Descartes’ Theorem from our “Zome Geometry” book

We decided to kick of 2015 with a Zometool project. Flipping through Zome Geometry last night I found a neat section on Descartes’ Theorem for polyhedra. This project made for a fun start to the year.

We built the 7 shapes that we’d need for the project before we started filming. I didn’t tell the boys what we’d be doing with the shapes, though, it was just prep work.

In the first part of our talk this morning I explained the procedure that we needed to follow to explore Descartes’ Theorem and then we worked through the calculation for both a cube and a tetrahedron. There was a little bit of confusion getting going with the process, but we were able to complete the calculation for both shapes. In addition to the geometry, there is lots of good arithmetic practice in this project!


For the next part of the project we completed the calculation in Descartes’ Theorem for a dodecahedron, an icosahedron, and an octahedron. The one little bit of extra work we had in this section was completing the calculation for the dodecahedron – it wasn’t obvious what the angles in a pentagon where, but we figured it out:


Next we moved to a slightly more complicated shape – a pyramid having a pentagon for a base. The difficulty here is that we don’t know the exact values of the angles. However, we do expect that following the procedure for Descartes’ Theorem should end up with a value of 720 degrees. We went through this computation and discovered the surprising fact that it doesn’t matter what the angles are. Nice one, Zome Geometry authors 🙂 :


The last part of our project was studying at a solid that could not deform into a sphere. This talk was interesting because it wasn’t super easy for the boys to see the shape. I probably should have used paper or something to make the shape solid, though maybe the discussion about what intersections in the Zome set were actually corners was helpful. Not sure how this one will work on video, but hopefully it is useful. Even with the slightly disjointed discussion, we did get to a different answer than we’d gotten to before. That was fun and it showed the boys that this shape was somehow different than the prior shapes, and a fairly simple procedure of counting angles could help us understand the difference:


So, another fun project from Zome Geometry. The Zome construction in this one isn’t too difficult, which is nice. This project gives lots of opportunities to expand on some basic 2d and 3d geometric knowledge and also plenty of opportunities to build on number sense. Fun way to start 2015.

A neat “last digit” problem from James Tanton

My younger son and I finished up the “last digit” chapter in our Introduction to Number Theory book today, and I was looking for a fun last exercise. Luckily James Tanton had my back this weekend:

I decided to use this problem as tonight’s exercise and even asked my older son to join in since it looked like we’d have a nice discussion.

We started by just talking through the problem a little bit. I asked each kid to rank the four problems from what they thought would be the easiest to what they thought would be the hardest. It was interesting that their ranking was not the same.

My younger son started off on what he thought was the easiest problem – the pattern in the last digit of the powers of 3:


Next up was the powers of 5. My older son recognized that this would be an easy problem. We talked for a little bit about why the last digit was always 5 and managed to squeeze out a pretty good (if short) discussion. After that we moved on to talking about the harder problems. Neither kid had any ideas about how to proceed with the larger numbers. Eventually my younger son suggested that it would be a good idea to move to the computer for the more difficult problems, so that’s what we did.


We ended the last movie trying to guess how long it would take the powers of 13 to repeat the last two digits. Both kids thought the repetition would happen pretty quickly – about every 2 or every 4 times. We’ve mostly been talking about last digits, so I guess that conditions you to believe that repetitions will happen pretty frequently.

We saw immediately that the repetition didn’t occur quickly. My younger son realized that we should be looking for a last two digits of 01 to singal the repetition – I was happy to hear that since that showed he was starting to understand what to look for.

At the end of this movie we were talking about moving on to look at 113. My older son thought there might be a pattern going from 3 to 13 to 113. Love when they have ideas like that.


The last thing we did was look at the last three digits of powers of 113. It turns out that my son’s guess for the length of the repeating pattern was right. He was pretty excited about that! To see if his patterned continued we looked to see if the last four digits of 1113 cycled every 500 powers.

