Education ∪ Math ∪ Technology

Geometric Constructions as puzzles

Geometric constructions are amongst my favourite things to teach in Geometry. Why? I see each geometric construction as a puzzle to be solved and I love watching children solve puzzles and share their solutions to those puzzles.

Puzzle #1: Given a line segment, draw a circle with its radius as the line segment.

Many constructions build on earlier constructions so that as students figure out how to do earlier constructions, they build the pieces they need to figure out more complex constructions. Further, more complex constructions embed all sorts of opportunities for practicing earlier constructions.

Puzzle #2: Draw another line segment with the same length as the given line segment with an endpoint on either A or B.

The invention of dynamic geometry software, like Geogebra, means that students can learn these earlier constructions without their early challenges using a compass and straightedge interfering with their ability to learn the mathematical ideas behind the constructions.

Puzzle #5: Draw a line segment that is exactly three times the length of the given line segment.

It is super helpful for students to have their prior work with constructions visible for themselves as examples to work from, and so once students have figured out how to do a construction with the digital tool, I have them transfer their construction to paper (ideally in their notebooks for reference) so they can access it later.

Puzzle #10: Draw three overlapping circles on the same line such that two of the circles have their centers on the middle circle.

Another advantage of the digital geometry tools is that you can provide partial constructions for students. This way students can work on the part of the construction that is new. This doesn’t give students practice with the earlier part of the construction but it is a subtle way to give hints to students for particularly complex constructions.

Puzzle #11: Use the circle below to help you draw a regular six-sided shape (regular hexagon).

When a student shares their constructions with the class, I usually call up a volunteer that is not that student to come up to the front of the room and perform the construction, following the verbal instructions from the first student. This means that the pace of the construction is likely to better match the pace other students can follow or copy the construction which leads to more students understanding the construction. After the construction is complete, I can ask another student to restate the instructions while I annotate important features of the construction.

Puzzle #17: Construct an octagon (regular 8 sided shape).

The last few constructions, I provide the least amount of support for students since a goal of mine is to see if students can do these constructions independently. However, note that in the instructions for the constructions, I try to make sure that the language of the constructions isn’t a barrier for my students.

Constructions #1 through #17 are available in this Geogebra book and a paper copy of these instructions is available as Lesson 2 of this Core Resource.

Let me know if you have any questions and please share other ideas you have about introducing students to constructions.