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Today's lesson is on properties of numbers. The first property that we will look at today
is the commutative property. This property works for both addition and
multiplication. The commutative property says that when you
are adding, you may add in any order and the answer will be the same. That is also true
for multiplication. So let's look at some examples.
3 plus 5 is equal to 8 but so is 5 plus 3. So you can see that the order doesn't matter.
And how about 5 times 10, that's 50 and that equals the same thing as 10 times 5. You can
see for multiplication and addition, it really doesn't matter what order we add or multiply.
Next we are going to look at the Associative Property. Again that is for addition and multiplication.
The associative property says that we can group the numbers in an addition problem or
a mulitiplication problem in any way and we will still get the same answer. I'm going
to show you some examples now. 2 plus 3 plus 4. If we group the 2 plus 3,
we'll have to add those first so we will get 5 plus 4. We want to see if we get the same
thing is we group the 3 plus 4. In this case, we are going to add the 3 plus 4 first to
get 7. 5 plus 4 equals 9 and 2 plus 7 equals 9. We can see that they are equal.
This also works for multiplication. In our example here, 2 times 5 times 3 equals 10
times 3 which we know equals 30. Here we are grouping the 5 times 3 which we know is 15
and 2 times 15 also equals 30. So we can see that they are both equal.
Next we are going to talk about the Identity Property.
We've talked about this in our number talks. The identity property for addition says that
the sum of any number and zero is that number. In other words, we can add zero to anything
and we will still get the same result. For instance 5 plus 0 is still 5, and 1057
plus 0 is still 1057. Adding zero does not change the value of the number.
The identity property for multiplication says that the product or the answer to a multiplication
problem of 1 and any number is that number. In other words, we can multiply any number
times one and the value of the number doesn't change. So 6 times 1 equals 6 or 352 times
1 equals 352. You can pick any number and multiply by 1 and it won't change. It will
keep its identity. Next we are going to look at the Distributive
Property but before we do that, let's talk about some important words. A sum is an answer
to an addition problem . Each number that you add together in an addition
problem is called an addend. And then you see another word underlined in
this explanation, a product. A product is an answer to a multiplication problem.
The Distributive Property sounds kind of complicated but I'll show you what it means. If you are
multiplying a sum times a number, you also multiply each addend or each of those numbers
by the number and then add their products. This is what I mean by that.
Let's say you have 5 times 20 plus 2. You could add those two together using order of
operations and say 5 times 22. But another way that you can do it is to use the disgtributive
property and the distributive property says that you can multiply 5 times the first addend,
20 and 5 times the second addend of 2 and then just add those together. So this is kind
of easy because we know that 5 times 20 is 100 and 5 times 2 is 10. So sometimes we use
the distributive property to make our math easier because now we know that 100 plus 10
is 110. It is a little bit easier than trying to do 5 times 22.
The distributive property is helpful when you are trying to do multiplication mentally.
Let's look at an example. How about 6 times 37. We can do that, but may it would be easier
to break that 37 into a sum. Let's make some real friendly numbers. 6 times 30 plus 7.
Now we can use the distributive property that says we can multiply 6 times the first addend
of 30 and 6 times the second addend of 7 and then just add those products together. So
6 times 30 is pretty easy to do in my head. I know that is 180. 6 times 7 is also easy
because I know my multiplication tables. So 6 times 7 is 42, and now all I have to do
is add those two. When I do that, I find out that the answer is 222.
You may be wondering why we even bother to learn properties and when they can be useful.
Well,they can be useful because they can make your math easier. So let me show you a really
quick example. Let's look at this problem, 11 plus 25 plus 9. Order of Operations tells
me to add from left to right. But I also know that I can switch the order because of the
Commutative Property and sometimes it is easy when I see numbers that I can group together
that are easy to add. So let's rewrite this as 25 plus 11, just switching the order of
the 25 and the 11, plus 9. And, if I do Order of Operations, I am going to add from left
to right, but if I group the 11 and 9, which I know is really easy because 11 plus 9 is
20. Then I have 25 plus 20 and that equals 45. So in the first step, I changed the order
so I used commutative property so it was a little bit easier to think about and then
when I grouped these numbers right here, I used the associative property so it was really
easy for me to do it mentally. So that's it for this lesson and we are going
to do some practice problems later.