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McGraw Hill My Math Grade 5 Chapter 8 Lesson 2 Answer Key Greatest Common Factor

All the solutions provided in McGraw Hill Math Grade 5 Answer Key PDF Chapter 8 Lesson 2 Greatest Common Factor  will give you a clear idea of the concepts.

McGraw-Hill My Math Grade 5 Answer Key Chapter 8 Lesson 2 Greatest Common Factor

Math in My World

Example 2 Find the GCF of 60 and 54. Make an organized list of the factors for each number. Then circle the common factors. 60:1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 54: 1, 2, 3, 6, 9, 18, 27, 54 The common factors are 1 , 2, 3, 6. So, the greatest common factor, or GCF, of 60 and 54 is 6.

Guided Practice

Find the GCF of each set of numbers.

Question 1. 8, 32 8: _______ 32: _______ The common factors are ___, ___, ___, and ____ So, the GCF of 8 and 32 is ____. Answer: The factors of 8 = 1, 2, 4, 8 The factors of 32 = 1, 2, 4, 8, 16, 32 Explanation: The common factors are 1 , 2 , 4 and 8 So, the GCF of 8 and 32 is 8

Question 2. 3, 12, 18 3: ____ 12: ____ 18: ____ The common factors are ___ and ____ So, the GCF of 3, 12, and 18 is _____. Answer: Factors of 3 = 1, 3 The factors of 12 = 1, 2, 3, 4, 6, 12 The factors of 18 are 1, 2, 3, 6, 9, 18 Explanation: The common factors are 1 and 3 So, the GCF of 3, 12, and 18 is 3

Independent Practice

Question 3. 24, 60 ____ Answer: 12 Explanation: The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The factors of 60 = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. The common factors are 1, 2 , 3 , 4 ,6, 12 So, the GCF of 24 and 60 is 12

Question 4. 12, 18 ____ Answer: 6 Explanation: The factors of 12 = 1, 2, 3, 4, 6, 12. The factors of 18 = 1, 2, 3, 6, 9, 18 The common factors are 1, 2, 3 and 6. So, the GCF of 12 and 18 is 6

Question 5. 18, 42 ____ Answer: 6 Explanation: The factors of 18 = 1, 2, 3, 6, 9, 18 The factors of 42 = 1, 2, 3, 6, 7, 14, 21, 42 The common factors are 1, 2, 3 and 6. So, the GCF of 18 and 42 is 6

Question 6. 30, 72 ____ Answer: 6 Explanation: The factors of 30 = 1, 2, 3, 5, 6, 10, 15, 30 The factors of 72 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72 The common factors are 1, 2, 3 and 6. So, the GCF of 30 and 72 is 6

Question 7. 4, 10, 14 ____ Answer: 2 Explanation: The factors of 4 = 1, 2 and 4 The factors of 10 = 1, 2, 5 and 10 The factors of 14 = 1, 2, 7 and 14 The common factors are 1, 2 So, the GCF of 4, 10, 14  is 2

Question 8. 14, 35, 84 ____ Answer: 7 Explanation: The factors of 14 = 1, 2, 7 and 14 The factors of 35 = 1, 5, 7 and 35 The factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84 The common factors are 1, 2 , 7 So, the GCF of 14, 35, 84  is 7

Question 9. 9, 18, 42 ____ Answer: 3 Explanation: The factors of 9 = 1, 3 and 9 The factors of 18 = 1, 2, 3, 6, 9, 18 The factors of 42 = 1, 2, 3, 6, 7, 14, 21, 42 The common factors are 1, 3 So, the GCF of 9, 18, 42  is 3

Question 10. 16, 52, 76 ____ Answer: 4 Explanation: The factors of 16 = 1, 2, 4, 8 and 16 The factors of 52 = 1, 2, 4, 13, 26 and 52 The factors of 76 = 1, 2, 4, 19, 38, and 76. The common factors are 1, 2 , 4 So, the GCF of 16, 52, 76  is 4

Problem Solving

Question 12. Twelve pens and 16 pencils will be placed in bags with an equal number of each item. What is the most number of bags that can be made? Answer: 4 Explanation: The factors of 12 = 1, 2, 3, 4, 6, 12 The factors of 16 = 1, 2, 4, 8 and 16 The common factors are 1, 2 , 4 Four bags can be made as 12/4 is 3 and 16/4 is 4 Each bag could have 3 pens and 4 pencils The most number of bags that can be made are 4 with an equal number of each item.

