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.

Building number sense using ideas from number theory

I love using Art of Problem Solving’s “Introduction to Number Theory” book as a way to help my younger son build number sense. We went through the book together a few years ago and he’s going through it on his own now.

Yesterday he worked on a section discussing perfect, deficient, and abundant numbers. These concepts are something that a kid can understand, but also come into play in unsolved problems in number theory. For a 4th grader I think the fun is in understanding the new ideas rather than their connection to the Riemann hypothesis:

Superabundant numbers on Wikipedia

For me, though, the ideas are a sneaky way to build numbers sense.

So, is 60 perfect, abundant, or deficient?

Here’s my son explaining what those terms mean:

Here’s his answer to the question:

Using AoPS’s “Intro. to Number Theory” to build number sense

I love the books from Art of Problem Solving.  Not just because of the subjects that the books cover, but also because the books give me a great chance to secretly review other subjects.

The Introduction to Number Theory book is amazing all by itself.  What makes me love it, though, is the opportunity it gives my younger son to build number sense.

Here’s a pretty typical example from this morning.  The problem (from the review section in Chapter 4) asks you to list the common divisors of 84 and 132.   I assume that the point of the problem is for the student to find the greatest common divisor, but instead my son lists all of the divisors and gets some great arithmetic practice.

Here’s the first part of the problem:

and here’s the 2nd half.

It is fun to watch him build up his number sense. There will be plenty of time to talk about greatest common divisor later!

A nice “possible / impossible” problem from the 2007 AMC 8

Ran across this problem from the 2007 AMC 8 today:

Screen Shot 2016-03-08 at 7.31.41 AM

Here’s a direct link to the full exam on Art of Problem Solving’s site:

The 2007 AMC 8 on Art of Problem Solving’s website

I thought the problem would make for a great discussion this morning. We began by discussing the first two choices:

and here’s a discussion of the last three choices:

I love using old contest problems with my kids. One reason is that I’m not that great at coming up with good problems on my own, and the old AMC problems tend to be really good. I especially like this problem because it gets kidsthinking about numbers and talking about mathematical reasoning. Fun little project!

Building place value and number sense with letter problems

I saw this wonderful problem from the 2009 AMC 8 last night:

Problem #24 from the 2009 AMC 8

Here’s the problem:

The letters A, B, C, and D represent digits. If AB + CA = DA and AB – CA = A, what is the value of D?

Many questions similar to this one appear on math contests for younger kids. At first these problems didn’t really stand out from all of the other problems, but lately I’ve come to see them as a neat way to get kids thinking about place value and number sense in general.

I also find the difference in approach between my older son (6th grade) and younger son (4th grade) to be fascinating.

Here are their approaches from last night.

My older son’s initial approach is to try to calculate the value of the individual digits:


After he finds the value of the various digits, I asked him to find a different approach that didn’t require so much calculation. Knowing the answer already helped, of course, but moving away from calculation also allowed him to see the place value ideas more clearly:


Now for my younger son – his initial approach involves much less calculation. He doesn’t have quite as much mathematical sophistication as his older brother (since he’s 2+ years younger) and he struggles a little to communicate the ideas that he’s seeing:


At the end of the last video he arrived at the idea to try out a few numbers. Once he starts down that path the place value ideas sort of emerge from the shadows and he finds his way to the end of the problem relatively quickly:


So, hopefully a nice example of how kids approach this type of arithmetic problem. Hopefully the example also shows how this type of problem can help kids think about place value and build number sense.

Two books that I secretly used to build number sense

[sorry this is written quickly and not really edited – I wrote it while my son was at a school event, so limited time.]

Saw this really great post today from Geoff Krall (via a Tracy Johnston Zager retweet):

It made me think of the two books that I’ve gone back to every now and then to help build a little number sense –


I guess I’ve used one of the books so much that it is no longer recognizable!! Here’s where you can buy them:

Art of Problem Solving’s Introduction to Number Theory

Art of Problem Solving’s Introduction to Counting & Probability

For me the point of these books was never about learning number theory or probability, though I’m sure the boys picked up a few things here and there. Instead the point was finding ways to build number sense by talking through either (i) some interesting properties of numbers, and (ii) some neat counting problems.

Here’s one example project from number theory:

Using Divisibility Rules to Build Number Sense

The last part of this project uses a really neat divisibility rule for 7 idea that I found on Tanya Khovanova’s website.

One other project mentioned in this blog is learning to do arithmetic in binary using duplo blocks. Here’s one of our addition projects which I thought was a great way for kids to see numbers in a slightly different light:

We were even able to use some of the ideas we learned in binary to reinforce some ideas about decimals:

Writing 1/3 in binary

Here are a few number sense examples inspired by ideas in the Introduction to Counting and Probability book:

Counting Arrangements around a Table

The hockey stick theorem and some fun geometry in Pascal’s Triangle

The discussions that we’ve had over the years about Pascal’s triangle sometimes let the kids find Pascal’s triangle in surprising places – :

Talking about Pólya’s Urn with kids – inspired by Jim Propp’s blog post

Again, the point of using these books for me wasn’t to teach number theory or probability, but rather just to find some fun problems that would hopefully help to build some number sense. The idea from Geoff Krall’s post that really reinforced this idea for me was this one:

Also, we’re not talking about shutting everything else down classroom-wise, lest you’re worried about losing precious class time. While coverage is overrated, let’s put that aside for now, shall we? We’re talking 10-20 minute activities and discussion here, maybe a couple times a week.

