Puzzles: How to modify problems – an example
Prove that the midpoint quadrilateral (a quadrilateral obtained by connecting consecutive midpoints of sides) of an isosceles trapezoid is a rhombus.
and invited readers to offer more than one proof.
I found more than one proof, and then asked: how can I modify this question to be appropriate for younger students? for older students? for weaker students? for stronger students?
Generate good, lousy, in between … toss out the bad, … sometimes stumble on something nice. … Ask yourself, “Let’s change the problem” and then ask “Is this new problem any good for … my students? … For anyone I know? ” And then change it again.
(Lots of variations below the fold. Harder problem for puzzle hounds at the bottom) —->
This was a little puzzle for me, not a big “Problem!” Technically, I might it fell on the line for me between problem and exercise; while I had never seen the question, I knew that it would collapse in the face of coordinates. I also know a euclidean (but high school style) proof that the midpoint quadrilateral of an arbitrary quadrilateral is a parallelogram (including, btw, the degenerate quadrilateral with 3 colinear vertices!), so I knew I could extend that for the second proof.
I distinguish between “problem” and “exercise” in the following way: If the solver has a ready algorithm that will work, if the solver has an approach that will get to the ‘answer,’ — this is an exercise. If the solver does not know how to start, or is unsure if an approach will lead to a solution, that would be a problem. It’s a fluid boundary. With experience, exercises may become problems. A problem for some students will be an exercise for others.
Dave Marain wants to know why I call things puzzles and not problems. I never thought about that. I think that problems (or even challenging exercises) that I work on, I call puzzles, because they are recreational, I enjoy them.
New Problems – just a little easier
- Include a diagram – always makes a difference. This applies to every other variant problem that follows.
- Add information that pushes the student in the ‘right’ direction: include (one? both?) diagonal(s) to prompt the euclidean proof, or instructions to use coordinates for the coordinate proof.
- Name some steps involved: “Using our theorem about the segment connecting midpoints of a triangle…” or “Place an isosceles trapezoid on a coordinate grid, and, using the midpoint formula and the distance formula, prove…” There are other options that accomplish the same steering, but leave plenty of work for the student.
Related problems – easier
- Supply coordinates (2a,0), (2b,2c) etc.
- Supply coordinates with numbers instead of variables: “Show that the midpoint quadrilateral of the isosceles trapezoid with coordinates (8,0),(6,4),(-6,4),(-8,0) is a rhombus”
- For easier theorem-based work, supply lengths: “Show that the midpoint quadrilateral of the isosceles trapezoid with sides of length 8,10,20,and 10 is a rhombus” A diagram with at least one diagonal drawn would help.
Related problems – even easier
- Give the lengths of the sides (as above) and ask for the area of the trapezoid and the area of the midpoint quadrilateral – this can be done without discovering that a rhombus is involved, but it might be noticed along the way. The sides I offered give easy to work with 6/8/10 triangles to make finding altitudes easier.
- Same as above, but with coordinates. As one standard coordinate-based way of finding irregular looking areas is to box them and subtract unwanted triangles, it is quite likely that this question would force the discovery that the midpoint quadrilateral is a rhombus. (I like this one! I could follow up by redoing it with variable coordinates – corresponds roughly to the extra proof I offered on e’s page).
- Find the perimeter of …. – Always easier for kiddies to cope with finding a number than with “showing” or “proving”
- Give them all the coordinates – trapezoid and rhombus, and ask them to verify that each figure is as claimed. Can they generate another example? – Again, numbers are easier. Proof is hard.
More difficult variations
- The question includes a conclusion. We could try something more open: Describe the midpoint quadrilateral (a quadrialteral obtained by connecting consecutive midpoints of sides) of an isosceles trapezoid. Prove that your description is correct
- We could ask for a ratio. Even strong students tremble when they see that word: Given isosceles trapezoid with M and N the midpoints of the bases and X the midpoint of another side, find the ratio MX:XN.
Even more difficult – letting the question range
- Without mentioning rhombus, express the area of the midpoint quadrilateral of an isosceles trapezoid in terms of its bases, B and b, and its other side, a (or, and its height, h. Less challenging, but still hard.) Hmm. Finding the area of any trapezoid in terms of its sides, without the height, might be a challenge.
- Same thing, but look for the perimeter. (In general, perimeters are easier to work with than area, but in this case it looks to be the opposite)
- Given isosceles trapezoid with midpoint of one base M, midpoints of the two congruent sides J and K, and intersection point of the diagonals, L, prove JLMK is a kite. – unfamiliar figure adds to the difficulty. Denying a diagram does as well.
Finally, for the puzzle-seekers who try their skills on harder stuff here
- The midpoints of a quadrilateral form a rhombus. Describe the quadrilateral. (what must it be, what can it not be, what might it be).
Now, I do want to point out, many of these proposed variations are mediocre. But when I work on a problem or puzzle, part of (per Polya) “Looking Back” is to generate lots and lots of new problems. Generate good, lousy, in between, don’t worry about quality at first, toss out the bad, try out the ones that look interesting, and sometimes stumble on something nice. It is a habit. Ask yourself, “Let’s change the problem” and then ask “Is this new problem any good for me? For my students? For other students? For anyone I know? ” And then change it again.