Tag arithmetic

A fun project from Art Benjamin and the Museum of Math

Yesterday Art Benjamin gave a talk at the Museum of Math. One neat tweet from from the talk was this one:

It is a pretty neat problem and I thought it would make a fun project for the boys today. I didn’t show them the tweet, though, because I wanted to start by exploring the numbers with increasing digits:

 

Next we tried to figure out what was going on. My older son wanted to try to study the problem in general, but then my younger son noticed a few things that at least helped us understand why the sum should be divisible by 9.

 

For the third video we started looking at the problem in general. The computations here tripped up the boys a bit at first, but these computations are really important not just for this problem but for getting a full understanding of arithmetic in general.

 

For the last part of the project we looked at two things. First was returning to a specific example to make sure that we understood how borrowing and carrying worked. Next we applied what we learned to the slightly different way of multiplying by 9 -> multiplying by 10 first and then subtracting the number.

 

After the project I quickly explored Dave Radcliffe’s response to MoMath’s tweet:

It took a bit of thinking for the boys to see what “works in any base” meant, but they did figure it out.

I love this Benjamin’s problem – it makes a great project for kids!

Dave Radcliffe’s “unit fraction” tweet

Saw a neat tweet from Dave Radcliffe a few weeks ago:

I’d played around with it a bit on Mathematica and the code was still up on my computer screen when we were playing with base 3/2 yesterday, so the kids asked about it.

Radcliffe’s proof is a bit too difficult for kids, I think, but the general idea is still fun to explore. I stumbled through a few explanations throughout this project (forgetting to say the series should be finite, and saying “denominator” rather than “numerator” at one point), but hopefully the videos are still clear.

I started by explaining the problem and looking at a few simple examples:

Next we looked at how it could be possible for a finite sum of distinct numbers of the form 1 / (an integer) could add up to 100, or 1000, or some huge number:

Now that we understood a bit about the Harmonic series, we jumped to Mathematica. I sort of half explained / half skipped over the “greedy algorithm” procedure that Radcliffe uses in his paper. I thought seeing the results would explain the procedure a bit better.

We played around with adding up to 3 and then a couple of numbers that the boys picked.

After playing around with a sum adding up to 3, we tried 4 and the boys got a big surprise. We then tried 5 and couldn’t get to then end!

After we turned off the camera we played around with the sum going up to 5 a bit more sensibly and found that there are (from memory) 102 terms and “n” in the last 1/n term has 142,548 digits!

So, a little on the complicated side, but still a fun math fact (and computer project!) for kids to explore.

Revisiting the Collatz Conjecture

Last week we made “math biographies” for the boys:

Math Biographies for my kids

When I asked them about their favorite unsolved problem, they both mentioned the Collatz conjecture. Unfortunately they couldn’t remember the details, but that made the choice of topic for today’s Family Math project easy!

I decided to approach the problem using sound much like we did when we looked at John Conway’s version of the Collatz conjecture:

The Collatz conjecture and John Conway’s “Amusical” variation

Before diving in to the sound, though, we reviewed the details of the Collatz conjecture:

 

Next we moved to Mathematica to listen to (a version of) the sound of the Collatz Conjecture. Sadly the camera was way out of focus here. I didn’t notice until the movies were published. So, sorry about that, but at least the sound comes through ok.

 

Next I asked the boys to change the procedure a little. My older son’s suggestion was to change the procedure from “multiply by 3 and add 1” to “multiply by 3 and add 3.”

My younger son noticed from the sound that the loop didn’t start with 1 – that was really fun to hear! Maybe a good thing, too, since the video is so out of focus 🙂

 

Finally, we made one more change to the procedure – this time “divide by 2” was replaced by “divide by 2 and then add 4.” We saw some new patterns again.

 

So, I love playing around with the Collatz conjecture with kids. First, it is always really fun to be able to show kids unsolved math problems. Lior Patchter has an incredible blog post about various different unsolved problems to share with kids at each grade level if you want more than just the Collatz conjecture:

Unsolved Problems with the Common Core

One thing that is really nice about playing with the Collatz conjecture is that you get to sneak in lots of arithmetic practice.

