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Using Properties of Addition and Multiplication to Solve Problems

Before we solve any problems, let’s take a minute to review the properties of addition and multiplication. Here’s a list of the properties we’re talking about.

The commutative property tells us that the order in which two numbers are added or multiplied does not change their sum or product. 𝑎 plus 𝑏 equals 𝑏 plus 𝑎. 𝑎 times 𝑏 equals 𝑏 times 𝑎. Commutative like the word commute has to do with the exchange of places. It’s a helpful way to remember what the commutative property is.

The associative property says that the way in which three numbers are grouped when they are added or multiplied does not change their sum or product. When you think of the associative property, you should remember that it’s about grouping, or the way things are grouped together. With the associative property of addition, taking 𝑎 and adding that to 𝑏 plus 𝑐 is the same thing as first adding 𝑎 and 𝑏 together and then adding 𝑐 afterwards. This is also true with multiplication.

The identity property is next. The identity property of addition says that the sum of an addend and zero equals the addend. In this case, 𝑎 is the addend plus zero equals 𝑎, the addend. The identity property of multiplication says that the product of a factor and one is the factor. Here, we have 𝑎, being our factor, multiplied by one equals 𝑎, which is the factor that we started with. When we use the identity property, what we’re saying is adding zero to our original addend or multiplying our original addend by one does not change the value of that expression.

Our last property to review is the distributive property. The distributive property is a little bit different because it actually is a property that combines addition and multiplication. The distributive property says that to multiply a sum by a number, you multiply each addend of the sum by the number outside the parentheses. To multiply a sum, in this case 𝑏 plus 𝑐, by a number, multiply each addend by the number on the outside of the parentheses. We’re taking our 𝑎, or a number outside the parentheses, and we’re distributing the multiplication to the 𝑏 and the 𝑐, or each addend of the sum.

Here are the properties we’ve been reviewing. Now, we’re gonna take these properties and solve some problems. Here’s an example.

Rewrite the expression 19 times 34 plus 21 using the distributive property.

Remember what the distributive property says. We’re going to need to take this 19 and distribute it across the 34 and the 21, each addend of the sum. It will look like this. 19 times 34 plus 19 times 21. Here’s our next question.

Jeff walked from home three blocks to the movie theatre. After the movie, he walked seven blocks to an ice cream shop. Write a mathematical sen10ce to show that the distance from Jeff’s home to the ice cream shop is the same as the return walk home. What number property illustrates that?

First, we wanna know what is this question asking us to do. We know that it wants us to write a mathematical sen10ce. That’s one thing. It’s also asking us to find a number property. I’m gonna mark in yellow the information that we were given, three blocks to the movie theatre, seven blocks to the ice cream shop. We’re trying to compare the distance from Jeff’s home to the ice cream shop to the distance of his return walk home.

The first thing this question is asking us to do is to write a mathematical sen10ce. Let’s start there. The first thing that Jeff does is walk three blocks, so that’s our first piece of information. After that, he walked seven blocks. This will be the distance from Jeff’s home to the ice cream shop. And we want to show that his return walk, his walk home, is the same distance.

His return walk will be the same as his walk from his home to the ice cream shop except in reverse. Three plus seven equals seven plus three. Jeff’s walk to the ice cream shop was the same distance as his walk home. This is an illustration of which property? The commutative property of addition. We’re only changing the order in which we’re adding these two addends. That doesn’t change the value of either side. Here, walking three blocks and then walking seven blocks is the same total distance as first walking seven blocks and then walking three more.

Rewrite the expression 10 times 47 minus 10 times five using the distributive property.

First, in order to rewrite this expression, you need to remember what the distributive property says. It says 𝑎 times 𝑏 plus 𝑐 is the same thing as saying 𝑎 times 𝑏 plus 𝑎 times 𝑐. In this problem, the 10 has already been distributed to the 47 and the five. We wanna change it so that the 47 and five are being added together before the multiplication happens.

So, first, we’ll take the 47 and the five and put them in a group with parentheses, 47 plus five. And then, we multiply that sum by 10. This is our new expression, 10 times 47 plus five. Here’s our last example.

Calculate 75 plus 16 plus five.

There’s a property of numbers we can use here to make solving this problem a little bit easier. Can you recognise what property would be helpful here? The associative property of addition will be very helpful here. Remember that the associative property deals with groupings and how we group members when we add them. Because of the associative property, I can change the grouping here. I can group 75 plus five together and then add 16.

Mentally adding 75 plus five is easier than mentally adding 75 plus 16. When I add the 75 and five together to get 80, then I can add the 16 easily. 96 is the answer here. Of course, you can add 75 plus 16 first and then add the five, but it’s going to take you longer than changing the grouping and then adding. And now, you have all the tools you need to try it for yourself.