# Triangle and construction review

My younger son has been studying Art of Problem Solving’s Introduction to Geometry book this year. He’s been doing most of the work on his own. Every now and then, though, I decided to check in and see how things are going. Today the topic he was looking at when I checked in was constructions. The two questions on the table when I stopped by were:

(i) Construct the incircle of a triangle, and

(ii) Construct the circumcircle of a triangle

Here’s how things went.

(i) The incircle

He has the basic idea for how the construction works, but misses one important idea. That idea is that the tangents of the incircle are not the feet of the angle bisectors. My guess is that this is a fairly common point of confusion for kids learning this topic:

(ii) the circumcircle

Here his solution was completely correct. He had a tiny bit of trouble in the beginning figuring out to construct the perpendicular bisectors, but he worked through that trouble fairly quickly.

It is always interesting to hear the ideas that kids have when they talk through a mathematical process. For me it was especially nice to hear that most of the ideas he’s learning as he works through this geometry book are sinking in pretty well.

# Exploring triangle congruence with Patty Paper Geometry

I asked the kids what they wanted to do for our Family Math project today and they told me they’d like to do another project from Patty Paper Geometry:

We found a neat way to explore triangle congruence in Chapter 8, so we dove into that project this morning. Here’s the introduction and a brief discussion about the difference between “congruent” and “equal”:

Next we prepared for the project with the camera off. The boys drew a random triangle on one piece of patty paper and then copied (and cut out) the side lengths on a different piece. The main part of this project is to explore whether or not you can make a non-congruent triangle out of the three side lengths.

Finally, we wrapped up by looking the same problem using a compass and straight edge. That part went find and the boys seem to believe in side-side-side congruence after these two exercises.

What was a fun surprise here is that my younger son wondered whether or not side-side-side-side would be a congruence theorem for quadrilaterals. My older son came up with a counterexample 🙂

So, another fun project from Patty Paper Geometry! I really love this book 🙂

# More Patty Paper geometry + another viral math problem

Earlier this week Michael Pershan was asked for ideas about how to introduce constructions. One suggestion was using patty paper:

It has been a while since our first Patty Paper Geometry project, so I thought we’d give it another try 🙂

Flipping through the book last night I found a project in section 4.1 about finding the sum of angles in a polygon. I’d actually never seen this approach before and it looked really fun. We started with a triangle (and I’m sorry the pencil on the patty paper didn’t show up super well in the videos):

Next, as suggested by my younger son (and also in the book 🙂 ) we moved on to a quadrilateral:

Now we had a few conjectures and tried the same experiment with a pentagon to see if our conjectures held. It was pretty neat to hear the language (both in this video and in the last one) that the kids used to describe what was happening with the pentagon’s angles.

With that patty paper exercise as preparation, we looked at a viral geometry problem from last week. Here’s one link about the problem:

VCE Further Mathematics ‘50 cent’ question leaves Australian students confused

The problem gave the kids a bit of trouble – here’s their conversation about it:

I wanted the kids to puzzle about the problem a bit more, but they needed to head out the door, so I decided to simplify the picture for them so help them see what was going on:

So, a fun project. Using the patty paper to see why angles in a triangle add up to 180 degrees was really neat. As I said above, I’d never see the idea presented this way before. The extension to polygons with more sides was really natural and also led to some great conjectures from the kids.

I also enjoyed seeing the kids struggle with the 50 cent problem at the end. Not being around groups of kids working on math problems all that much, I usually have a difficult time knowing when (or why) problems will be difficult. Seeing the difficulty first had is always a great lesson.

# Patrick Honner’s construction challenge shows the beauty of math

Last week Patrick Honner posed this clever geometry question on twitter:

There were many wonderful solutions posted on Twitter and on Google+, and I wanted to highlight 3 of them as wonderful examples of mathematical reasoning. These are the kinds of fantastic ideas that really highlight the beauty in math to me.

So, in the order that I saw them:

(1) Kate Nowak’s solution:

This solution uses two (related) ideas from geometry: similar triangles and ratios. There are two reasons that this solution was appealing to me. First, it is a fantastic use of abstraction – the point X that is part of the similar triangles and also part of the original triangle is not on the page, but we can still use properties involving these triangles in the solution.

Second, one of my great memories from learning geometry in Mrs. Whitney’s class (in, gulp, 1984 . . . ) was learning how to use ideas from geometric constructions to divide. Who knows why, but that idea just blew me away. In this solution, Nowak uses the ideas that Mrs. Whitney showed my class to divide a segment into two pieces of just the right proportion. The especially cool thing about Nowak’s construction is that we don’t know what the value of the ratio, only that the two line segments are divided into two pieces that are in exactly the same proportion. What a great, and super instructive, solution.

(2) Alexander Bogomolny’s construction:

I liked this solution because the approach is a clever twist on the ideas that Nowak’s used. Instead of dividing segments into equal, but unknown, proportions, Bogomolny divides two specific line segments in half. To achieve this goal he rotates a line about a point.