Question 13. Oliver has 14 chocolate chip cookies and 21 iced cookies. Oliver gives each of his friends an equal number of each type of cookie. What is the greatest number of friends with whom he can share the cookies? Answer: 7 Explanation: The factors of 14 = 1, 2, 7 and 14. The factors of 21 = 1, 3, 7 and 21. The common factors are 1, 7 So. the GCF of 14 and 21 is 7 7 friends each get 2 chocolate and 3 iced.

HOT Problems

Question 15. ? Building on the Essential Question How can you find the greatest common factor of two numbers? Answer: The greatest common factor is the greatest factor that divides both numbers. To find the greatest common factor, first list the prime factors of each number. Explanation: The factors of 12 = 1, 2, 3, 4, 6, 12. The factors of 18 = 1, 2, 3, 6, 9, 18 The common factors are 1, 2 , 3, 6 So, the GCF of 12 and 18 is 6

McGraw Hill My Math Grade 5 Chapter 4 Lesson 2 My Homework Answer Key

Question 1. 21, 30 ____ Answer: 3 Explanation: The factors of 21 =1, 3, 7 and 21. The factors of 30 = 1, 2, 3, 5, 6, 10, 15 and 30. The common factors are 1, 3 So, the GCF of 21, 30 is 3

Question 2. 12, 30, 72 ___ Answer: 6 Explanation: The factors of 12 = 1, 2, 3, 4, 6, 12. The factors of 30 = 1, 2, 3, 5, 6, 10, 15 and 30. The factors of 72 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72 The common factors are 1, 2 , 3, 6 So, the GCF of 12, 30, 72 is 6

Question 4. Mathematical PRACTICE 3 Justify Conclusions The GCF of any two even numbers is always even. Determine whether the statement is true or false. If true, explain why. If false, give a reason. Answer: Even Explanation: All even numbers have two as a factor. That means that any set of even numbers will have at least a two as a common factor. Since 2 will be a part of the greatest common factor, it will have to be even.

Vocabulary Check

Question 5. Circle the correct term that makes the sentence true. The (greatest, least) of the common factors of two or more numbers is the (greatest, least) common factor of the numbers. Answer: greatest , greatest Explanation: The ( greatest ) of the common factors of two or more numbers is the ( greatest ) common factor of the numbers.

Test Practice

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Chapter 4, Lesson 2: Greatest Common Factor

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Greatest Common Factor

The highest number that divides exactly into two or more numbers. It is the "greatest" thing for simplifying fractions!

Let's start with an Example ... 

Greatest common factor of 12 and 16.

• Find all the Factors of each number,
• Circle the Common factors,
• Choose the Greatest of those

So ... what is a "Factor" ?

Factors are numbers we can multiply together to get another number:

A number can have many factors:

Factors of 12 are 1, 2, 3, 4, 6 and 12 ... ... because 2 × 6 = 12, or 4 × 3 = 12, or 1 × 12 = 12.

(Read how to find All the Factors of a Number . In our case we don't need the negative ones.)

What is a "Common Factor" ?

Say we have worked out the factors of two numbers:

Example: Factors of 12 and 30

Then the common factors are those that are found in both lists:

• Notice that 1, 2, 3 and 6 appear in both lists?
• So, the common factors of 12 and 30 are: 1, 2, 3 and 6

It is a common factor when it is a factor of two (or more) numbers.

Here is another example with three numbers:

Example: The common factors of 15, 30 and 105

The factors that are common to all three numbers are 1, 3, 5 and 15

In other words, the common factors of 15, 30 and 105 are 1, 3, 5 and 15

What is the "Greatest Common Factor" ?

It is simply the largest of the common factors.

In our previous example, the largest of the common factors is 15, so the Greatest Common Factor of 15, 30 and 105 is 15

The "Greatest Common Factor" is the largest of the common factors (of two or more numbers)

Why is this Useful?

One of the most useful things is when we want to simplify a fraction:

Example: How can we simplify 12 30 ?

Earlier we found that the Common Factors of 12 and 30 are 1, 2, 3 and 6, and so the Greatest Common Factor is 6 .

So the largest number we can divide both 12 and 30 exactly by is 6 , like this:

The Greatest Common Factor of 12 and 30 is 6 .

And so 12 30 can be simplified to 2 5

Finding the Greatest Common Factor

Here are three ways:

• find all factors of both numbers (use the All Factors Calculator ),
• then find the ones that are common to both, and
• then choose the greatest .

