Tag algebra

Finding Cos(72)

My older son is learning trig out of Art of Problem Solving’s Precalculus book this year. Yesterday he was working on the “sum to product” section, which derives rules for expressions like Cos(x) + Cos(y). It reminded me of one of my all time favorite math contest problems:

Today I thought I would show him my solution to that problem. What we go through probably isn’t the best or easiest solution, but I think it is an instructive solution for someone learning trig.

We started by talking about the problem and how some of the ideas he was currently learning could help solve it:

At the end of the last video we’d found a nice equation that we derived from the original problem:

\cos(36^o) - \cos(72^o) = 2 \cos(36^o) * \cos(72^o)

Now we used the double angle formula to simplify even more and find a cubic equation satisfied by Cos(36):

Now we tried to find the solutions to the cubic equation we found in the last video. This part gave my son a bit of trouble, but he eventually got there.

Now we were almost home! We just had to compute the value of Cos(72) and we’d be able to solve the problem. That involved one last application of the double angle formula:

I think solving this problem from scratch would be far too difficult for just about any kid just learning trig. But, the fun thing about this problem is that the ideas needed to solve the problem are all within reach using elementary trig identities. So, I think that working through the solution to this problem is a nice exercise for kids.

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The cube root of 1

After a week of doing a little bit of practice for the AMC 8, my older son has returned to Art of Problem Solving’s Precalculus book. The chapter he’s on know is about trig identities.

Unrelated to his work in that book, the cube root of 1 came up tonight and he said “that’s just 1, right?” So, we chatted . . .

First we talked about the equation x^3 - 1 = 0:

For the second part of the talk, we discussed the numbers \frac{1}{2} \pm \frac{\sqrt{3}}{2} and their relation to the equation e^{i\theta} = \cos(\theta) + i\sin(\theta)

Finally, I connected the discussion with the double angle (and then the triple angle) formulas that he was learning today. You can use the same idea in this video with \cos(5\theta) to find that \cos(75^{o}) = (\sqrt{5} - 1)/4:

So, a lucky comment from my son led to a fun discussion about some ideas from trig that he happened to be studying today 🙂

Using 3d printing to help explore a few ideas from introductory algebra

Last spring I was playing around with some different 3d printing ideas and found a fun way to explore a common algebra mistake:

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

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

Today I decided to revisit that project. We started by looking at the same idea from algebra:

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

At first we talked about the two equations using ideas from algebra and arithmetic.

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Now I asked the boys for their geometric intuition and then showed them the 3d printed graphs of the two functions.

This part ran a little long while my younger son was stuck on a small but important point about the graph z = (x + y)^2 – I didn’t want to tell him the answer and it took a couple of minutes for him to work through the idea in his mind.

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Next I showed them 3d prints of x^3 + y^3 and (x + y)^3 and asked them to tell me which one was which. It is really neat to hear the reasoning that kids use to go from shapes to equations.

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For the last part of the project I asked the boys to come up with their own algebra “mistakes” for us to explore. My older son chose to compare the graphs of \sqrt{x^2 + y^2} and x + y.

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My younger son chose the two equations x^2 - y^2 and (x - y)^2. Changing the + to a – in our first set of equations turns out to have some pretty interesting geometric consequences – “it looks sort of like a saddle” was a fun comment.

One especially interesting idea here was exploring where x^2 - y^2 = 0. We used Mathematica’s ContourPlot[] function to explore those two lines because those lines weren’t immediately obvious on the saddle.

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I’m happy to have had the opportunity to revisit this old project. I think exploring simple algebraic expressions is a fun and sort of unexpected application of 3d printing.

Talking inverse functions with my older son

My older son was stuck on a question about inverse functions from his Precalculus book:

Let f(x) = \frac{cx}{2x + 3}

Assume that f(f(x)) = x for all x \neq -3/2. Solve for c.

We began by talking about what was giving him difficulty and then moved on to solving the problem.

Next we moved on to looking at the function on Mathematica. It was a little unlucky that the scale was different for the x- and y-axes, but I think the pictures still got the point across.

After we finished talking I posted about the problem on twitter and John Golden made a neat Desmos version of the problem:

Starting AoPS’s Precalculus book this year

Since we stopped home schooling about 2 years ago I’ve mostly been doing projects with my kids rather than covering new content. This year I wanted to get back into content and have decided to do a slow walk through Art of Problem Solving’s Precalculus book with my older son.

