## Playing with Arithmetic

Addition II

### Annabella Wood

29 June 2006

Problems to explore ...

1) | 156 | 2) | 578 | 3) | 396 | 4) | 274 | 5) | 537 | 6) | 1154 |

| __ + 39 __ | | __ + 102__ | | __ + 851__ | | __ + 154__ | | __ + 623__ | | __ + 3296__ |

Start with problem #1.

We notice that it has a 39 in it. Since 39 is nearly 40, and 40 is easier to add to another number than is 39, let's use 40 to do our calculation. Then we can switch back to 39 after the calculation is done. Since 39 is 1 less than 40, we could say 156 + 39 = 156 + 40 - 1. So add 156 + 40. What did you get? I got 196. Now subtract 1 from 196 and we end up with 195. Notice that that is the same result we would have gotten with either of the two other methods we've used so far.
Can you find something about #2 that would make it easier to calculate? Could we make both numbers round numbers that would give us the same answer as if we had worked with the original numbers?
Here's a suggestion. 578 is 2 less than 580. 102 is 2 more than 100. Would you agree that it would be really easy to add 580 + 100 ? You're right...and you can...and here's why...
If you *add and subtract* the same number from anything, it leaves you with the original thing. So if you take problem #2, *add* 2 to the top number and then *subtract* 2 from the bottom number it will leave you with the original problem. This means that if you add (578 + 2) + (102 - 2) you will get the same result as you would by adding 578 + 102. So add 578 + 102 any way you want. What did you get? I got 680. Now add 580 + 100. Notice, it gives you the same result, but the calculations were easier.
Now, let's look at the last four problems. Are any of them like #2 in that you can add or subtract a number from the top line and subtract or add the same number from the bottom line to wind up with two round numbers to work with? How about problem #5?
How would you approach this problem?...
Here's what I would do. I would add 3 to the top line, and subtract 3 from the bottom line. That leaves me with 540 + 620. That gives me 1160. Is the result the same as if you worked the problem in other ways?

Let's work through #6 together.
What number could we use to make #6 an easier addition problem? ____

Do you add or subtract it from the top line? ____________

Do you add or subtract it from the bottom line? __________

What is our new addition problem? _________ + _________ = _____________.

Does the result come out the same if we solve the problem in other ways? If so, we're doing great.
Now let's look at #3.
Can you make this a simpler problem by adding and subtracting? _______

Why not? _________
Could we make it easier by using a round number? ____

Is problem #3 similar to #1? ______ If so, in what way? ______

What would you suggest doing to solve this problem? (There is more than one correct answer to this question, of course.) _____

Go ahead and explore #4 on your own...
Did you need to use the original numbers? _______________

Can you write down two different ways you can change the numbers so that it is easier to solve than the original?

1) __________________ + ___________________ = __________________

2) __________________ + ___________________ = __________________
How many ways can you do this problem? ______________

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Congratulations!
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Now, put down your pencils, and let's think a little bit about what you've discovered, in arithmetic and ...
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