Download Rules of Divisibility

Math 8 Notes – Unit 2: Rational Numbers
Syllabus Objective: (2.1) The student will write rational numbers in equivalent forms.
Rational Number: any number that can be written as a fraction
n
, where n and d are integers
d
and d ≠ 0.
Relatively prime: numbers that have no common factors other than 1. The goal to simplifying
fractions is to make sure that the numerator and denominator are relatively prime.
Rules of Divisibility
Now would be a good time to review the Rules of Divisibility. It is beneficial for students to be
familiar with rules of divisibility. When familiar with these rules, it allows students to be focused
more on the concept and skill of the lesson rather than to be bogged with the arithmetic. Students
already know some of these rules.
Chances are students already know if a number is divisible by 2, 5 or 10. For instance, if asked to
determine if a number is divisible by two, students can tell you that it has to be even. Divisibility
rules for 10 and 5 are also familiar to students.
RULES OF DIVISIBILITY: A number is divisible
By 2, if it ends in 0, 2, 4, 6 or 8
By 5, if it ends in 0 or 5
By 10, if it ends in 0
By 3, if sum of digits is a multiple of 3
By 9, if sum of digits is a multiple of 9
By 6, if the number is divisible by 2 and by 3
By 4, if the last 2 digits of the number is divisible by 4
By 8, if the last 3 digits of the number is divisible by 8
Simplifying Fractions
You can often simplify fractions by dividing both the numerator and denominator by the same
nonzero integer. You can show the students how to use the divisibility rules to find a common
factor, or review how to find the GCF.
Greatest Common Factor (GCF) – the greatest whole number that divides evenly into each
number.
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Math 8, Unit 2: Rational Numbers
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Strategy 1
To find the GCF, list all the factors of each number. The largest factor that is on both lists is the
GCF.
Example: Find the GCF of 24 and 36.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
GCF is 12, the greatest factor on both lists.
Strategy 2
To find the GCF, write the prime factorization of each number and identify which factors are
common.
Then multiply the common factors.
Example: Find the GCF of 24 and 36.
Prime Factorization of 24 = 2 2 2 3
Prime Factorization of 36 = 2 2 3 3
The common factors are 2, 2, and 3. The GCF is 2 2 3 = 12
Strategy 3
To find the GCF, create a factor tree for each number. Start with the GREATEST factor
common to each number. Then multiply the common factors.
Example: Find the GCF of 24 and 36
24
36
6
4
2
6
2
6
2
GCF = 6(2)= 12
OR
24
36
4
3
6
3
4
2
9
3 3
GCF = 4(3)=12
Multiple strategies are illustrated, but only use the ones you believe are appropriate for your
students.
Helpful Hints:
 If the smaller number divides evenly into the larger number, the smaller number is the
GCF. Example: 9, 27 Since 9 divides evenly into 27, then 9 is the GCF of 9 and 27.
 If the numbers do not share any factors greater than 1, then the GCF is 1.
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Math 8, Unit 2: Rational Numbers
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Decimals and Fractions
Batting averages are great examples of how decimals and fractions are related. For instance, if Joe
3
goes up to bat 10 times and gets 3 hits, his batting average is
or .300. If Jane goes up to bat 8
10
3
times and gets 3 hits, her batting average is or .375. These examples show students that decimals
8
or fractions can be used to represent the same number.
It is also helpful to have students learn and memorize the following common fraction/decimal
conversions. Have students create flashcards for the following and refer to them throughout the year
as LTMR activities. If students are able to recognize and convert these common fractions and
decimals, it will increase their confidence and math skills.
Common Fraction to Decimal Conversions
Converting Decimals to Fractions
Algorithm for:
Converting Decimals to Fractions:
1. Determine the denominator by counting the number
of digits to the right of the decimal point. That place
value will be the denominator.
2. The numerator is the number to the right of the
decimal point.
3. Simplify.
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Math 8, Unit 2: Rational Numbers
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Example: Convert 0.52 to a fraction. (“52 hundredths”)
52
0.52 
Don’t forget to simplify. You can use
100
this example to review the
divisibility rules or GCF.
52  4 13