Today this problem gave him a little trouble:

Find the domain and range of f(x) = 1 / (1 + \frac{1}{x})

We talked about the problem when he got home from school tonight. Here’s what he thought about the domain:

Next we talked about the range which is a more complicated problem (at least I think so anyway):

So, I’m happy that he was able walk through this explanation after school today. WE spent a while talking about it this morning and I was hoping that the ideas wouldn’t slip out of his mind during the day.

A mistake that led to a great conversation

My older son had a homework problem that asked him to find the area of the region bounded by the two equations:

(i) | 2x + 3y | \leq 6, and

(ii) | x - 2y | \leq 4

Mathematica’s picture of that shape is here:

Equation1

He told me that he used Pick’s theorem and found that the area of the shape was 13 square units.

There’s just one small problem – you can’t use Pick’s theorem to find the area of this shape since the corners of the shape are not lattice points of the grid.

What to do . . . .

I wrote a quick little program that picked 100 million random points in the 10×10 square centered at (0,0) and tested whether or not they were part of the shape. That program found that 13.71% of the points were part of the shape – that was enough to convince him that the area might be larger than 13 square units.

Next I had him re-read Pick’s theorem to see what went wrong. He saw pretty quickly that the shape didn’t meet the condition of having the corners lie on lattice points.

I really wanted to try to find a way to make Pick’s theorem work with this shape.  I had him determine the y-coordinate for the far right corner.  The value was y = -2/7.

After finding that value, we had a good talk about scaling.  To make the new grid larger we had to *divide* the x and y coordinates in the equations by 7.  Here’s Mathematica’s picture of the new shape and grid (note that the x and y values run from -25 to 25 in this picture):

GridPick.jpg

With this shape we are able to use Pick’s theorem to calculate the area.   We counted 40 grid points on the boundary without too much difficulty.   Counting the ones in the middle was a little bit more of a pain, so we wrote a short program to perform that calculation for us.  Note that we have to change the “less than or equal to” from the original equations to “strictly less than” since we want to be inside the shape:

LatticeCounting
So, we have 653 lattice points in the inside and 40 on the boundary.  Pick’s theorem tells us that the area is equal to the number of lattice points on the interior plus half the number of lattice points in the boundary minus 1.  That’s 672 units.    In the picture above, 1 unit is equal to 1/49 of a unit in the original picture, so the original area is 96 / 7 or 13 5/7.   Close to what he found originally, but not equal!

Along the way we also talked about alternate ways to find the area – the easiest being dividing the shape into two triangles with a vertical line through the middle.

I’m really excited about the discussion that we had tonight.  Funny how many important ideas in math can come up from a problem about absolute value and inequalities.

 

 

 

Lessons from a great geometry homework problem

My older son had a terrific homework problem in his enrichment math class. I wanted to walk through the problem again today so that my younger son could see it and also to highlight some of the lessons in the problem.

To start the project we revisted a fun geometry problem that will make a surprise appearance at the end of the homework problem:

Next I introduced the homework problem. My older son is familiar with this problem, but my younger son is seeing it for the first time. In this video my older son highlights the main ideas that we need to solve the problem (well . . . see the next video for the one we forgot!):

Here’s the one extra piece that we missed from the last video:

Next with the triangles labelled properly, we worked to see how we can use the Pythagorean theorem to help solve for the values of the two unknowns. I used this section of today’s project to give my younger son a little algebra practice:

Now comes the task of simplifying the two complicated equations. Hopefully that will help us make some progress towards solving them.

After the simplifying in the last video we are now ready to take a crack at solving for the radius of the smaller circles. Solving the equation involves solving a quadratic and that gave us a chance to talk about factoring.

Finally, we went back to the picture from the homework problem. We hadn’t solved for x in the project, but now we can use the pictures to help us find x’s value. We see an 8-15-17 triangle and also a 3-4-5 triangle. We also see the 5-5-8 triangle from the beginning of the project!

So, a fun project connecting a neat geometry problem that we’ve studied before with a new homework problem.