In all honesty, I don’t know why this pattern is happening, but the kids really liked it. Even if it was just a guess, it was neat to see how happy my son was that he’d guessed the pattern correctly.


So, another really fun math conversation with the kids from a James Tanton tweet. I’ve really enjoyed the section in our number theory book on last digits – it seems to have really helped my younger son build on his existing number sense. The problems with the last two, or three, or more digits are fun, too. They also are great, and relatively straightforward, ways to introduce a little computer math to kids. Definitely a fun little conversation tonight.

Terry Tao’s MoMath lecture part 3: The speed of light and parallax

[sorry if this doesn’t read too well.  woke up sick today and am getting sicker.  boo 😦 Didn’t have the energy for too much editing. ]

In the last few weeks I’ve been writing about Terry Tao’s incredible public lecture delivered at the  Museum of Math over the summer and how that lecture provides many great examples you can use to talk about math with kids.  The first two posts are here:

Part 2 of using Terry Tao’s MoMath lecture to talk about math with kids – Clocks and Mars

Part 1 of using Terry Tao’s MoMath lecture to talk about math with kids – the Moon and the Earth

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

Today we were talking about the piece of the talk starting around 1:04:30 – how physicists obtained the first estimate for the speed of light and also  how astronomers measured the distance to nearby stars.

We began by watching Terry Tao’s presentation and then discussing the boy’s reaction to the video.  They seemed to have a reasonably good understanding of how the measurement of the speed of light was done.  Their ability to understand the talk is why I think Tao’s lecture is so great for kids to see – his explanations are incredibly easy to follow.  We had to clarify a few points, but after those clarifications we were able to repeat the calculation.

Also, working through the calculation is a nice exercise in place value and division for kids.

Following that conversation we moved on to the discussion of how astronomers measured the distance to the nearest stars.  The portion of Tao’s lecture that discusses parallax is amazing, but one really interesting thing that his pictures don’t really illustrate are all of the distances involved.  I draw a similar picture to the one Tao used in the talk and then mention what the proper proportions would be at the end of the video.

Also, at the beginning of this video my younger son was confused by the distances I had written down for the radius of the Earth and the radius of the Sun.  I’m not sure exactly what was bothering him, but since a critical point for understanding parallax is understanding distances, we spent a few minutes at the beginning of the video making sure he understood those distances properly.

Finally, we went out to the back yard to demonstrate the relative distances involved in the measurement of the distance to Alpha Centauri.  We used a small balloon with radius about 2.5 inches for the sun and a grain of salt for the Earth.  At this scale the radius of the Earth’s orbit around the sun is about 50 ft.    Also at this scale, if we were standing in New York City Alpha Centauri would be in Los Angeles!   Sorry for all the coughing in this one – I’m a little sick today 😦

So, one more neat project for kids coming from Terry Tao’s lecture.  It is a little hard to go into the details of how the angles were measured since you need trigonometry for that, but the geometry is easy enough to understand.   Attempting to “draw” the picture to scale in our back yard was really fun, too.  The calculation of the speed of light really just requires a little arithmetic and is a nice example to show to help build up number sense.

Definitely a fun morning!

What learning math sometimes looks like part 3: Multiplying in binary

We’ve spent the last week or so in our number theory book talking about arithmetic in bases other than 10.  One of my favorite activities to help kids learn about other bases is using Duplo blocks to model arithmetic in binary.  This activity has seemed to help both of the boys get a little bit better understanding of place value.

The section we were covering in the book today was multiplication in other bases.  Unfortunately what I thought was going to be one last short example in subtraction took more time than I expected.  That problem made our discussion of multiplication shorter than it needed to be.    When I got home tonight I thought it would be good to do revisit multiplication so we took out the Duplo set to work through examples of multiplication in binary.

