 ÷ 3 since that tells how many
8 8
3
6
groups of 3 of something are in 7 of that thing. For –
÷ –, we calculate 3 ÷ 6 ,
8 8
6
3
1
which is –
, since that tells how much of the –
ÀWVLQ–
.
2
8
8
‡ ,IWZRIUDFWLRQVKDYHdifferent denominators, we can use a model to see how
PDQ\WLPHVRQHIUDFWLRQÀWVLQDQRWKHURUZHFDQXVHHTXLYDOHQWIUDFWLRQVZLWK
the same denominator and divide numerators.
1
1
1
1
For example, for –
÷ –, the picture shows that the –
VHFWLRQÀWVLQWRWKH–
section
2 3
3
2
1
1–
times.
2
1
3
1
3
2
1
We could write –
÷ – as –
÷ – = – or 1–
. It is the same as representing the model
2 3
6 6 2
2
using equivalent fractions.
Using a Unit Rate
‡ $QRWKHUZD\WRWKLQNDERXWGLYLVLRQLVWKLQNLQJDERXWXQLWUDWHV)RUH[DPSOH
if we can drive 80 km in two hours, we think of 80 ÷ 2 as the distance
we can travel in one hour.
1
1
If a girl can complete –
of a project in two days, we divide –
¸WRÀJXUHRXW
3
3
1
of
how much she can complete in one day. Similarly, if she can complete –
3
1
1
1
a project in –
day, we divide –
÷ –WRÀJXUHRXWKRZPXFKVKHFDQFRPSOHWH
2
3 2
1
in 1 day. Since you know that we could also multiply –
× 2; it makes sense
3
1
1
1
that –
÷ – = – × 2.
3 2 3
36
September 2011
© Marian Small, 2011
Fractions (IS)
Dividing Fractions
(Continued)
2
1
‡ 6XSSRVHZHNQRZWKDW–
3 of a can of paint can cover –
RIDVSDFH7RÀJXUH
4
1
2
÷ –.
out how much of the space one full can of paint can cover, we calculate –
4 3
2
1
1
1
1
If –
of a can covers –
of a space, then –
of a can covers –
of the space –
÷2
3
4
3
8
4
3
(
3
1
(
)
1
2
3
)
, which is 1, can covers –
of the space 3 × –
. That means –
÷ – = –.
and then –
3
8
8
4 3 8
1
2
3
8
3
Since we know that –
÷ – = –– ÷ –– = –, this makes sense.
4 3 12 12 8
1
What we did was divide the –
by 2 and multiply by 3.
4
1
2
1
1
3
So –
÷–=–÷2×3=–
× –.
4 3 4
4 2
We reversed the divisor and multiplied by the reciprocal, the fraction we get
by switching the numerator and denominator.
‡ $QRWKHUZD\WRWKLQNDERXWWKLVLV
1
1
1÷–
= 3 since three are 3 groups of –
in 1 whole.
3
3
2
3
2
1÷–
= – since there are only half as many groups of –
in 1 whole as there
3 2
3
1
.
would be groups of –
3
1 2 1 3
2
1
– ÷ – = – × – since there are only one fourth as many groups of – in –
4 3 4 2
3
4
2
in
a
whole.
as groups of –
3
‡ Improper fractions can be divided in the same way as proper fractions.
‡ Mixed numbers are usually written as improper fractions to divide them.
1. How would you calculate each? Explain why your strategy makes sense.
6
a) –
÷4
8
4
b) –
÷3
9
5
1
2. a) How does this model show –
÷ –?
8 3
b) Estimate the quotient.
37
September 2011
© Marian Small, 2011
Fractions (IS)
Dividing Fractions
(Continued)
3. What division is being modelled?
a)
0
1
_
3
1
2
_
3
4
_
3
2
5
_
3
b)
0
1
_
4
2
_
4
3
_
4
1
5
_
4
6
_
4
2
7
_
4
9
_
4
1
4. Estimate each quotient as closer to –
RURU&LUFOH\RXUFKRLFH
2
3 7
a) –
÷–
8 8
closer to –
2
5 2
÷–
b) –
4 3
closer to –
2
11 13
÷ ––
c) ––
3
4
closer to –
2
2
1
1
1
1
1
÷ 4–
d) 2–
3
2
closer to –
2
closer to 1
closer to 2
closer to 1
closer to 2
closer to 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.
