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Art of Problem Solving: Associative Property of Addition
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The associative property of addition allows us to add the numbers in any order we want to get the same answer.
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Art of Problem Solving: Associative Property of Addition
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The community property tells us that when we add two numbers, it doesn’t matter what order we add them in. But what if we have three numbers?

That’s where the Associative Property comes in.

Now once again I’ll start an example adding some hats. But unfortunately, I mean three types of hats and I only have two hands.

Got an another place to put some hats.

So start by putting some hats on my head, there we go, floppy hats on the head.

Looks good? Thank you.

And fuzzy caps here, and I got some ball caps.

So I got five ball caps, three fuzzy caps and two of these floppy hats. How many hats do I have?

交换律告诉我们当我们在计算两个数的和的时候,加的顺序是不影响结果的。但如果我们有三个数呢?这就是用到结合律的地方。

现在再一次我会举出帽子加法的例子。但不幸的是,这里有三种帽子,我却只有两只手。

哈,找到放帽子的地方啦。

那先把一些帽子放在我头上。软帽在我头上,看起来还不错?谢谢。

然后这边是毛帽,这里是鸭舌帽。所以我有五个鸭舌帽,三个毛帽,两个软帽。我一共有多少帽子呢?

Well you could start off by counting the hats in my hands, and adding other two hats on my head. That would sound like: 5 plus 3, and we put that in parentheses, the parentheses tell us we add those first, plus the two floppy hats on my head.

Of course that’s not the only way we could add them up. We could start, well, add the five caps over here and then three plus two, fuzzy hats and floppy hats over here. Now we look like this, five plus the combination of three and two. Again the parentheses tell us we add those two first.

Now of course it doesn’t matter what order we add these hats in, we’re gonna end up in the same number hats.

So these two have to be equal.

你可以先算我手上的帽子,再把头上的两个加起来。这就像5加3,我们把它用括号括起来,括号告诉我们我们先加的是它们,再加2个头上的软帽。

当然这不是唯一把它们加起来的方法。我们也可以先把3个毛帽和2个软帽加起来,5个帽子在这边。那现在就像这样,5加上3与2的和,括号还是表示先加了括号里的两个。

当然,我们加的顺序并没影响,终究会得到同样数目的帽子。

所以这两个应该是相等的。

This is ridiculous. I’m gonna take these off. This one on, here we go.

Now it’s not special about five three and two. You can do this with any three numbers. We use variables to show that.

We take three numbers, a b and c. We add a and b first, and then c. We’re gonna get the same thing as we start off with a and added that to the total of b and c.

These two have to be the same.

And that’s what the Associative Property of Addition tells us.

It tells us it doesn’t matter if we add first two first, or last two first, you’re gonna get the same sum. What that means is you can get rid of these annoying parentheses, and just write, a plus b plus c because it doesn’t matter if we add the first two first or the last two first.

这看起来滑稽死了。我要把这些摘掉,戴上这个。开始吧。

现在并不是532这三个数字有什么特别之处。你可以用任意数字。我们用变量来表示。

我们有三个数,abc。先加ab,再加c。当我们用a加bc之和时会得到同样的答案。这两个一定相等。

这就是结合律告诉我们的。

它告诉我们,先加前两个,先加后两个,你都会得到同样的结果,意思是说你可以去掉这些讨厌的括号,直接写,a加b加c因为先加哪两个都没关系。

Then we combine it with community property which tells us we can reverse, we can flip the order of any two of these. We’re gonna write a plus b plus c as b plus a plus c. Alright we can flip these two and then a plus c plus b. This tells us that it doesn’t matter what order we add these three numbers in. We’re gonna get the same sum. And we can extend it to any number of numbers. Four numbers or five numbers and we use the associate property and community property to tell us we can add those numbers in whatever order we want.

And if you are anything like me, you like the one things anyway you want to.

现在我们把它和交换律联系起来。交换律告诉我们可以对调任意两个数字的顺序。我们可以写a+b+c,也可以b+a+c。我们可以对调这两个数,然后a+c+b。这告诉我们加数的顺序并不重要,我们会得到同样的和。我们可以把它扩展到任意数量的数字,四个,五个,我们可以应用结合律和交换律,用我们想用的任意顺序去做加法。

如果你和我一样,你会喜欢用你喜欢的方式做事情的。

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