We got off to an interesting start when my son choose the example 101_2 times 100_2.  He recognized that multiplication by 100_2 just added zeros to the original number.  This was an interesting observation since we’d not talked about that specific idea this morning.  I wanted to try out another example to see what would happen when this trick wasn’t there to help.  Turned out that the trick was getting in the way a little:

I think my son was quite surprised to see that his method at the end of the last movie didn’t work.  One of the things about multiplying in other bases is that you lose your number sense a little bit and it isn’t easy to see when you’ve arrived at the wrong answer.  That’s at least part of what makes these exercises in other bases such a nice way to build up the ideas of place value – that’s really the only thing you can focus on in these problems!

We looked at the problem again and tried to figure out where things had gone wrong the last time.  Going through it a bit more slowly helped see that several numbers last time were accidentally combined into one.   Having found our way through this problem, I gave him one last problem to work through.  He seemed to have a little better sense of the multiplication process by the end of the exercise:

This was a really interesting process to watch.  A little trick that he learned was limiting his ability to understand how to multiply.  It was hard for him to see that this trick wasn’t helping, though, since the wrong answers aren’t so easy to see when you are working in different bases.   When we walked through the problem the second time it was a little easier for him to see what went wrong since he knew that *something* had gone wrong.  It was nice to see him work through the last problem completely on his own after all of this work.

Finally, just for completeness, here are two videos where we do addition and subtraction in binary with duplo blocks:

Adding in binary with Duplo blocks

Several years ago I was talking about adding in binary with my older son.  On a whim we started using Duplo blocks to see how a “binary adding machine” would work.  It was a really fun exercise and I returned to it today with my younger son when we started the section in our book about adding in other bases.

I like talking about adding in binary with Duplo blocks for a couple of reasons.  First, it helps reinforce the idea of place value.  Second, it shows that you can add numbers in bases other than 10 without first converting them back to base 10.  Finally, both of these nice features happen in a setting that is pretty fun and surprising for the kids.

Our project from this morning went pretty well:

So well, in fact, that my son asked if we could do more tonight, so we did this second project.  This time we added 4 binary numbers, or was it 100 numbers for you binary fans 🙂

We’ll cover subtraction the same way, too, which is even more fun – you just need a new color to represent -1.  Can’t wait for that!

All about that base – a fun exercise from Art of Problem Solving

Last night I stumbled on a great little exercise from Art of Problem Solving’s Introduction to Number Theory book, though I’d didn’t dawn on me how neat the problem was until later in the evening.

The problem itself is pretty easy to state:  Convert 100 from base 10 to base 9  (or from base 9 to base 8, or base 8 to base 7, and etc.)

What I realized late last night is that working through this problem allows us to find fun ways to connect arithmetic, algebra, and geometry.   So we revisited the exercise this morning,  starting off with a quick review of the original problem:

After the introduction, I wanted to reinforce a basic problem solving idea by starting with an easier problem.  This problem I had in mind was showing how the number 10 in one based could be represented in other bases.  My son suggested looking at the number 1 first, so we did both:

After talking about the easier problems for a little bit we switched from the whiteboard to the floor to see if snap cubes could help us see some geometry.  We reviewed converting the number 10 in one base to one base below to make sure that we understood how the snap cubes represented the different place values.  At the end of the video we took a quick peek at now to represent three digit numbers with the snap cubes.

Now comes the fun!  We know that 100 in base 5 converts to the number 121 in base 4.  Can we see that relationship in a geometric way with our snap cubes?

