I'm tutoring a 6th grader in math at the moment and because she never has a ton of homework I like to give her some interesting extra problems to do. It seems she really enjoyed a problem I showed her the other day where we counted up how many ways there were of getting between two points in a city without actually counting them all up.

I think the thing she really liked about it was that we had a picture in front of us (a roadmap for a madeup town with ridiculously few roads) and could use some simple math to conclude things about it.

Are there any other visual problems like this that you guys recommend that only involve arithmetic skills (not required but if it involves fractions or decimals all the better)? Or is there a website or book that has several of these types of problems? Thanks in advance!

  • $\begingroup$ I also posted it at matheducators.SE as I wasn't sure which site was more appropriate. $\endgroup$ – user282321 Oct 20 '15 at 22:44
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    $\begingroup$ Not sure if this fulfills the "involving fractions" part in the desired way, but how about showing her the visual proof of the identity $1=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots$? Or perhaps that of $1+3+5+7+9+\ldots+(2n-1)=n^2$? There are many visual proofs out there that should be understandable by an attention-paying sixth-grader. Of course, these suggestions aren't as concrete and down-to-earth as the problem you mentioned, which might be a tad off-putting at worst... $\endgroup$ – MonadBoy Oct 20 '15 at 23:01
  • $\begingroup$ Thanks for the suggestions @A.Sh. I had actually already considered the first one (great minds and all that). :) $\endgroup$ – user282321 Oct 20 '15 at 23:17
  • $\begingroup$ I see :) Now that I think about the counting problem you mentioned, combinatorics is a very fun subject, concrete, and filled to the brim with possibilities for visualization and picture-drawing (without risking becoming as dull as, say, geometry). It can also make mathematics seem truly powerful (it did so to me, at the very least), as those simple counting principles can give such a great payoff in finding out "worthwhile" answers without much actual work. Perhaps some elementary discrete mathematics is to be recommended? Perhaps coupled with basic probability, for the fractions. $\endgroup$ – MonadBoy Oct 20 '15 at 23:29

I know this is obviously late, but just in case: I would suggest some basic combinatorial math that you could use almost any visual to assist in. This would be doubly visual because it opens up use of the visual tool of Pascal's Triangle, and therefore all the creative uses for it one could find.

One off the top of my head example:

Let's say you want to have a variety of sandwiches for lunch and you want to see how many days you can go without repeating the exact sandwich, with a limited amount of ingredients, but you are not too opinionated on what "defines" a sandwich; in this case, just that there should always be bread.

The ingredients you have are only one kind of bread, mayo, mustard, lettuce, onions, turkey, swiss cheese and cheddar cheese.

There is a formula that you could use to determine that you could have 127 different sandwiches before repeating, or you could use Pascal's Triangle, which could also tell you how many 1 ingredient sandwiches you would have, how many 2 ingredient and so on, by going down the "line" at 7.


This particular example is also "useful" in real life with things such as the never ending pasta bowl and "endless" burger combinations at friendly restaurants etc...

If your student is interested in these kinds of things, you can introduce other basic systems such as fibonacci numbers etc.

I hope you are still working at this kind of thing and this provides some ideas.


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