This gives rise to a nice question for a geometry student -> how do you do that with a compass and a straight edge?

Next he uses a neat property of trapezoids – the line connecting the midpoints of the two bases also passes through (i) the intersection of the diagonals, and (ii) the intersection of the two legs (when extended).

That give rise to a second great question for geometry students – prove that this statement about trapezoids (with non-parallel legs) is true!

So, I love the way that this proof not only answers the question posed by Patrick Honner, but also can be used to expose geometry students to a few other ideas and challenges.

(3) Patrick Honner’s solution:

Finally, the solution that Honner provided to his own problem is terrific:

The surprising (to me) idea Honner uses in this solution is that the altitudes of a triangle intersect at a single point. At first glance it is not at all clear how you might use this idea to solve the problem, but Honner shows how to find a triangle with X as vertex whose altitudes intersect at the point P. This solution shows a beautiful way in which two seemingly unrelated ideas in geometry are connected.

As I mentioned above, there were many solutions given on Twitter to Honner’s problem, but these three really stood out for me. I love that this problem illustrated how different people can approach a problem in different ways, and all of the different ways can be instructive.

Also, in a week when the internet spent way too much time discussing the differences between 5 + 5 + 5 and 3 + 3 + 3 + 3 + 3, it was nice to get this wonderful reminder from Nowak, Bogomolny, and Honner that math is about beautiful ideas and not just about getting an answer.

# Finding the center of a circle

Fun little inquiry project today – how do you find the center of a circle? We touched on this question in a project a few weeks ago, but it was nice to revisit it in a little more depth today.

We started with an introduction to the question and quick review of what we talked about last time. My younger son remembers that you can find the diameter of the circle by finding the longest chord in the circle with a ruler. My older son thinks about the idea of folding the circle in half to find the diameter. We’ll explore that second idea a little later in the project.

At the end of the last video my older son suggested that we could find the diameter of the circle by drawing a 90 degree angle on the circle. That seemed like an idea worth exploring more carefully.

After we find the diameter with the 90 degree angle the next question is how to find the center?

In the next part of the project I wanted to show a new way to find the center of the circle with a compass and straight edge. I had one way in mind, but my older son thought of a clever way by inscribing any triangle and finding the intersection of the perpendicular bisectors. This idea was fun to explore:

The last part of the project was playing around with the folding idea using our patty paper. This approach is actually a project in our Patty Paper Geometry book:

I had a hard time communicating the geometric idea going on with the patty paper – the idea is that two diameters intersect in the center of the circle. We talked around the idea for a while before getting to it.

So a fairly relaxed project. Lots of different ways to approach the problem – simple measuring, some basic construction ideas, and then a little folding. Hopefully a fun way to for kids to see that math problems can be approached from many different angles.

# Patrick Honner’s rectangle dissection problem

Got a happy surprise last night when Patrick Honner sent me a draft of a geometry project to look through. I’d just finished up a chapter in our Geometry book yesterday and was looking for a project to do with my older son today, and, hey, a project just fell into my lap!

This one is a little longer than normal – I essentially filmed our entire 30 minutes this morning to provide a comprehensive review for Patrick. I think that my son enjoyed this project and learned some interesting ideas in geometry along the way. There’s a nice 3D printing project hiding in here, too, but as we are in the process of moving houses right now, I’m not sure that I have everything I would need for that project readily at hand.

Here’s the introduction to the activity along with step 1:

Step 2 is a basic construction. I’m just making sure that he understands the instructions at this point.

Step 3 is a second basic construction. In this part you have to remember how to measure a length with a compass.

Step 4 is the last construction required to set up the problem. Pretty similar to step 3.

Step 5 is the first problem in the project. We have to prove that two triangles are congruent. There’s a nice little “aha” moment around the 2:05 mark.

Step 6 is the first of two steps where we prove two line segments have the same length:

Step 7 is the second of the two steps where we prove that two line segments have the same length:

Step 8 is the doozy – we are going to try to show that the original rectangle that we’ve chopped up can be rearranged into a new rectangle with different side lengths. This part gave my son quite a bit of trouble. I had to chop up the conversation here into three pieces, and also had to give lots of help to get him to the end of this one. Around 1:50 In the third video in this sequence he sees how to slide the pieces into place:

The last part of this project is showing that this picture isn’t just geometry – there’s a neat algebraic identity hiding here, too. We looked at the geometry behind this algebraic identity in two different ways.

So, I really enjoyed the 30 minutes we spent on this project today. The basic ideas in geometry – some simple constructions and congruences – were things that my son was able to tackle without too much difficulty. The idea of rearranging the pieces to form a new rectangle was quite a challenge, though. But, not so challenging that he lost interest or got frustrated. It was also neat to be able to look at the connection between geometry and algebra at the end. Obviously it was especially nice to hear my son say that the project was fun.

As I said that the beginning of this post, it was a nice bit of luck that this project fell into place today.

