Download 1 Addition and Subtraction 2 Mixed numbers and improper fractions

Worsheet for 5.3: Operations with Fractions
1
Addition and Subtraction
Adding/subtracting when there is a common denominator Draw a picture to
work out each problem. Write your answer in lowest terms; if you are
able to reduce, use your picture to show this too.
1.
2.
3.
1
8
3
4
2
5
+
−
+
4
8
1
4
4
5
For each problem below:
Adding/subtracting when you need to find a common denominator
a picture of each fraction.
• Draw
• Find the least common denominator for the fractions.
• Draw a picture to show how to rename each fraction using the least
common denominator.
• Add or subtract and draw a picture of the result.
• If it is possible to reduce the result, use your picture to show how to
do this. Give your answer in lowest terms.
1.
2.
3.
2
1
2
2 + 3
2
5
3 + 6
3
3
10 − 4
Mixed numbers and improper fractions
improper fraction: numerator at least as big as denominator
• Fractions with the same number on top and bottom are equal to 1:
2 3 4
2 , 3 , 4 , etc.
• Fractions where the numerator (top) is bigger than the denominator
(bottom) have a value more than 1: 32 , 45 , 22
5 , etc.
mixed number: has a whole number part and a fraction part. The fraction part
must be proper. A mixed number represents an addition problem:
whole number + proper fraction
• Example: 3 12
3 12 MEANS 3 PLUS
1
2
1
2.1
Converting between improper fractions and mixed numbers
Algorithm for converting a mixed number to an improper fraction:
algorithm :
A bc =
Example :
3 12 =
( A×c)+b
c
(3×2)+1
2
=
6+1
2
=
7
2
Why it works (discussed using the example above):
• 3 12 represents the addition problem
3+
1
2
• Recall that we can write any whole number as that whole number
over 1. So our addition problem can be written as
3 1
+
1 2
• The LCD is 2.To rewrite 31 with a denominator of 2, we multiply both
the top and bottom by 2.
3
3×2
3×2
=
=
1
1×2
2
We know this is 26 , but to understand the algorithm we will write it
as 3×2 2 . Our addition problem becomes
3×2 1
+
2
2
• Since the denominators are the same, we keep the denominator and
add the numerators.
3×2 1
(3 × 2) + 1
6+1
7
+ =
=
=
2
2
2
2
2
2
Algorithm for convertin an improper fraction to a mixed number:
algorithm:
1. Write the fraction as a long division problem.
a
b becomes b a
2. Do the division. The quotient becomes the
whole number part of the mixed number.
The remainder becomes the numerator of a
fraction with the original denominator.
example:
11
4
Step 1:
Step 2:
4
4
11
2
11
8
3
Answer:
← remainder
2 43
Why it works (discussed using the example above.)
• A fraction is a division problem. If we had 11 brownies to divide
evenly among 4 children, each child would get 11
4 brownies. Step 1
just writes the division problem a different way.
• The quotient tells us the number of whole brownies we can give to
each child. If we were “dealing out” brownies (like we deal playing
cards) we could go around 2 times, giving each child one brownie
each time, until we didn’t have enough brownies to go around another full time. At this point, we would have 3 leftover brownies.
Since there are 4 children, we could cut each brownie into 4 pieces
(fourths) and give each child one piece (one fourth) from each of the
3 leftover brownies. Thus, each child would get 2 whole brownies
plus 3 fourths of a brownie. This is
2+
3
3
3
=2
4
4
Your turn!
1. Write 3 25 as an improper fraction. Justify your work 3 different ways:
(a) Using an area model.
(b) Using a number line.
(c) By writiing out the steps in the algorithm given above.
2. Write
13
4
as a mixed number. Justify your work 3 different ways:
(a) Using an area model.
(b) Using a number line.
(c) By writing out the steps of the algorithm given above.
2.2
Adding and subtracting with mixed numbers
Work out each problem below 2 ways:
• By keeping each number as a mixed number and regrouping as necessary.
• By converting each number to an improper fraction, doing the addition
or subtraction, and then converting back to a mixed number.
For both methods, you should give your answer as a mixed number.
2.
2 12
1.
4 23
2
+ 33
+ 3 23
3.
