# The Number System

### Unit 3 - Mrs. Jenkins' Math

The vocabulary terms you need to know for this unit.

## 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)?

Practice finding the Greatest Common Factor. Make sure you are logged in on your Khan Academy!

## 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?

Practice finding the Least Common Multiple. Make sure you are logged in on your Khan Academy!

## 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?

## GCF Real World Problem

## LCM Real World Problem

Practice word problems involving the GCF and 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?

Try practice problems involving the distributive property! Make sure you are logged in to your 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?

## Read About It! - Simplifying Fractions

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**.

Practice Simplifying Fractions! Make sure you are logged in on your Khan Academy!

## Read About It! - Multiplying 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.

Practice Multiplying Fractions! Make sure you are logged into your Khan Academy!

## Read About It! - Adding Fractions

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.

## Read About It! - Subtracting Fractions

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**.**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**.**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**.**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**.

Practice Subtracting Fractions! Make sure you are logged into your Khan Academy!

## Understanding the Operations

**- Combining two or more quantities to get a total (sum)**

__Addition__- 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** - Repeated addition

- 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."

Practice solving word problems involving adding and subtracting fractions.

Practice solving word problems involving multiplying fractions.

## 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?

Practice using models to divide fractions!

Practice dividing fractions! Make sure you are logged into your Khan Academy!

## Adding and Subtracting Decimals

**Learning Target:**I can add and subtract decimal numbers.

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

Practice Adding and Subtracting Decimals! Make sure you are logged into your Khan Academy!

## Multiplying Decimals

**Learning Target:**I can multiply decimal numbers.

**Essential Question:** How do you multiply decimal numbers?

Practice Adding Fractions! Make sure you are logged into your Khan Academy!

Practice Multiplying Decimals! Make sure you are logged into your Khan Academy!