And another example:

2 . Or we can find the prime factors and combine the common ones together:

3. Or sometimes we can just play around with the factors until we discover it:

But in that case we must check that we have found the greatest common factor.

Greatest Common Factor Calculator

OK, there is also a really easy method: we can use the Greatest Common Factor Calculator to find it automatically.

Other Names

The "Greatest Common Factor" is often abbreviated to GCF , and is also known as:

• the "Greatest Common Divisor" or GCD
• the "Highest Common Factor" or HCF

6.1 Greatest Common Factor and Factor by Grouping

Learning objectives.

By the end of this section, you will be able to:

• Find the greatest common factor of two or more expressions
• Factor the greatest common factor from a polynomial
• Factor by grouping

Be Prepared 6.1

Before you get started, take this readiness quiz.

Factor 56 into primes. If you missed this problem, review Example 1.2 .

Be Prepared 6.2

Find the least common multiple (LCM) of 18 and 24. If you missed this problem, review Example 1.3 .

Be Prepared 6.3

Multiply: −3 a ( 7 a + 8 b ) . −3 a ( 7 a + 8 b ) . If you missed this problem, review Example 5.26 .

Find the Greatest Common Factor of Two or More Expressions

Earlier we multiplied factors together to get a product . Now, we will reverse this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring .

We have learned how to factor numbers to find the least common multiple (LCM) of two or more numbers. Now we will factor expressions and find the greatest common factor of two or more expressions. The method we use is similar to what we used to find the LCM.

Greatest Common Factor

The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.

We summarize the steps we use to find the greatest common factor.

Find the greatest common factor (GCF) of two expressions.

• Step 1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
• Step 2. List all factors—matching common factors in a column. In each column, circle the common factors.
• Step 3. Bring down the common factors that all expressions share.
• Step 4. Multiply the factors.

The next example will show us the steps to find the greatest common factor of three expressions.

Example 6.1

Find the greatest common factor of 21 x 3 , 9 x 2 , 15 x . 21 x 3 , 9 x 2 , 15 x .

Find the greatest common factor: 25 m 4 , 35 m 3 , 20 m 2 . 25 m 4 , 35 m 3 , 20 m 2 .

Find the greatest common factor: 14 x 3 , 70 x 2 , 105 x . 14 x 3 , 70 x 2 , 105 x .

Factor the Greatest Common Factor from a Polynomial

It is sometimes useful to represent a number as a product of factors, for example, 12 as 2 · 6 2 · 6 or 3 · 4 . 3 · 4 . In algebra, it can also be useful to represent a polynomial in factored form. We will start with a product, such as 3 x 2 + 15 x , 3 x 2 + 15 x , and end with its factors, 3 x ( x + 5 ) . 3 x ( x + 5 ) . To do this we apply the Distributive Property “in reverse.”

We state the Distributive Property here just as you saw it in earlier chapters and “in reverse.”

Distributive Property

If a , b , and c are real numbers, then

The form on the left is used to multiply. The form on the right is used to factor.

So how do you use the Distributive Property to factor a polynomial ? You just find the GCF of all the terms and write the polynomial as a product!

Example 6.2

How to use the distributive property to factor a polynomial.

Factor: 8 m 3 − 12 m 2 n + 20 m n 2 . 8 m 3 − 12 m 2 n + 20 m n 2 .

Factor: 9 x y 2 + 6 x 2 y 2 + 21 y 3 . 9 x y 2 + 6 x 2 y 2 + 21 y 3 .

Factor: 3 p 3 − 6 p 2 q + 9 p q 3 . 3 p 3 − 6 p 2 q + 9 p q 3 .

Factor the greatest common factor from a polynomial.

• Step 1. Find the GCF of all the terms of the polynomial.
• Step 2. Rewrite each term as a product using the GCF.
• Step 3. Use the “reverse” Distributive Property to factor the expression.
• Step 4. Check by multiplying the factors.

Factor as a Noun and a Verb

We use “factor” as both a noun and a verb:

Example 6.3

Factor: 5 x 3 − 25 x 2 . 5 x 3 − 25 x 2 .

Factor: 2 x 3 + 12 x 2 . 2 x 3 + 12 x 2 .

Factor: 6 y 3 − 15 y 2 . 6 y 3 − 15 y 2 .

Example 6.4

Factor: 8 x 3 y − 10 x 2 y 2 + 12 x y 3 . 8 x 3 y − 10 x 2 y 2 + 12 x y 3 .

