rounding off whole numbers © math as a second language all rights reserved next #5 taking the fear...

Rounding Off

Whole Numbers

Rounding Off

Whole Numbers

© Math As A Second Language All Rights Reserved

next

#5

Taking the Fearout of Math

next


The Art of Estimating Whole Numbersnext

It is important to be able to makereasonable 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

next


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 mostlikely to make a careless error.

3,895,567,893+ 4,793,874,997

1

0

34

next


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

1To paraphrase the National Rifle Association’s slogan, “Calculators don’t make mistakes. The people who enter the numbers do!”

next


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

nextnext

next


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

4,000,000,000 5,000,000,000

next


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

4 billion 5 billion

7 billion 9 billion

but no greater than 9 billion.

next

next


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

next


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

nextnext

next


If we are counting by billions, we might ask the question…

““What number in our list of multiples of a What number in our list of multiples of a billion is closest in value tobillion is closest in value to

3,895,567, 893?3,895,567, 893?”.”.

nextnext

The common mathematical wording for this question is the statement…

““Round off the number 3,895,567, 893 to Round off the number 3,895,567, 893 to the nearest billionthe nearest billion.”.”

next

next


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

nextnext

next


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 half half More than half

3,000,000,000 3,500,000,000 4,000,000,0003,895,567,893

Since 3,895,567,893 is more than half way between 3 billion and 4 billion, it is closer

to 4 billion.

next

next


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 half half More than half

4,000,000,000 4,500,000,000 5,000,000,0004,793,774,997

Since 4,793,874,997 is more than half way between 4 billion and 5 billion, it is closer

to 5 billion.

next

next


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+

next

next


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)

3,895,567,893 4,000,000,000

5,000,000,000+ 4,793,874,997+

next

9,000,000,000

next


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+

next

underestimate

actual

number

upper bound

overestimate

3 billon

4 billon

4 billon

5 billon

<

<

<

<

< 9 billon7 billon < actual sum

nextnext

next


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

next


Doing the actual computation we see that…

nextnext

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 < 8,689,442,890

The answer is reasonable in the sense that it is within the range of our estimate.

next


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

next


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

next


One moral of this story is that we do not have to know the

correct answer to conclude that some answers are incorrect!

next

next


The Rote Method for Rounding Offnext

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…

next


The Rote Method for Rounding Offnext

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*

next


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.

next

*

next


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.

next


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.

next


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.

next

5 7,

And we add 1 to 56 and get 57.

5 6, 9 8 2

next

next


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).

next


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

next


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

next


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

next


In other words, when we are given a “rule of thumb” to follow, it should be tempered by our own number sense.

next


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 52nd multiple and of a

hundred (5200) but less than the 53rd 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

next


5 2 0 0 (52nd 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 (53rd multiple of 100)

next

In the next presentation, we will talk about

unadding (subtraction).


rounding 1,875 2,000

rounding off whole numbers © math as a second language all rights reserved next #5 taking the fear...

Documents

math slide

numbers math

rights reserved notice

adjectivenoun theme

language of place value

terms of numbers

large numbers

addition algorithm