# Going through a Kate Nowak exercise

Tonight on twitter Kate Nowak published an exercise she was planning to use in her geometry class tomorrow. I may need to, um, borrow this exercise for my son tomorrow since I may need to run off to work early. I decided to go through it carefully just in case I get any questions at work. That short exercise helped me see the problems much better than just reading the list. Here’s are my thoughts, but first the twitter post:

(a)

For this one I used the ruler to draw the 3 inch side and the compass to draw the range of points 5 inches away from one of the endpoints of that side. Lots of possible triangles and hopefully a good start to the lesson.

(b)

Here I started with the 3 inch side at the bottom of the page. That was lucky because 5 inches and 6 inches are longer on the paper that I expected. If you put the 3 inch side in the middle of the paper, you’ll not have room (on 8 x 11 paper anyway!). I used my compass to draw a 5 inch circle centered at one of the endpoints of the 3 inch side and then used it again to draw a 6 inch radius circle centered at the other endpoint. One of the intersection points for these two circles was on my paper and allowed me to draw the triangle.

I used the compass again only for problems (d) and (i).

(c)

Had to use the protractor for the first time on this one. Lots of possible choices of triangles here.

(d)

I like this one. After drawing the 120 degree angle it took me a bit to figure out where to put the 5 inch side. Then I remembered the compass and picked a random point on one of the sides and drew a 5 inch radius circle. Just reading this one I missed that it would be more challenging than some of the other problems.

(e)

Having measured the 120 degree angles in the last problems in the right hand side of the protractor, I did the same thing here with the 30 degree angle at the start and accidentally began by nearly drawing a 150 degree angle. Oops! I’ll be interested to see if the students notice the similar triangles from the different groups on this one.

(f)

Now that I was a pro at drawing acute angles, this one went pretty smoothly 🙂

(g)

This one also went pretty smoothly.

(h)

I found it easier on this one to draw the 100 degree angle and then the 30 degree angle on the other side of the 6 inch side. I sort of feel like I cheated, but I didn’t see where the intersection point with the 50 degree angle would be right away so I solved a different problem.

(i)

Hey, two possible solutions!! I’d like to hear what the kids have to say about this one, too!

So, fun little exercise. If I do have to run off to work early tomorrow, I’ll definitely give the worksheet to my son and have him attempt to create all of these various triangles.

# Geometric Constructions with Origami

Yesterday I saw a great tweet from Evelyn Lamb:

I’d never see constructions using origami before, so the idea that you could trisect an angle using paper folding was brand new to me. One of Zsuzsanna Dancso’s comments in the Numberphile video made me believe that you could also solve the “doubling a cube” problem using paper folding. Sure enough that construction is here:

How to double a cube with origami from Cutoutfoldup.com

The combination of that construction and the construction in the Numberphile video made for a great Family Math topic for today.

I started by talking through some basic ideas of compass and straight edge constructions from Euclidean geometry. My older son and I have touched on this topic in our geometry work, but my younger son has never seen it. Because this topic was new to my younger son I didn’t want to go that much into detail. The main idea for today was to introduce the three famous impossible constructions: (1) doubling a cube, (2) trisecting an angle, and (3) squaring a circle and then show how to solve (1) and (2) in the last two videos.

Our first origami construction was solving the “doubling a cube” problem by constructing $\sqrt[3]{2}.$ The directions I linked above are really easy to follow, so each of us made our own version of this one:

The last thing we did was attempt the “trisect an angle” origami construction that Zsuzsanna Dancso demonstrates in the Numberphile video. This construction is a little bit more difficult than the doubling a cube one, so we worked on this one together rather than making three separate constructions. I also used a ruler to draw in some of the lines just to speed things up, but it is easy to see that we could make the same lines by folding.

So, not as much mathematical detail in this one, but some fun history and some fun constructions. I wasn’t aware of the idea of origami constructions before seeing the Numberphile video, so this project had a little bit of extra fun for me because the kids and I got to learn something new together!

# Surprises you get watching kids do math

My older son is finishing up a chapter about right triangles in his Geometry book.  The last section in the chapter discusses how to construct perpendicular lines.  This topic has come up a few times previously, so I dedicated a bit more time to the previously section – Heron’s formula – before starting in with the new section this morning.  I also left one of the examples in the book for him to work through as part of his homework.

One of the other homework questions was to construct a segment with length $\sqrt{3}$ given a segment of length 1.  I thought this might be a little bit of a challenge and was pretty interested to see how he would approach it.

When I got home from work this afternoon I got a nice surprise:

Me: Were you able to construct the 30-60-90 triangle?

Him:  That wasn’t one of the homework questions.

Me:  Oh, I thought there was one about constructing a length equal to $\sqrt{3}.$

Him:  I made a 1 – $\sqrt{2}$$\sqrt{3}$ triangle for that one.

Funny how focused your mind gets when you “know” how to do something.  Wouldn’t have thought of this approach in a million years.  Nice little construction!