1. Rounding Off Whole Numbers © Math As A Second Language All Rights Reserved next #5 Taking the Fear out of Math

2. next © Math As A Second Language All Rights Reserved The Art of Estimating Whole Numbers next It is important to be able to make reasonable estimates when we use the traditional pencil-and-paper algorithms for addition, especially when the addition involves adding very large numbers. The traditional algorithm has us doing the arithmetic from right to left. Thus, we would first add the 3 ones and 7 ones to get 10 ones and our partial answer would then look like… 3,895,567,893 + 4,793,874,997 1 0

3. next © Math As A Second Language All Rights Reserved The point is that we get to the most significant denomination (in this case, billions) after we’ve been working the longest, and hence, the most likely to make a careless error. 3,895,567,893 + 4,793,874,997 1 0 3 4

4. next © Math As A Second Language All Rights Reserved next With the abundance of calculators now in use, one might argue that there is no need to go through learning to become an expert on the addition algorithm. However, it is possible that digits were entered incorrectly. 1 note 1 To paraphrase the National Rifle Association’s slogan, “Calculators don’t make mistakes. The people who enter the numbers do!”

5. next © Math As A Second Language All Rights Reserved So even with a calculator, one should have enough number sense to be able to estimate the answer even before beginning to perform the actual computation. 2 note 2 Another problem with a calculator is that it can only store a certain number of digits. So if the number is too great, the calculator will not be able to store all the digits. For example, consider the following addition problem… 3,895,567,893 + 4,793,874,997 next

6. © Math As A Second Language All Rights Reserved Even if we were to use a calculator to find the answer, we should first observe that 3,895,567,893 is greater than 3 billion but less than 4 billion; next 3,895,567,893 + 4,793,874,997 while 4,793,874,997 is greater than 4 billion but less than 5 billion. 3,000,000,000 4,000,000,000 5,000,000,000

7. next © Math As A Second Language All Rights Reserved From the above diagram, we see that the sum has at least 7 billion, next 3,895,567,893 + 4,793,874,997 3 billion 4 billion 5 billion 7 billion9 billion but no greater than 9 billion. next

8. © Math As A Second Language All Rights Reserved Notice that we think of 3,895,567,893 as being a whole number. However, if the noun is “billions”, it is not a whole number. It is more that 3 billion but less than 4 billion. In other words, in terms of numbers being adjectives that modify nouns, suppose we are “counting by billions,” which means that instead of counting “1, 2, 3…” with no noun specified, we count “1 billion, 2 billion, 3 billion…” Note on Adjective/Noun Theme

9. next © Math As A Second Language All Rights Reserved These numbers are called multiples of a billion. Note on Adjective/Noun Theme In the language of place value, multiples of a billion end in nine 0’s. Thus, the first six multiples of a billion are… 1,000,000,000 2,000,000,000 3,000,000,000 4,000,000,000 5,000,000,000 6,000,000,000 next

10. © Math As A Second Language All Rights Reserved If we are counting by billions, we might ask the question… “What number in our list of multiples of a billion is closest in value to 3,895,567, 893?”. next The common mathematical wording for this question is the statement… “Round off the number 3,895,567, 893 to the nearest billion.”

11. © Math As A Second Language All Rights Reserved Since the number 3,895,567,893 is closer in value to 4 billion than to 3 billion, the answer to this question is 4 billion; or in the language of place value 4,000,000,000. For example, let’s return to our original addition problem. 3,895,567,893 + 4,793,874,997 next

12. © Math As A Second Language All Rights Reserved In the statement, “When rounded off to the nearest billion, the number 3,895,567,893 becomes 4,000,000,000”, we mean that among all possible multiples of one billion, the multiple that gives the most accurate approximation for 3,895,567,893 is 4,000,000,000. next Less than halfhalfMore than half3,000,000,0003,500,000,0004,000,000,000 3,895,567,893 Since 3,895,567,893 is more than half way between 3 billion and 4 billion, it is closer to 4 billion. next

13. © Math As A Second Language All Rights Reserved next In a similar way, if we round off 4,793,874,997 to the nearest billion it becomes 5,000,000,000. That is, 4,793,874,997 is greater than 4 billion, less than 5 billion, but closer in value to 5 billion. Less than halfhalfMore than half4,000,000,0004,500,000,0005,000,000,000 4,793,774,997 Since 4,793,874,997 is more than half way between 4 billion and 5 billion, it is closer to 5 billion. next

14. © Math As A Second Language All Rights Reserved So by rounding off each number to the nearest billion… next 3,895,567,893 …the addition problem becomes much less cumbersome. 4,000,000,000 5,000,000,000 + 4,793,874,997 +

15. © Math As A Second Language All Rights Reserved This tells us, even before we begin to do the actual arithmetic that the answer to the addition problem… next …should be “around” 9 billion (or in place value notation, 9,000,000,000) 3,895,567,8934,000,000,000 5,000,000,000+ 4,793,874,997+ next 9,000,000,000

16. © Math As A Second Language All Rights Reserved With respect to the above note, we are able to find upper and lower bounds for the sum 3,895,567,893 + 4,793,874,997 by observing that… next 3,895,567,893 lower bound 4,793,874,997 + underestimate actual number upper bound overestimate 3 billon 4 billon 4 billon 5 billon < < < < < 9 billon7 billon < actual sum

17. © Math As A Second Language All Rights Reserved The above upper and lower bounds tell us that in addition to our estimate that the sum is “around” 9 billion, that the correct answer has to be greater than 7 billion but less than 9 billion. Summary And we know all of this before we even begin to perform the actual addition (either by hand or with the aid of a calculator). next

18. © Math As A Second Language All Rights Reserved Doing the actual computation we see that… next 3,895,567,893 lower bound 4,793,874,997 + underestimate actual number upper bound overestimate 3 billon 4 billon 5 billon < < < < < 9 billon7 billon < 8,689,442,890 The answer is reasonable in the sense that it is within the range of our estimate.

