Random Math Question
 

So a surprising debate has erupted on my Facebook page about the following math question someone's 3rd grader had on a test. I say surprising, because I think the answer is 100% obvious.

The school library bought a new bookshelf for $285 and a new table for $234. To the nearest ten, about how much more was the bookshelf than the table?

My answer =

The question is poorly worded. We are taught to teach estimate by rounding then doing the operation, which would be 290-230, but it doesn't say by rounding to the nearest ten, just to the nearest ten. As a teacher, I would have to count both answers.

10th grade dropout question, why would you round first?

I always hated those rounding questions*. Just do the effing math.


* Yes, when I was a kid. Sorry (not sorry), I was good at math.

It's a fair question. Most text and rules say that you round first and then add (or subtract). It's not an exact science by any means as the above example shows you.
I will say that our curriculum wants us to teach so many useless things when it comes to estimation. We've got to teach overestimation and underestimation and the real fun one, compatible number estimation in division. By the time you do all the rules, it's simpler just to do the damn operation. :D

edit: grammar

Why would you round first and then add/subtract? That just makes it even more estimated estimate.

When I do the math in these situations, I'm seeing (280+x)-(230+y), do 280-230 first and finally figure out the difference between +x and +y.

That said, given that 290-230 logic, I can see the argument for both $50 and $60 being correct.

Yup, given the wording, both should be considered correct. It is an estimate after all. The intent is probably just to drop the last digit and do the subtraction (the whole point here is to teach the kids shortcuts so that when they do a precise problem, they have some tools to quickly verify that it's a reasonable answer or that they should go back and redo it), but the wording is confusing, because "to the nearest 10" makes it sound like you should do the real subtraction, and then round it, which is useless.

I don't think it's useless. If I need to give you roughly the difference between those two objects in cash I want to give you $50 not $60. The fact that $60 could be considered 'close enough' in this sort of scenario to justify doing ridiculous pre-operation rounding makes my head hurt.

This problem seems designed to lead to this debate. Which might be what it is trying to do.

I'm the only dumbass who picked 60. I had to see that everyone got 50, and realized 85-34 = 51 then round... ah ha.

The student in question answered 50 and got it wrong. The teacher said it was indeed 290-230=60. If that's what they're looking for, then the question is poorly worded. "To the nearest ten" isn't brought up until the 2nd sentence, which is joined with the request for the difference.

would be pretty pissed if my kid said 50 and got that wrong. If you are estimating how much closer it is to a round nember it is much closer to 50 than 60, duh!

Hard to be angry if they were taught in class to round first then compare.

Edit: for the record, I am of the opinion that $60 is incorrect.

Fair point, why let common sense and real world application get in the way.

Depends on the application. In my line of work, depending on the parameter, 100 would be a totally reasonable answer.

We arent talking about your line of work.

In the real world, if someone told me they bough a bookself and chair for those amounts, and they cost about $60 apart, I would look at them like they are a fucking idiot.

+1

The test question appears to be designed specifically to point out that the lesson was to round the separate numbers first and to punish those that did what I did above.

I suppose a better test question would have been: estimate to the nearest 10 what the total cost was. Then, I'd side with $290+$230=$520 being correct and $280+$230=$510 being incorrect. And quickly realize that in the real world we'd be satisfied if the student would be able to estimate it at "$500 and a bit".

Seemed to me you were making a pretty general statement so I was saying that in some situations it may be acceptable. In the context of your second point I totally agree.

Your first mistake is to bring logic into the educational system. :)

I've seen these types of questions before as if they are trying to catch the student into doing the order of estimation backward. I've always seen it as round first, then operation. I've never seen a text book do it the other way, yet I can't think of an example off hand where the taught way would be more accurate than the other way.

And then there's front end estimation, which in this case would give us an estimate of $0.

I love teaching math, but so much of estimation is complete crap.

i'm glad someone brought up front end estimation. they taught that to my 5th grader last year. i have a degree in math, and when she brought that home i wrote a note to the teacher asking her why they were even teaching this. she agreed that it was useless, but its what the curriculum had. thanks to the common core, i now have a hard time helping my kids with their math homework even though that is my line of study and i work as an application developer.

i did the same thing you did...rounded first and then subtracted.

Point being that the question doesn't ask you to estimate at all. It asks what the closest 10 is to the difference of the 2. It basically asks for the answer and then asks what 10 is closest to that number.

If it wanted the answer to be 60 it should have said to estimate the answer to the closest 10. I think that the teacher is wrong. I'd have gone head to head over that one.

If you can quickly and exactly make the calculation using the actual numbers, why would you need to round or estimate afterwards at all, since you have the EXACT number at that point. Isn't the point of estimating and rounding be to be able to more quickly make that calculation using round numbers?

Edit: If I've added up my grocery purchases as I go and know I've spent $131.43, then I do not need to round or estimate, because I am just that awesome at math. Realistically though, I'm going through the store going, "OK, that's $5 for that, $10 for that, $20 for that" - because it's easier to get a quick estimate of how much I'm spending that way.

So I think I get why estimating before the calculation is more practical. The question is still stupid as worded though.

Round to the nearest ten gives 310-210=100, which is then rounded to the nearest ten making the answer 110.

I feel like that is just as correct an answer as 60.

As someone who has estimate in his job description I think the teacher is dumb.

The idea of rounding first to make calculations easier makes complete sense. My problem is that's not what the question asks. Based on the specific wording of the question, there is no other correct answer but $50.

But the question is exactly why I hate standardized tests. As someone mentioned earlier, this question is clearly designed to trap a student who would do the calculation first and then round. What's stupid is by setting up the question this way, they picked a calculation that's extremely easy to do without rounding, so it doesn't even really teach the concept well.

