# The Number System

## Greatest Common Factor

Textbook Pages 31-32, 35

Learning Target: I can find the greatest common factor (GCF) of two or more numbers.

Essential Question: How do you find greatest common factor (GCF)?

Greatest common factor explained | Factors and multiples | Pre-Algebra | Khan Academy
"Factor Up" a song about finding greatest common factor (GCF)

## Least Common Multiple

Textbook Pages 37-40

Learning Target: I can find the least common multiple of two or more numbers.

Essential Question: How do you find the least common multiple of two or more numbers?

Easiest Way To Find The Least Common Multiple!
The LCM & GCF Song

## GCF and LCM Word Problems

Textbook Pages 31-32, 35-40

Learning Target: I can use the GCF and LCM to solve real world problems.

Essential Question: How do you use the GCF and LCM to solve real world problems?

## LCM Real World Problem

Real Life GCF & LCM

## The Distributive Property

Textbook Pages 33-36

Learning Target: I can re-write a sum as a product using the distributive property.

Essential Question: What is the distributive property and how do you use it to re-write sums as products?

How to use the distributive property to factor out the greatest common factor | Khan Academy

## Adding, Subtracting, and Multiplying Fractions

Textbook Pages 79-84.

Learning Target: I can apply the GCF and LCM to fraction operations.

Essential Question: How can you use the GCF and LCM to help with fraction operations?

Some fractions can describe large quantities. Simplifying a fraction can make it easier to understand its value. To simplify a fraction, divide the numerator and the denominator by a common factor. A fraction that has been simplified by the greatest common factor is in simplest form. Remember that the greatest common factor (GCF) is the greatest factor that two or more numbers have in common.

Here is a fraction: 48/60

• Simplify the fraction to better understand its value.
• First, find a common factor for the numerator and denominator.
• 48 – 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
• 60 – 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
• Then, divide by the GCF common factor. The GCF for 48 and 60 is 12
• 48÷12=4
• 60÷12=5
• The simplest form of 48/60 is 4/5.

Comparing fractions with different denominators can be difficult. Some fraction may look similar, but not be equivalent fractions. You can compare fractions with different denominators by comparing them in their simplest form.

Here are two fractions: 3/6 and 4/8

• Let’s simplify 3/6. To do this, divide the numerator and denominator by the GCF. The GCF of 3 and 6 is 3.
• 3÷3=1
• 6÷3=2
• So, 3/6 = 1/2

• Let’s simplify 4/8. To do this, divide the numerator and the denominator by the GCF. The GCF of 4 and 8 is 4.
• 4÷4=1
• 8÷4=2
• So, 4/8 = 1/2
• Both 3/6 and 4/8 are equivalent to 1/2. Therefore, 3/6 and 4/8 are also equivalent fractions.

Guided Practice

Simplify the fraction.

27/36

• First, find the GCF of 27 and 36. The GCF for 27 and 36 is 9.
• 27 – 1, 3, 9, 27
• 36 – 1, 2, 3, 4, 6, 9, 12, 18, 36
• Then, divide both the numerator and the denominator by 9.
• 27÷9=3
• 36÷9=4
• The simplest form of 27/36 is 3/4.

Examples

Simplify each fraction.

1. 4/20

• First, find the GCF of 4 and 20. The GCF for 4 and 20 is 4
• Then, divide both the numerator and the denominator by 4.
• The simplest form of 4/20 is 1/5.

2. 8/16

• First, find the GCF of 8 and 16. The GCF for 8 and 16 is 8
• Then, divide both the numerator and the denominator by 8.
• The simplest form of 8/16 is 1/2.

3. 5/15

• First, find the GCF of 5 and 15. The GCF for 5 and 15 is 5
• Then, divide both the numerator and the denominator by 5.
• The simplest form of 5/15 = 1/3.
Math Antics - Simplifying Fractions

Before you multiply, you can write another equivalent,

simpler fraction in the place of a fraction.

For example: 3/6 x 5/8

• 3/6 is simplified to 1/2 before multiplying.
• Then new problem will be 1/2 x 5/8.

Another example: 3/7 x 4/10

• In this example, 4/10 is simplified to 2/5 before multiplying.
• So the new problem is 3/7 x 2/5.

Why does this work? Obviously, we can write 1/2 instead of 3/6, or 2/5 instead of 4/10, since they are equivalent.

You can also simplify “criss-cross.”

For example: 7/6 x 3/9

• We simplify 3 and 6, writing 1 and 2 in their place. Think of it as the fraction 3/6 being simplified into 1/2, but the 3 and 6 are across from each other.
• So the new problem will be 7/2 x 1/9.