Factor: 15 x 3 y − 3 x 2 y 2 + 6 x y 3 . 15 x 3 y − 3 x 2 y 2 + 6 x y 3 .

Factor: 8 a 3 b + 2 a 2 b 2 − 6 a b 3 . 8 a 3 b + 2 a 2 b 2 − 6 a b 3 .

When the leading coefficient is negative, we factor the negative out as part of the GCF.

Example 6.5

Factor: −4 a 3 + 36 a 2 − 8 a . −4 a 3 + 36 a 2 − 8 a .

The leading coefficient is negative, so the GCF will be negative.

Factor: −4 b 3 + 16 b 2 − 8 b . −4 b 3 + 16 b 2 − 8 b .

Try It 6.10

Factor: −7 a 3 + 21 a 2 − 14 a . −7 a 3 + 21 a 2 − 14 a .

So far our greatest common factors have been monomials. In the next example, the greatest common factor is a binomial.

Example 6.6

Factor: 3 y ( y + 7 ) − 4 ( y + 7 ) . 3 y ( y + 7 ) − 4 ( y + 7 ) .

The GCF is the binomial y + 7 . y + 7 .

Try It 6.11

Factor: 4 m ( m + 3 ) − 7 ( m + 3 ) . 4 m ( m + 3 ) − 7 ( m + 3 ) .

Try It 6.12

Factor: 8 n ( n − 4 ) + 5 ( n − 4 ) . 8 n ( n − 4 ) + 5 ( n − 4 ) .

Factor by Grouping

Sometimes there is no common factor of all the terms of a polynomial. When there are four terms we separate the polynomial into two parts with two terms in each part. Then look for the GCF in each part. If the polynomial can be factored, you will find a common factor emerges from both parts. Not all polynomials can be factored. Just like some numbers are prime , some polynomials are prime.

Example 6.7

How to factor a polynomial by grouping.

Factor by grouping: x y + 3 y + 2 x + 6 . x y + 3 y + 2 x + 6 .

Try It 6.13

Factor by grouping: x y + 8 y + 3 x + 24 . x y + 8 y + 3 x + 24 .

Try It 6.14

Factor by grouping: a b + 7 b + 8 a + 56 . a b + 7 b + 8 a + 56 .

Factor by grouping.

• Step 1. Group terms with common factors.
• Step 2. Factor out the common factor in each group.
• Step 3. Factor the common factor from the expression.

Example 6.8

Factor by grouping: ⓐ x 2 + 3 x − 2 x − 6 x 2 + 3 x − 2 x − 6 ⓑ 6 x 2 − 3 x − 4 x + 2 . 6 x 2 − 3 x − 4 x + 2 .

Try It 6.15

Factor by grouping: ⓐ x 2 + 2 x − 5 x − 10 x 2 + 2 x − 5 x − 10 ⓑ 20 x 2 − 16 x − 15 x + 12 . 20 x 2 − 16 x − 15 x + 12 .

Try It 6.16

Factor by grouping: ⓐ y 2 + 4 y − 7 y − 28 y 2 + 4 y − 7 y − 28 ⓑ 42 m 2 − 18 m − 35 m + 15 . 42 m 2 − 18 m − 35 m + 15 .

Section 6.1 Exercises

Practice makes perfect.

In the following exercises, find the greatest common factor.

10 p 3 q , 12 p q 2 10 p 3 q , 12 p q 2

8 a 2 b 3 , 10 a b 2 8 a 2 b 3 , 10 a b 2

12 m 2 n 3 , 30 m 5 n 3 12 m 2 n 3 , 30 m 5 n 3

28 x 2 y 4 , 42 x 4 y 4 28 x 2 y 4 , 42 x 4 y 4

10 a 3 , 12 a 2 , 14 a 10 a 3 , 12 a 2 , 14 a

20 y 3 , 28 y 2 , 40 y 20 y 3 , 28 y 2 , 40 y

35 x 3 y 2 , 10 x 4 y , 5 x 5 y 3 35 x 3 y 2 , 10 x 4 y , 5 x 5 y 3

27 p 2 q 3 , 45 p 3 q 4 , 9 p 4 q 3 27 p 2 q 3 , 45 p 3 q 4 , 9 p 4 q 3

In the following exercises, factor the greatest common factor from each polynomial.

