Truncation vs Rounding

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mathdad

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Say you are a teacher in grade school. How would you explain the difference between truncation and rounding to your fifth grade class?

I will give you two samples to play with.

A. Truncate 20.98752 to two decimal places.

B. Round 20.98752 to two decimal places.
 
The difference between the two terms is a bit subtle. In my opinion, explaining the difference in words could easily get confusing, so I would use a visual demonstration, accompanied by a brief oral explanation of each step as I do it.

When truncating, you simply remove all digits after the desired level of accuracy and leave the remaining digits alone. This could be demonstrated by writing the number on the board and erasing every digit after the hundredths place.

When rounding, you need to pay attention to the digit in the decimal place directly following the desired level of accuracy. If this digit is 5 or more, increase the preceding digit by 1, increasing the digit preceding that if increment the digit "turned over" to a 0; else leave it alone. As before, I would demonstrate by erasing all the digits after the thousandths place. Then I would circle the new last digit (i.e. in the thousandths place), explaining that because 7 is 5 or more, we would round up. Then I would erase the 8 in the hundredths place and replace it with a 9, finishing up by erasing the 7 in the thousandths place.

In a more complicated example, say rounding 20.99752 to 2 decimal places, I would do exactly the same as before, except after I replaced the 9 in the hundredths place with a 0, I would then circle this number too, explaining that because we "rolled over" to a 0, we now need to increment the previous digit as well. I would then do that, circling the new 0, again reminding of the roll over rule, to indicate that we now need to increment the last digit before the decimal place. I would finish up by erasing the 7 in the thousandths place, explaining that we leave the .00 at the end to indicate that we did indeed round to two decimal places, even though they're both now 0.
 
Say you are a teacher in grade school. How would you explain the difference between truncation and rounding to your fifth grade class?
Why would I want to explain it? Truncation is a less exact simplification.
 
Say you are a teacher in grade school. How would you explain the difference between truncation and rounding to your fifth grade class?

I will give you two samples to play with.

A. Truncate 20.98752 to two decimal places.

B. Round 20.98752 to two decimal places.
I would start by pointing out that "truncation" means literally "cutting off" extra digits, and is the same as "rounding down" to the next lower number with the given precision. So 20.98752 becomes 20.98; I'd show this on a number line: 20.98752 is between 20.98 and 20.99, so when we truncate we are just moving to the left. I might also discuss situations where this is the appropriate way to round.

Then I'd say that "rounding" as we commonly think of it is really "rounding to the nearest number" with the given precision. Initially, I'd look at the number on the number line, pointing out that it is closer to 20.99 than to 20.98, so the former is the "nearest". Then I'd explore with them the fact that 20.985 is exactly between the two numbers, so anything to the left rounds down to 20.98, while anything to the right rounds up to 20.99.

Eventually we'd get to the question, what should 20.985 itself round to? But that's a separate discussion (and very interesting).
 
The difference between the two terms is a bit subtle. In my opinion, explaining the difference in words could easily get confusing, so I would use a visual demonstration, accompanied by a brief oral explanation of each step as I do it.

When truncating, you simply remove all digits after the desired level of accuracy and leave the remaining digits alone. This could be demonstrated by writing the number on the board and erasing every digit after the hundredths place.

When rounding, you need to pay attention to the digit in the decimal place directly following the desired level of accuracy. If this digit is 5 or more, increase the preceding digit by 1, increasing the digit preceding that if increment the digit "turned over" to a 0; else leave it alone. As before, I would demonstrate by erasing all the digits after the thousandths place. Then I would circle the new last digit (i.e. in the thousandths place), explaining that because 7 is 5 or more, we would round up. Then I would erase the 8 in the hundredths place and replace it with a 9, finishing up by erasing the 7 in the thousandths place.

In a more complicated example, say rounding 20.99752 to 2 decimal places, I would do exactly the same as before, except after I replaced the 9 in the hundredths place with a 0, I would then circle this number too, explaining that because we "rolled over" to a 0, we now need to increment the previous digit as well. I would then do that, circling the new 0, again reminding of the roll over rule, to indicate that we now need to increment the last digit before the decimal place. I would finish up by erasing the 7 in the thousandths place, explaining that we leave the .00 at the end to indicate that we did indeed round to two decimal places, even though they're both now 0.

Good notes here. Thanks.
 
I would start by pointing out that "truncation" means literally "cutting off" extra digits, and is the same as "rounding down" to the next lower number with the given precision. So 20.98752 becomes 20.98; I'd show this on a number line: 20.98752 is between 20.98 and 20.99, so when we truncate we are just moving to the left. I might also discuss situations where this is the appropriate way to round.

Then I'd say that "rounding" as we commonly think of it is really "rounding to the nearest number" with the given precision. Initially, I'd look at the number on the number line, pointing out that it is closer to 20.99 than to 20.98, so the former is the "nearest". Then I'd explore with them the fact that 20.985 is exactly between the two numbers, so anything to the left rounds down to 20.98, while anything to the right rounds up to 20.99.

Eventually we'd get to the question, what should 20.985 itself round to? But that's a separate discussion (and very interesting).

Interesting and to the point.
 
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