FRACTIONOLOGY
©2013 Judo Math Inc.
7th Grade
Rational Numbers Discipline: Blue Belt Training
Order of Mastery (7.NS.3)
1. Converting & Simplifying Fractions
2. Adding & Subtracting Fractions
3. Multiplying & Dividing Fractions
4. Ordering & Comparing Fractions
5. Fraction FUN!
Welcome to the Blue Belt – Advanced Fractionology
Fractions, they strike fear in the hearts of many… but why?!?! It’s just one number on top of
another number, right? Well, not really, which is I guess why people start to hate them so much.
Before you start this discipline, I would like
for you to take a deep breath and throw all
of your old feelings about fractions out the
window and enter this belt with an open
mind. You CAN understand fractions with
the help of some judo math friends, your
teacher, and a little bit of a positive
attitude! We believe in you master
fractionologist!
Good Luck Grasshopper.
Standards Included:

7.NS.3 Solve real-world and mathematical problems involving the four operations with rational
numbers.1
©2013 Judo Math Inc.
Quick Fraction Introduction
Fractions are nothing more than a part of a whole, that is a part (parts) of a greater whole (total
parts). For example, you know when you order a pizza and it is sliced into 8 total slices. You
decide to eat 5 of the 8 total slices, you just ate 5/8 of the pizza. 5 represents the part and 8 is
your total parts or whole.
5
8
5 parts in the Numerator
8 parts whole in Denominator

Fractions can be turned into decimals which can be turned into percents-they are just different
forms of the same thing!
5/8 is turned into a decimal by 5 divided by 8 which is .625 that is 62.5% of the pizza.
Now, a word of advice about Mixed Fractions, these are whole numbers combined with a
fraction, say something like 2
3
The 2 is the whole number and 3/5 is the fraction. It is
5
sometimes best to turn these fractions into Improper Fractions in order to work with them. To
do this, multiply the denominator 5 by the whole number 2 and then add the numerator 3. That
is, 5 * 2 = 10 plus
3 and you get an improper fraction of
13
They are the same number but just
5
represented in different forms! I know you’ve probably heard all of this before, fraction master,
I just wanted to remind you!!!
1. Converting and simplifying fractions

There are some simple rules for converting and simplifying fractions that you most definitely
learned before, but in case you’ve forgotten, here are a few quick tips!
Fraction
Decimal - Divide the top by the bottom
Decimal
= 3/20)
Fraction - Use place value and simplify ( 0.15 is 15 hundredths or 15/100
Percent
Decimal - Divide by 100 or move the decimal two places to the left
Decimal
Percent - Multiply by 100 or move the decimal two places to the right.
Percent
Fraction – Put the percent over 100 and simplify
Fraction
Percent – Find an equivalent fraction that has 100 for the denominator
1
Fraction
Decimal
Percent
2/3
.65
35%
.4
35/125
.135
12/10
95.6%
1.25
1
5
8
Now place all of the numbers from the previous page on this number line: (equivalent fractions,
decimals, and percents should be in the same place!)
1.25
1¼
125%
1. Roy Halliday threw 41 strikes out of 73 pitches, what was his percentage of strikes?
(what is your part and what is your whole)
2
2. Last year, Tiger Woods birdied an amazing 338 holes out of 1150 holes he played. What
was his percentage? (remember, part / whole)
Equivalent Fractions: Anything multiplied by 1 is ITSELF! This comes in very handy when finding
equivalent fractions because you can simply multiply a fraction by 1… or at least some form of 1.
What do I mean by this? 3/3, 4/4, 5/5, 12/12…. These are all versions of 1 and if you multiply a
fraction by them, you get an equivalent fractions
½ x 2/2 = 2/4
1/2 x3/3 = 3/6… 2/4 and 3/6 are all equivalent to ½!
Find at least 4 fractions that are equivalent to each of these by multiplying both the numerator
and denominator by the same number
Fraction
½
4 Equivalent Fractions
2/4, 3/6, 4/8, 9/18
3/5
8/9
1½
3/7
4/13
¾
3
Simplifying Fractions: Simplifying fractions is the opposite of finding equivalent fractions.
Instead of multiplying the numerator and denominator by the same thing, you find a number
that evenly goes into both the numerator and denominator and divide by that. If you’re having
trouble, find the greatest common factor of them!
Write these fractions in simplest form:
15
=
18
1.
4.
36
=
54
2.
12
=
27
3.
44
=
66
5. 12
54
=
99
6.
8.
36
=
72
9. 3 =
4
=
8


7.
2
=
24
9
10. Brianna had 4/12 of a pizza and Alan had 5/18 of a pizza. Who had more?!
4
11. Sam had 3/9 of a pie and Jamie had 2/5 of a pie. Who had more?!
12. Damien spent 4/13 of his $20 allowance and a pie and Sandra spent 1/3 of her $20
allowance. Who spent more?
2.Adding/Subtracting Fractions
Check out this adding problem and explain what’s wrong with it…
1 1 2
 
2 3 5
_______________________________________________________________________
________________________________________________________________________
________________________________________________________________________
If your answer above was something like adding/subtracting requires a common denominator
then you are exactly right! Look at this example to refresh your memory and then write the
steps for adding fractions on the lines
1. ___________________________________________
2. __________________________________________
3. __________________________________________
4. ___________________________________________
5. ___________________________________________
5
1.
7 5
 
12 16
2.
11 23


4 18
3. 
15 10


24 27

4.
3 4
5  
8 9

5.