However, I don't have a problem with Common Core, which I think does a great job of teaching practical uses of numbers. The grouping concept works extremely well for the 2nd and 3rd graders I used to tutor. It's amazing to me how many of my high school students will say the difference between (8,2) and (-4,2) is 4 and not understand why that's impossible. I teach them to get to 0 and then go the rest of the way. I think Common Core concepts will make that idea easier to understand.

the point of this is to teach you how to estimate. obviously we can all do this math in our heads, but if these were large numbers you would round first and then do the math.

$60 is the right answer. all the other answer proves is that you can round $51

The way I do it is none of the above. To estimate, I make the last digit(s) the same, thus easier to subtract. In this case, I change 234 to 235 which is 50 from 285. I do it frequently for difference between two years: like one of my ancestors was born 1618 and died 1694. I turn 1694 to 1698 and estimate that he lived about 80 years. For dates crossing the century, I find the difference from 100 and then add it to the other.

This is actually a good point. More complicated numbers on the front end of this problem would be a decent fix to a poorly worded question.

I should clarify: hard to get angry at the teacher for marking the answer wrong if the student failed to apply the process as taught. Perfectly reasonable to be angry at the teacher for having a poor lesson.

Exactly. A point I should have made earlier. When I introduced compatible numbers the other day I told them estimation is not rounding. Estimation is how we make problems simpler to do. Most of the time it involves rounding, but doesn't have to.
And yes, the teacher in the OP wanted them to estimate by rounding first, but it isn't worded that way. I looked through our math today and every time it either says "estimate by rounding to the greatest place" or "estimate by rounding to the ______ (whatever place value)"

I came up with 50.

But, when I do the grocery shopping, I round each item.

So I obviously have no idea how this problem is suppose to be correctly done. :lol:

I'm an engineer at heart... $51 is the right answer. Why the heck would you need to round to the nearest 10? It serves no purpose other than a hypothetical.

As an illustration of both the ability to round and the ability to identify the tens place of a given number?

Common core confounds me... why not just do the actual math like the good ole' days ;)

To what end? What idiot student needs help rounding by applying a theory to the equation?

(I jest... I am probably not even using theory or equation correctly in the above sentence. Just hand the darn kid a calculator already!!)

:)

Yeah, because teachers are in charge of picking out what and how they have to teach these days. If you wanna fix it, call your representative, not say a teacher is dumb for following what they're legally required to do.

+1 on round to the nearest 10.

Ten isn't really a unit anyone would ever round to, especially since numbers go infinitely far in both directions. Hell, you might be able to argue that any answer between 41.00000...1 (i.e. barely above 41) and 60.9 (with 9 repeating) should be counted as correct. Who says a unit of ten has to go 0, 10, 20, 30, etc?

So yeah, just a poorly worded question. 50 is probably the "best" answer though based on the wording.

Edit: I'm not thinking logically tonight. If you're rounding to the nearest unit of 10, the number you round to would need within 5 of the actual answer (assuming you round at the end).

While I agree with your premise, it is very likely that the teacher did, in fact, write the test question and did so lazily and poorly, imho.

$50. 285-234 = 51, and 50 is the nearest ten for determining approximate price differential.

This rounding and then doing the operation can make sense with larger numbers as someone else pointed out, but in this example, it makes no sense to do so.

Come to think of it, the only times in my life I can recall rounding and then operating is in figuring out lifespan, like Bucc mentions, and for the purposes of figuring out how much cap space I'll have in free agency for a DDS league.

...unless you are a 3rd grader. and I would argue it makes no sense to round a number after performing an exact calculation. Why would you do that?

again, the question isn't "how do you round a number" it's "how do you estimate an answer". you estimate an answer by rounding the numbers in question and then doing the operations.

Let's say I go out to eat with a buddy.

Appetizer: $8.35
My entree: $12.95
My drinks: $7.99, $4.95
His entree: $14.95
His drinks: $9.99, $4.95

Total: $70.06 (including tax)

I give him $30 ($4 + $13 + $8 + $5) for the food and $6 for the tip/tax.

There are plenty of reasons/places to round before you add things together. Just because the initial question was simple doesn't mean that all applications of such are going to be just as simple.

Same here. I see 280-230 --> 50 which is a quick estimate and then I check the unit figures and see they are close so I don't even compute them, so 50 is the natural correct estimate to me.

I thought a bit more about this and realized that I ended up at $50 because I'm usually rounding numbers to the nearest 5 when I see them and remember that the numbers were rounded. For estimation purposes, 3, 4, 5, 6 and 7 all become '5' (and 8, 9, 0, 1 and 2 are '0'). The second step is the correction for the actual digits 5 minus 4. In essence, I'm doing what the teacher in question wants to happen, expect that I round to 5's rather than 10's and rounding difference tricked me into the 'wrong' answer.

What brought me back here, is that yesterday at work I was looking at monthly production numbers for January through August (they were all in a range between 4.0 and 14.5), presented in a format that didn't allow copy-paste into excel. I decided to add up in my head, rushing things and ended up making an error, writing down 77 while the actual sum was 67 (oh irony, I was off by 10). For the task at hand, the difference between 67 and 77 wasn't a big deal. In retrospect, I think I should have estimated first - rounding to the nearest 5, like I usually do - and would have ended up with 65 to realize that 77 was too much.

What third grader is buying $519 in furniture??


:devil:

/looks up France vs USA in rankings list
/looks up Netherlands vs USA in rankings list

/decides to trust Euros on this one

:D