# "Long-division puzzles" can help middle-grade-level students become actual problem solvers, but what should solution look like?

This is my first post. I hope it's acceptable.

EDIT Since there are people to whom such notation is foreign, I will point out that the problem represents KRRAEE / KMS, where PEI is the quotient and KH is the remainder. KHHR represents P times KMS and ALH is KRRA minus KHHR; the E is "brought down" per division algorithm. The next two hunks are defined similarly.)

Because I want math students to become problem solvers in a manner than does not rely heavily on the "template-based" problem solutions that permeate math education at the middle-grade level, I would be interested in seeing written, easily-followed, mathematically- and logically-valid derivations of the solution to such division puzzles as the one below, such presentation being one that a reasonably competent "Algebra I" student could both follow and produce. (Maybe I just have a bad case of OCD, but I daily find joy in solving these puzzles, taking care to produce a presentation that just such a student could both follow and produce. Part of the joy stems from the belief that a summer project devoted to such problems could greatly enhance students' problem-solving abilities.)

Spoiler alert: Answer follows. But since the requirement is that the solution be an actual derivation, go ahead and look. It's hard enough without it.

Having given this a lot of thought, I think you might find the tableau below useful in organizing thoughts. It amounts to each non-vanishing "subtraction column" with blanks on either side of possible loaning and borrowing.

Before I post a solution I've arrived at, here is an outline that guides solution, whether writing it or following it.

The 3-deep-dashes symbol can be read "is congruent to (modulo 10)" [i.e, it says that the units digit of the multiplication on its left-hand side is found on its right-hand side.]

I find it hard to believe that, with long division's future uncertain, this problem type will be easily swallowed; but the inherent mathematics and logic may turn the tide of opinion.

This is a particularly-easy problem of its type, chosen mainly because modular arithmetic, accessible to the target student audience, makes solution easier.

It’s not hard to write out a solution in more readable form. Here’s a sketch, giving the key steps.

The little $HHL$ column shows that $L$ is either $0$ or $9$. Suppose that $L=9$. The $RHL$ column then shows that $R=H-1$. However, this means that $KRRA<KHHR$, which is impossible, so $L=0$, and $R=H+1$. From the $RHA$ column we see that $A=1$, and it follows from the $ARH$ column and the fact that $R=H+1$ that $H=5$ and $R=6$. We now have this:

           PEI
KMS)K661EE
K556
----
105E
M5K
----
K01E
1PMM
----
K5


It’s clear from the last subtraction that $K=2$:

           PEI
2MS)2661EE
2556
----
105E
M52
----
201E
1PMM
----
25


The middle subtraction shows that $E=1+2=3$:

           P3I
2MS)266133
2556
----
1053
M52
----
2013
1PMM
----
25


The last subtraction now shows that $M=8$ and hence that $P=9$, and $9\cdot 28S=2556$ shows that $S=4$. Finally, we must have $284\cdot I=1988$, so $I=7$:

           937
284)266133
2556
----
1053
852
----
2013
1988
----
25

• Yeah, I guess I'm backing off saying it would be ugly. But would it be clear to and/or appeal to a 13-year-old? I'm just asking, because I've never taught at that level. It could be a goal to make it become clear and appealing. I'll add another puzzle that isn't as easy as this one. Jul 19 '15 at 21:52
• And I agree that the solution I presented isn't particularly easy to follow, colors and outline notwithstanding. It makes me question the whole solution presentation method I've settled on. I like @Brian's method of inserting known values, but it does lengthen the amount written, which I'd hoped to minimize since so many kids don't want to write ANYTHING to show their thought process. And that the solution above is just a "sketch", even more writing (I assume) would be required. Jul 19 '15 at 22:25
• I think my main objection to @Brian's presentation is that it doesn't show the base equations from which solution follows and that the notation KRRA<KHHR isn't clear in its meaning to just anyone--in fact that construct crossed my mind early on and I rejected it in favor of more math-looking stuff that eventually evolved into the imperfect tableau with the boxes. I'm not nit-picking. Oh, maybe I am, but I can pick a zillion nits out of my own solution. Which is why I'm here. Jul 19 '15 at 23:19

Here's the solution I arrived at one day recently. It's in a format that I found, after much experimentation, to be one that MAYBE the target audience could both follow and use as a guide to writing their own solutions.

If the requirement were to write a truly-mathematical-looking derivation (ordered, with many words, from first equation down the page to last), NOBODY (including ME!) would be interested.

I truly believe that students at ANY level would become better problem solvers if given these (drum roll) PUZZLES--math FUN; math at its best? At least for young'uns. (And me!)

The use of color and the a), b), ... guide could make this not only easier but even MORE fun--AND ACCESSIBLE.

Here's another problem, significantly harder than the previous, but still accessible to target audience. The statement of the problem (i.e., northwest corner only) comes from a PennyDell puzzle magazine:

• I don't like posting the second problem as "an Answer" but I don't know how else to keep this thread flowing. It seems editing the original post to contain this new puzzle would be as bad in a different way. Jul 19 '15 at 22:23
• M.SE isn't a forum, so this isn't a thread that needs to be kept flowing. You should post this puzzle as a separate question (with links between it and the current one; use the "Share" item to get the appropriate URLs). Moreover, you should probably edit the question title(s) to reflect the fact that you're looking for clear and accessible presentations of solutions to such puzzles. ("Can [such puzzles] help [...]?" seems somewhat beside the point.) You might consider posting instead to Puzzling.SE, since users there may have useful strategies to share.
– Blue
Jul 19 '15 at 22:57
• As @Blue suggested, I've moved the 2nd puzzle here. I also edited this question's title. But I don't think I'll take the questions to Puzzling.SE since there's more math and pedagogy inherent in these problems than typical puzzles. (I'm still open to going there, though.)(I also don't want to delete this "Answer" since doing so appears to also delete these comments.)(As it should?) Jul 19 '15 at 23:12