1
7 1 
6
6. - 5
5
3
1 
18 4

Challengers:
7.
3
5 3
2 

10
7 8
8.  5
7
1  27
 12 

8
3
7
6
9. Your little brother in 4th grade is just learning to add fractions. He has the problem ½ plus ¼
for homework. Draw pictures and write an exact explanation for how he should solve this
problem!
_______________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
3. Multiplying& Dividing Fractions
Multiplying Fractions: One way to look at multiplication is repeated addition. By this definition,
we can see that 4 x ½ = ½ + ½ + ½ + ½ = 2.
Furthermore: 6 x 1/3 = 1/3 + 1/3 + 1/3 + 1/3 + 1/3 + 1/3 = 6/3 = 2.
When you multiply a number it seems to get smaller. Why is this? Explain using the above
examples or some of your own:
_______________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
One way to think of multiplying is “of” and so when we do 4 x ½ it means ½ of 4!
3 4
4 3
 means of .
9 8
8 9
7
Using what you know about multiplying so far, estimate the following products (don’t solve!)
1. 5 x
1
=
3
2. 10 x
3.
4.
1
=
2
1 2
x
=
5
3
4 1
x =
5 2
5. 12 x
1
=
3
6. 10 x
1
=
2
The simplest way to multiply fractions is to multiply the numerators and multiply the denominators;
then reduce. Solve these multiplication of fraction problems and simplify!
5.

6.
3 1 2
  
8 9 5

8.

1 5
 
4 6
7.
1
2
5 
5
7

1 4 4
  
3 5 9
9.

3 49
2

6

7 12
5
10.
35x 9y
1

4 
72y 7x
5

8
11. 3/8 of 42 is what number?
12. 2/9 of 3 ¼ is what?
13. 3 2/5 of 5 ½ is what?
14. Dale made 5/8 of the amount of money that Sally made at the annual carwash. What are some
possible amounts of money that the two could have made?
9
Dividing Fractions:
Whenever you see the division sign, you should say to yourself: How many ____ are in ____.
Example: 5  1/2 means how many ½ are in 5! Look at this picture to answer the question:
Answer: ___________
By now in your life, you have probably learned the short cut for dividing fractions… find the reciprocal
(the flip or its inverse) of the fraction after the division sign, and then change the sign to multiply.
Ex.
3 4 3 9 27
   
4 9 4 4 16
7th grade challenge: WHY does “flip and multiply work?” Write down your own ideas or do a little
research on the internet to find a “proof” that works. Document your understanding on the lines below:
_______________________________________________________________________
________________________________________________________________________
Find the reciprocal:
1.
7

25

2.
4

7
4. 2
3. 8 =

1

3

Divide the Fractions: FIRST write it in the form “how many __ are in ___” then solve!
14.
2 1
 
5 4
How many
1
2
are in ?
4
5
2 4 8
3
  1
5 1 5
5
10
15.
5 12


6
19
16.
3 1


8 8

17.
1
4 
6
18.

3
7
33
21

19. Try this?!
xyz
yz
d 
abc
c

20. A cabinet maker needs 1 ½ pieces of wood. How many 1 ½ foot pieces can he get out of 12 ¼ foot
section of wood?
21. How many 1/8 (one-eighths) are there in 5 ¾?
11
22. Is there a difference between multiplying 2/5 by 5 or dividing 2/5 by 1/5? Why or why not?
Explain….
23. Any number times its reciprocal always equals what? “Prove” your solution with at least 5
examples!
Challenger:
7xy 2 8 3y 3 12y 6