After seeing the neat geometric connection we returned to the board to talk about the algebraic connection.  This felt like  a really natural way to talk about some basic algebra, and my son seemed to be comfortable with the basic algebra we discussed here:

The next thing we talked through is the question that my son wondered about in the beginning – is there a relationship between these base conversions and  Pascal’s triangle?    We have actually done a few similar projects before.  See here, for example:

Pascal’s Triangle and Powers of 11

I didn’t want to go into too much depth since we were already 30 minutes in to this talk, but we did spend some time looking to see if the next line of Pascal’s triangle – 1 3 3 1 – had a relationship with converting cubes of numbers to different bases:

Having found that Pascal’s triangle did indeed seem to help us understand how to convert cubes into different bases, we went back to the snap cubes to see if there was a geometric connection here, too:

So, a fairly innocent looking question from our Art of Problem Solving book leads to a really fun project connecting arithmetic, geometry, and algebra.  Super fun way to spend a morning.  The more time we spend in this book, the more I’ve come to appreciate how a little introductory number theory can be a neat way to build up number sense.

A neat coincidence with place value problems

Over the last few days I’ve been working through some pretty difficult place value problems with my younger son.  These problems come from the “Algebra with Integers” section of Art of Problem Solving’s Introduction to Number Theory book.  As I wrote yesterday part of the difficulty is that we haven’t really covered any algebra yet.

Despite the lack of familiarity with algebra, my son seems to really like these problems.  He is absolutely fascinated when we arrive at the answers and is amazed that some of the problems have more than one answer.  The problems are all like the one in the video below:

Tonight I tried to throw in a little curve ball – how many two digit integers are 11 times the sum of their digits?  I wasn’t sure if he would see that this situation was impossible right away.  He didn’t, which was lucky because I really wanted to see how he would react to the algebra telling him there were no solutions.  Initially that math gave him a little trouble, but I loved seeing him realize that something was wrong without being quite sure what it was:

Definitely a fun couple of days.  I like these problems both for the easy introduction to algebra and for the place value review.   They also serve as a surprise example of a “low entry / high ceiling” problem.   I hadn’t really thought about this feature of these problems until I saw this challenge problem involving digits and decimal representation from Christopher Long on Twitter yesterday:

as well as a few fun follow ups like this one (and check Chris’s twitter feed for plenty more):

Pretty sure that the first time I saw a problem of this type was in the book Lure of the Integers by Joe Roberts which shows this interesting result:


The problems on Twitter are obviously a little harder than the ones I’m covering with my son, but I think older kids would probably like them a lot.   Even just showing some of the solutions might catch a kid’s interest since It is pretty surprising to see how large the solutions are.   I love simple to state problems that turn out to be a little more difficult than you might initially think!

Always happy to see the math I’m covering with the boys overlap in some small way with the math people are sharing on Twitter!

A neat place value exercise: what about fifty ten?

My younger son and I started a new chapter in our Introduction to Number Theory book today – “Algebra with Integers.”  It is big step up in difficulty from the prior chapters, and even more of a step up because I have really not covered any algebra with him at all.  Even though the discussions are a little longer now, many of the problems and examples remain accessible to him.  Sadly, that’s not going to remain true for much longer, so our little tour through introductory number theory will come to an end soon.

One of the first problems that we looked at today focused on place value:

Find a two digit integer that is equal to three times the sum of its digits.

Even getting going with this problem is a challenge if you’ve not had algebra, but a few concrete examples like 34 and 87 helped get the ball rolling.  Eventually he was able to write down an equation that would help solve the problem.  If the 10’s digit of the number is A and the 1’s digit of the number is B, we know that:

(1) 10*A + B = 3*(A + B),

and so 7A must be equal to 2B.  It was really interesting for me to see the leap from “the number is AB” to “the number is equal to 10A + B” happen right in front of me.

Solving this equation is no small task  and really is only possible after you recognize that A and B have to be single digit positive integers.  Once you do recognize that important piece, though, it is not super hard to see that the solution to the original problem is A = 2 and B = 7.  Thus, the number we are looking for is 27.

With that problem as a warm up we were ready for another challenge: find all two digit positive integers that are equal to four times the sum of their digits.

If we proceed as above we find that the equation we need for this problem is:

(2) 10*A + B = 4*(A + B),

which simplifies (a little more than the first one) to 2A = B.  Instead of just one solution here, there are several:  12, 24, 36, and 48 come to mind quickly once you know that B = 2A.