−
4 14
3 34
4.
−
4
5 34
3 13
3
Multiplication
3.1
Multiplying by a whole number
1. Show that 5 × 23 = 10
3 by drawing a picture and using the fact that multiplication represents repeated addition.
2. Carol has 4 plates. On each plate, she has put half a cookie. How many
cookies does she have in all?
(a) Identify the common core problem type for this problem.
(b) Solve the problem with a picture.
3. Max has a piece of wiood that is 4 feet long and
the area of Max’s piece of wood?
2
3
of a foot wide. What is
(a) Identify the common core problem type for this question.
(b) Solve the problem with a picture.
4. Suri has 21 apples. Frank has
apples does Frank have?
2
3
as many apples as Suri. How many
(a) Identify the common core question type for this question.
(b) Solve the problem by drawing a picture.
3.2
Multiplying by a fraction of the form
number
1
n
where n is a whole
1. Fill in the blanks: Multiplying by 14 is the same as dividing by
2. Draw a picture to demonstrate the above fact for
1
2
× 14 .
3. Write a story problem to go with the number sentence: 5 ×
3.3
.
1
4
=?
Multiplying by a fraction of the form ba , where a and b are
whole numbers and b 6= 0
1. Fill in the blanks: Multiplying by ba is the same as multiplying by
which divides the given amount into “b” equal parts, and then multiplying by
, which corresponds to repeated addition, giving
us “a” copies of the resulting pieces.
2. Draw a picture to illustrate the above fact with the problem
5
1
2
× 43 .
,
3.4
Connecting multiplication to equivalent fractions, cancelling,
and multiplying by improper fractions and mixed numbers
Example
3
2
3
2
3·2
×
=
×
=
=
4 15
2·2 3·5
2·2·3·5
1
2·5
1
3·2
1
=
=
3·2
2·5
10
| {z }
1
Your turn!
Work the following multiplication problems by first cancelling, justifying
your cancellation as shown in the example above. If necessary, write your answer as a mixed number, not an improper fraction.
1.
12
27
×
18
48
2.
25
32
×
24
15
3. 1 13 × 4 21
6
4
Division
1. In this problem, we consider what happens when we divide by a whole
number.
(a) Draw a picture/sequence of pictures to illustrate the division problem 32 ÷ 4.
(b) Fill in the blanks: dividing by 4 is equivalent to multiplying by
. Dividing by a whole number n is equivalent to
multiplying by
.
2. In this problem, we consider what happens when we divide by a fraction
of the form n1 , where n is an integer.
(a) Draw a picture/sequence of pictures to illustrate 1 ÷ 14 .
(b) Draw a picture/sequence of pictures to illustrate 3 ÷ 14 .
(c) Draw a picture/sequence of pictures to illustrate
5
4
2
3
÷ 14 .
(d) Draw a picture/sequence of pictures to illustrate ÷ 14 . (Hint: even
though we don’t need a common denominator to divide, a common
denominator will be helpful in illustrating this problem.)
(e) Fill in the blanks: dividing by 14 is equivalent to multiplying by
. For a whole number n, dividing by n1 is equivalent to multiplying by
.
3. In this problem, we put the pieces together. Fill in the blanks. Dividing
by 23 is the same as dividing by 2 and then dividing again by 13 . Dividing
by 2 is equivalent to multiplying by
, and dividing by
1
is
equivalent
to
multiplying
by
.
Putting
these pieces
3
together, dividing by 32 is equivalent to multiplying by
.
In general, dividing by a fraction ba is equivalent to multiplying by
We call this the
of ba .
4. Work the following problems by using the standard algorithm for dividing by a fraction (which you just described in problem 3). If necessary,
give your answer as a mixed number rather than an improper fraction.
(a)
(b)
3
4
2
7
÷
2
5
÷ 89
(c) 3 54 ÷ 2 13
7
.
5
Children’s mistakes
The following are part of question #3, p. 284 of your text. For each problem,
describe the mistake the child made and how you would demonstrate this to
the child.
d
7 25
− 3 45
4 25
g.
2
3
l.
8 81 ÷ 2 14 = 4 12
×
3
4
= 89
8