Mr. Piotrowski
Chapter 4 Operations with Fractions
Add, Subtract, Fractions & Mixed Numbers - Pretest
4-1 Students will add and subtract both fractions and mixed numbers with like
denominators page 156
If fractions have the same ‘name’ (denominator) count-up the top of the fraction (numerator).
Think about the bar models.
1
example:
4
1
1
3
4
4
4
+ + =
Three –fourths of a bar is shaded or we can say one-fourth is not shaded.
When the numerator is larger than the denominator it is called an improper fraction. That means it has
to be simplified into a mixed number. Mixed numbers are written with a whole number and fraction.
Notice below when you add
5
4
4
4
+
1
4
=
5
4
is an improper fraction.
1
Simplify it to a mixed number 1 . There are one and one-fourth shaded bars.
4
5
Another example with addition:
12
8
Can be written as
8
8
8
7
+8=
12
8
4
+8
8
4
4
4
1
Since is one whole, the answer is 1 . Simplified this is 1 ÷ = 1
8
8
8
4
2
Notice that dividing by one does not change the answer… make sure to divide the top and bottom of 4/8
by 4… so it stays an equivalent fraction.
Subtraction example:
3
1
2
1
18 − 8 = 18 = 14
Subtraction with regrouping from the whole number:
1 3/8 – 5/8 =
8
3
5
(8 + 8) – 8
Answer:
3
8
8
Here you changed one whole into 8
3
6
3
+ 8 = 8 simplified to 4
so you have enough to take 5/8 from.
Mr. Piotrowski
Other Skills & Concepts to practice:


Practice writing mixed numbers as improper fractions. See notes and Mr. P’s handout.
‘Joe Trella, Fraction Fella’ story and practice worksheet (Improper fractions & mixed #s).
4-2 Students will add mixed numbers and add fraction that have unlike
denominators. Page 158
When using line models, denominators must be multiples of each other for easier comparison. Make
equivalent fractions with a common denominator.
Skill Example:
1
1
+ 1 =?
2
3
First find a multiple of 2 and 3 that is in common. Let’s use 6 as the common denominator. How do we
make fractions with a 6 in the bottom? Answer: the third multiple of 2 is six. Whatever you do to the
top, do to the bottom; multiply by
3
3
1 3 3
× =
2 3 6
1
Next write an equivalent fraction for 3
1
3
Solution:
Fraction sense using models:
1
13
1
+2
5
= 16
2
2
2
6
× =
3
6
2
5
+ 16 = 16
Mr. Piotrowski
𝟏
𝟑
4
9
13
𝟏
Another example: 𝟒 𝟑 + 𝟐 𝟒 = 4 12 + 2 12 = 6 12 = 𝟕 𝟏𝟐
If you were using thirds and fourths model pieces, exchange the pieces for twelfths first, then add.

Practice: Play “Spin for the Sum” on page 161 in text.
4-3 Students will subtract fractions and mixed numbers that have different
denominators. Page 162
The steps to Subtracting are similar to adding fractions. You need a common denominator first.
However you might need to turn the first term into an improper fraction.
With any subtraction problem, sometimes you need to ‘borrow’ or rename some part of a whole
before you subtract. Here is an example using whole numbers. Take from the 100’s column so there
is enough in the ones column to subtract 5.
402 – 5 = 300 + 90 +12 – 5 = 397
Fractions Example:
before subtracting.
1
3
4 5 − 1 10 = 𝑛
4
2
=
10
1
2
12
10 2
+
10 10
3
−1
10
_________
7
2
2 + simplified = 2
3
10

2
Change 4 5 into 4 10 then make 4 10 into 3 10
10
9
10
‘Billy Doogan, Roving Weather Man’ story problem and worksheet.
4-4 Problem-Solving: students will solve problems that have more than one step.
Multiply, Divide - Fractions & Mixed Numbers
4-5 Students will multiply fractions page 168.
4-6 Students will multiply mixed numbers (first do 4 x ½ = ½ + ½ + ½ + ½ page170.
 Multiplying two fractions together result in a smaller number for the product.
 Multiplying a fraction with a mixed number gives you a number bigger than the fraction alone.
 Multiplying a mixed number with a mixed number gives you a larger number answer.
Mr. Piotrowski
Skill: Multiply straight across the numerators, then the denominators. It is not necessary to make
common denominators before multiplying (but you can if you really would like to!). Notice that the
answer does have a denominator that is a multiple of both 2 and 6.
1 5
5
× =
2 6 12
Concept: What is one-half of five-sixths? John has a candy bar but he already ate one-sixth of it. Now
he wants to share half of what is left with his sister. How much of the candy bar will he give her?
What is one-half of five-sixths?
If you were to make an array or rectangle measuring 2 x 6 (using the denominators you are breaking
up one whole bar model into six pieces and 2 pieces in the other direction making 12 pieces total.)
Now for identifying the numerator (the count being considered):
1
Color one of the two rows (2) orange.
5
And shade or fill five of the six columns ( ) with a pattern.
6
The drawing will show where the color rows and shaded columns have crossed. This is the answer
5
(12) as shown with the orange color that is shaded. One half of five sixths is identified as orange with
shading.