At this point my son told me that he thought it was interesting that these numbers were all multiples of 12 but then dropped that thought to tell me that he thought that there were no more solutions.

“Why not?”

“Because the next one would start with 5, but double 5 is 10 and 5(10) isn’t a two digit number.”

“What would it be if it was a number?”


“But the 5 is the tens digit, not the hundreds digit.”

” . . . . fifty ten . . . ?”

“Interesting.  What do we usually call that number?”


“What is the sum of the digits of fifty ten?”


“What is 4 times 15?”

“60 . . . hey, it works!”

“What about sixty twelve.”

“. . . . yes, 18 X 4 is 40 + 32 = 72 = sixty twelve.”

I left it there, but thought it was a fun conversation because it got him thinking about place value.  Just like our conversation last week about the relationship between the rows of Pascal’s triangle and powers of 11:

A day in the life: Building and Extending Number Sense

I really like being able to sneak in some ways to think about number sense and place value while talking about problems that (to him) aren’t obviously related to those topics.

Made for a fun morning.

A day in the life: building and extending number sense

I’m not entirely sure why but I’ve been spending a lot of time recently thinking about different ways to build up number sense.  About a week ago I started a chapter on similar triangles with my older son, and the problems in that chapter have helped me gain a better understanding of the importance of building “algebra sense” (for lack of a better phrase) too.    I’m surprised how many opportunities there are to focus on both of these topics now that I’m actively paying attention to them.   An odd coincidence today made me want to write up the conversations I had with my kids this morning.

But first I want to back up to a coincidence from yesterday.

As I mentioned above I’ve been studying similar triangles with my older son for a week or so.  The bit of math that seems to be giving him the most difficulty isn’t the geometry, it is working with the ratios that arise in the problems about similar trirangles.   Here’s one of the problems we worked through yesterday just to give an example of the ratios that come up in these problems:

I felt that it would be good to review some of the algebra behind these ratio equations before finishing the similar triangle section and found several sets of practice problems on Khan Academy that provided more or less exactly the review I was looking for.  Here’s one set for example:

Khan Academy Ratio problems

Although people have widely differing views about Khan Academy, I think one nice advantage that it has is that the problem sections are great for this type of review.

Interestingly last night Steven Strogatz posted this picture on twitter:

The similarity between the homework I wanted my son to do and the homework assignment in Strogatz’s tweet got me thinking about context.  The motivation to learn more about geometry was enough for my son to understand the purpose of the Khan Academy problems.  Actually, he even asked to do more.   I don’t know the context of the other homework assignment, but do think that without proper context that assignment could seem quite dull.  This coincidence from yesterday reminds me to be careful to be clear about why I’m asking the boys to do the homework I give them.

Now on to today . . . .

This morning my younger son and I were talking about palindromes (section 6.5 in Art of Problem Solving’s Introduction to Number Theory book).  We began with several simple examples – numbers like 11, 454, 34543 – and then he stopped me:

kid:  “I know a long list of palindromes.”
me:  “what is it?”
kid: [ writes the first 4 rows of Pascal’s triangle on the board ]

This example is definitely a fun one for looking at palindromes, but it also turns out to be a great one for building on number sense.  The connection I wanted to focus on was how the rows related to powers of 11, and how that connection seems to break down in the row:  1 5 10 10 5 1.

My first question to him was whether or not this specific row was a palindrome.  He surprised me by saying that although the number you get by putting all of the terms together, namely 15101051, was not a palindrome, you could get a palindrome you looked only at the last digits, so 150051.  Interesting observation.  We’ll have to return to this topic later when we talk about modular arithmetic!