3
a) –
÷4
5
5 1
b) –
÷–
6 3
c) 8 ÷ –
3
7
9
d) –
÷ ––
8 10
14 2
e) ––
÷–
5
3
f)
2
2
2
2–
÷ 3–
5
3
1
FXSVRIÁRXUWRGLYLGHLQWRHTXDOEDWFKHVIRUIRXUUHFLSHV
6. You have 3–
3
a) $ERXWKRZPXFKÁRXULVDYDLODEOHIRUHDFKEDWFK"
b) +RZPXFKÁRXULVWKDWSHUEDWFK"
38
September 2011
© Marian Small, 2011
Fractions (IS)
Dividing Fractions
(Continued)
3
2
7. You can tile –
RIDÁRRUDUHDLQ–
of a day.
5
4
a) +RZPXFKRIWKHÁRRUFDQ\RXWLOHLQRQHIXOOGD\"
b) What computation can you do to describe this situation?
3
1
8. Write a story problem that would require you to divide –
by –
to solve it. Solve
5
8
the problem.
9. a) &KRRVHYDOXHVIRUWKHEODQNVWRPDNHWKLVWUXH
3
–– ÷ –– = –
3
4
b) List another possible set of values.
3
–– ÷ –– = –
3
4
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
––
September 2011
© Marian Small, 2011
Fractions (IS)
Dividing Fractions
3
(Continued)
1
9
1
3
8
11. Kevin noticed that –
÷ – is –, but –
÷ – = –.
8 3 8
3 8 9
a) ([SODLQZK\LWPDNHVVHQVHWKDWWKHÀUVWTXRWLHQWLVJUHDWHUWKDQEXWWKH
second one is less than 1.
b) Explain why it also makes sense that the quotients are reciprocals.
5
6
12. You have to explain why dividing by –
is the same as multiplying by –
. What
5
6
explanation would you use?
40
September 2011
© Marian Small, 2011
Fractions (IS)
Relating Situations to Fraction Operations
Learning Goal
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Open Question
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Make the problems as different as you can. You do not have to solve them.
3 1
–+–=
5 3
3 1
–––=
5 3
3 1
–×–=
5 3
3 1
–÷–=
5 3
‡ ([SODLQ\RXUWKLQNLQJIRUHDFK
41
September 2011
© Marian Small, 2011
Fractions (IS)
Relating Situations to Fraction Operations
(Continued)
Think Sheet
It is important to be able to tell what fraction operations make sense to use for
solving fraction problems.
Addition
Addition situations always involve combining. For example:
1
1
of my essay done. I did another –
RIWKHHVVD\+RZPXFKRILWLVÀQLVKHG
I had –
3
2
1
1
5
now? Answer: –
+ – = – of the essay
3 2 6
Subtraction
Subtraction situations could involve take away, comparing, or deciding what
to add.
3
1
‡ $QH[DPSOHRItake away is: I had –
cup of juice. I poured out –
cup. How
4
3
3
1
5
much of a cup of juice is left? Answer: –
– – = –– cup
4 3 12
5
1
‡ $QH[DPSOHRIcomparingLV,ÀQLVKHG–
8 RIP\SURMHFW$QJHODÀQLVKHG–
of her
4
5
1
3
SURMHFW+RZPXFKPRUHGLG,ÀQLVK"$QVZHU–
8 – –
=–
4 8
5
‡ $QH[DPSOHRIdeciding what to addLV,KDYHÀQLVKHG–
of my project. How
8
5
3
=–
much more is left? Answer: 1 – –
8 8
Multiplication
Multiplication situations could involve counting parts of sets, determining areas of
rectangles, or applying rates.
1
‡ $QH[DPSOHRIcounting parts of setsLV-HIIÀOOHGFXSVHDFK–
of the way.
3
,IKHKDGÀOOHGWKHFXSVWRWKHWRSLQVWHDGKRZPDQ\FXSVZRXOGKHÀOO?
