Shape & Symmetry

Kajitsu puzzles are for students learning rotational & mirror symmetry. Download instructions and an extensive set of these beautiful puzzles here.

Give practice to your students in calculating area and perimeter.

Give your students practice identifying mirror and rotational symmetry. I’m 95% happy with this puzzle… the mild negative issue is that some students see the 2D shapes as 3D shapes and this confuses their perception of mirror symmetry. If you are prepared for this issue it does generate a good mini discussion.

Circus Tent Puzzle The best tents are the ones that have the smallest perimeter. Both of these tents have a perimeter of 7. That's the best you can do for $17. Each triangle is worth $3. Each square is worth $4.

Given a tent price find the smallest perimeter. Let's start by finding the smallest perimeter possible for a $6 tent. Try it.

There is only one way to stitch together $3 triangles and $4 squares to make a $6 tent. This is it and the perimeter is 4. What about a $7 tent?

Again, there is only one way to stitch it. The perimeter is 5.

These simple initial ones may be skipped through quickly if the class is advanced - otherwise, this is a great place for students to link geometry and arithmetic.

Let's fast forward to a $16 tent...

These are both good designs giving the minimal perimeter of 8.

These are both good designs giving the minimal perimeter of 8.

For $19 the lowest perimeter I could find was 9. I'm choosing a unitless measure for perimeter, but in your classroom you should use yards or meters - whatever is standard in your country.

For $20 the lowest perimeter I could find was 8. By no means should you trust that I have found the lowest perimeter. As the price gets higher and higher my guesses get less and less certain.

If some of your students would find it inspiring to be asked to jump ahead and work on $30 to $40 tents, let them. It is often children in the middle or bottom of a class that will feel most inspired by being ask to go forth and blaze new territory.

This just looks like an ugly solution for the $22 tent. Let's keep looking for something with a smaller perimeter. Let's remove 3 squares and add 4 triangles. That will keep the cost fixed.

That's better. If your students do end up beating my answers (which is inevitable) I'd love you to brag to me about it 😉

It is surprising that all shapes with minimal perimeter are not convex. Some are concave. The way I would describe "concave" to a class is to say that two wasps can play hide and seek in a concave tent like the ones above. They can find positions where they cannot see each other. This is not true of the next few convex tents...

This looks great, but...

This is better.

Let's fast forward to $30. My favourite part of this puzzle was from $30 to $40 where things were just difficult enough to make me feel uneasy...

There are actually lots of equally good solutions for 27...

I'm starting to really enjoy the minimalist beauty of some of these creations.

This probably has the least perimeter. I'm sad because it looks a lot less cool than the fox mask on the previous slide.

This can't be optimal, but I have not found anything with a smaller perimeter.

This escape pod is my favourite shape yet. I don't mind if students start to diverge from the tent theme into a star wars theme. That shows that they understand the malleability of the superficial story whilst preserving the mathematics underneath.

... but before I get excited about a score of 10... Glen Whitney asked (July 3rd, 2021) if the shape created has to be solid. If a hollow shape is allowed he can do better than the space pod on the previous slide. He can get a perimeter of only nine. What do you think? The great thing about playful mathematics is that you get to make the rules!

Your students should be expected to ask big questions for which they are ill-equipped to find the answers. That natural curiosity to wonder beyond their ability is great to behold...

I'm in the same place. I ask questions - most of which are beyond my ability or perseverance to answer: I wonder if triangles will eventually take over nearly all of the lowest-perimeter answers. Of course some squares are needed for those prices which are not a multiple of three.

...but perhaps from here on, multiples of three only contain triangles...

If I find a particularly pleasing shape like the one above, I'd like to know if it is the lowest perimeter solution for other square and triangle prices. For example, if triangles are $5 and squares are $7 - is this shape a minimal perimeter shape for $68. Let's jump ahead...

Now I'm just enjoying jumping ahead to beautiful solutions... forgetting the base ones in-between. Flip this on your students - and for a homework assignment - ask them to find "the most beautiful tent" under $100. This is a great homework assignment where the arithmetic and simple geometry is consumed by a loftier aesthetic.

We have enough experience now that we can guess that this won't work...

...and we would be right. But this doesn't look that great either... but I'm too lazy to figure out right now if this is as low a perimeter as we can achieve.

I'll wager a lot of money that this is the best solution for $72.

But I'd wager only a little that 18 is the smallest perimeter for $132... especially since it can be made with all triangles. I'm off to make another puzzle. Enjoy exploring this one.

Glen Whitney asks - what is the smallest perimeter that can contain a set of polygons costing $n. An n-sided polygon costs $n. This problem gets tougher a lot faster than our puzzle.

“Venn for Polygons” is a highly engaging puzzle that gives students ages 8+ practice with properties of polygons. A special puzzle-sheet must be printed for color blind students. “Mishap at Venn Zoo” is a full presentation of a polyomino variant that includes rotational symmetry. This is good for students as young as 8 or 9. “ToothPick Polygons” has got a tough second question, but the first part is as easy as “Mishap at Venn Zoo.”

Try finding the largest and smallest rhombus, rectangle, trapezoids etc. so their vertices include a special point and the other vertices lie on the lattice points of a grid.