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
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 sentence 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
First, we wanna know what is this
question asking us to do. We know that it wants us to write a
mathematical sentence. 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
The first thing this question is
asking us to do is to write a mathematical sentence. 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 numbers 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.