Making equivalent expressions using distributive property and GCF: Understand how to find the GCF in a set of numbers

Well, what is a GCF? GCF acrostically stands for the greatest common factor and is found when listing the factors of a number. A factor is a number that can be multiplied by another to get a number. For example, the number twelve, (12), consists of the factors 1,2,3,4,6, and 12. 1 times 12 equals 12, 2 times 6 also equals 12, as well as 3 times 4. These are called factor pairs.

In order to find the GCF, we need a set of numbers. Let’s start off with the two numbers 14 and 49. Before I explain, use the explanation in the paragraph above to attempt the question.

Before we find the GCF, we need to list out the factors of 14 and 49.

14: 1,2,7,14

49: 1,7,49

Look at the factors of the two numbers and see what they have in common. Have you found the common factor? If you found that 7 was in both sets, then great! This concludes that 7 is the GCF.

Now that you know about the GCF, it’s time to create an equivalent with distributive concepts.

Let’s take the expression -90-60w. We need to find an expression that is equivalent to this set of numbers. The first step is finding the GCF of 90 and 60, so let’s list the factors of the numbers.

90: 1, 2, 3, 5, 6, 9, 10, 16, 18, 30, 45, 90

60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Now, let’s see if we can discover the greatest common factor. Got your answer? If you got 30, good job! This is our GCF.

Since we know 30 is our greatest common factor, we are going to start building our equivalent expression.

-30(3-2w)

Well, where did 3, 2w, and -30 come from? Look at the list of factors. The GCF is 30, but the equation has a negative sign on both numbers (-90, -60w.) Therefore, our outer number, (next to the parenthesis), will be -30. We also need to find out what number times -30 equals -90 and -60.

-30 times 3 equals -90.

-30 times 2 equals -60

Now we know this information, it’s a good start.

Because 30 times 3 equals -90 and -30 times 2 equals -60, we can use this information to create a stable expression.

-30(3-2w)

Let’s see if -30(3-2w) is equivalent to -90-60w. First, we can check by multiplying -30 by 3, then 2.

Why can’t we do 3-2w instead? Aren’t we supposed to do the operations in the parenthesis first?: We cannot subtract 3-2w because they are not like terms. This means that they do not share similarities they need to have in order to subtract each other. Instead, we will multiply because parenthesis, ( ), act as multiplication symbols.

-30× 3=-90 We also need to multiply –30 by 2w.

-30× 2w=-60w

Reminder: We cannot subtract nor add unlike terms, but we can bend the rules for multiplication and division.

So, -30(3-2w) is equivalent to -90-60w.

Although -30(3-2w) is equivalent to -90-60w, there are many other expressions that have the same value. For example, 10(-9−6w) and -3(30+20) are both different equations that have the same value as -90-60w.

Now that you know how to make equivalent expressions with distributive property, try some questions on your own. You can look back if review how I solved the problem if you forget or do not understand.

1: Make an expression equal to the expression -4+16w (Use the greatest common factor.)

Remember, you need to find the GCF first. Do you have your answer? If you got 4(-1+4w), good work!

2: Is this 2(2+w+17) equivalent to 4-2w+32?

If you got true, that’s unfortunately not correct. This is because 2(2+w+17) simplifies to 4+2w+34. This is not equivalent to 4-2w+32.

3. Make an expression equal to the expression 3y+9a-12o (Use the greatest common factor.)

Since the numbers 3,9, and 12 all have a common factor of 3, the equivalent expression should be 3(y+3a-4o.)

You’ve gotten the hang of finding and writing equivalent expressions with distributive property. Keep up the good work and stay tuned to learn more!