100  4 25
Example: Convert 8.32 to a fraction. (“8 and 32 hundredths”)
Since there are two digits to the right of the decimal, the denominator will be 100. The numerator is
the number to the right of the decimal point, 32.
32
8.32  8
100
8
32  4
8
8
100  4
25
Converting Fractions to Decimals
One way to convert fractions to decimals is by making equivalent fractions.
1
to a decimal.
2
Since a decimal is a fraction whose denominator is a power of 10, look for a power of 10 that 2 will
divide into evenly.
1 5

2 10
Example: Convert
Since the denominator is 10, the decimal will end in the tenths place which means there will be only
one digit to the right of the decimal point. The answer is 0.5.
3
to a decimal.
4
Again, since a decimal is a fraction whose denominator is a power of 10, look for powers of 10 that
will divide evenly. Ten is not divisible by 4, but 100 is divisible by 4. By making equivalent
fractions,
Example: Convert
3 75
3
3 75

Therefore,
is converts to 0.75 because 
.
4 100
4
4 100
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Math 8, Unit 2: Rational Numbers
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Not all fractions can be written with a power of 10 as the denominator, so it is necessary to look at
another way to convert a fraction to a decimal. That is, to divide the numerator by the denominator.
3
to a fraction.
8
Since 8 is not a factor of any power of 10, you must divide.
Example: Convert
3  8  0.375
Comparing and Ordering Fractions
To compare fractions with different denominators, find a common denominator, make equivalent
fractions, and then compare the numerators.
Common Denominators
Let’s say there are two cakes, one chocolate, and the other vanilla. The chocolate is cut into thirds
and the vanilla into fourths as shown below. Ashton eats one piece of chocolate cake and one piece
of vanilla cake, as shown by the gray pieces.
Since two pieces of cake have been eaten, is it correct to say that Ashton ate
2
of a cake?
7
Let’s re-define a fraction. The numerator tells how many equal pieces have been eaten and the
denominator states how many equal pieces make one whole cake. Since the pieces are not equal,
2
of a cake does not fit the above definition of a fraction, and clearly, 7 pieces do not make one
7
1
1
whole cake. Therefore, trying to add
to and coming up with does not fit the definition of a
4
3
fraction.
The key is to cut the cakes into equal pieces.
The dark lines indicate additional cuts to each cake.
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Math 8, Unit 2: Rational Numbers
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By making additional cuts on each cake, both cakes are now made up of 12 equal pieces. That’s good
news from a sharing standpoint – everyone gets the same size piece. Mathematically, the concept of a
common denominator has just been introduced.
Here is the way the additional cuts were made: Cut the second cake the same way the first was cut
and the first cake the same way the second was cut as shown in the picture. Now, that’s a piece of
cake!
Clearly, it is not convenient to make additional cuts in cakes every time there are cakes with unlike
pieces. It is necessary to find a way that will allow students to determine how to make sure all the
pieces are the same size. That process is called “finding a common denominator.”
Least Common Denominator: the least common multiple of the denominators.
Three Methods for Finding a Common Denominator
1. Multiply the denominators.
2. Write multiples of each denominator, find the
common multiple.
3. Use the reducing method, especially for larger
denominators.
1
1
and .
3
4
Multiply the denominators, namely 3  4  12 . Therefore, the common denominator is 12.
Method 1: Find the common denominator between
2
1
and .
3
4
Write out the multiples of 3 and the multiples of 4. Then find a common multiple.
Method 2: Find the common denominator between
3: 3, 6, 9, 12, 15, 18, …
4: 4, 8, 12, 16, 20, 24, …
Since 12 is a multiple of 3 and 4 it is a common multiple and can be used as a common
denominator.
Method 3: This method is used when the denominators are large and the other two methods aren’t
convenient.
Find the common denominator between
5
7
and
.
18
24
Use the two denominators to write a new fraction, and then simplify.
Holt: Chapter 2
Math 8, Unit 2: Rational Numbers
18 3