6 m + 9 6 m + 9

14 p + 35 14 p + 35

9 n − 63 9 n − 63

45 b − 18 45 b − 18

3 x 2 + 6 x − 9 3 x 2 + 6 x − 9

4 y 2 + 8 y − 4 4 y 2 + 8 y − 4

8 p 2 + 4 p + 2 8 p 2 + 4 p + 2

10 q 2 + 14 q + 20 10 q 2 + 14 q + 20

8 y 3 + 16 y 2 8 y 3 + 16 y 2

12 x 3 − 10 x 12 x 3 − 10 x

5 x 3 − 15 x 2 + 20 x 5 x 3 − 15 x 2 + 20 x

8 m 2 − 40 m + 16 8 m 2 − 40 m + 16

24 x 3 − 12 x 2 + 15 x 24 x 3 − 12 x 2 + 15 x

24 y 3 − 18 y 2 − 30 y 24 y 3 − 18 y 2 − 30 y

12 x y 2 + 18 x 2 y 2 − 30 y 3 12 x y 2 + 18 x 2 y 2 − 30 y 3

21 p q 2 + 35 p 2 q 2 − 28 q 3 21 p q 2 + 35 p 2 q 2 − 28 q 3

20 x 3 y − 4 x 2 y 2 + 12 x y 3 20 x 3 y − 4 x 2 y 2 + 12 x y 3

24 a 3 b + 6 a 2 b 2 − 18 a b 3 24 a 3 b + 6 a 2 b 2 − 18 a b 3

−2 x − 4 −2 x − 4

−3 b + 12 −3 b + 12

−2 x 3 + 18 x 2 − 8 x −2 x 3 + 18 x 2 − 8 x

−5 y 3 + 35 y 2 − 15 y −5 y 3 + 35 y 2 − 15 y

−4 p 3 q − 12 p 2 q 2 + 16 p q 2 −4 p 3 q − 12 p 2 q 2 + 16 p q 2

−6 a 3 b − 12 a 2 b 2 + 18 a b 2 −6 a 3 b − 12 a 2 b 2 + 18 a b 2

5 x ( x + 1 ) + 3 ( x + 1 ) 5 x ( x + 1 ) + 3 ( x + 1 )

2 x ( x − 1 ) + 9 ( x − 1 ) 2 x ( x − 1 ) + 9 ( x − 1 )

3 b ( b − 2 ) − 13 ( b − 2 ) 3 b ( b − 2 ) − 13 ( b − 2 )

6 m ( m − 5 ) − 7 ( m − 5 ) 6 m ( m − 5 ) − 7 ( m − 5 )

In the following exercises, factor by grouping.

a b + 5 a + 3 b + 15 a b + 5 a + 3 b + 15

c d + 6 c + 4 d + 24 c d + 6 c + 4 d + 24

8 y 2 + y + 40 y + 5 8 y 2 + y + 40 y + 5

6 y 2 + 7 y + 24 y + 28 6 y 2 + 7 y + 24 y + 28

u v − 9 u + 2 v − 18 u v − 9 u + 2 v − 18

p q − 10 p + 8 q − 80 p q − 10 p + 8 q − 80

u 2 − u + 6 u − 6 u 2 − u + 6 u − 6

x 2 − x + 4 x − 4 x 2 − x + 4 x − 4

9 p 2 + 12 p − 15 p − 20 9 p 2 + 12 p − 15 p − 20

16 q 2 + 20 q − 28 q − 35 16 q 2 + 20 q − 28 q − 35

m n − 6 m − 4 n + 24 m n − 6 m − 4 n + 24

r 2 − 3 r − r + 3 r 2 − 3 r − r + 3

2 x 2 − 14 x − 5 x + 35 2 x 2 − 14 x − 5 x + 35

4 x 2 − 36 x − 3 x + 27 4 x 2 − 36 x − 3 x + 27

Mixed Practice

In the following exercises, factor.

−18 x y 2 − 27 x 2 y −18 x y 2 − 27 x 2 y

−4 x 3 y 5 − x 2 y 3 + 12 x y 4 −4 x 3 y 5 − x 2 y 3 + 12 x y 4

3 x 3 − 7 x 2 + 6 x − 14 3 x 3 − 7 x 2 + 6 x − 14

x 3 + x 2 + x + 1 x 3 + x 2 + x + 1

x 2 + x y + 5 x + 5 y x 2 + x y + 5 x + 5 y

5 x 3 − 3 x 2 + 5 x − 3 5 x 3 − 3 x 2 + 5 x − 3

Writing Exercises

What does it mean to say a polynomial is in factored form?

How do you check result after factoring a polynomial?

The greatest common factor of 36 and 60 is 12. Explain what this means.