1
8
Answer: 8 × –
= –.
3 3
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?
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September 2011
© Marian Small, 2011
Fractions (IS)
Relating Situations to Fraction Operations
(Continued)
1
‡ $QH[DPSOHRIdetermining areas of rectangles is: One rectangle is –
as long
3
2
1
64
2
as wide as a 4 × 8 rectangle. What is its area? Answer: –
× – × 4 × 8 = ––
and –
3
3 3
9
2
‡ $QH[DPSOHRIapplying rates is: Jane can paint a wall in –
of an hour. How
3
1
1
2
5
long would it take her to paint 2–
walls? Answer: 2–
× – = – hours
2
2 3 3
Division
Division situations could involve sharing, determining how many groups, or
determining unit rates
2
‡ $QH[DPSOHRIsharing is: –
of a room must be painted. Four friends are going
3
to share the job. What fraction of the room will each paint if they work at the
2
2
÷ 4 = ––
of the room
same rate? Answer: –
3
12
5
of a
‡ $QH[DPSOHRIdetermining how many groups is: You want to measure –
3
1
FXSRIÁRXUEXWRQO\KDYHD–
2 FXSPHDVXUH+RZPDQ\WLPHVPXVW\RXÀOOWKH
5 1
1
1
– cup measure? Answer: – ÷ – = 3– times
2
3 2
3
1
2
of an hour to clean –
of
‡ $QH[DPSOHRIdetermining unit rate is: It takes you –
5
2
2
1
4
the house, how much of the house could you clean in 1 hour? Answer: –
÷–=–
5 2 5
of the house in 1 hour.
More Than One Option
Sometimes the same situation can be approached using more than one
operation. For example, if you can solve a problem by dividing, you can also solve
it by multiplying.
1
2
The last problem above could be solved by thinking: –
× –– = –
.
5
2
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43
September 2011
© Marian Small, 2011
Fractions (IS)
Relating Situations to Fraction Operations
(Continued)
1. Tell what operation or operations you could use to solve each problem. Write the
equation to represent the problem.
1
2
a) &\QWKLDKDV–
containers of juice. Each smaller container holds –
as much
5
2
DVDODUJHFRQWDLQHU+RZPDQ\VPDOOFRQWDLQHUVFDQVKHÀOO?
3
1
b) About –
of the athletes in a school play basketball. About –
of those players are
4
4
in Grade 9. What fraction of the students in the school are Grade 9 basketball
players?
1
1
c) Stacey read –
of her book yesterday and –
of it today. How much of the book
5
3
has she read?
2
2
d) It takes Lea’s mom about 1–
hours to drive to work in the morning and 1–
hours
3
3
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?
7
e) The gas tank in Kyle’s car was –
full when they started a trip. Later in the day,
8
1
the tank registered –
full. How much of the tank of gas had been used?
4
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September 2011
© Marian Small, 2011
Fractions (IS)
Relating Situations to Fraction Operations
f)
(Continued)
2
1
You can travel –
of the way to your grandmother’s home in 1–
hours. How
3
2
much of the way can you travel in 1 hour?
2. a) What makes this a multiplication problem?
3
1
Fred has 2–
times as much money as his sister. Aaron has –
as much money as
2
4
Fred. How many times as much money as Fred’s sister does Aaron have?
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
How many times will you check the turkey?
c) What makes this a subtraction problem?
3
1
Ed has 3–
rolls of tape. He uses about 1–
rolls to wrap gifts. How many rolls of
2
4
tape does he have left?
3. Write an equation you could use to solve each problem in Question 2.
a)
b)
c)
45
September 2011
© Marian Small, 2011
Fractions (IS)
Relating Situations to Fraction Operations
(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
did you workout?
5. How would you keep part of the information in Question 2b, but change some
of it so that:
a) it becomes a multiplication question
b) it becomes a subtraction question
6. Finish this problem so that it is a division problem:
1
Alexander has 3–
dozen eggs.
2
7. What hints can you suggest to decide if a problem requires subtraction to solve it?
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September 2011
© Marian Small, 2011
Fractions (IS)
Fraction Tower
1
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September 2011
© Marian Small, 2011
Fractions (IS)