Now we'll talk about number sense. Number sense consists of good intuition about numbers and 解决问题解决方案的良好本能。这是一种熟悉数字模式的感觉 他们的关系。显然，在整个定量方面都有良好的数字感觉将是一个很大的帮助 section, but what if somebody doesn't have it, how do you get it?

所以首先，考虑一些以前的策略课程，除以五，加倍和减半，平方捷径。 这些人的高度具体建议是从数量意义上蒸馏出来的。 In other words, we at Magoosh, we have number sense, and we picked up these patterns and said, look, here's a useful pattern, 除以五，加倍和减半。

我们非常明确向您展示这些模式。在这些课程中，您基本上递过了一些有用的模式。 Number sense is about seeing more patterns on your own. The folks with good number sense tend to be the ones who love math. Especially the people who in addition to all the required math also learn more on their own, and maybe even do recreational math.

现在，为一些人完全荒谬的乐趣做数学的想法，但相信它， 世界上有些人为乐趣做一些数学。现在讽刺是良好的数字感觉将是最有帮助的 the folks who don't like math, who avoided it and never did anything more than they absolutely had to.

如果他们被迫在枪口这样做，他们只会做数学。但当然避免数学不会增强数字意义。 因此，不幸的是，不喜欢数学的人的行为是最不可能开发数量的。 不幸的是，数字意义上没有规则，这对数字模式来说是一种直观的意义。

它无法明确沟通。换句话说，我们无法列出每个模式 that exists in mathematics. Developing this intuition requires experience, and part of this 涉及一种关于数字模式的开放性好奇心。这真的很重要，特别是对于从未喜欢的人 数学，看看你是否至少可以到达那个开放的好奇心。

一些很酷的模式是什么?当你莫ve through the math lessons, 你会了解一些单独的模式。当我们谈论整数属性或时，我们将指出更多的模式 代数或类似的东西。对那些模式感到好奇， 用计算器或纸张探索它们。

换句话说，做比你必须更多的数学，看看你是否可以自己注意到更多的模式。 When you do a problem and read solutions, get curious about what if questions. What if this or that aspect of the problem were different? 这种差异如何改变问题？理解有时改变问题，例如制作特定的问题 number bigger or smaller, might make the problem unsolvable, but that's also an important thing to understand.

为什么一个问题这种方式可解决，但是当我们制作这种改变时，它就变得无法解决？ 在许多方面，关于数字及其模式的好奇是进入数量的意义。 For example, what patterns do the multiples of seven follow? And what is similar or different about the patterns of the multiples of 17, 或27,37，以7中结束的任何其他数字。

What's the connection, say, between the patterns of the same multiples of two numbers that have a sum of ten? 如三个倍数与七个倍数相比，四个倍数与六倍相比，类似的东西。 我还建议你可以玩数字的游戏。我意识到，如果你不喜欢数学，那么游戏的想法可能听起来不寻常， but it's only by frequent exposure to numbers, and even playing with them, that you will develop intuition for these patterns.

所以这是一个非常简单的游戏，第一步，你会随机挑选四个单位数字。 You might roll a die four times, you might pick four cards from a deck of cards, until you have four single digit numbers. Then you're gonna use those four numbers in any combination. You can use addition, subtraction, multiplication, division, exponents, 括号，您可以使用它们，每个数字，一次，只能生成1到20的所有数字。

所以假设你刚刚给出这个起始集，2,3,4,5并结果是一个非常方便的开始集。 If you're just beginning, don't worry about getting all the numbers from 1 to 20, just see how many different ways you can combine those four numbers and 结果是什么。我们可以组合四个数字的不同方式是什么？

For example, we can certainly add all four of them together, all right, that would result in 14. 我们可以成对乘以它们，然后添加它们。例如，2 x 3 + 4 x 5，那是6 + 20，即26。 We could pair them in a different way, (2x5) + (3x4) = 10 + 12 = 22.

So those seem to wind up with relatively big values. We could also just multiply a single pair and then add the other two. 如例，（2x3）+ 4 + 5，那是6 + 9，即15.我们也可以配对2和4，（2x4）+ 3 + 5， 那是8和8，这是16.我们也可以用指数愚弄。

So for example, 2 to the 3, which is 8 + 4 + 5 = 17, or 3 squared + 4 + 5 = 18. Notice that in the course of these calculations, we've already gotten numbers 14, 15, 16, 17 and 18. So we've gotten some of the numbers in a row already. We could subtract in pairs.

For example, we could subtract 3- 2, which is 1, 5- 4, which is 1, and then just add those, 1 + 1 = 2. Or instead of adding them, we could change that addition to a multiplication, that is 1 times 1, which is 1. 我们可以将其它不同的方式配对，5- 2是3，加4- 3，为1,3 + 1 = 4。

Or instead of adding those two pairs, we could multiply them. And so these are some ways that we get the numbers 1, 2, 3, and 4. So we're starting to get different combinations by doing it. And of course there are many, many more ways to combine the numbers. So at first you can just explore, see how many different ways you can get. See how many different numerical results you can get from combining them in different ways.

And if you stumble onto more than one way to produce the same numerical result, that's also wonderful. If you're more ambitious, then you could be more systematic and try and get all the numbers from 1 to 20. 在本课程下面，实际上存在于PDF的链接，其中我展示了如何使用此集合获取1到20的所有数字，并且具有更加难的设置。

这是一个练习问题，暂停视频，然后我们会谈论这个。所以这是一个代数问题，而且 we'll see a lot more of this when we get to the algebra section. But I'll say here that part of number problem is just noticing patterns. So there are a few different patterns to notice here. So notice first of all, 6N times whatever is in parentheses, that's gonna be divisible by 6, so it must be an even number.

马上我们可以消除所有奇数答案，好吗？现在，请注意，如果我们采取此表达，那就是我们的等式， notice that we have an N factored out in this expression here. And so what we're gonna do is factor out an N in this equation down here. So we factor out, and we get N(20N + 7) = 201.

Well, that's starting to look a lot like the thing that we wanna find the value of. In fact, all we have to do is multiply both sides by 6, then we get 6N(20N + 7). Well, that has to equal 6 x 201, 这不错，6 x 201 = 6（200 + 1）。嗯，6 x 200是1200,6 x 1是6， 1200 + 6 = 1206。

当然，如果数字感觉是新的东西，那种方法看起来很慢，艰难，效率低下。 这有点像骑自行车。如果你之前从未骑过自行车，那么你骑自行车的第一天， 你要脱掉，你不会很快。但当然，当我们都知道，一旦你擅长骑自行车， you're gonna be able to go much faster than walking.

和众多的方式，你必须习惯于命名。起初，它会缓慢而褶皱，你会犯错误，但是 over time you'll get incredibly efficient. Curiosity and a sense of play are the best states of mind for noticing details and picking up patterns. That's in mathematics, in fact, that's in general.

如果您可以培养这些态度，他们将帮助您注意到数字和互连的数字。