It is also fun to turn the numbers into music just to give the kids a slightly different way of experiencing the pattern in the numbers.

A great question from Dave Radcliffe

During a twitter conversation earlier in the week, Dave Radcliffe presented this question:

The question is a really deep and really challenging one for kids. Truthfully it is probably a little over the head of my kids, but I thought I’d give it a try anyway. I’ll revisit this one (hopefully!) several times over the course of this school year – although the question confused my kids a little bit, I really like it.

Here’s my older son’s (started 7th grade today!) thoughts on Dave’s question:

Here’s my younger son’s thoughts – he’s in 5th grade. I took a little extra time at the beginning with him to work through some examples with numbers so that the abstract symbols wouldn’t be so confusing:

Does 1 – 1 + 1 – 1 + 1 . . . . . = 1/2

This morning, for a little first day of school fun, we played with Grandi’s Series.

I’ve seen the series pop up in a few places in the last few days – first in part of a little note I wrote up inspired by a Gary Rubenstein talk:

A Talk I’d live to give to calculus students

and then a day or two later in this tweet from (the twitter account formerly known as) Five Triangles:

So, what’s going on with this series? What would the boys think?

Here’s their initial reaction:

And here’s their reaction when I showed them what happens when we assume that the series does sum to some value x:

We have touched a little bit on this series (and my favorite math term “Algebraic Intimidation”) previously:

Jordan Ellenberg’s “Algebraic Intimidation”

It is fun to hear the boys struggle to try to explain / reconcile the strange ideas in Grandi’s series. I’m also glad that they are learning to think through what’s going on rather than just believing the algebra.

What a kid learning math looks like – a challenging base problem

My younger son has been working through Art of Problem Solving’s Introduction to Number Theory book this summer. The topic for the last few weeks has been arithmetic in different bases. Today he can across a problem that gave him a lot of trouble. He worked on it alone for about 15 minutes and then we talked about it.

Here’s the problem:

In a certain base (12)*(15)*(16) = 3146. What is that base?

Here’s the first part which summarizes his initial thoughts on the problem. The work he does here shows that the base used in this problem must be lower than 10. Once he discovers that fact we talked about a few other ways that we could have seen that the base wasn’t 10.

Next we tried to see how we could identify the base we were looking for using some of the ideas from the last video. We used the last digit idea to eliminate 7 and 8, but the last digit idea told us that base 9 might actually work.

Then we did the arithmetic to show that we were indeed looking for base 9.

So, a really challenging problem, but a fun talk for sure. Working through problems like this one are a great way to review arithmetic and a neat way for kids to learn some basic ideas in number theory.

Dave Radcliffe’s polynomial activity day 1

Saw this really fun tweet from Dave Radcliffe yesterday:

This looked like a fun project for kids, though it wasn’t obvious how to get started. It turns out that Mathematica has a handy function called PolynomialMod[] that tells you what a polynomial looks like modulo an integer – so that made life easier!

I decided that for today’s project we’d explore (1 + x)^n using Mathematica and see what patterns we could find. The introduction to today’s project involved introducing basic polynomial multiplication. Luckily, a natural way to multiply polynomials looks a lot like multiplying 2-digit numbers. I used that connection to introduce the project:

After the introduction I had the boys play on Mathematica and compute various powers of (1 + x)^n starting with (1 + x)^0. We got a little confused between Fibonacci numbers and Pascal’s triangle, but here is what they saw:

For the last part of the project today we used PolynomialMod[] to look at the various powers of (1 + x)^n in mod 2. I wanted to get them used to this Mathematica function to make it easier to explore (1 + x + x^2)^n mod 2 tomorrow. After they explored the powers of (1 + x)^n mod 2 up to n = 8, we talked about patterns in the numbers:

So, a fun little computer math project. It was fun to hear the kids talk about the patterns and also fun to talk about some basic ideas like polynomial multiplication and modular arithmetic. Definitely excited to explore some of the more complicated patters tomorrow.

What a kid learning math can look like – different bases edition

My younger son is working his way through Art of Problem Solving’s Introduction to Number Theory this summer. He’s currently learning about different bases and how to convert from base 10 into those other bases.

