A surprise square root of 2 discussion

I’m having a bit of a roller coaster ride through square roots with my younger son. Sometimes things that I think will be hard about square roots are easy for him, and sometimes things I think will be easy are hard. Today was the latter case as some initial discussions about the square root of 2 led to more confusion than clarity. So I decided to scrap the overall plans for today and just talk about about \sqrt{2}. The new goal was to see why it was not rational.

So, we started off by discussion why it wasn’t an integer:

 

Next we tried to see if the square root of 2 was a fraction. I intended to talk about the standard proof by contradiction here, but about 30 seconds in my son remembered our continued fraction approximation for \sqrt{2}. That surprise memory led to a quick review of the first couple of convergents in the continued fraction expansion. We saw some fractions that were nearly equal to \sqrt{2} but none of them were exactly equal.

He understood that if we could write \sqrt{2} as a fraction, the continued fraction expansion would eventually stop (it may be a stretch to say that he understands this, but he at least has the intuition that this fact would be true). Since the continued fraction expansion goes on forever, there must be no rational number that is exactly equal to \sqrt{2}.

 

Finally, we covered what I intended to cover in the last little talk – the usual proof by contradiction that \sqrt{2} is not rational. We end the conversation by mentioning some other numbers that are not rational – some for the same reason as \sqrt{2} and some for other reasons.

 

So a fun and unplanned discussion about the square root of 2. Hopefully these little side discussions end up building up his number sense a little and help him gain a better understanding of square roots.

A fun number theory / math history project for kids

We’ve come to the end of our Introduction to Number Theory book. The last thing that we looked at today proved to be pretty interesting, though, so I thought I’d write it up. It is a fun way for kids to learn a little math history, get a little arithmetic practice, and also see (and understand a little bit) of pretty deep math.

The topic begins with Lagrange’s four squares theorem – every positive integer can be written as the sum of four squares:

 

Next we move forward a little over 100 years to the work of Ramanujan. He found a total of 53 ways to write the positive integers using perfect squares. We explore one of Ramanujan’s formulas here:

 

Finally, I introduce the work of John Conway and Manjul Bhargava, though we don’t work through any examples. Instead after this short video I showed my son the video that Quanta Magazine made when Bhargava won the 2014 Fields Medal:

 

Here’s the Quanta magazine article and video is here. If you can choose only one video about mathematicians to show to kids, pick this one – so many points in it to draw in young kids:

Quanta Magazine’s article on Manjul Bhargava

My son has really loved learning about number theory. I’m really happy that we were able to go through this book together. He told me last night that he’s decided that he wants to learn more from our old Prealgebra book, so that’s going to be the next topic. My plan now is to make number theory the topic for a relaxed summer math course for both kids – can’t wait for summer to come!

Grothendieck, Heron, and Brahmagupta

My son is taking the AMC 8 today so I thought we’d have sort of a light morning with math.  The section of the book we were meant to cover today was Heron’s formula for the area of a triangle.  Not exactly a “light” subject if you delve into the proof!  Instead, I took today as an opportunity to talk a little math history.

Quite a bit after Heron, in the 600’s actually, the Indian mathematician Brahmagupta found an amazing generalization of Heron’s formula.  Brahmagupta’s formula calculates the area of a cyclic quadrilateral and the triangle is a special case when one of the sides has length zero.

Though I know very little of the mathematics that Grothendieck actually studied, I took the example of Brahmagupta finding Heron’s formula as a special case of his formula as an example of solving a problem via generalization.   In the pieces I’ve read about Grothendieck in the last few days, his ability to find the right generalization of a given problem seemed to be one of his great gifts that the writers focused on.  The analogy with Heron’s formula and Brahmagupta’s formula  appeared to my best shot at mentioning Grothendieck’s contribution to math to my son.

The first video introduced Heron’s formula and discussed a little bit of history.

The next part of our talk introduced Brahmagupta’s formula and talked about why the triangle is (possibly!) a special case of this formula.  For it to be a special case, we just need to show that you can circumscribe a circle about any triangle.

In the next video we try to see if it is possible to always circumscribe a circle about a triangle.  This led to a discussion of a slightly easier problem – how do you find the set of points that is an equal distance from two given points?  Solving that problem leads to the idea that it may, in fact, always be possible to draw a circle that hits all three verticies of a triangle.

Finally, we jump over to the kitchen table to try to construct the circumcircle of a given triangle.  It is a neat construction and it is always a nice surprise when you get the three perpendicular bisectors to intersect in a single point!

So, a fun morning showing a little math history and a couple of really amazing geometry formulas.   Nice to have a light day every not and then.

Terry Tao’s MoMath Talk Part 2: Clocks and Mars

Last week I wrote about finding Terry Tao’s incredible public lecture delivered at the  Museum of Math and how that lecture provides many great examples you can use to talk about math with kids:

Terry Tao’s MoMath Lecture Part 1: The Earth and the Moon

for ease, the direct link to the Terry Tao lecture  is here:

Today I wanted to use a second example from that lecture for a little math talk with the boys.  This topic comes from approximately 42:30 into the video when Tao discusses Copernicus’s calculation of how long it took Mars to orbit the sun.   This calculation is an incredible scientific achievement, especially when you consider that telescopes hadn’t even been invented yet!