24 4
Page 6 of 17
Now cross multiply: 24  3 or 4 18 . It doesn’t matter because either way you get 72.
Therefore, the common denominator is 72.
Being able to find common denominators will be useful when ordering and comparing fractions.
Example: Use < or > to compare the fractions.
3
5
2
3
Find a common denominator and make equivalent fractions, then compare the
numerators:
3
2
9
10
5
3
15
15
Since 9 < 10, we have
3 2
 .
5 3
Example: Order the following numbers from least to greatest.
3 7 2
, ,
4 10 3
To order the three fractions, find a common denominator for all three fractions and make equivalent
fractions using the common denominator. Then, compare the numerators to rewrite the fractions from
least to greatest.
The common denominator is 60. Methods 1 and 3 are not practical to find the common denominator
of three fractions. Use method 2 to find the common denominator. Have students notice that one of
the denominators is 10. Which multiple of 10 is also a multiple of 4 and 3? 60 is the first multiple
that all three denominators have in common.
Using 60 as the common denominator write equivalent fractions.
3 45
7 42
2 40
 ,

,

4 60
10 60
3 60
Now you can order the fractions based on the equivalent fractions written with a common
denominator.
2 7 3
,
Answer: ,
3 10 4
.
Adding/Subtracting Fraction with Like Denominators
Syllabus Objectives: (2.2) The student will add rational numbers.
(2.3) The student will subtract rational numbers.
In order to add or subtract fractions, we have to have equal pieces. Suppose we have another cake
that is cut into 8 equal pieces. Three pieces are eaten by Ashton and four pieces are eaten by
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Math 8, Unit 2: Rational Numbers
Page 7 of 17
Catherine. This means 7 pieces of cake, or
write
7
of one cake has been eaten. Mathematically, we would
8
3 4 7
  .
8 8 8
Notice, only the numerators are added because the numerator represents how many equal pieces are
eaten. Why are the denominators not added? Remember that fractions are defined where the
denominator tells how many equal pieces make one whole cake. If the denominators were to be
added that would mean there are 16 equal pieces of cake, and that is not the case for this example.
Algorithm for:
Adding/Subtracting Fractions with Like Denominators
1. Add/subtract the numerators, and keep the
denominator.
2. Write your answer in simplest terms.
Example: 3
7
3
1
10 10
Step 1: Subtract the whole numbers.
Step 2: Subtract the fractions.
Step 3: Simplify.
3 1  2
7
3
4


10 10 10
2
4
2
2
10
5
Borrowing When Subtracting Mixed Numbers
The concept of borrowing when subtracting with fractions is typically a difficult area for kids to
1
5
4
master. For example, when subtracting 12  4 , students usually answer 8 when they subtract
6
6
6
this problem incorrectly. In order to ease the borrowing concept for fraction, it would be a good idea
to go back and review borrowing concepts that kids are familiar with. Be sure to remind students
that they are borrowing one whole from the units place and those fractions have the same value in the
numerator and the denominator.
In this problem students
1
5
will need to borrow 1
12  4
whole from the 12 which
6
6
6
can be written as .
6
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Math 8, Unit 2: Rational Numbers
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1 6
5
11   4
6 6
6
7
5
11  4
6
6
11  4  7 and
7
7 5 2
 
6 6 6
2
1
which simplifies to 7
6
3
Example: Evaluate a 
5
1
when a  .
18
18
Remember substitution???
Now simplify.
1
5
6
 
18 18 18
6 1

18 3
You can also take the opportunity to make a connection between operations with rational numbers
and operations with integers. It is helpful to review the rules for adding/subtracting integers as they
also apply to rational numbers.
Example: Show 2  ( 5) is similar to
Since 2   5  3 , then
2
5
   .
7
7
2  5
3
     .
7  7
7
This is a good time to remind students
about the following “rule”: if the signs
are different, find the difference and
keep the sign of the largest absolute
value. It doesn’t matter what “type” of
number you are working with the rules
are the same.
Example: Catherine had 6 cups of chocolate chips. She used 2
1
cups to make cookies. How many
3
cups of chocolate chips does she have now?
1
62
3
3
1
2
5 2 3
3
3
3
There are 3
Holt: Chapter 2
2
cups of chocolate chips are left.
3
Math 8, Unit 2: Rational Numbers
Page 9 of 17
Multiplying Rational Numbers
Syllabus Objective: (2.4) The student will multiply rational numbers.
Multiplying fractions is pretty straight forward. Here is the algorithm for multiplying fractions:
Algorithm for Multiplying Fractions
1.
Make sure that you have only proper or improper fractions. Mixed numbers are not
allowed.
2. Multiply the numerators.
3. Multiply the denominators.
4. Simplify.
Or…you may choose to simplify prior to multiplying if appropriate.
Example:
1 6