What is the GCF of y 4 , y 5 , y 4 , y 5 , and y 10 ? y 10 ? Write a general rule that tells you how to find the GCF of y a , y b , y a , y b , and y c . y c .

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!

…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential - every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help?Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.

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Greatest Common Factor (GCF)

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Greatest Common Divisor (GCD)

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How to Find the GCF

Step 2. find the factors of the first number., step 3. find the factors of the second number..

Greatest Common Factors

With our Greatest Common Factors lesson plan, students learn how to find the greatest common factor of two whole numbers under 100.

Included with this lesson are some adjustments or additions that you can make if you’d like, found in the “Options for Lesson” section of the Classroom Procedure page. One of the optional additions to this lesson is use divisibility rules to help students find factors (all even numbers will include 2 as a factor, numbers ending in 5 or 0 will have 5 as a factor, etc.).

Description

Additional information, what our greatest common factors lesson plan includes.

Lesson Objectives and Overview: Greatest Common Factors teaches students how to find the greatest common factor of two numbers. At the end of the lesson, students will be able to find the greatest common factor of two whole numbers less than or equal to 100. This lesson is for students in 5th grade and 6th grade.

Classroom Procedure

Every lesson plan provides you with a classroom procedure page that outlines a step-by-step guide to follow. You do not have to follow the guide exactly. The guide helps you organize the lesson and details when to hand out worksheets. It also lists information in the blue box that you might find useful. You will find the lesson objectives, state standards, and number of class sessions the lesson should take to complete in this area. In addition, it describes the supplies you will need as well as what and how you need to prepare beforehand.

Options for Lesson

Included with this lesson is an “Options for Lesson” section that lists a number of suggestions for activities to add to the lesson or substitutions for the ones already in the lesson.  One optional addition to this lesson is to allow students to search for a pattern within the list of factors for a number, to help reinforce their pattern recognition. You can also use divisibility rules to help students find factors (all even numbers will include 2 as a factor, numbers ending in 5 or 0 will have 5 as a factor, etc.). If you have more advanced students, they can use numbers larger than 100 to factor. Finally, you can use random numbers on display in the classroom, like parts of phone numbers, addresses, and more, for students find the GCF of.

Teacher Notes

The teacher notes page includes a paragraph with additional guidelines and things to think about as you begin to plan your lesson. This page also includes lines that you can use to add your own notes as you’re preparing for this lesson.

GREATEST COMMON FACTORS LESSON PLAN CONTENT PAGES

What is a factor.

The Greatest Common Factors lesson plan includes three content pages. Students likely already know how to divide numbers and have some multiplications facts memorized. For example, they likely know that 3 x 7 = 21, so 3 and and 7 can evenly divide into 21 without a remainder. 3 and 7 are factors of 21, or numbers that divide evenly into other numbers. You can multiply factors together to get another number.

3 and 7 are factors of 21, and so are 1 and 21. Every number has itself and 1 as factors. For the number 11, for example, the only factors are 1 and 11 because they’re the only two numbers that you can multiply together to get 11.

Many numbers have more than two factors. The number 12, for example, has 1, 12, 2, 6, 3, and 4 as factors (because 1 x 12, 2 x 6, and 3 x 4 all equal 12). For the number 9, the factors are 1, 3, and 9. For 16, the factors are 1, 2, 4, 8, and 16. And for 28, the factors are 1, 2, 4, 7, 14, and 28.

Being able to find factors of numbers helps you be better at multiplication and division. It also helps you reduce fractions. In order to reduce fractions, you need to find the greatest common factor or two or more numbers.

Finding the Greatest Common Factor

Some numbers have some of the same factors as other numbers. We call these the common factors. In the earlier examples, the numbers 16 and 28 have 2 and 4 as common factors.

Let’s look at another example. Say we wanted to find the common factors of 15 and 30. First, we need to list out the factors of each number. The factors of 15 are 1, 3, 5, and 15, while the factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. Therefore, their common factors are 1, 3, 5, and 15. The greatest common factor (GCF) of 15 and 30 is 15, because it is the largest factor that they have in common.

For some numbers, the only factor they have in common is 1. Other times, two numbers might have many common factors. To find the GCF, you can always list out the factors of each number and compare. However, an even easier method to try first is to check if the smaller number can divide into the larger number without a remainder. If it can, the smaller number is the GCF! For example, for the numbers 6 and 18, we know that 6 can divide into 18 evenly. Therefore, 6 is the GCF.