A project for kids inspired by Nassim Taleb and Alexander Bogomolny

I woke up yesterday morning to see this problem posted on twitter by Alexander Bogomolny:

About a two months ago we did a fun project inspired by a different problem Bogomolny posted:

Working through an Alexander Bogomolny probability problem with kids

It seemed as though this one could be just as fun. I started by introducing the problem and then proposing that we explore a simplified (2d) version. I was excited to hear that the boys had some interesting ideas about the complicated problem:

Next we went down to the living room to explore the easier problem. The 2d version, |x| + |y| \leq 1, is an interesting way to talk about both absolute value and lines with kids:

Next we returned to the computer to view two of Nassim Taleb’s ideas about the problem. I don’t know why the tweets aren’t embedding properly, so here are the screen shots of the two tweets we looked at in this video. They can be accessed via Alexander Bogomolny’s tweet above (which is embedding just fine . . . .)

Taleb1

Taleb2

The first tweet reminded the boys of a different (and super fun) project about hypercubes inspired by a Kelsey Houston-Edwards video that we did over the summer:

One more look at the Hypercube

The connection between these two projects is actually pretty interesting and maybe worth an entire project all by itself.

Next we returned to the living room and made a rhombic dodecahedron out of our zometool set. Having the zometool version helped the boys see the square in the middle of the shape that they were having trouble seeing on the screen. Seeing that square still proved to be tough for my younger son, but he did eventually see it.

After we identified the middle square I had to boys show that there is also a cube hiding inside of the shape and that this cube allows you to see surprisingly easily how to calculate the volume of a rhombic dodecahedron:

Finally, we wrapped up by using some 3d printed rhombic dodecahedrons to show that they tile 3d Euclidean space (sorry that this video is out of focus):

Definitely a fun project. I love showing the boys fun connections between algebra and geometry. It is also always tremendously satisfying to find really difficult problems that can be made accessible to kids. Thanks to Alexander Bogomolny and Nassim Taleb for the inspiration for this project.

Steve Phelp’s 3d pentagon

Sorry that this post is written in a bit of a rush . . . .

I saw a neat tweet from Steve Phelps earlier in the week:

The shape sort of stuck in my mind and last night I finally got around to making two shapes inspired by Phelp’s shape. My shapes are not the same as his – one of my ideas for this project was to see if the boys could see that the shapes were not the same.

So, we started today’s project by looking at the two shapes I printed overnight. As always, it is really fun to hear kids talk about shapes that they’ve never encountered before.

Next we looked at Phelp’s tweet. The idea here was to see if the boys could see the difference between this shape and the shapes that I’d printed:

Finally, we went up to the computer so that the boys could see how I made the shapes. Other than some simple trig that the boys have not seen before, the math used to make these shapes is something that kids can understand. We define a pentagon region by 5 lines and then we vary the size of that region.

I’m not expecting the boys to understand every piece of the discussion here. Rather, my hope is that they are able to see that creating the shapes we played with today is not all that complicated and also really fun!

This was a really fun project – thanks to Steve Phelps for the tweet that inspired our work.

A problem about cones for kids courtesy of Dan Anderson

Saw a fun tweet from Dan Anderson when I got up this morning:

Here’s a direct link to the CNN article:

The artificial glacier growing in the desert

The article is interesting all by itself, and the mathematical question Dan is asking was the subject of our project this morning.

First I asked the boys to read the article – here’s what they thought:

I was happy that the idea about the cone having the least surface area for a given volume came up when the boys were summarizing the article. We now moved on to investigating that question.

We first looked at a cube:

The calculations for the cube were pretty easy. Now we moved on to a slightly more complicated shape -> half of a sphere.

Working through the various volume and surface area formulas is a nice introductory algebra exercise for kids:

Now we moved on to looking at cones. Looking carefully at cones is quite a bit more complicated than looking at cubes or spheres. So, first we played with the formulas and reduced the surface area formula to one variable. We got that formula at the end of this movie:

The formula we found in the last video was a bit complicated, so we moved to Mathematica for a bit of help. The graph of the surface area for different values of radius of the cone is a shape that the boys haven’t seen before.

It was fun to talk about how this shape could be helpful in studying the question that Dan asked in the tweet.

It was also fun for me to hear how they thought about ways to zoom in on the minimum.

Definitely a fun project – would be especially good for a calculus class, I think.