The area of a triangle is one-half the length of its base times its height; the area of a rectangle is just its length times its width. This week’s post is about an activity that gives students puzzles that can be fairly challenging, but which are built on these two simple rules. Beyond that, after working some of these puzzles, students can create their own puzzles and challenge one another. This post is the next in our series on problem factories.
The puzzles in this post use the area formulas for two simple shapes to compute the area of a fairly wide set of shapes. The obvious reason for doing this is to practice finding areas; solving these puzzles also improves problem solving skills by demonstrating that you can go pretty far with even a couple of simple rules. There is a deeper layer to this. There are far more than the two simple area formulas we use here taught in geometry classes or used in calculus classes. Most of those formulas can be recovered using the techniques needed to solve the puzzles in this post. Working the puzzles in this post sets up your students to find geometry, or parts of calculus, much easier to master. In addition, the team and whole-class activities at the end of the post are a good arena to develop teamwork skills. The problems in this post can be solved in many different ways, and working on them in teams will also help with discovery and insight.
Geometry puzzles on grids.
All of the puzzles have the instructions “Find the shaded area”. The areas are all displayed on grids, which are made up of squares of side length one. All of the examples in this post can be solved by adding, or subtracting, rectangles and triangles, all of which have easy-to-compute areas. Occupy Math also includes a solution with each puzzle — if you are posing these as problems, be sure to trim off the answer. Let’s take the first problem apart by using colored outlines on its elements to make the basic idea clear.
Here is the decomposition of the problem into two triangles and a square. Compare these with the solution.
Some of the puzzles can be solved by just counting blue squares; others cannot. For all the puzzles, it is less work to solve the problem by adding and subtracting areas of squares and triangles. This leaves room for a lesson on the value of thinking things through over a brute force approach. It is also good to discuss the notions of positive (shaded) space as opposed to negative (white) space. Sometimes the best solution comes from subtracting the areas of negative spaces from a large positive space.
A catalogue of puzzles
Scroll past the puzzles for comments on making your own puzzles, having students do them, and other ways to use the puzzles.
Making your own problems
The examples in the post show that the trick is drawing a shape on a grid that can be explained with rectangles, triangles, and sometimes “half squares” which is just another way to deal with a triangle. To make your own puzzle, you get some grid paper, lightly outline a shape, and check that you can see a path to solving it. This is not difficult — but the cleverness required to solve the puzzle varies quite a lot, as the examples show. This is the problem factory aspect of these puzzles.
A nice class activity is to have groups of students compose puzzles and write out solutions. These puzzles are handed in and the groups is then graded on the correctness of their solutions. The instructor compiles the puzzles and the teams work them all. You may or may not want to have a group work the puzzle they contributed; Occupy Math favors this as it serves as a check on submitting excessively ornate puzzles. A way to enhance this activity is to have the class vote on which of the puzzles were best (they should not be allowed to vote for their own puzzle) and perhaps have bonus points for being voted best.
Occupy Math usually gives appropriate grade levels with an activity post. Depending on your school system, these puzzles start being useful in third or fourth grade, possibly later, depending on when the rules for areas of triangles and rectangles are taught. There is a caveat here — some of the puzzles Occupy Math decided not to put in the post are actually more like a grade eight difficulty level. If you make your own puzzles, try not to go overboard.
Other activities with these puzzles
As currently phrased, these puzzles are an exercise in exact computation. They can also be used for exercises that build estimation skills. The easier activity in this direction is to pick two puzzles with a similar area (not the same, that kind of violates the spirit of estimation) and ask the students to determine which is larger, quickly, or without access to a calculator or pencil and paper. A harder activity is to pick several of these puzzles and ask the students to sort them into ascending order of size. Having some close areas and some that are obviously different is good for the sorting activity — estimation solves the easy cases, the hard ones require calculation.
There are some obvious ways to augment this activity — if you are studying circles, you can add these into the mix, for example, though this disrupts the property of having whole number answers. Occupy Math designed the example puzzles to have whole number answers so that the focus would stay on the puzzles and the geometric properties used to solve them, rather than getting into the weeds of arithmetic. That said, the problems you construct for yourself are likely to have fractional areas, which is a perfectly good thing. Occupy Math is publishing a book soon that includes more of these puzzles. Drop him a line at dashlock@uoguelph.ca if you would like a note when the book is published.
I hope to see you here again,
So remember to wear your mask!
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics
Occupy Math 