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Number Sense

Developing Number Sense. Number sense consists of good intuition about numbers and good instincts for numerical problem solving. It is a sense of familiarity with the patterns of numbers and their relationships. Obviously, having good number sense would be a huge help throughout the Quant section, but what, but if one doesn’t have it, how does one get it?

The folks with good number sense tend to be the folks who love math, especially the people who, in addition to all the required math, also learned more math on their own, and maybe even did recreational math. I realize that’s a foreign idea to some people, but there are people in the world who do recreational math. The irony is that good number sense would be the most helpful to the folks who don’t like math.

And these folks typically avoided math and never did any more than they absolutely had to, but of course that behavior does not enhance number sense. And so what if you’re one of these people in particular? You took some math in high school, you didn’t take more math than you had to take. You avoided math in college.

You were hoping it would go away and never come back again, and now you’re facing a test and you’re gonna have to do more math. And so number sense would be helpful to you. How do we get some number sense here? Unfortunately, there are not really rules for number sense. It’s an intuitive sense of the patterns about numbers.

It really can’t be communicated. One pattern that we did discuss in the last video, for example, is the whole thing about if you square n and then you add n and then add n plus one, then you get n plus one squared. That’s an example of a particular pattern. That’s a good pattern to know.

Number sense is about patterns in general. So that is one pattern of number sense that would be good to know about. Developing number sense in general requires experience. And part of this involves a kind of open-minded curiosity about the patterns of numbers and the patterns of math in general. As you move through the math lessons here, you will learn about individual patterns.

Throughout all the content areas, you will be seeing different patterns. Get curious about these patterns. Explore them a little more with a calculator or on paper. Play with the math a little bit. Gee, they said in the math lesson that such and such is true. Is that really true?

Let me fool around with it. Let me try some example problems. Let me try some numbers and see if it actually works the way that they say that it works. Get curious about it. When you do problems and read solutions, get curious about what if questions.

What if this or that aspect of the problem were different? How would that difference change the problem? Understand that sometimes changing a problem in a certain way makes it un, unsolvable, but that is also an important thing to understand. Why is a problem this way solvable, but a problem with this one change is unsolvable?

People who write test questions have, obviously have to understand that very carefully, what problems are solvable, what problems are unsolvable. When you start thinking in terms of that, you’re actually starting to think in terms of the people who write the math problems. And the more you can understand that world view, the more successful you will be attacking the math section.

I’ll also suggest one game you can play with numbers. 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. So here’s the game I’m suggesting.

Step one, pick four single digit numbers at random. You might roll four dice or you might pick cards from a deck of cards until you have picked four single-digit numbers. So you just want, at random, four starting values. Then you’re gonna take those starting values, those four numbers, and in any combination of addition, subtraction, multiplication, and division, you’re gonna use them to produce all the numbers from 1 to 20.

Each of the four starting numbers must be used once and only once each time. And so the question is how are you gonna use these four numbers to produce 1? How are you gonna use these four numbers to produce 2, and so forth? So I’ll just demonstrate to you to get a sense of this. So just for the sake of argument, I’m gonna start with the numbers 2, 3, 4, and 5.

This is a relatively easy starting set I’ll start with here. And using these four numbers, each time I have to use all four of them once and only once, I have to produce all the numbers from 1 to 20. Now here, I’m gonna show going through in order from 1 to 20. You don’t have to do it in order. For example, you might start out by saying, well, gee, what happens if I add up the four numbers?

That would be a very easy number to get first. Or what happens if I add up three and subtract one? Or what happens if I multiply two and then add the other two? So you can kinda do it out of order and, and pick off the easy combinations first. And then once you’ve done that, then you just have to fill in the rest. But I’m gonna show it starting from scratch in order.

So 1, I’ll just use the fact that 2 is next to 3 and 4 is next to 5. So obviously, 3 minus 2 is 1. 5 minus 4 is 1. 1 times 1 is 1. And if I have 1 times 1 to get 1, I might as well recycle that and just do 1 plus 1 to get 2.

So part of this game is seeing how you can change patterns slightly to get a different result. And then, then you get, as it were, two birds for the price of one, or sometimes even three birds for the price of one. Because you’re using the same pattern tweaked a little bit to get a few different values.

For 3, I’m gonna do it this way. I’m gonna do 4 plus 2, which is 6, minus 5 is 1, and 1 times 3 is 3. For 4, I’m gonna use the fact that 2 and 4 are two apart, and 3 and 5 and two apart. So 4 minus 2 is 2. 3 minus, 5 minus 3 is 2.

And I’m just gonna do 2 plus 2. I could also do 2 times 2. That would also get me 4. For 5, we have a bunch of different choices. One way that I’m gonna do it, just to share something different, 4 plus 5 is 9, divided by 3 is 3.

2 plus 3 is 5. Once you practice this more and more, you’ll actually realize that for each number, there are several different ways you could do it. And that can be part of the game as you get better at it. How many different ways can you produce each number with the four starting numbers?

6, I’m just gonna add 3 plus 4 plus 5 is 12, divided by 2. And that gives me 6. 7, I’m gonna multiply 2 times 3. And then I’m just gonna add 1, which is 5 minus 4. 8, I’m gonna add 4 plus 2. And then at 5 minus 3, 5 minus 3 is another 2.

For 9, I’m gonna add 4 and 5 to get 9, and then just multiply it by 1. 3 minus 2 is 1. For 10, I’m gonna do 2 times 5 and multiply by 1, 4 minus 3. And I’m gonna recycle that. For 11, I’m just gonna do 2 times 5 plus 1. For 12, I’m gonna do 2 plus 4, which is 6, times 5 minus 3, which is 2.

And that would get me 12. For 13, I’m gonna do 4 minus 2, which is 2. 2 times 5 is 10, and just add 3. For 14, as I pointed out, if we just add the numbers together, we get 14. So that probably, if you were, for starters, you should always just find that first, because that one is just taken care of.

That’s easy. For 15, I’ll do 2 times 3, which is 6, plus 4 plus 5. Notice that I’m taking advantage of the fact here that 2 times 3 is 1 bigger than 2 plus 3. And so that’s a trick that can be helpful many times when you’re doing this game.

For 16, I’m gonna do 5 minus 3, which is 2. 2 plus 2 is 4. And then 4 times 4 is 16. For 17, I’m gonna get 15 from 3 times 5. And then just add 4 minus 2, which is 2. And 2 plus 15 is 17.

18, here I’m gonna use an exponent. And once you get bigger, you might wanna start thinking about how squares and exponents might help you. So 3 to the power of 2 is 9. 9 plus 4 plus 5, that’s 9 plus 9 is 18. For 19, I’m going to do 4 plus 5, and then minus 3 plus 2.

Or in other words, plus, or, or minus 3 minus 2, minus 1 essentially. That gives me 19. And then similarly, 20, it’d just be 4 times 5 times 1. So there we are. I’ve got all the numbers from 1 to 20.

It can be done. Curiosity and a sense of play are the best states of mind for noticing details and picking up patterns. And so that’s why I’m suggesting this game. If there’s somebody else who’s sort of at your similar level in terms of math ability, play this game together.

You can start out by cooperating together, trying to get all the numbers from 1 to 20. At a certain point when you feel that you’re starting to get better, you can compete against each other, race against each other. Who can get from 1 to 20 first, or who can find the most numbers in 15 minutes, something like that.

Adding a little bit of competition to it can actually make it fun. If you can cultivate these attitudes, these will help you notice the patterns and interconnections that go into number sense.

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