Tag algebra

Comparing x^2 + y^2 and (x + y)^2 with 3d printing

Yesterday we did a project exploring a common algebra mistake -> assuming that (x + y)^2 = x^2 + y^2. That project is here:

Does (x + y)^2 = x^2 + y^2

Today I thought it would be fun to explore the same idea using 3d printing. During the day I made prints of the two surfaces

(i) z = x^2 + y^2, and

(ii) z = (x + y)^2.

Here they are:

For the project I asked the boys to try to figure out what the two graphs looked like over the domain -2 < x < 2, and -2 < y < 2, and they showed them the shapes. Though not really by design, the choice of a square for the domain turned out to lead to an interesting discussion at the end.

Here's how the project went:

(1) What does z = x^2 + y^2 look like?

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(2) What does z = (x + y)^2 look like?

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(3) What about the actual shapes surprises you?

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I really enjoyed the combination of these two projects. Hopefully seeing the shapes of the surfaces becomes one little extra reminder that the two commonly confused expressions x^2 + y^2 and (x + y)^2 are not the same.

Does (x + y)^2 = x^2 + y^2

In a few projects that we’ve done over the last couple of days my younger son has gotten a little confused on some basic algebra. Not something I’m worried about as ideas like does:

(i) (x + y)^2 = x^2 + y^2, or

(ii) \sqrt(x^2 + y^2) = x + y

(in case that latex isn’t displaying properly, the entire expression is supposed to be under the square root.)

are questions that confuse everyone when learning algebra.

Today we did a short project to talk about these equations. We ended up spending most of the time on (i) just because it was a little easier to talk about. First, though, was just a quick look at both equations:

Now we looked at (x + y)^2 = x^2 + y^2 more carefully. You can see my younger son’s confusion at the beginning. To help get past that confusion we looked at what (x + y)^2 actually means.

As we were talking during the last project I noticed a bunch of snap cubes near by (from one of last week’s projects). Rather than move on to the square root example I thought it would be better (and also fun) to view the square example from a geometric perspective.

This was a fun discussion and I especially enjoyed seeing the boys find a few different geometric approaches to the problem.

Connecting arithmetic and geometry

I’ve been kicking around a few ideas about connecting arithmetic and geometry. My first thoughts were revisiting an idea we’ve played with a few times before:

So, today I decided to look at these two sums with the boys:

1 + 2 + 3 + 4 + \ldots + n, and

1^2 + 2^2 + 3^3 + \ldots + n^2.

We got off to a slower than expected start because my younger son didn’t remember the formula for the first sum correctly. I’m trying hard to break through the idea of relying on remembering formulas, so I was actually happy to review where the formula came from, though.

After the introduction we moved on to studying the first series using snap cubes. What is the geometry hiding behind the formula.

This part of the project to a totally unexpected turn, though:

I decided to keep going with the new sum that added up to n^2 to see if we could make another connection. The boys did remember that the sum of odd integers connects to perfect squares, so I challenged them to find the connection between that formula and the new one they just stumbled on.

Finally we moved on to the sum of squares formula. Lots of fun questions from the boys here, including if the idea extended to 4 dimensions!

The shape here is more difficult to build that it initially seems, but they got through it and now hopefully have a better idea of where the formula comes from.

We wrapped up by looking very briefly at pyramids.

I’d like to do more projects like this one and develop a bunch of different ways to share connections between arithmetic and geometry with kids.

Using 3d printing to review properties of lines

As a follow up to our last two projects with 3d printed triangles, I thought it would be fun to try out a similar project. The point of the 2nd project wasn’t the geometry, though, it was to use the process of making the shapes as a way to review some basic ideas about lines.

So, I started by showing my older son the basic idea – we wanted to write down the equations of the three lines that border a 5-12-13 triangle. Since we have a right triangle the equations aren’t too difficult, but are still useful for a simple review:

Next we went upstairs to Mathematica to make our 3d template using the function RegionPlot3d:

After the prints finished we played around with them a bit to see which triangles we could make with the same area:

I think that making little shapes like this might be one of the best educational uses of 3d printing. Kids get an opportunity to apply some basic math knowledge and create some fun shapes!

Exploring some fun 3d transformations

Today’s project with the boys was exploring some simple (to code!) transformations. The question was how would the shapes change under those transformations.