A few days ago he wondered about how multiplication would work. Yesterday we worked through one example and was a great reminder about what a kid learning math can look like:

Today we looked at multiplication in different bases more carefully. First I picked a simple example to show him that the same process you use in base 10 works in other bases (so, you don’t have to convert to base 10 first):

Next I asked him to pick two numbers to multiply. Somewhat unluckily he picked two 4 digit numbers in base 2 so the calculation is a little tedious. Still, though, it all worked!

This was a really fun project – I can’t wait to work with him a little more on arithmetic in different bases.

Summer math talks

I’ll be giving two talks at math camps this summer. The first is at the east coast Idea math camp at the beginning of July and the second is at a math camp at Williams college in the middle of July.

I’m super excited to be able to have the opportunity to give these talks and can’t wait for the chance to interact with the students at the two camps.

The topics I’ll cover – not surprisingly – will come from some of the projects I’ve done with the boys this year. The talk at Williams is about 45 minutes shorter than the Idea math talk, so one or two of the topics below will get cut out.

Here’s are the ideas I’d like to cover:

(1) Larry Guth’s “No Rectangles” Problem

Larry Guth’s “No Rectangles” problem

I covered the 3×3 and 4×4 cases with the 2nd and 3rd graders at my younger son’s school as part of Family Math night and the kids loved the problem. With high school students I’d like to try to explore some of the larger cases and also discuss why this is a difficult problem for computers to solve.

Patrick Honner also showed me this related problem which I’ll leave as a challenge for the students 🙂

(2) Ann-Marie Ison’s Math Art

Our projects with Ann-Marie Ison’s art

I’m still waiting to hear what sort of projection capability I’ll have at the two events, but oh do I hope I have the ability to share this program with the students:

The explorations you can do with this simple modular arithmetic idea are incredible.

 

(3) A problem from Po-Shen Loh’s MoMath talk

What I love about Loh’s talk is that he takes an extremely difficult problem – one from the 2010 International Mathematics Olympiad – and turns it into a talk that is accessible to the public.

His approach is so accessible that I talked through the first part of the problem with my 4th grade son:

I’m very excited to hear the different guesses that the students have for the answer to Loh’s two questions.

(4) Bjorn Poonen’s N-Dimensional Sphere problem

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Here’s the problem and our project on the problem:

A strange problem I overheard Bjorn Poonen discussing

Bjorn Poonen’s sphere problem

I’m guessing that not all of the kids will have seen geometry beyond 3 dimensions, so this problem will take a little bit of setting up.  Luckily the only complicated bit of math that they need to understand it is the Pythagorean theorem and I’m guessing that all of them will know that theorem.

I was blown away by the answers to Poonen’s questions when I finally worked through them.  This was also one of the most enjoyable projects that I’ve done with the boys this year.

I hope I have enough time to show the students the fun relationship between \pi and e hiding in this problem, too:

A fun surprise in Bjorn Poonen’s n-dimensional sphere problem

 

Can’t wait to talk about these problems with the kids!

A fun way to estimate e with kids

Saw this tweet from John Allen Paulos earlier in the week (and thanks to Patrick Honner for reminding me!):

I thought the activity would be fun for kids so we tried it this morning for our Family Math activity.

We started by looking at rolling 6 6-sided dice, which sort of built on our activity from last week:

An introductory dice statistics project for kids

I hoped this introductory activity would be a good way for the boys to get a feel for why some squares on the chessboard in the main activity would be blank:

Next we tried out the activity with a 12-sided dice – how many of the 12 numbers do we expect to not come up in 12 rolls?

Finally we moved on to John Allen Paulos’s activity – 64 random numbers. We had Mathematica give us the 64 random integers from 1 to 64 and put snap cubes on the squares of the chessboard to represent those numbers. All of this was off camera (don’t worry!) – here’s what the boys had to say when we were done:

I think this is a great activity for kids. Even the simple part of finding the right square to mark is a nice math challenge for them. The point isn’t for kids to understand what the number e is, rather the point is the surprise that we can estimate the number of blank squares fairly accurately. A fun extension would be estimating the number of squares with exactly 1 cube (the result might surprise you if you don’t know it!).