Why are we allowed to simplify in such a manner?

Compare the above problem to this one: 7/9 x 3/6 (It's almost the same problem isn't it?)

Surely you can see that in this problem, we could simplify 3/6 to 1/2 before multiplying. And, these two multiplication problems are essentially the same problem, because they both lead to the same expression and the same answer: the first one becomes

• 7 × 3 = 21
• 6 × 9 = 54
• So, 21/54 (without simplifying)
• and the second one becomes
• 7 × 3 = 21
• 9 × 6 = 54
• So, 21/54 (without simplifying).
• Therefore, since you can simplify 3/6 into 1/2 in the one problem, you can do the same in the other also.

You can even simplify criss-cross several times before multiplying.
Multiply Fractions: Cross Simplification

Fractions that have the same denominator have a common denominator. To add fractions with a common denominator, you find the sum over the numerators over the common denominator. Not all addition problems will involve fractions with common denominators.

Here is an addition problem: 1/2 + 1/4

One-half and one-fourth have different denominators and represent different quantities of a whole.

To add fractions with different denominators, you will need to rewrite the fractions so that they have a common denominator before finding the sum.

• The first step is to the find the least common multiple (LCM) of the denominators, 2 and 4. Remember that the LCM is the smallest multiple that is shared by the numbers being compared. This LCM will become the lowest common denominator (LCD) for the fractions.
• List the multiples of 2 and 4.
• 2: 2, 4, 6, 8, 10 . . .
• 4: 4, 8, 12, 16 . . .
• The least common multiple of 2 and 4 is 4.
• Then, rewrite each fraction with the common denominator of 4. Multiply the numerator and the denominator of 1/2 by 2 to find the equivalent fraction: 1/2=2/4
• The second fraction, 14, is already written in terms of fourths so nothing needs to be done.
• So now the problem is 2/4 + 1/4 and you can add the fractions with common denominators.
• 2/4 + 1/4 = 3/4
• Finally, simplify the fraction, if possible. The 3/4 is a fraction in simplest form.

You can add any number of fractions with unlike denominators as long as you rewrite the fractions with a common denominator.

Guided Practice

Find the sum.

1. 2/7 + 3/9 =

• First, check for a common denominator. The denominators are 7 and 9 and are not common. Find the LCD using the LCM of 7 and 9.
• 7: 7, 14, 21, 28, 35, 42, 49, 56, 63
• 9: 9, 18, 27, 36, 45, 54, 63
• The LCD is 63.
• Then, rewrite the fractions. Find the equivalent fractions with the denominator 63.
• 2/7 = 18/63
• 3/9 = 21/63
• 18/63 + 21/63
• Next, add the fractions. Add the numerators over the common denominator.
• 18/63 + 21/63 = 39/63
• Finally, simplify the fraction. The GCF of 39 and 63 is 3. Divide the numerator and the denominator by 3.
• 39 ÷ 3 = 13
• 63 ÷ 3 = 21
• The sum is 13/21.

Examples

Find the sum. Answer in simplest form.

1. 1/2 + 2/6 =

• First, check the denominators. The denominators are 2 and 6. The LCD is 6.
• Then, rewrite the fractions with the common denominator.
• 1/2 = 3/6; nothing needs to change with 2/6 because the denominator is already 6.
• 3/6 + 2/6
• Next, add the fractions. 3/6 + 2/6 = 5/6
• The fraction is in simplest form. The sum is 5/6.

2. 2/3 + 1/9 =

• First, check the denominators. The denominators are 3 and 9. The LCD is 9.
• Then, rewrite the fractions with the common denominator.
• 2/3 = 6/9; nothing needs to change with 1/9 because the fraction already has a 9 in the denominator.
• 6/9 + 1/9
• Next, add the fractions. 6/9 + 1/9 = 7/9
• The fraction is in simplest form. The sum is 7/9.

3. 4/5 + 1/3 =

• First, check the denominators. The denominators are 5 and 3. The LCD is 15.
• Then, rewrite the fractions with the common denominator.
• 4/5 + 1/3 = 12/15 + 5/15
• Next, add the fractions. 12/15 + 5/15 = 17/15
• The fraction is in simplest form. The sum is 17/15.

You can add fractions with different denominators by rewriting the fractions with a common denominator before adding. The same step is taken to subtract fractions with different denominators.

To find the difference of two fractions with different denominators, rewrite the fractions with a common denominator before subtracting.

Here is a subtraction problem: 6/8 − 1/4 =

The fractions in this problem have different denominators, 8 and 4.

