Patrick Honner’s challenging parallelogram problem

[post writing note – not much editing on this one. Long day today – I’m going to take a bubble bath instead of editing . . . .]

After taking a day away from our Geometry book to work through a really nice set of exercises that Kate Nowak posted on twitter:

Going through a Kate Nowak Exercise

we started a new section in our Geometry book today – parallelograms.

Well, sort of. Some problems from an old AMC 10 distracted us (in a good way) for about 45 min, so we really only spent about 10 minutes talking about parallelograms. For homework I asked my son to work through the example problems in our book, some of which were pretty challenging. For example, one problem asked you to prove that if you had a quadrilateral ABCD with AB = CD and AD = BC, then ABCD must be a parallelogram.

I posted our daily video on twitter today and mentioned that we talked about parallelograms. Patrick Honner responded by mentioning a really nice challenge problem:

The problem itself is this: If a quadrilateral has a pair of opposite, congruent sides and a pair of opposite, congruent angles, is it a parallelogram?

I’d rate this problem as being more challenging than any of the example problems that my son worked on today, but I asked him to take a look at it during the day anyway so that we could talk it through tonight.

Here’s the first part of the talk – about 4 min long. He has a little bit of trouble getting going, but we do get to talk about some interesting properties:

(1) He knows that the statement would be true for a parallelogram,
(2) He gets a little confused about “the same” meaning “parallel”
(3) Next he draws the picture in a configuration where the two remaining sides are perpendicular to one of the other sides. He is able to see that the two sides that are supposed to be equal, couldn’t be equal in this configuration (without being parallel).

Our conversation continued after this, but I broke it into two pieces for ease of watching:


We continued the conversation with a new picture. In this piece

(1) We start with a little confusion about the remaining two angles. He initially believes that they must be the same.
(2) After we clear up that confusion there’s a nice little coincidence – he has very nearly drawn the picture you need to solve the problem.
(3) The lucky picture allows us to take a guess that the shape is not required to be a parallelogram.


We took a break for dinner, and after dinner we returned to the project using our Zometool set. I wasn’t sure how this part was going to go, but it actually went really well. My fear was that the Zometool set would not plug in properly in the configurations that we needed (which is what did happen), so I purposely started with some large side lengths to mask the potential problems created by the pieces not plugging together properly. At first we tried to make the two opposite angles really small. We found that there was a way to make a parallelogram, but it seemed like no other quadrilateral would work with these two small opposite angles.

Oops – reviewing the movie I see that I missed an opportunity to talk about a non-convex quadrilateral. Next time . . . .

The next angle we tried turned out to be 60 degrees just by luck of how the blue struts plug together. Here we were able to find a configuration that seemed to work. Unluckily it did not work in a configuration where all of the blue struts plugged into each other, but it is still clear that the configuration satisfies the conditions of the problem.


Although this problem was a little over my son’s head, I enjoyed talking about it and hearing his thoughts about how to approach it. He’s having a little bit of trouble with the common math mistake of assuming the things you want to be true are already true. Hopefully working through problems like this helps him learn how to avoid that sort of mistake.

I was also really happy to see how much insight our Zometool set provided in this problem. A more typical use of the Zometool set for us has been building elaborate 3D shapes, but here the use was providing us with two side and two angles that we knew were equal. Having those easy-to-make manipulatives allowed us to get to the heart of the geometry really quickly. Fun project!


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