In summary, there are four easy steps that you can use to find the greatest common factor of two numbers. First, find and list the factors of the first number. Second, find and list the factors of the second number. Third, circle all of the common factors. Fourth, identify the greatest common factor. The more you practice, the easier it will be!

GREATEST COMMON FACTORS LESSON PLAN WORKSHEETS

The Greatest Common Factors lesson plan includes three worksheets: an activity worksheet, a practice worksheet, and a homework assignment. You can refer to the guide on the classroom procedure page to determine when to hand out each worksheet.

FINDING THE GCF ACTIVITY WORKSHEET

Students will work with a partner to complete the lesson activity. Each pair will cut out and fold each of the numbers printed on the worksheet. They will then randomly choose two or three of the numbers, writing them in the “numbers” column on the worksheet. Next, they will find the factors of each number and write the GCF in the space provided.

LIST THE FACTORS PRACTICE WORKSHEET

For the practice worksheet, students will list the factors for different numbers. They will also circle the common factors and GCF for some of the factors.

GREATEST COMMON FACTORS HOMEWORK ASSIGNMENT

Like the practice worksheet, the homework assignment asks students to list the factors for different numbers. They will also circle the common factors and GCF for some of the factors.

Worksheet Answer Keys

This lesson plan includes answer keys for the practice worksheet and the homework assignment.  If you choose to administer the lesson pages to your students via PDF, you will need to save a new file that omits these pages. Otherwise, you can simply print out the applicable pages and keep these as reference for yourself when grading assignments.

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• Lesson 4: GCF Factoring and Factoring by Grouping

Hi Everyone!

On this page you will find some material about Lesson 4. Read through the material below, watch the videos, and follow up with your instructor if you have questions.

Lesson 4: GCF Factoring and

Factoring by Grouping

Table of Contents

In this section you will find some important information about the specific resources related to this lesson:

• the learning outcomes,
• the section in the textbook,
• the WeBWorK homework sets,
• a link to the pdf of the lesson notes,
• a link to a video lesson.

Learning Outcomes .

• Know what a GCF is.
• Be able to  factor out a GCF.
• Be able to factor by grouping
• Know the limits of factoring by grouping
• Communicate effectively using written and oral means.

Topic . This lesson covers

Section 4.5: Greatest Common Factor and Factoring by Grouping.

WeBWorK . There is one WeBWorK assignment on today’s material:

GCF-Grouping

Lesson Notes.

Video Lesson.

Video Lesson 4 (based on Lesson 4 Notes)

Warmup Questions

These are questions on fundamental concepts that you need to know before you can embark on this lesson. Don’t skip them! Take your time to do them, and check your answer by clicking on the “Show Answer” tab.

Warmup Question 1

In the product $2\cdot 5 =10$, the numbers $2$ and $5$ are factors of $10$. We also have $1\cdot 10=10$, so $1$ and $10$ are also factors of $10$. Can you find all positive factors of 20?

Show Answer 1

$$1,2, 4, 5, 10, 20$$

Warmup Question 2

(a) Find all factors of $30$.

(b) Find all factors of $50$.

(c) Find all common factors of $30$ and $50$.

(d) Find the GCF (greatest common factor) of $30$ and $50$.

Show Answer 2

(a) $\pm 1, \pm 2, \pm 3, \pm 5, \pm 6,\pm 10, \pm 15,\pm 30$

(b) $\pm 1,\pm 2, \pm 5, \pm 10, \pm 25,\pm 50$

(c) $\pm 1,\pm 2, \pm 5, \pm 10$

(d) The GCF of $30$ and $50$ is $10$.

Warmup Question 3

$$3x^2y^4(5x^7-2xy^2+4).$$

Show Answer 3

\begin{align*}& 3x^2y^4(5x^7-2xy^2+4)\\=&15x^9y^4-6x^3y^6+12x^2y^4\end{align*}

Quick Intro

This is like a mini-lesson with an overview of the main objects of study. It will often contain a list of key words, definitions and properties – all that is new in this lesson. We will use this opportunity to make connections with other concepts. It can be also used as a review of the lesson.

A Quick Intro to the GCF Factoring and Factoring by Grouping

Key Words. Terms, factor, GCF (greatest common factor), factor by grouping

$\bigstar$ The terms of the polynomial $2x^3-4x^2+6x$ are $2x^3$, $-4x^2$ and $6x$.

The GCF (greatest common factor) is the greatest factor of all terms.

