Continued Fractions and the quadratic formula day 2

Yesterday my younger son and I did a fun project on continued fractions and the quadratic formula. He really seemed to enjoy it so we stayed on the same topic today.

I started by asking him to recall the relationship that we talked about yesterday and then to make up his own (repeating) continued fraction:

After he’d chosen the continued fraction to study, we looked at the first few approximations to get a feel for what the

Finally he used the quadratic formula to solve for the value of the new continued fraction – it turned out to be (3 + \sqrt{17}) / 2!

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A fun quadratic formula project with continued fractions and Fibonacci numbers

My son had a 1/2 day of school today due to the snow storm. Instead of having him work through problems form Art of Problem Solving’s Algebra book, I thought it would be fun to do a quadratic formula project since we had more time than usual.

I was a little brain dead as the spelling in the project will show, but still this was a nice project showing a neat application of the quadratic formula.

We started by looking at a common continued fraction and seeing how the Fibonacci numbers emerged:

Next we tried to see how to find the exact value of this continued fraction – here is where the quadratic formula made a surprise appearance:

Finally, we tried to decide which of the two roots of our quadratic equation were likely to be the value of the continued fraction. We had a slight detour here when my son thought that \sqrt{5} was less than 1, but we got back on track after that:

It was definitely fun to show my son how the quadratic formula can appear outside of the problems in his textbook 🙂

Sharing a problem from Gil Strang’s new linear algebra book with kids

I was fortunate to get Gil Strang’s new linear algebra book last week – it really is terrific:

Though it is hardly a book for kids, one problem from the first section jumped out as ones kids could actually work through, so I shared it with the boys today.

Here’s my younger son (in 7th grade) talking through the problem – he struggles a bit, but eventually finds the right idea to get to the end of the problem:

Next my older son (in 9th grade) talked through the problem. He was able to work through the problem without too much difficulty, so I asked him to explain his solution in more detail at the end:

I’m really excited to go through this book on my own over the next few months – hopefully will find a few more problems here and there to pull out and share with kids.

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.