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Lesson by
**
Mike McGarry
**

Magoosh Expert

Magoosh Expert

Number sense, here we'll talk about this very curious idea of number sense. Number sense consists of good intuition about numbers and good instincts for numerical problem solving. It's a sense of familiarity with the patterns of numbers and their relationships. Now why would an ACT student need this sense of the patterns in mathematics?

Most problems on the ACT math test can be solved in a routine way, just like ordinary high school math. Here's the problem, find the answer. Some of them, though, lend themselves to speedy shortcuts that become apparent through number sense. If you're comfortable with number sense this will save you time on maybe half a dozen questions on the ACT math test.

Including some of the hardest questions, some of the questions in the 50s and above on the ACT math test, and those can be the most challenging problems, but if you have some number sense, they can go much more quickly. 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 learned more on their own.

And maybe even did recreational math, that is to say, they fooled around with numbers purely for fun because they found it enjoyable. And of course if you do that, you pick up all kinds of intuitions about numbers. Now the irony is that good number sense would most help the folks who don't like math, who avoided it and never did any more than they absolutely had to do. But of course, that avoidant behavior does not enhance number sense at all, and so this video is especially directed to all those people who actually weren't so fond of math and never really investigated math all that much.

Learning even a little bit of number sense could help you enormously on the ACT math test. Unfortunately, there are not really rules for number sense. It's more an intuitive sense about the patterns of numbers. It can't really be communicated explicitly. Developing this intuition requires a lot of experience with numbers, experience with problems.

And part of it also involves a kind of open-minded curiosity about the patterns of math. So what are the patterns? For example, what are the patterns of the multiples of seven? How are they related to the multiples of 17 or the multiples or 37 or something like that?

What are the patterns of different combinations of multiples? What are the patterns of primes? What are patterns you notice when you add and subtract? All of these are patterns that you can observe? And the more you study math the more you'll see these patterns. As you move through the math lessons here, you will learn about some individual patterns, in some of the content videos we talk about individual patterns.

And get curious about those patterns, explore them a little more with calculator or on paper. When you do problems and read solutions, get curious about what if questions, what if this aspect of the problem were a little bit different? How would that difference change the problem? Understand that sometimes making a small change in a problem makes the problem unsolvable.

But that is also important thing to understand. Why is it that when we make this one little change we can solve that problem, but if we make another little change the problem becomes unsolvable? And if you really understand that you really understand the problem very deeply. I'll also suggest one game you can play. If number patterns are something that you really have never really studied, this is a game that you can play.

You can play this by yourself, you can play this with friends, and the more game like you can make it, the more curiosity and fun you can bring to it. The more it'll enhance your learning experience. I realize that if you are not fond of math, the idea of a game might sound unusual, but it's only by frequent exposure to numbers, even playing with them, that you will develop intuition for these patterns.

And again, playfulness and curiosity are optimal states for learning. And so the more you can cultivate those things intentionally, the more you're actually going to learn. So here's the game very simple. Step one, you're gonna pick four single digit numbers at random. You might roll four dice, you might pick cards from a deck of playing cards until you have picked four single digit numbers.

You just have to get four single digit numbers at random. Then, you're gonna use those four numbers in any combination of addition, subtraction, multiplication, division, and exponents to produce all the numbers from 1 to 20. Each of the four numbers must be used once and only once, each time. And so I'm gonna get those four numbers, and I'm gonna have to use all four of those numbers.

I'm gonna have to use each one exactly once to produce 1. And then use each one exactly once to produce 2 and so forth. So I'll just start with a very simple sense. This is a very simple set. There're no repeats, it's just 2, 3, 4, 5. And notice that the sum of those numbers incidentally is 14.

So right there 14 is kind of a bonus we'll get that for free. We wanna use those numbers to build all the numbers from 1 to 20. And so first of all, let's build 1. Well, notice 3 minus 2, that's 1. 5 minus 4, that's also 1. What we have two ones, obviously we can't add them, but we could multiply them, and so 1 times 1 gives us 1.

Or instead of multiplying those two ones I could add those two ones, and so that's a little number sentence right there. That to get one I could multiply those two ones, to get two I could add the two ones. To get three one thing I can do, is multiply three and then I can build one from six minus five, and so three times one is three. I can get four, by creating two, twos, and I could either add the two two's, I can also multiply the two two's, and I'll point out that for many of these, there's many, many ways that you could get.

I'm just showing one particular way that you can get each number. But in fact you could probably come up with four or five ways to get each number. And in fact if you're playing with your friends you could even make a competition. How many different ways can you produce three using these four numbers, that sort of thing. All right to get 5, one way to get it, and this is kind of an odd way to get it, I'm going to get the 4 + 5 making 9, divided by 3, well that's 3 and that 3 + 2 gives me 5.