8 7
1 6
1 6
6
3
 


8 7
87
56
28
Don’t
forget to
simplify.
your
answer.
3
This example
was simplified on
the diagonals.
.prior to the
multiplication.
1 6
1 6
3
 


8 7
7
28
4 8
OR:
2
 3
Example: 6  
6 2
12
2
6    
4
1 3
3
 3
 1
 3
Example: 2  5 
2 16
32
2
 1
2  5  
 
 10
1 3
3
3
 3
Note: Once again you can take the time to help students make the connection between operations with
integers and operations with rational numbers. The more practice that they students have with negative
numbers, the better they will be.
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Math 8, Unit 2: Rational Numbers
Page 10 of 17
Remind students to always
follow the rules for
multiplication (i.e. the
product of two negatives is
positive) even with rational
numbers.
 6  5
 7  8 
Example: ( 6)( 5) is similar to       .
( 6)( 5)  30
30
 6  5
    
56
 7  8 
Which is
15
.
28
Multiplying Decimals
You may want to start with some fraction problems to lead you into the decimal multiplication rules.
Example: 7 0.3
3
10
7
7
25
100
175
100
21
10
2
Example: 7 0.25
1
or 2.1
10
1
75
or 1.75
100
Once you complete a number of these examples students may notice the connection between the number
of digits to the right of the decimal point in the problem and the number of digits to the right of the
decimal point of the answer. At that point you can move to the algorithm.
Algorithm for Multiplying Decimals
1. Line the numbers up vertically.
2. Multiply as if the numbers were whole numbers.
3. Count the number of digits to the right of the decimal point in the problem and move
the decimal point in the answer that number of places to the left.
Example:
There are two digits to
the right of the decimal
point in the problem, so
there must be two digits
to the right of the
decimal point in the
solution.
0.51
2(0.51)
 2
1.02
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Math 8, Unit 2: Rational Numbers
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Example:
You can drop the zeros at
the end. I have kept them
in the product to illustrate
that there are 3 digits after
the decimal point, just as
expected.
 3.75
( 0.4)( 3.75)
  0.4
1.500
Dividing Decimals
Syllabus Objective: (2.5) The student will divide rational numbers.
A visual representation of division of fractions would look like the following.
Example:
We have
1 1

2 8
1
which looks
2
like:
1
1
' s are there in , we need to cut this entire
8
2
diagram into eighths. Then count each of the shaded one-eighths.
Since the question we need to answer is how many
As you can see there are 4 shaded pieces therefore
Example:
We have
1 1
  4.
2 8
5 1

6 3
5
which looks like:
6
Since we are dividing by
1
, we need to take the original diagram and “cut” it into thirds. This looks like:
3
1
As you can see there are 2
Holt: Chapter 2
1
1
2
1
5
thirds shaded from our original .
2
6
Math 8, Unit 2: Rational Numbers
Page 12 of 17
Be careful to choose division examples that are easy to represent in visual form.
Well, because some enjoy playing with numbers, they found a quick way of dividing fractions.
They did this by looking at fractions that were to be divided and they noticed a pattern. And here
is what they noticed.
Algorithm for Dividing Fractions
1.
2.
3.
4.
5.
6.
Make sure you have proper or improper fractions. No mixed numbers.
Invert the divisor (the second number).
Simplify on the diagonals (if possible).
Multiply the numerators.
Multiply the denominators.
Determine your sign and simplify if necessary.
Example: 
3 2