1
When using operations with fractions the word of means multiply. If you take 2 𝑜𝑓 12 you will
1
12
12
get 6. This can be written 2 × 1 = 2 = 6
Another example:
2
1
2
1
× 12 = 36 remember to simplify your answer to 18.
3

Mixed number should be turned into improper fractions before multiplying. The reason is that
you must multiply all parts of the number to all parts of the other number.
1
3
7
7
49
2 3 × 1 4 = 3 × 4 = 12
Simplify your answer by dividing 12 into 49. Or figure out how many times 12 can be subtracted
from 49. Your remainder should be written in twelfths.
49 12 12 12 12 1
𝟏
=
+
+
+
+
=𝟒
12 12 12 12 12 12
𝟏𝟐
Mr. Piotrowski
4-7 Students will find the reciprocal of a number. (link to inverse operation)



Concept: If you want to undo a multiplication answer, use division. Likewise you may undo a
division answer by using multiplication. 10 divided by 2 is 5. So 5 times 2 is 10.
Skill: If you have a fraction 5/8 then flip it upside down to make 8/5 this is called a reciprocal or
inverting the fraction making the numerator the denominator. If you multiply these two
5 8
40
numbers together, you get 1.
∗ = =1
8
5
40
You may never divide by zero. Division by zero is ‘undefined’.
0
It is okay to say 6 This is like saying John shared 0 pieces of his candy bar with his sister.
6
It is NOT possible to say 0 since you cannot give away 6 pieces of something that does not exist!
Another example why you cannot divide by zero:
12 divided by 6 is 2
6 times 2 is 12
because
12 divided by 0 is x
0 times x = 12
would mean that
You cannot find an answer (x) to multiply with zero to make a 12. Therefore dividing by zero is
not allowed.
4-8 Students will divide one fraction by another fraction.

Concept: The words ‘can cover’ are a clue that division with fractions is involved. Use mouse
sheets.
How many half circles can cover a whole circle? Answer 2 half circles
1
1
2
2
 Skill: 1 ÷ 2 = 1 × 1 = 1 = 2
Example:
How many sixth pieces can cover
and found one-half of five-sixths)
5
Invert the second term of the equation and multiply.
5
12
? Answer 2 ½ (look back to drawing in 4-6 where we multiplied
1
Using formula: 12 ÷ 6 = remember to invert the second fraction before multiplying:
5
12
6
1
× =
30
=
12
30 12 12 6
=
+
+
=
12 12 12 12
2
1
2
Mr. Piotrowski
6.4-9 Students will divide a fraction by a whole number and a whole number by
a fraction.
1
2
3
6
1
1
1
Examples: 2 ÷ 3
= 1×1=1=6
If you had two pizzas and each was cut into thirds. How many third size pieces of pizza could cover
two whole pizzas? (answer: 6)
1
Example: 3 ÷ 2
=
×2=6
3
If there was only one-third slice of pizza left and you split it with your sister, how much of a whole
pizza would you get? (Answer: a one-sixth size piece for each of you)
6.4-10 Students will divide one mixed number by another mixed number.
Example:
𝟏
𝟏
𝟏 ÷𝟏 =
𝟑
3
1
+
3
3
4
5
÷4
3
4
4
×5
3
𝟏
𝟏
𝟏𝟓
𝟒
÷
4
1
+4
4
=
Rewrite as an improper fraction
=
16
= 15 =
Invert second term and multiply
Simplify the improper fraction into a mixed number if possible.
One and one-fourth can cover One and one-third; one whole time with a little left over.
1
4
2 ÷5 =
3
5
7 5
35
×
=
3 29 87
Five and four-fifths can only partially cover two and one-third. Actually, only thirty-five eightysevenths of it!
 One shortcut to consider before multiplying fractions:
You may cross cancel before multiplying two fractions. Find a factor that goes into the numerator of the
first fraction and denominator of the second fraction. Do same with the denominator of first and
5
2
1
1
1
numerator of second if possible.
× 25 = 8 × 5 = 40
16
how many 5’s in 5 and how many 2’s in 2
how many 2’s in 16
how many 5’s in 25
It is easier than multiplying 16 x 25 then having to simplify the fraction further.
6.4-11 Students will solve a problem by first solving a simpler problem. (‘of’
means to multiply when it comes to fractions or decimals and finding a fractional part of a whole. The
words “will cover” means to divide. E.g. how many paper clips did Mark use to make a chain?)
Mr. Piotrowski
“Editor for the Day”
Students will edit a printed math paper for computation and spelling errors.
Customary Length, Capacity, & Weight REVIEW
Students use what they learned about fractions and connect to fractions of measurement.
6.4-12 Students will change between units of length in the customary system. (use 3, 12, 36 lesson_)
6.4-13 Students will change capacity and weight measures in the customary system.
6.4-14 Students will solve problems involving measurement. (perimeter)
Fractions of whole groups:
A team of 18 kids are attending the soccer game. Coach says that the entire team must travel in
equal sized groups of two or more. How many different ways can this group travel? Show this using
counters or circles representing the groups.
Answer: What are the factors of 18? Set them up in an array or model of counters.
1x18 means one whole group of 18 kids is okay.
000000000000000000
18x1 is NOT okay. Coach does not want 18 groups with just one kid in each group.
Rest of the answer: 2x9, 9x2, 3x6, 6x3
000000000
000000000
00
00
00
00
00
00
00
00
00
000000
000000
000000
000
000
000
000
000
000
Another question:
What is one-third of 18?
1
18
1
6
1
6
Answer: of means multiply when dealing with fractions. 18 × 3 = 1 × 3 = 1 × 1 = 1 = 6
Notice that since the denominator 3 is a factor of 18, you can look at the array above and see one-third
of 18 is six. 18 can divide up into three equal groups of six. Also, as a side note: Two-thirds is left. That is
12.