25.
8
5x 56
25x 2
3x 2 12x 3


24.
4 y 3 48y 5


12
4. Ordering and Comparing Fractions
Order the fractions from least to greatest. One way to do this is to compare by finding a common
denominator for all the fractions and then ordering by size of the numerator. What other ways can you
think of to do this?!
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
1.
2.
4
6
,
3
3.
5
,
5
9
2
,
6
1
,
7
4.
11
3
,
12
5.
4
7
,
10
4
8
11
,
7
1
,
4
8
3
7
2
6.
,
11
6
10
3
,
12
4
,
5
1
,
11
8
13
Which one is bigger? Are they the same?
Put a >, < or = sign in between each set of fractions to tell us which is bigger.
1.
2.
3.
5
6
8
6
4
8
11
9
10
11
12
24
4.
5.
6.
3
5
5
1
1
4
9
8
15
3
9
5
7.
8.
9.
3
26
6
2
11
1
31
27
17
11
14
17
10.
11.
12.
7
42
1
4
1
8
15
18
16
12
17
18
13.
14.
15.
4
2
8
10
7
4
5
7
12
15
25
26
14
5. Fraction FUN!
**optional TEAM ACTIVITY!! This is a puzzle. By using your fraction skills, you will piece together new
words that will spell out a sentence. Once your team gets the sentence, bring it up to your teacher to
verify that it’s correct. Don’t say it out loud!
1. The first half of food + the last quarter of door.
____________________
2. The last third of hat + the first 2/5 of heavy.
____________________
3. The second 1/3 of office + the last 1/4 of door + the first 1/3 of street.
____________________
4. The last half of go + the last 1/2 of done.
____________________
5. The last 1/8 of elephant + the first 1/5 of order.
____________________
6. The first 3/4 of fine + the last 3/4 of dish.
____________________
7. The last 1/6 of cement + the first 3/7 of history.
____________________
8. The last half of bath + the first 1/3 of end + the last 2/7 of require.
____________________
9. The first 2/5 of water + the last 3/4 of fits.
____________________
10. The last 1/6 of Glenda.
____________________
11. The first 1/3 of principal + the first half of zero.
____________________
12. The first 1/7 of instant + the first third of fat.
____________________
13. The first 2/5 of young + the first 1/10 of understand.
____________________
14. The first 1/4 ugly + the first 1/5 of settlement.
____________________
15
15. The first 1/4 of youthful + the last half of pour.
____________________
16. The first 1/4 of hesitate + the last 2/3 of sad.
____________________
17. The first 1/3 of permanent + the first half of iodine.
____________________
18. The first 2/6 of clover + the last 2/4 of blue.
____________________
19. The first 1/4 of Mark + the last 3/5 of stars.
____________________
20. The last 1/4 of Meri + the first 1/5 of Susan.
____________________
21. The first 3/5 of dirty + the last 3/7 of perfect + the first 2/5 of Lynda.
____________________
22. The first 3/4 of bent + the last 2/3 of breath.
____________________
23. The first 1/3 of Thomas + the first 1/8 of Endicott.
____________________
24. The first 3/5 of sound + the last 2/9 of Aylsworth.
____________________
25. The first quarter of positive + the first two thirds of Lee.
____________________
26. The first 4/9 of periscope + the last 2/5 of blood.
____________________
27. The first third of get + the second fourth of Jody.
____________________
28. The first half of loud + the last half of book.
____________________
16
Now for some more fun:
1. Alima, Dylan, Alexis, Gautum and Matty participate in the semester lock-in. The advisors purchase
1
hamburgers from In-n-Out for dinner. Before bed, Alima is starving and eats 4 of the hamburgers.
1
4
During the night, Dylan wakes up hungry and eats of the remaining hamburgers. Even later that night,
1
Alexis wakes up and eats 3 of the remaining hamburgers. In the morning, Gautum is the first to wake up
1
3
1
2
and eats of the hamburgers for a delicious breakfast. Finally Matty wakes up and eats of the
hamburgers Gautum left. After Matty, there are 2 hamburgers remaining. How many hamburgers did
the advisors originally buy?
2. Mark loves cookies. So for his birthday, 10 of his relatives sent him 12 cookies each. One-third of
the cookies were peanut butter, half of the cookies were chocolate chip, and the rest of the cookies
were oatmeal. How many oatmeal cookies were there?
17
And now for the ultimate fraction challenge…
4. The basement of High Tech Middle flooded and now there is 2.5 inches of water in it. Last time when it
flooded there was
3
8
inch of water, and it took our pump 45 minutes to pump it out.
a. How long will it take this time?
b. When I started the HTM pump, I realized that the pump had only enough gasoline for one hour
of use. According to the pump manual, the gasoline tank can hold 0.5 gallon, which is sufficient
for 5 hours of use. What is the least amount of gasoline I need to add to the HTM pump to
ensure that it can pump all of the water out of our basement?
18
1
c. Unfortunately, I have only 5 gallon of gasoline to add to my pump. High Tech High has a
1
portable pump with the same capacity as the High Tech Middle pump. It has 4 gallon of
gasoline. Using all of the gasoline available, would the two pumps be able to pump 100% of
the water out of the High Tech Middle basement? If there were enough gasoline and if the two
pumps were working together, how long would it take them to pump 100% of the water out of
my basement? (Note: Two pumps have the same capacity if they pump the same amount of
water in the same amount of time.)
4
d. Suppose the High Tech High pump has only 5 the capacity of the High Tech Middle pump. If the
two pumps were working together, how long would it take to pump all of the water out of the
basement? I f the two pumps were working together , how many inches of water would be
pumped out of the basement at any given moment from the start of their working together?
19