The nice thing about these two books from Art of Problem Solving is that they are full of neat problems that you can use for 10-20 minutes here and there for a little non-standard (and hopefully fun!) number sense building 🙂

A pretty neat counting problem from Mathcounts

The boys came back from a 2 day camping trip today. I had some afternoon meetings, but luckily they got back early enough for us to do a little counting project.

Without anything in particular planned, I just picked the first problem in the challenge problem section at the end of chapter 2 in our Introduction to Counting and Probability book. The problem is a pretty neat case by case counting problem (even though it looks fairly dull at first glance):

How many positive integers between 24 and 125 have a digit sum that is a multiple of 7?

The boys did a nice job of breaking the problem up into cases right away. In the first video we look at the cases where we have a 2 digit integer. I really liked hearing my younger son talk about the two digit numbers whose digit sum is 14 – it is fun to hear a young kid bring together the ideas you need to work through this problem. One of the nice surprises about this problem is that there’s some interesting number sense ideas hiding inside of it!


In the next video we look at the three digit numbers from 100 to 125 whose digit sum is a multiple of 7. There’s a little surprise in this part, too, since there are no numbers in the range with a digit sum is 14. It takes a moment for the boys to realize that the largest digit sum isn’t necessarily the largest number, so that’s a another neat math idea hiding in this problem:


So a nice problem with a couple of neat ideas beyond the case by case counting. Fun little project to do in the lucky extra 15 minutes I had with the kids today.

Count like an Egyptian part 3

This morning we looked at Chapter 3 in Count like and Egyptian. This chapter discusses how to calculate areas of triangles and the area of a circle using Egyptian ideas of multiplication and division.

Our prior two blog posts about the book are here:

Going through Count Like an Egyptian with the Boys

Count Like an Egyptian Part 2

You can purchase the book here:

Count Like and Egyptian by David Reimer

And, of course, a hat tip to Evelyn Lamb who pointed out this book to me about a month ago:

Since it had been a couple of weeks since we last looked at the book, we started with a quick review of Egyptian multiplication. Most of the ideas had stayed with the boys, which was actually pretty nice to see, but one little piece of the process got reversed in their mind. There’s more detail on this process in our first project from the book linked above.


To get going with the ideas in chapter 3 of the book, we spent a little bit of time talking through how to divide by 2. My younger son listed some procedures that he knows for dividing by 2 – long division, for example – and my older son showed how to reduce a complex division problem into pieces that you already knew how to do. This second approach is pretty similar to the approach discussed in the book:


One time you might find yourself dividing by two is when you are calculating the area of a triangle. We work through several examples of using Egyptian multiplication to calculate the area of a triangle:


The last part of the project was using Egyptian multiplication to find the area of a circle. The book claims that the Egyptians used the approximation \pi \approx 3 \frac{1}{8}, so in order to calculate (or approximate, I guess) the area of a circle we need to learn how to divide by 8.

We talk through how to do that building off of dividing by 2 and then find an approximate value for the area of a circle with radius 10.


The math history that we are learning in this book is really fun. What I really like about going through this book with kids, though, is all of the conversations about arithmetic help them build up their number sense. I’d definitely recommend this book to anyone looking for fun and different ways to talk about arithmetic with kids.

Introduction to divisibility rules

This is the first of two short blog posts today.

My younger son and I started the section on divisibility rules in our Introduction to Number Theory book. I don’t remember the context, but we have talked a little bit about divisibility rules before. He knows some of the rules, but now we are going to learn how to understand these rules through the lens of modular arithmetic.

Last night we talked about divisibility rules without even looking at the book. I just wanted to hear what he had to say:


Today we talked a little more in depth about some of the basic rules – namely divisibility by 2, 5, and 10. He seemed to be able to understand the ideas and gave a really nice explanation of why the divisibility rules for these numbers work. It was fun to hear his explanations (despite my stumbling explanation of the problem that we were working on . . . .):


I remember being fascinated by these divisibility rules as a kid, though I’m sure that I just learned the rules without really understanding why they worked. Learning the ideas behind these rules isn’t too complicated, though, and hopefully helps build up number sense and a little bit of sense about place value, too. Definitely a fun little project.

An incredible article about data science

If you or your students are interested in understanding ways that math can be applied to problems outside of academic / school settings, this recent article from “I Quant NY” is an absolute must read. Hat tip to Patrick Honner for pointing it out to me:

So much of what is important in mathematical problem solving is on full display in the piece – noticing, wondering, basic number sense, and tons and tons of persistence.

oh, and no equations more complicated than calculating a 20% tip.

I’d guess that students ranging in age from middle school to graduate school can get something – and probably quite a lot – out of this article. The analysis, methods, and conclusions shared in the article provide such valuable lessons that I honestly can’t think of a better starting point to understand what quantitative analysis can bring to the table in the mythical “real world.”

If you want one little sound bite / takeaway, let it be this passage:

When Doing Data Science, Look at Your Raw Data. If there is one thing I have learned doing data science, it is to always look closely at your raw data in addition to your aggregate statistics. It would not have been possible to figure this out without looking at a subset of individual rides.”

Bravo I Quant NY!!