• Rewrite the fractions so they share a common denominator. The least common denominator (LCD) is the lowest common multiple (LCM) of 8 and 4.
• 8: 8, 16, 24 . . .
• 4: 4, 8, 12, 16 . . .
• The LCM is 8. Find the equivalent of each fraction with the denominator 8. 6/8 is already in terms of eighths so it does not need to be changed. Multiply the numerator and denominator of 1/4 by 2.
• 1 × 2 = 2
• 4 × 2 =8
• So, 1/4 = 2/8
• 6/8 − 1/4 = 6/8 − 2/8
• Now you can subtract the fractions with the common denominator. Subtract the numerators over the common denominator. 6/8 − 2/8 = 4/8
• Finally, simplify the fraction by dividing the numerator and the denominator by the greatest common factor (GCF). The GCF of 4 and 8 is 4. Divide the numerator and the denominator by 4.
• 4 ÷ 4 = 1
• 8 ÷ 4 = 2
• So, 4/8 = 1/2. The difference is 1/2.

Guided Practice

Find the difference.

1. 3/4 − 6/12 =

• The fractions do not have a common denominator. The denominators are 4 and 12.
• First, find the LCM of 4 and 12. The LCM is 12.
• 4: 4, 8, 12, . . .
• 12: 12, 24 . . .
• Then, rewrite the fractions with the common denominator of 12.
• 3/4 = 9/12; 6/12 does not need to be changed since it already has a denominator of 12.
• 3/4 − 6/12 = 9/12 − 6/12
• Next, subtract the fractions. Subtract the numerators over the common denominator. 9/12 − 6/12 = 3/12
• Finally, simplify the fraction. 3/12 = 1/4. The difference is 1/4.

Examples

Subtract the following fractions. Answer in simplest form.

1. 5/6 − 1/3 =

• First, find the LCM of 6 and 3. The LCM is 6. Then, rewrite the fractions with a common denominator.
• 1/3 = 2/6; 5/6 does not need to change since 6 is already in the denominator.
• Next, subtract the fractions. 5/6 − 2/6 = 3/6
• Finally, simplify the fraction. 3/6 = 1/2. The difference is 1/2.

2. 1/2 − 4/9 =

• First, find the LCM of 2 and 9. The LCM is 18.Then, rewrite the fractions with a common denominator.
• 1/2 = 9/18
• 4/9 = 8/18
• 1/2 − 4/9 = 9/18 − 8/18
• Next, subtract the fractions. 9/18 − 8/18 = 1/18. The fraction is in simplest form. The difference is 1/18.

3. 4/5 − 1/4 =

• First, find the LCM of 5 and 4. The LCM is 20. Then, rewrite the fractions with a common denominator.
• 4/5 = 16/20
• 1/4 = 5/20
• 4/5 − 1/4 = 16/20 − 5/20
• Next, subtract the fractions. 16/20 − 5/20 = 11/20
• The fraction is in simplest form. The difference is 11/20.
Subtracting Fractions Using LCM

## Understanding the Operations

Addition - Combining two or more quantities to get a total (sum)
• When adding two fractions, you must first make sure that the denominators are the same. Then, add only the numerators. Be sure to simplify!

Subtraction - Taking away a part/quantity from a total to find a missing part or the difference between the total and part/quantity

• When subtracting two fractions, you must first make sure that the denominators are the same. Then, subtract only the numerators. Be sure to simplify!

• Multiplication problems are always set in the same way: (# of groups) x (# in each group) = Total (Product)
• For example: 6 x 4 means 6 groups of 4, or 4 + 4 + 4 + 4 + 4 + 4.
• Another example: 4 x 6 means 4 groups of 6, or 6 + 6 + 6 + 6.
• 6 x 4 and 4 x 6 represent two different scenarios although they have the same result of 24.
• When multiplying two fractions, you are finding a fractional part of another fraction.
• A clue to determine if a word problem's operation is multiplication is that it will have the key word "of."
Example of adding fractions with unlike denominators word problem | Pre-Algebra | Khan Academy
Multiplying Fractions Word Problems (ex 1)
Multiplying Fractions - Solving Word Problems

## Understanding the Operations

What is division?

## Dividing Fractions

Learning Target: I can divide and model division of fractions.

Essential Question: How do you model division of fractions? How do you divide fractions?

Dividing Fractions with Fraction Models
Modeling Division of Common Fractions
dividing fractions using the area model

Learning Target: I can add and subtract decimal numbers.

Essential Question: How do you add and subtract decimal numbers?