What’s the mistake? How can you help the kid?

When you’re done with this post, go check out Mary Dooms’ blog and go follow her on twitter. Thanks for the submission, Mary!

  • mr bombastic

    Well, it appears that he/she mistook all the zeros in 165000 as “place holders” or whatever it is they are called. So the next row has 4 “placeholders” when it should only have 3.

    Worrisome that the student feels the need to write 08.25 instead of 8.25. Also worrisome that the student doesn’t seem to realize that 8 x 12 is 96 which doesn’t round up somehow to 842. And, to top it off, the student seems not to understand the difference between computing volume and area.

    Interesting that the student carried out so many fairly elaborate steps succesfully while producing complete nonsense for an answer in every way possible.

    Not a fan of the context of this problem if the intent is hand computation.

  • Whoa, two different mistakes here? It looks like (1) they are confusing surface area and volume and (2) they’re not doing a good enough job lining up the place value of things. I’m especially worried about those intermediate steps where the decimal point is in the wrong place — if those were in the right spot, then maybe each partial product would make sense and the place values would have stayed correct. Estimation is a good thing for keeping these working! With the decimal points where they belong, the last step should be “a bit more than 8 times 10, so around 80” and the answer here of 825 would be clearly off.

    Maybe I’d like to see an extra partial product for the 0, too. (Maybe this kid puts TWO extra 0s on the end at each line instead of just one?)

    I do like the fact that those intermediate steps get decimal points at all — I often see the “ignore the decimal points until the end and then stick them back in the right place” algorithm.

  • Joshua, Can you elaborate more on your final statement? I really try to stress place value and estimation in the context of finding where to place the decimal in a decimal multiplication problem, but fight against last year’s 5th grade teachers who teach kids to stack the decimals (adding 0’s if necessary) and then multiply without decimals, then count the total place value and put the decimal back in. I would love any tips, resources, etc that you can throw my way to help me add to this. It’s my first year trying to emphasize the WHY and not the HOW for decimal operations, and I think it’s gone MUCH better with division, but still struggling with multiplication.

    THanks!

  • What I’m saying is that for 8.25 x 12.09, I’d like to see .09 times 8.25, and then 0 times 8.25, and then 2 times 8.25, and then 10 times 8.25. The point being that each of the partial products should actually be a partial product rather than some work that you have to put together by an arcane method to get the actual result.

    I’m pretty sure that due to the font this won’t line up correctly (and I don’t know if there’s an HTML tag like “code” or something that would work to give me a fixed-width font, so I’m going to try to fudge it with this font), but it would look something like this:

    8.25
    x 12.09
    ——-
    .7425
    0
    16.5000
    82.5000

    I might even have the 0 in the second row, the last two 0s in the third row, and the last three 0s in the fourth row, marked some special way.

    Another approach to all this is to use lattice multiplication, which has the definite advantage of keeping track of place value everywhere if you read it correctly.

    Or, you can use an area model: draw a rectangle that’s 8.25 by 12.09 (probably not to scale) and then cut the edges into pieces. The work above corresponds to not cutting the 8.25 edge, and cutting the 12.09 edge into pieces of 10, 2, 0, and .09. But you coud do pieces of 8 and .25, and 12 and .09, or cut everything into single digits (which is more like lattice multiplication), or whatever. Then you fill in all the partial products, like 8*12 and 8*.09 and .25*12 and .25*.09, and then add those up.

    Which in turn reminds me of the multiplication tables I suggested that my friend David Millar designed: http://thegriddle.net/educators

  • I submitted this mistake and I’m thinking this is a reading issue. Most of my advanced math 6th graders had trouble with this problem. It wasn’t intended to be a “human calculator” problem. I wanted to know if they understood the problem and knew how to solve it. Many kids were paralyzed. They “kind of” knew it was an area, not a volume, problem, but they didn’t know where to start. I’m thinking I need to do much more reading strategies in math class. A simple beginning of drawing a sketch with dimensions should have been the norm.

    Do others see reading issues in math that could be masked as mistakes? How do you handle them?

  • mr bombastic

    By reading issue, do you mean the student’s do not bother to really read and think about the problem? Instead they just look at the 3 numbers in the problem and do something with them? If so, tell them they are wrong and to try again.

    I find it hard to believe that advanced 6th graders would not understand the meaning of this problem. I think it is possible they do not have a good understanding of how the idea of area can relate to 3 dimensional shapes.

  • Mr. Bombastic, sorry for the delay in getting back to you. I think part of the problem is that some students don’t think they need to read closely, and others don’t know how to read closely. This same group also didn’t do well on explaining, using estimation, a decimal division problem. The quotient was given, minus the decimal, and they were to place the decimal. The problem was 29.644 divided by 9.27. It was presented as long division and 32 was the quotient. I got elaborate answers about “ball to the wall”, which I did not teach, yet only two students responded appropriately using estimation.

    In my standard classes, there is a greater diversity of learners and the mistakes are more rampant. Here’s a problem that confounds students: Given the perimeter and base of a rectangle, find the area. Are there too many steps? Can they not transfer and apply their understanding to a new situation?