To get 6 I'm just gonna add 3 + 4 + 5 is 12 and divide it by 2. To get 7 I'm gonna multiply 2*3 to get 6 and then add 1. To 8 I'm gonna add 4 + 2 + a subtraction that equals 2 4+2+2=8. To get 9, I'm gonna multiply 9*1 to get the 4+5 to get 9 and multiply it times 1, to get 10 I'm gonna multiply 2*5 and then times 1.

Well that's just how I'm gonna get 11. I'm gonna get 2*5+1. To get 12 one way to get it is 6*2. So anything that's a product of smaller numbers it's easy to build that as a product. 13 I'll get by producing 10 2*5 and then just adding 3.

14 is the one we're given for free that's where we just add all four numbers so that's a bonus. 15 I'll get by 6 2*3 is 6, + 4 + 5 16 I'll get, by 4 *4 17 I'll get from 15 + 2. 18 I'll get, here I use an exponent.

3 to the 2 is 9 + 4 + 5 is another 9 so that's 18. 19 I'll get from making 20, and then essentially subtracting 1 from 20, and then 20 I'll get 4 times 5 and then just multiplying by 1. So it can be done, and so here I went through these very quickly, but of course the challenge is to try and do this with yourself. And probably the best way to do this is corporate with friends until you all get better at building all the numbers from 1 to 20 with any four starting members.

And this was a set that was particularly easy 2, 3, 4, 5. It starts to get trickier if you start to have repeats for example if you had 3, 3, 7, 7 or something like that. And then you have to build all the numbers, that would be a little bit trickier, but that would be a good challenge. Once again, I'll say that curiosity and a sense of play are the best states of mind for noticing details and picking up patterns.

And it's really very important to appreciate. Curiosity is not really something that purely comes from outside, you have the power to generate curiosity. And the more genuine curiosity you can generate, the more you will learn. And this is actually true across disciplines. It's a really important thing to appreciate.

So if you can cultivate these attitudes and notice the patterns and connections then you can start to develop this elusive thing called number sets. As with all the mental math videos, number sense takes time and practice to build. So when you first start out, you're not gonna have much number sense and it's not gonna save you much time. You really have to invest in it, and the more invest in it, it will help you not only the ACT math test, it will also help you in every math class you ever take for the rest of your academic career.

And so there isn't much of the immediate payback but the rewards of developing number sense are significant as one grows into these skills.

Show TranscriptMost problems on the ACT math test can be solved in a routine way, just like ordinary high school math. Here's the problem, find the answer. Some of them, though, lend themselves to speedy shortcuts that become apparent through number sense. If you're comfortable with number sense this will save you time on maybe half a dozen questions on the ACT math test.

Including some of the hardest questions, some of the questions in the 50s and above on the ACT math test, and those can be the most challenging problems, but if you have some number sense, they can go much more quickly. 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 learned more on their own.

And maybe even did recreational math, that is to say, they fooled around with numbers purely for fun because they found it enjoyable. And of course if you do that, you pick up all kinds of intuitions about numbers. Now the irony is that good number sense would most help the folks who don't like math, who avoided it and never did any more than they absolutely had to do. But of course, that avoidant behavior does not enhance number sense at all, and so this video is especially directed to all those people who actually weren't so fond of math and never really investigated math all that much.

Learning even a little bit of number sense could help you enormously on the ACT math test. Unfortunately, there are not really rules for number sense. It's more an intuitive sense about the patterns of numbers. It can't really be communicated explicitly. Developing this intuition requires a lot of experience with numbers, experience with problems.

And part of it also involves a kind of open-minded curiosity about the patterns of math. So what are the patterns? For example, what are the patterns of the multiples of seven? How are they related to the multiples of 17 or the multiples or 37 or something like that?

What are the patterns of different combinations of multiples? What are the patterns of primes? What are patterns you notice when you add and subtract? All of these are patterns that you can observe? And the more you study math the more you'll see these patterns. As you move through the math lessons here, you will learn about some individual patterns, in some of the content videos we talk about individual patterns.

And get curious about those patterns, explore them a little more with calculator or on paper. When you do problems and read solutions, get curious about what if questions, what if this aspect of the problem were a little bit different? How would that difference change the problem? Understand that sometimes making a small change in a problem makes the problem unsolvable.

But that is also important thing to understand. Why is it that when we make this one little change we can solve that problem, but if we make another little change the problem becomes unsolvable? And if you really understand that you really understand the problem very deeply. I'll also suggest one game you can play. If number patterns are something that you really have never really studied, this is a game that you can play.