In the case of $2x^3-4x^2+6x$, the GCF is $2x$. By factoring $2x$ out, we obtain

$$2x(x^2-2x+3).$$

$\bullet$ Another example:

$$2(x-9)-x(x-9).$$

Here the terms are $2(x-9)$ and $-x(x-9)$. The GCF is the binomial $x-9$. By factoring $x-9$ out, we obtain

$$(2-x)(x-9).$$

$\bullet$ Factoring by Grouping

This method applies to four-term polynomials. First, factor the GCF out of the four terms, if any. Then factor the GCF out of the first two terms. Factor the GCF out of the last two terms. If the two remaining factors share a common binomial factor, factor it out.

\begin{align*}&2acx^2+2adx+2bcx+2bd\\=&\underbrace{2}_{\stackrel{GCF \;of\;2acx^2,}{ 2adx,\; 2bcx \; and \;2bd}}(acx^2+adx+bcx+bd)\\=&2(\underbrace{ax}_{\stackrel{GCF \;of \;acx^2\;}{ and\; adx} }(cx+d)+\underbrace{b}_{\stackrel{GCF\; of \; bcx \;}{ and \; bd}}(cx+d))\\=&2(ax+b)\underbrace{(cx+d)}_{\stackrel{GCF \;of \; ax(cx+d) \;}{ and \; b(cx+d)}}\end{align*}

Video Lesson

Many times the mini-lesson will not be enough for you to start working on the problems. You need to see someone explaining the material to you. In the video you will find a variety of examples, solved step-by-step – starting from a simple one to a more complex one. Feel free to play them as many times as you need. Pause, rewind, replay, stop… follow your pace!

A description of the video

In the video you will see the following

• $3x^2y^4(1-2xy^2+3x^2y^3)$
• the GCF of $2\cdot 3^2\cdot 5^2$ and $2^2\cdot 3^3\cdot 5$
• the GCF of $x^2$, $x^3$ and $x^4$
• the GCF of $3x^2y^4$, $6x^3y^6$ and $9x^4y^7$
• the GCF of $10x^2y^3$ and $15x^3y$
• factorization of $3x^2y^4-6x^3y^6+9x^4y^7$
• factorization of $x^2+3x+2x+6$
• factorization of $12x^2+10x-18x-15$

Try Questions

Now that you have read the material and watched the video, it is your turn to put in practice what you have learned. We encourage you to try the Try Questions on your own. When you are done, click on the “Show answer” tab to see if you got the correct answer.

Try Question 1

Factor out the GCF $$10x^2y^3-15x^3y.$$

The GCF of $$10x^2y^3-15x^3y$$ is $5x^2y$. So

\begin{align*}&10x^2y^3-15x^3y\\=&5x^2y(2y^2-3x).\end{align*}

Try Question 2

Factor out the GCF $$5x^6y^9-10x^7y^6+5x^3y^5.$$

The GCF of $$5x^6y^9-10x^7y^6+5x^3y^5$$ is $5x^3y^5$. So

\begin{align*} &5x^6y^9-10x^7y^6+5x^3y^5\\=&5x^3y^5(x^3y^4-2x^4y+1).\end{align*}

Try Question 3

Factor by grouping $$27x^2+18x-6x-4.$$

\begin{align*}&27x^2+18x-6x-4\\=&27x^2-6x+18x-4\\=&3x(9x-2)+2(9x-2)\\=&(3x+2)(9x-2)\end{align*}

You should now be ready to start working on the WeBWorK problems. Doing the homework is an essential part of learning. It will help you practice the lesson and reinforce your knowledge.

It is time to do the homework on WeBWork:

When you are done, come back to this page for the Exit Questions.

Exit Questions

After doing the WeBWorK problems, come back to this page. The Exit Questions include vocabulary checking and conceptual questions. Knowing the vocabulary accurately is important for us to communicate. You will also find one last problem. All these questions will give you an idea as to whether or not you have mastered the material. Remember: the “Show Answer” tab is there for you to check your work!

• What is a GCF?  Give an example.
• How do you factor out a GCF?
• When should you look to factor by grouping?  Is it always possible?

(a) Factor the GCF out $$100x^3y^2-6xy.$$

(b) Factor by grouping $$10x^2+5x-4x-2.$$

Show Answer

(a) The GCF is $2xy$.

\begin{align*} &100x^3y^2-6xy\\=&2xy(50x^2y-3)\end{align*}

\begin{align*}&10x^2+5x-4x-2\\=&5x(2x+1)-2(2x+1)\\=&(5x-2)(2x+1)\end{align*}

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