Word problem with decimals

[Mermaid500]

Hi everyone!

I got such great help when I made my first post, I thought I would ask if someone could double check a problem for me. My 10 year old has completed the problem but we are not sure if it is correct.

Omar measures one machine part with length 4.392 centimeters and another part with length
6.82 centimeters. What is the difference in length? Use estimation to check your answer for
reasonableness.

We did 6.82 - 4.392 = Our answer was 2.428.

We tried using estimation to check our answer and we got 3.0. Because 6.8 is rounded up to 7.0 and 4.3 is rounded down to 4.0, this is how we got 3.0. When we subtracted 7.0- 4.0 = 3.0

Even though we do not have to include the 3.0 to our answer, I'm wondering if we did something wrong because 2.428 should be rounded down to 2.0, not up to 3.0.

Thoughts? Comments would be very appreciated. Thanks in advance!

 

[Dr.Peterson]

Hi everyone!

I got such great help when I made my first post, I thought I would ask if someone could double check a problem for me. My 10 year old has completed the problem but we are not sure if it is correct.

Omar measures one machine part with length 4.392 centimeters and another part with length
6.82 centimeters. What is the difference in length? Use estimation to check your answer for
reasonableness.

We did 6.82 - 4.392 = Our answer was 2.428.

We tried using estimation to check our answer and we got 3.0. Because 6.8 is rounded up to 7.0 and 4.3 is rounded down to 4.0, this is how we got 3.0. When we subtracted 7.0- 4.0 = 3.0

Even though we do not have to include the 3.0 to our answer, I'm wondering if we did something wrong because 2.428 should be rounded down to 2.0, not up to 3.0.

Thoughts? Comments would be very appreciated. Thanks in advance!

In a sense, an estimate can never really be wrong, because by definition it is an intentionally wrong answer! (It's intended to make a tradeoff between accuracy and speed, so an accurate but slow "estimate" would hardly deserve the name.)

Sometimes you will be taught to do exactly what you did, which is a first approximation; if so, that could be the answer they expect. But then you should be taught how to improve the accuracy of an estimate. When you subtract, rounding the numbers in opposite directions results in a greater error than intentionally rounding both in the same direction (unless neither number is changed much when rounding). So here I would tend to round both in the same direction, getting an estimate of 7 - 5 = 2.

And you'll notice that the actual answer is between the two estimates.

Further, since you are just using the estimate to check for reasonableness, you should consider your answer reasonable if it is within 1 or 2 of your estimate, based on your knowledge of the likely error due to rounding. This check is intended to catch errors like accidentally subtracting 68.2 - 4.392 = 63.808. (There are many errors it would miss.)

Finally, you could also round more closely, e.g. 6.8 - 4.4 = 2.4, if you care enough to do a little more work.

Oh, one more thing: Since you rounded to the nearest whole number, you shouldn't write your answer as 3.0, which implies the tenth place means something. Just call it 3, as I did. Then it doesn't feel so wrong.

And one more, since someone will probably mention it: Your answer is really 2.428 cm. Units matter.

 

[JeffM]

First, I agree with everything Dr. Peterson said.

Second, language matters. When you do a reasonableness check, do not say it “is” or “equals.” Say it “approximately is” or “approximately equals” or “is not far from.” Anything that reminds you that what you are trying to do is to avoid gross error.

 

[Mermaid500]

In a sense, an estimate can never really be wrong, because by definition it is an intentionally wrong answer! (It's intended to make a tradeoff between accuracy and speed, so an accurate but slow "estimate" would hardly deserve the name.)

Sometimes you will be taught to do exactly what you did, which is a first approximation; if so, that could be the answer they expect. But then you should be taught how to improve the accuracy of an estimate. When you subtract, rounding the numbers in opposite directions results in a greater error than intentionally rounding both in the same direction (unless neither number is changed much when rounding). So here I would tend to round both in the same direction, getting an estimate of 7 - 5 = 2.

And you'll notice that the actual answer is between the two estimates.

Further, since you are just using the estimate to check for reasonableness, you should consider your answer reasonable if it is within 1 or 2 of your estimate, based on your knowledge of the likely error due to rounding. This check is intended to catch errors like accidentally subtracting 68.2 - 4.392 = 63.808. (There are many errors it would miss.)

Finally, you could also round more closely, e.g. 6.8 - 4.4 = 2.4, if you care enough to do a little more work.

Oh, one more thing: Since you rounded to the nearest whole number, you shouldn't write your answer as 3.0, which implies the tenth place means something. Just call it 3, as I did. Then it doesn't feel so wrong.

And one more, since someone will probably mention it: Your answer is really 2.428 cm. Units matter.

Thank you so much! Those last two tips are really great information for me to know.

 

[Mermaid500]

First, I agree with everything Dr. Peterson said.

Second, language matters. When you do a reasonableness check, do not say it “is” or “equals.” Say it “approximately is” or “approximately equals” or “is not far from.” Anything that reminds you that what you are trying to do is to avoid gross error.

Good point, thank you!