I started with introducing the idea in 2d. It isn’t necessarily the simplest idea, and I had no intention to go into any details. The basic question I wanted them to think about was this – would a straight line stay straight under this transformation?

Next we looked at a tetrahedron (actually two tetrahedrons) under some similar 3d transformations:

Now for the punch line – what do the same transformations do to an octahedron?

Finally, I wasn’t planning on doing this part, but to clarify some of the ideas from the first part of the project we went up to the computer to show them what the transformations did to a line in 2 dimensions:

So, I think this is a fun way for kids to explore some 3d shapes and also begin to understand a little bit about how algebra and geometry are related

Learning math by studying 3d printing

My son spent the last couple of months preparing for the AMC 10. Now that the test is behind him I’m going to spend some time with him studying 3d printing.

Today we looked at some simple code in the F3 program:

sphere.jpg

The details of the code don’t matter that munch – all the code is doing is testing whether or not a point is inside of a sphere by checking whether or not the distance from that point to the center is greater than or less than the radius.

Immediately two ideas come to mind:

(i) how do we compute distance in 3 dimensions?

(ii) is that distance measure unique.

So, after 1 minute of looking at code we went to the whiteboard 🙂

Our previous 3d prints of the sphere and torus in different L^p metrics were still on the table, so I used those as props.

The first topic was distance in two dimensions:

The second topic was distance in three dimensions:

The last topic was how the L^p metrics vary as p varies – it was lucky we had the spheres handy 🙂

Today’s conversation was actually a nice surprise – I think there’s going to be quite a lot of fun math review that comes from studying 3d printing more carefully.

An AMC12 algebra problem that gave my son trouble

The problem below gave my son some trouble this morning:

Algebra.jpg

When he got home from school we talked about it in more detail and it seemed to make more sense for him than it did this morning. The problem is a nice introductory algebra / quadratic problem:

Next I showed him a similar solution, but where “x” represented a different number:

Finally – just for a completely different way of looking at the problem – I wanted to show him a way that we could use the choices to help us find the solution. This is sort of cheating, but he was very confused by the problem this morning and I wanted to show him a way to get a little un-stuck when you are stuck.

Also, we got interrupted by the guy servicing our furnace – so sorry the video jumps in the middle 🙂

Struggling through a challenging AMC 10 problem

This problem gave my son some difficulty yesterday – it is problem #19 from the 2011 AMC 10a

AMC10.jpg

Last night we talked through the problem. The talk took a while, but I was happy to have him slowly see the path to the solution. Here’s his initial look at the problem:

Next we looked at the equation y^2 - x^2 = 141. Solving this equation in integers is a nice lesson in factoring. Unfortunately by working a bit too quickly he goes down a wrong fork for a little bit.

In the last video we found that the original population of the town might have been 484, and it might have changed to 634 and then once more to 784. We had to check if 784 was a perfect square.

Finally, we needed to compute the approximate value (as a percent) of 300 / 484. The final step in this problem is a great exercise in estimating.

So, a really challenging problem, but also a great problem to learn from. We went through it one more time this morning just to make sure that some of the lessons had sunk in.

Why I love watching my kids learn math

Had a great night with the boys tonight. My older son was working on some old AMC 10 problems and we talked through one that stumped him for his movie:

It was #15 from the 2013 AMC 10a:

 

AMC10.jpg

Next I spent some time with my younger son. He’s been studying the basics of lines using Art of Problem Solving’s Introduction to Algebra book plus a little bit of Khan Academy (when I’ve been traveling for work). I asked him what he’d learned so far and loved his response. It was a great reminder of the joy of learning new math ideas for the first time:

What I was hoping for with the boys and math

When I started making math movies with the boys my goal was to show other kids what kids doing math can look like. There are examples everywhere of adults doing math, so kids can see those examples with no problem. There aren’t nearly as many examples of what it looks like when kids work through problems, though.

So, 5 years into it we are all pretty comfortable in front of the camera and my younger son – just by luck – is making exactly the videos that I was dreaming about in the beginning.

Below are the last two ones we’ve made. They show him working through algebra problems. Nothing fancy, nothing speedy, but really nice work through the problems. I love the way he thinks through problems and think that other kids might enjoy these examples showing what a kid doing math can look like.