19. next © Math As A Second Language All Rights Reserved For example, suppose that instead of entering 4,793,874,997 on the calculator we had erroneously entered 5,793,874,997. In that case the calculator would have given us 9,689,442,890 as the answer. A “Reasonable” Note While there are many numbers between 7 billion and 9 billion, 9,689,442,890 is not one of them! next

20. © Math As A Second Language All Rights Reserved Clearly, there are many whole numbers that are between 7 billion and 9 billion, and as a result it is still possible that in obtaining the above sum we made an error in the arithmetic. A “Reasonable” Note However, our estimate helps us to be sure that we have not obtained an unreasonable answer. next

21. © Math As A Second Language All Rights Reserved One moral of this story is that we do not have to know the correct answer to conclude that some answers are incorrect! next

22. © Math As A Second Language All Rights Reserved The Rote Method for Rounding Off next Too often students are given a rote “recipe” to follow which results in their obtaining a correct estimate but without properly understanding what has happened. For example, to round off 5,286 to the nearest thousand, the recipe is…

23. © Math As A Second Language All Rights Reserved The Rote Method for Rounding Off next Step 1: Locate the place to which you’re rounding off. In this example, we are rounding 5,286 off to the nearest thousand, so we locate the thousands place. Using an asterisk to locate the place, we obtain… 5, 2 8 6 *

24. © Math As A Second Language All Rights Reserved next Step 2: If the digit immediately to the right of the asterisk is less than 5, simply replace all of the digits to the right of the arrow by 0’s. 5, 0 0 05, 2 8 6 Our answer rounded to the nearest thousand would be 5,000. *

25. © Math As A Second Language All Rights Reserved next Notice that the “short cut” is simply a mechanical way to summarize the logical way. To review this in greater detail, we know that the multiples of a thousand end in three 0’s. Hence, the first few multiples of a thousand are 1,000, 2,000, 3,000, 4,000, 5,000, 6,000, etc.). 5,286 is greater than 5,000 but less than 6 thousand. Since 5,500 is halfway between 5,000 and 6,000, and 5,286 is between 5,000 and 5,500 it means that 5,286 is closer to 5,000 than 6,000.

26. © Math As A Second Language All Rights Reserved next Step2: However, if the digit immediately to the right of the asterisk is greater than 5, we still replace every digit to the right of the asterisk by 0, but this time we add 1 to the number that is left of the zeros. 5 6, 9 8 2 * For example, if we wanted to round off 56,892 to the nearest thousand, the asterisk would be over the 6.

27. © Math As A Second Language All Rights Reserved next The digit to the right of the asterisk is 9. 0 0 0 Therefore, the digits to the right of the asterisk become 0’s. * Our answer rounded to the nearest thousand would be 57,000. 5 7, And we add 1 to 56 and get 57. 5 6, 9 8 2

28. © Math As A Second Language All Rights Reserved next Notice that the “short cut” is simply a mechanical way to summarize the logical way. To review this in greater detail, we know that the multiples of a thousand end in three 0’s. Hence, the first few multiples of a thousand are 1,000, 2,000, 3,000, 4,000, 5,000, 6,000, etc. When we count by thousands 56,892 is between 56,000 and 57,000, and 56,892 is between 56,500 and 57,000 (and therefore closer in value to 57,000).

29. © Math As A Second Language All Rights Reserved When rounding off numbers between 5,000 and 6,000 to the nearest thousand, all the numbers less than 5,500 become 5,000 while all numbers greater than 5,500 become 6 thousand. Note on Rounding Up/Down A fine point occurs if the number is exactly 5,500 in which case we have to use our own judgment when we round off to the nearest thousand. next

30. © Math As A Second Language All Rights Reserved Some books advise that in this case we round up rather than down. That is, they would round 5,500 off to 6,000. This can be a bit “dangerous”. For example, suppose we were adding 5,500 ten times. The exact answer would be 55,000. However, if we round each term up to the next thousand, the sum becomes 60,000 and if we round each terms down to the next thousand, the sum becomes 50,000. next

31. © Math As A Second Language All Rights Reserved A better strategy would be to alternate between rounding up and rounding down. In fact, in this illustration if we rounded half of the terms up to 6,000 and the other half down to 5,000, we would get the exact answer as our approximation. The important thing to remember is that rounding off is just a tool for helping us make estimates, and in that sense, it is a supplement and not a replacement for us using our own judgment. next

32. © Math As A Second Language All Rights Reserved In other words, when we are given a “rule of thumb” to follow, it should be tempered by our own number sense.

33. next © Math As A Second Language All Rights Reserved How we round off often depends on the degree of accuracy that we require. There are times when we might want to round off 5,286 to the nearest hundred rather than to the nearest thousand. Counting by hundreds we see that 5,286 is greater than the 52 nd multiple and of a hundred (5200) but less than the 53 rd multiple of a hundred (5300). Moreover, 5,286 is closer in value to 5,300 than it is to 5,200. Hence, to the nearest hundred 5,286 rounds up to 5,300. next

34. © Math As A Second Language All Rights Reserved 5 2 0 0 (52 nd multiple of 100) Summary Using the short cut, since we are rounding off to the nearest hundred, our asterisk goes over the 2. The digit to the right of 2 (namely, 8) is greater than 5. Hence, we replace 52 by 53 and replace the remaining digits by 0’s. next 5 2 5 0 5 2 8 6 5 3 0 0 (53 rd multiple of 100)

35. In the next presentation, we will talk about unadding (subtraction). © Math As A Second Language All Rights Reserved rounding 1,8752,000