4 5
3 5
15

4 2
8
7
or  1
8

3
8
Example: 2  2
19 1 19

8 2 16
3
or 1
16
Holt: Chapter 2
Multiply 3•5, then
multiply 4•2. Make
sure to simplify your
answer, and watch
your signs.
In this example you first must change
the mixed number to an improper
fraction. Then multiply by the
reciprocal of 2, which is
Math 8, Unit 2: Rational Numbers
1
.
2
Page 13 of 17
Dividing Decimals
Algorithm for Dividing Decimals
1.
2.
3.
4.
In the divisor, move the decimal point as far to the right as possible.
In the dividend, move the decimal point the same number of places.
Bring the decimal point straight up into the quotient.
Divide.
Example: 0.12 0.456
0.12 0.456
Move the decimal point two places in
the divisor.
12 0.456
Move the decimal point two places in
the dividend.
3.8
12 45.6
Follow rules for long division.
0.456 100 45.6
By moving the
 decimal the same number of places to the
0.12 in the
100divisor
12 and the dividend, we are essentially
right
multiplying our original expression by one. We are
making equivalent fractions by multiplying the
numerator and denominator by the same number. If we
move the decimal point one place, we are multiplying
the numerator and denominator by 10. By moving it two
places, we are multiplying the numerator and
denominator by 100, etc.
Example: Evaluate
5.25
when n  0.15 .
n
5.25 5.25 100 525


 35
0.15 0.15 100 15
Holt: Chapter 2
Math 8, Unit 2: Rational Numbers
0.456 100 45.6

0.12 100
12
Since 0.15 has two
decimal places, use
to clear the
decimal, and then
divide.
Page 14 of 17
Adding and Subtracting with Unlike Denominators
Syllabus Objective: (2.6) The student will add fractions with unlike denominators.
Syllabus Objective: (2.7) The student will subtract fractions with unlike denominators.
Let’s add
1 1
2
to . Do you think that we will get ?
3 4
7
What does the picture tell us?
1
4
1
3
looks like
looks like
Note: When drawing your pictures be sure to cut one horizontally, and the other vertically.
The pieces are NOT EQUAL, so we cannot add them. We need to divide each shape into
equally sized pieces. Do this by using the horizontal cuts from the second visual on the first.
Then use the vertical cuts from the first visual on the second. (Both will now be divided into
twelfths.)
1
4
3
12
1
3
4
12
Now that we have a common denominator we can add
3
4
3 4
7
 . The sum of


.
12 12
12 12 12
Adding the numerators, a total of 7 equally sized pieces are shaded and 12 pieces make one unit.
If you did a number of these problems, you would be able to find a way of adding and
subtracting fractions without drawing the picture. It is best to work through several “nice”
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Math 8, Unit 2: Rational Numbers
Page 15 of 17
problems using the pictures to help the students understand the need for a common denominator.
Once the students are comfortable with this concept you can move to the algorithm.
Algorithm for Adding/Subtracting Fractions
1.
2.
3.
4.
5.
Example:
Find a common denominator.
Write equivalent fractions.
Add/subtract the numerators.
Keep the common denominator.
Simplify.
1 2

5 3
Step 1: Find a common denominator—
methods were previously explained
1 3
2 10

and 
5 15
3 15
3 10 13


15 15 15
Example: Evaluate n 
Step 2: Write equivalent fractions
Step 3: Add the numerators
Step 4: Simplify (not necessary in this
example)
2
3
for n  .
3
4
Don’t forget to
substitute. Then follow
the rules for
adding/subtracting
fractions.
3 2

4 3
9 8
1
 
12 12 12
Adding and Subtracting with Unlike Denominators
Syllabus Objective: (2.8) The student will solve equations with rational numbers.
In this section it will be important for you to make connections between problems that the
students have already worked with and this “new” concept of solving equations with rational
numbers. Students already know how to solve one-step equations, and now they know how to
work with rational numbers. This standard simply combines the two concepts.
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Math 8, Unit 2: Rational Numbers
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Compare equations that the students already know how to solve to equations with rational
numbers.
Example:
n49
Step 1: subtract
4  4
Step 2: simplify
n5
Example:
Step 1: divide
Step 2: simplify
4 9

5 10
4
4
 
5
5
1
x
10
x
3x  12
8.2 p  32.8
3x  3  12  3
8.2 p  8.2  32.8  8.2
x  4
p  4.00
Reinforce with the students that the work is the same regardless of the type of number in the
equation.
Syllabus objective: (2.10) The student will explain the relationship among equivalent
representations of rational numbers.
Throughout the unit you should provide your students the opportunity to explain their thinking,
steps, process, etc. Make sure that you provide the opportunity for students to explain the
relationship between equivalent representations of rational numbers. This can be done as part of
an activity, a whole group discussion, a homework questions, ticket-out-the-door, etc.
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Math 8, Unit 2: Rational Numbers
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