You can play this by yourself, you can play this with friends, and the more game like you can make it, the more curiosity and fun you can bring to it. The more it'll enhance your learning experience. I realize that if you are not fond of math, the idea of a game might sound unusual, but it's only by frequent exposure to numbers, even playing with them, that you will develop intuition for these patterns.

And again, playfulness and curiosity are optimal states for learning. And so the more you can cultivate those things intentionally, the more you're actually going to learn. So here's the game very simple. Step one, you're gonna pick four single digit numbers at random. You might roll four dice, you might pick cards from a deck of playing cards until you have picked four single digit numbers.

You just have to get four single digit numbers at random. Then, you're gonna use those four numbers in any combination of addition, subtraction, multiplication, division, and exponents to produce all the numbers from 1 to 20. Each of the four numbers must be used once and only once, each time. And so I'm gonna get those four numbers, and I'm gonna have to use all four of those numbers.

I'm gonna have to use each one exactly once to produce 1. And then use each one exactly once to produce 2 and so forth. So I'll just start with a very simple sense. This is a very simple set. There're no repeats, it's just 2, 3, 4, 5. And notice that the sum of those numbers incidentally is 14.

So right there 14 is kind of a bonus we'll get that for free. We wanna use those numbers to build all the numbers from 1 to 20. And so first of all, let's build 1. Well, notice 3 minus 2, that's 1. 5 minus 4, that's also 1. What we have two ones, obviously we can't add them, but we could multiply them, and so 1 times 1 gives us 1.

Or instead of multiplying those two ones I could add those two ones, and so that's a little number sentence right there. That to get one I could multiply those two ones, to get two I could add the two ones. To get three one thing I can do, is multiply three and then I can build one from six minus five, and so three times one is three. I can get four, by creating two, twos, and I could either add the two two's, I can also multiply the two two's, and I'll point out that for many of these, there's many, many ways that you could get.

I'm just showing one particular way that you can get each number. But in fact you could probably come up with four or five ways to get each number. And in fact if you're playing with your friends you could even make a competition. How many different ways can you produce three using these four numbers, that sort of thing. All right to get 5, one way to get it, and this is kind of an odd way to get it, I'm going to get the 4 + 5 making 9, divided by 3, well that's 3 and that 3 + 2 gives me 5.

To get 6 I'm just gonna add 3 + 4 + 5 is 12 and divide it by 2. To get 7 I'm gonna multiply 2*3 to get 6 and then add 1. To 8 I'm gonna add 4 + 2 + a subtraction that equals 2 4+2+2=8. To get 9, I'm gonna multiply 9*1 to get the 4+5 to get 9 and multiply it times 1, to get 10 I'm gonna multiply 2*5 and then times 1.

Well that's just how I'm gonna get 11. I'm gonna get 2*5+1. To get 12 one way to get it is 6*2. So anything that's a product of smaller numbers it's easy to build that as a product. 13 I'll get by producing 10 2*5 and then just adding 3.

14 is the one we're given for free that's where we just add all four numbers so that's a bonus. 15 I'll get by 6 2*3 is 6, + 4 + 5 16 I'll get, by 4 *4 17 I'll get from 15 + 2. 18 I'll get, here I use an exponent.

3 to the 2 is 9 + 4 + 5 is another 9 so that's 18. 19 I'll get from making 20, and then essentially subtracting 1 from 20, and then 20 I'll get 4 times 5 and then just multiplying by 1. So it can be done, and so here I went through these very quickly, but of course the challenge is to try and do this with yourself. And probably the best way to do this is corporate with friends until you all get better at building all the numbers from 1 to 20 with any four starting members.

And this was a set that was particularly easy 2, 3, 4, 5. It starts to get trickier if you start to have repeats for example if you had 3, 3, 7, 7 or something like that. And then you have to build all the numbers, that would be a little bit trickier, but that would be a good challenge. Once again, I'll say that curiosity and a sense of play are the best states of mind for noticing details and picking up patterns.

And it's really very important to appreciate. Curiosity is not really something that purely comes from outside, you have the power to generate curiosity. And the more genuine curiosity you can generate, the more you will learn. And this is actually true across disciplines. It's a really important thing to appreciate.

So if you can cultivate these attitudes and notice the patterns and connections then you can start to develop this elusive thing called number sets. As with all the mental math videos, number sense takes time and practice to build. So when you first start out, you're not gonna have much number sense and it's not gonna save you much time. You really have to invest in it, and the more invest in it, it will help you not only the ACT math test, it will also help you in every math class you ever take for the rest of your academic career.

And so there isn't much of the immediate payback but the rewards of developing number sense are significant as one grows into these skills.