Teaching Epsilon

March 14th is Pi Day in honor of the transcendental number that starts out 3.14... It is also Albert Einstein's birthday.

We have never done anything for Pi Day before, mostly because I don't bake so making a pie seemed insurmountable, but someone told me that cheesecake is easy. Epsilon and I made a basic cheesecake recipe yesterday and stuck it in the fridge overnight. The recipe was easy, as promised, and today we will decorate the cheesecake with a portion of the most famous irrational number.

With Epsilon's little brother poking his finger into the cheesecake, we had to do our pi calculation experiment quickly. I grabbed a roll of twine out of the drawer and first we measured the circumference of the cheesecake with a measure of twine. Then we used Epsilon's ruler to measure the twine. The cheesecake had a circumference of 72 cm. Using the same technique, we measured the diameter and found it to be 24 cm.

I wrote the decimal expansion of pi to 5 decimal places on scrap paper: 3.14159

Underneath that, Epsilon wrote 74/24 just before she grabbed my phone to check our measurements against the value of pi. The calculator told her that 74/24 = 3.083333. She gasped a little at how close we got with measuring twine.

Then we ate the cheesecake.

A few facts about pi:

The formula for calculating the circumference of a circle is $$2 \pi r$$ so by dividing the circumference by the diameter, you get the rough approximation of pi.

Curious about the digits of pi? This website has a webpage with the first million digits of pi.

Although all of the digits 1-9 appear in pi by the 14th digit, the first occurrence of 0 happens in the 33rd digit.

Archimedes was credited with giving the first approximation of pi, bounding it between 3 10/71 and 3 1/7.

Epsilon and I had about an hour to kill while riding the train so I decided to teach her about the Möbius function. This function is what I had in mind when I was teaching her about prime factorization. I was very excited to get started and see just how difficult a concept the Möbius function would be for her.

The lesson began with a review of prime factorization. Epsilon had only seen the topic once and I wanted to make sure she remembered it. She went through a few examples and I was sure she understood the general idea. It was rather late and I noticed that I could tell the different between tired-Epsilon and refreshed-Epsilon by the quality of her handwriting. While Epsilon wrote, I went through my plan on how to explain the Möbius function to a bright 7 year old.

A Snag

Oops. Epsilon doesn't know about functions. Before we could get started, we had to cover the basics of functions. So I started with the function box:

The function box takes numbers as input in the top, does the function transformation, and then spits out the number answer at the end. Starting with a couple of linear examples, I suggested the function x+1 and asked Epsilon what the function box would do to the numbers 5 and 7. She correctly responded with 6 and 8. Then I picked a slightly more complicated function, 3*x + 2, for the function box.

Linear examples are relatively intuitive and straightforward. However, the Möbius function maps all positive integers into only 3 outputs: -1, 0, and 1. I wasn't sure how Epsilon was going to handle that concept.

The Möbius function

Answer: It went well.

Starting with a piece of paper with 3 columns on it, with the numbers -1, 0, and 1 written at the top of each column. I told Epsilon that we were going to play a game with a special function. Starting with the column, with the number 0 at the top, I told her to put numbers in that column if the prime factorization has at least 2 of the same number. Some obvious examples are 4 and 9. But what about 75? Epsilon factored 75 to 3*25 and 25 to 5*5, so

$$75 = 3*5*5$$

Epsilon put 75 in the 0 column.

After many examples of integers that map to 0, we had to map numbers to -1 and 1. First we had to find a number that didn't map to 0, such as 15. Epsilon factored this to 3*5 and noticed that there are only distinct primes. The second part is easy. I had her count the number of prime factors. If the number of prime factors is even, she put the number in the 1 column, if the number of prime factors is odd, she put it in the -1 column.

This was the best game ever. She started demanding that I give her numbers. I had to carefully construct numbers so the columns were all well represented. This is easy if you take prime numbers, multiply them together, and then give the result to be then factored again.

Over the next several days, Epsilon non-systematically applied the Möbius function to all the integers between 2 and 100 so I counted this lesson as a success.

I remember being a kid in 6th grade, watching a video of Donald Duck learning math. I loved this video very much, even though the rest of my class cheered more for Bambi versus Godzilla. Hint: Godzilla won.

From time to time when I got older, I searched in vain for a copy of Donald Duck in Mathemagic Land. As technology progressed, a little site named Youtube was born. This website grew and grew and then one day, poof! Donald Duck in Mathemagic Land was available.

I saved this video for a day when I needed little Epsilon to be busy and showed it to her. She was entranced. I got my work done. Win/win.

Now that Epsilon has a good handle on the concept of a prime number, we can talk about some cool prime stuff. The flip side of prime integers are, of course, composite integers. So we formalized how to decompose composites into their prime factors.

