A fun connection between quadratic equations and continued fractions

My younger son is beginning to study quadratic equations in Art of Problem Solving’s Introduction to Algebra book. So far he’s essentially only seen quadratic equations that factor over the integers. For today’s project I wanted to show him that there are simple equations with fairly complicated (compared to integers!) roots.

We started with a problem similar to ones that he’s already seen:

Next I showed him a type of equations that he’s not see before and we spent 5 min talking about his ideas of how you could solve it:

Finally, for the specific equation we were looking at, I showed him how we could use continued fractions to solve it. As a bonus he remembered the connection between the Fibonacci numbers and the golden ratio and that got us to the exact solution!


What a kid learning algebra can look like


Section 9.2 of Art of Problem Solving’s Introduction to Algebra is one of my favorite sections in any book that my kids have gone through. The section has the simple title – “Which is Greater?”

One question from that section that was giving my younger son some trouble today was this one:

Which is greater 2^{845} or 5^{362}

I decided our conversation about the problem would make a great Family Math talk, so we dove in – his first few strategies to try to solve the problem resulted in dead ends, unfortunately. By the end of the video, though, we had a strategy.

Now that we’d found that 2^7 and 5^3 are close together, we tried to use that idea to find out more information about the original numbers.

I found his idea of approximating at the end to be fascinating even if it wasn’t quite right. It was also interesting to me how difficult it was for him to see that the two numbers on the left hand side of the white board were each bigger than the two corresponding numbers on the right hand side of the board. It is such a natural argument for someone experienced in math, but, as always, it is nice to be reminded that arguments like that are not obvious to kids.

Using Dan Anderson’s line art program to explore lines with my son

My younger son just started the section on lines in Art of Problem Solving’s Algebra book. By happy coincidence Dan Anderson shared this fun line math art program:

I thought it would be fun to have my son play around with the program and just see what his reactions were.

Here are his initial reactions to the program:

Next I had him manipulate a different part of the program – it turns out that what happens on the screen is pretty complicated to explain!

Finally, even though he hasn’t studied any trig, yet, I had him change the two starting shapes to see how the pictures changed. By luck the changes he made produced a really fun set of shapes to explore:

Dan’s program is a great tool to use to have kids play around with lines. Once we get to the section on equations of lines, I’m definitely going to come back to it to show how the lines are being drawn. I might also use this program to explore parametric curves when we get to that topic in my older son’s calculus class.

Introduction to derivatives of inverse functions

We’ve started a new chapter in our calculus book -> Inverse functions.

After wondering a bit about how to approach this topic, I tried starting out in a different book, Spivak’s Calculus, which has a slightly more theoretical approach.

Now not sure how much the extra theory helped, but we did have a nice discussion about inverse functions this morning.

Tonight I wanted to give a few concrete examples and avoid the theory as much as possible. After a brief discussion, I started with the example of y = x^2 and found the derivative of the inverse function:

Next we moved on to y = e^x and y = \ln(x). He already has seen a bit of discussion about e^x and its derivatives, so I let him play with the ideas about inverse functions to see if he could find the derivative of \ln(x) on his own:

Next up were the inverse trig functions. Today I chose to focus on y = \arcsin(x) and y = \arctan(x). I started by showing him \arcsin(x) and how basic trig relations produced a pretty surprising derivative:

Next up was \arctan(x). I let him try this one on his own, but I rushed into it too quickly and forgot to talk about the domain. That led to a bit of confusion at the end, but overall I was happy that he was able to get the general idea.

A neat geometry problem I saw from Catriona Shearer

I saw this tweet from Catriona Shearer last week:

It was a fun problem to work through, and I ended up 3d printing the rectangles that made the shape:

Today I managed to get around to discussing the problem with the boys. First I put the pieces on our whiteboard and explained the problem. Before diving into the solution, I asked them what they thought we’d need to do to solve it:

Next we move on to solving the problem. My older son had the idea of reducing the problem to 1 variable by calling the short side of one of the rectangles 1 and the long side x.

Then the boys found a nice way to solve for x. The algebra was a little confusing to my younger son, but he was able to understand it when my older son walked through it. I liked their solution a lot.

Now that we’d solved for the length of the long side, we went back and solved the original problem -> what portion of the original square is shaded. The final step is a nice exercise in algebra / arithmetic with irrational numbers.

Definitely a fun problem – thanks to Catriona Shearer for sharing it!

Introductory U-substitution

We’ve been doing a little bit of work in the integration chapter of our calculus book. Today the topic was u-substitution. I was expecting the topic to produce a great deal of confusion, but things went pretty smoothly. There will be plenty of time for confusing integration problems later, I guess, ha!

When my son got home from school tonight I had him try out three problems from the u-substitution section. The first two are straightforward and the third is a bit of a surprise.

Here’s the first one: \int (1 - 2y)^{13} dy

For the next problem I chose a trig integral: \int \cos(x)^4 \sin(x) dx

Finally, I chose a problem that isn’t obviously a u-substitution problem when you look at it: \int x \sqrt{x - 1} dx

Again, I was worried that we’d get off to a bit of a rocky start, but thinks went well today. Going to have him work through a bunch of problems tomorrow to get a little more u-substitution practice.

An exploration of Euler’s method

My older son is studying calculus and my younger son is studying algebra this year. ¬†I’d run across some problems on Euler’s method looking at old BC calculus exams, but we are still a long way from talking about Euler’s method in the calculus course. ¬†However, as my younger son begins to study lines, I thought it might be fun to do some visuals on slope fields and just touch on Euler’s method as a way of talking about slope.

Here’s how I introduced the topic:

Next I gave them a neat visual example -> wind speed and direction on the earth

Now I moved to Mathematica to talk about slope fields in general. The two specific slope fields we looked at here were (in calculus language) dy/dx = y and dy/dx = x.

Finally I had the boys choose their own fields and try to describe them ahead of time:

Definitely a fun project. The mathematical idea behind Euler’s method isn’t that hard and kids can understand the concept pretty easily. I really had a lot of fun exploring the ideas of slope fields with the boys this morning.