In the lecture Tao describes the remarkable story behind the calculation, but does not go into the details of the calculation itself.  To be clear, that’s not a criticism – the point of his lecture was to tell the story not to dive into the details.  Exploring the details of this particular calculation is a great topic to discuss with kids, though.  The only background material required is some basic knowledge about fractions.

We began this morning by watching the (approximately) 5 minute portion of the talk in which Tao describes how Copernicus calculated the time it took for Mars to Orbit the sun.  Following that we went to the whiteboard to talk about what we learned, and to head down the path of understanding the calculation in detail.   The starting point I chose for understanding the calculation is asking questions about the angles formed by the hands of clocks.

I will say at the start that it was a little harder for my kids than I was expecting.  The discussion and the explanations below are not at all flawless and have several false starts.  As I’ve said many times, that’s what learning math (and, in this case, a little physics) looks like.  Watching the films of this discussion prior to publishing this post has reinforced my feeling that Tao’s lecture  is a great spring board to talking math with kids.

Having looked at a few examples of when the angles between the hour hand and minute hand of a clock would be zero, in the next part of the talk we began to try to drill down on the math.  The starting point for the discussion here was the observation by my older son that the minute hand moves 12x faster than the hour hand.    In this video we try to write down some expressions that describe how fast the two hands of the clocks are moving:

The next step was writing down an equation that told us how far the hour and minute hands would move in “t” minutes.  In retrospect I wish I would have made a different choice in the approach here since jumping directly to the algebra made a simple idea a little harder than it needed to be.   If I could do it again I’d probably cover the ideas in this video nearly in reverse (and I’m annoyed with myself for getting frequency and period reversed, too.  Can’t get everything right . . . .)

However, even with the little bit of extra time that introducing the algebra at the wrong moment led to, the discussion here did get us to an equation that looked a lot like the equation Terry Tao had written down in his presentation slides.

At the end of the last video we got to an equation that helps us understand when the hands of a clock are exactly on top of each other – now we solve it!  Solving this equation is a great exercise for kids who have a little familiarity with fractions.  We sort of stumble out of the gates with the solution, but once we get on the right track we actually get to the end in sort of a neat way.

With all of this background out of the way we can return to the equation that Terry Tao had in his presentation.  We being this part by briefly talking about difference between our clock equation and the equation that Copernicus solved..  After that introduction we solve the equation and determine how long it takes for Mars to orbit the Sun!

I’m really excited about using more examples from Terry Tao’s lecture to talk math with kids.  There are so many great things about this lecture – for instance the incredible historical information and the great opportunity to see Terry Tao speak on an accessible topic – but for me the new examples the talk contains for talking  about some basic school math with kids is the best thing about this public lecture.    Who would have thought that calculating the orbit of Mars just boiled down to simple fractions?!?

Using Terry Tao’s MoMath public lecture to show math to kids

Recently I became aware that the Museum of Math in NYC has videos of more than 50 lectures given at the Museum in the last year (or maybe few years).  The lectures discuss the use of math in an astonishing variety of fields ranging from pure theory to every day life (a recent lecture was about math and cooking, for example, though I don’t know if that one is online yet).  The collection of lectures is here (click on the “Math Encounters” link at the top of this video to see the list):

One of the lectures that caught my eye was Terry Tao’s “The Cosmic Distance Ladder.”    That talk is here:

It caught my eye for a  couple of different reasons.  First, Tao is one of the top mathematicians in the world and it is a rare treat to see him speak.  An even rarer treat to see him deliver a lecture designed (quite successfully) to be accessible to the public.   Having watched this talk several times now, I think many parts of it provide fun and exciting examples for kids of how math has been used to advance science.

I hope to use many pieces of his lecture to show my own kids some important math and science.  I began with two fun, and frankly amazing, examples today:

(1) The approximation of the radius of the Earth by Eratosthenes  (beginning around 17:20 in the video of the lecture), and

(2) The approximation of the radius of the orbit of the Moon by Aristarchus (beginning around 25:45)

Both of these are obviously remarkable scientific achievements, and Tao’s lecture does a wonderful job of explaining the ideas behind the discoveries.   The lecture doesn’t dive too deep into the calculation of these results, though.  That was not the point of the lecture – not at all – so my point isn’t even remotely a criticism.  Rather I took it as an exciting opportunity to use the videos to teach.  Exploring the calculations in the lecture a little more carefully seems to me to be a great way to use this lecture to help kids learn a bit of math, a bit of physics, and a bit of history.  Only some basic geometry is needed to understand the calculations.  Diving in a little deeper into the math this morning with the boys was really fun.

First off is the approximation of the radius of the Earth:

Second is the approximation of the radius of the orbit of the Moon:

The calculations we do are slight simplifications (as is noted in the original lecture), but I think the important mathematical ideas are here.  Discussing the limitations of these calculations and ways to improve them could be a fun student project.

I’m really happy to have stumbled on this collection of  lectures at the Museum of Math, and am super excited to spend some time over the next few months trying to figure out fun ways to used them to help kids see interesting examples of how math is used (and has been used) in the world.