On one of my many notebooks, I wrote down the number 15 and asked Epsilon if it is prime. No. OK, so what 2 numbers multiply together to get 15? 3 and 5. Is 3 prime? Is 5 prime? Yes and yes. This is what my paper looked like: (my.handwriting.is.not.so.neat.)

The next number we factored was the product of more than 2 primes, so our technique took multiple steps. I asked Epsilon what 2 numbers multiplied together equal 30. She came up with 5*6. Note: there is more than one answer to this question.

This time, when I asked her if 5 and 6 are prime, I correctly heard a yes and a no. So we weren't done. Since 5 is prime, we just carried it down to the next line. All that was left was factoring 6. In the end, this is what Epsilon came up with:

With this technique, start with 2 factors and repeat until the answer to "is this prime" is yes for all factors. Depending on the size of the number, several iterations may be required, but the problem is only as hard as factoring integers.

To check that Epsilon fully understood the concept, I sent her off with a few integers, which she correctly wrote as a product of primes. The largest integer she factored was 100.

$$100 = 2*2*5*5$$

I am confident Epsilon understood the technique and this week I will teach her motivation for the technique. Or in simple terms "what do we use this for?"

I showed Epsilon this blog and she wanted to write a post.

In school, I did a math hour and we had to find shapes in a painting. We got to pick a card with our partner. And on the card I found lots of squares and lot of triangles. At home, I picked a painting to find shapes. This is the painting I picked.

I sea a bunch of circles and lots of rektangles. I sea squares and disks. I sea a rhombus.

I learned what a rhombus is and the difference between a disk and a circle. I liked looking for the shapes.

(This painting is for sale by the artist.)

Epsilon was doing her school math homework - multiplication - when she discovered a trick. She came up to me, saying that she knew a faster way to multiply by 5. I looked at her expectantly and she said that if she had an even number, she'd divide the number by 2 and then multiply the result by 10. She went on to say that if she had an odd number, she'd subtract 1 and then she'd have an even number. Then divide by 2 and add 5 to the result. I was rather impressed she noticed that she could use different cases depending on whether she was multiplying by an odd or even number.

Rather than leave her with her new trick, I told her to prove it.

We grabbed a piece of paper and started out with an even number, 8.

$$8*5$$

Her first step was to divide 8 by 2, so I left some space and wrote

$$=    *5$$

Epsilon rewrote 8 as 2*4 so in the blank spot, she wrote

$$=2*4*5$$

But that is not quite right because we need to multiply 2*5 to get 10. We had talked about how multiplication is commutative, ie x*y = y*x so our product became

$$=4*2*5$$

$$=4*10$$

That shows it works for n=8, but what about when n=2k, where k is an integer. That is the definition of an even number and we have to show that Epsilon's trick works for all k for this to truly be a proof. This is where I glossed over formality a little for my 7 year old audience. We looked at several more examples and talked about how an even number can always be written as 2 times another whole number.

We briefly talked about odd numbers but because that involves the distributive law and order of operations, I decided to hold off on the proof until another day. Epsilon did notice that an odd number was just an even number plus one, which I consider a success for the day.

Thus, we have begun an introduction to mathematical proofs.

Epsilon and I were on the train to pick up a couple of school supplies. She was clearly bored and did not want to be on the train but fortunately she had brought a small notebook and pen with her. So I took the notebook and drew 8 points and connected the 8 points with lines. What did I have? An octagon.

I asked Epsilon if, using only straight lines, could she turn this octagon into only triangles. She took the pen and started drawing straight lines. It is interesting to see what she comes up with and the holes in my instructions. Her first attempt looked like this:

We looked at her drawing and both of us saw a lot of triangles but the center is not a triangle and when she is done, I would like every part of the octagon to be a triangle. So I clarified that all straight lines should be drawn from one vertex to another. To simplify the problem, I decided to use a square. Epsilon added 2 lines and turned the square into a collection of triangles.

While Epsilon did turn the square I drew into several triangles, this was not the answer I was looking for. I wanted her to draw the minimum number of lines possible and while she did it in 2 lines, it is possible to triangulate a square in 1. So I challenged her to triangulate the square with 1 line. Success!

We moved from the square to a hexagon, a six-sided figure. Epsilon triangulated the hexagon with 3 straight lines.

From this point, she clearly got the idea and quickly triangulated the octagon and pentagon I drew. Just in time too, our train pulled into the station as she finished the last figure. There are some reasons that I  asked her to triangulate polygons in this way but she isn't there yet. For now, she just thinks of it as a puzzle.

A note to parents: the polygons I drew were not necessarily regular polygons. The important thing for this lesson is the polygons are all convex. Read about convex versus concave polygons at Wikipedia.