My next question for him was about the powers of 11.  Starting at 11^0, the powers of 11 are 1, 11, 121, 1331, 14641, and 161,051.  Why did we lose the connection to Pascal’s triangle when we computed 11^5?  This led to a wonderful conversation about place value and eventually to showing why we did not actually lose the connection to Pascal’s triangle at all.  Really fun, and I think a neat way to talk through place value while getting in a little arithmetic practice, too.

Later in the morning my older son got tripped up on this problem from the 2006 AMC 8:

2006 AMC 8 problem 24

The problem has a really lucky connection to palindromes since an important observation in solving it is that one number is equal to another number multiplied by 101.  Talking through this problem also led to a good conversation about place value.  Luckily the notes from the conversation about Pascal’s triangle and place value happened to still be up on the board when this second conversation took place.

Seeing some of the earlier work that was on the board my older son said that he thought you could make the row 1 5 10 10 5 1 into a palindrome by working in base 11.  Ha – another unexpected response, but also now a wide open door to talk a little about what I’m calling “algebra sense.”

We quickly reviewed the place value conversation I had with my younger son about how the rows connect to powers of 11, but then looked at what happens in base 11.  Surprise –  powers of 12!!  Don’t think he saw that coming 🙂    Now maybe 5 to 10 minutes of conversation about what the polynomials (x + 1)^n and (x + y)^n look like and we’ve quite unexpectedly done some neat work that helps build up familiarity with algebra and algebraic expressions.

So a fun morning.  As I have the goal of working on number sense in the back of my mind, I’m excited to see all of opportunities that come up to work on it.  Algebra sense, too, but Strogatz’s post from yesterday reminds me to be extra careful about context.  It is fun to take advantage of the lucky times like this morning when that context appears almost by magic!

Introduction to Number Theory with my younger son

I saw this neat interview with Terry Tao yesterday:

In the first paragraph he mentions that he thinks that number theory isn’t likely to become an important subject in school math because it doesn’t have lots of applications.  I’m sure he is right, but agreeing with the idea doesn’t mean I have to like it!  I’m working through Art of Problem Solving’s “Introduction to Number Theory” book with my younger son this year and we are absolutely having a blast.   I’m obviously not suggesting a trip through Hardy and Wright, but the basic introduction to number theory in this book is so engaging, so fun and so useful for building up basic arithmetic skills, that I would happily suggest it for any kid looking to learn a little extra fun math.

The full talbe of contents is listed on the Art of Problem website here:

Go there for the chapters and subsections, but if you want a quick taste of the book the chapter titles are:

1. Integers:  The Basics

2. Primes and Composites

3. Multiples and Divisors

4. Prime Factorization

5. Divisor Problems

6. Special Numbers

7. Algebra with Integers

8.  Base Numbers

9. Base Number Arithmetic

10. Units Digits

11. Decimals and Fractions

12. Introduction to Modular Arithmetic

13. Divisibility Rules

14. Linear Congruences

15. Number Sense

I went through this book with my older son (also when he was in 3rd grade) and stopped after chapter 13.  I will probably stop at the same place here.

Maybe the Terry Tao interview from yesterday planted the seed in my mind, but the work we did this morning got me so excited that I wanted to write about it.  The problem we were tackling seemed pretty innocent to me at first:

Problem 4.7:

(a) Find the prime factorization of 45.

(b) Find the prime factorization of each of the four smallest multiples of 45 larger than 45:  90, 135, 180, and 225.

(c) What is the relationship between the prime factorizations from (b) and the prime factorizations from (a).

Yesterday we talked a little bit about factor trees and part (a) just reviews that topic.  He writes that 45 is 5 x 9 and then 5 x 3 x 3.    Not much to discuss, so we move to part (b).    I should say that he’s not writing out the products in the same way that I am here, he’s writing factor trees like the picture below.   Not sure how to format those trees in WordPress (or if taking the time to figure it out would improve the post!!).

Factor Trees

Next up  90 = 9 x 10 = 3 x 3 x 2 x 5.    I was expecting to see that he’d write 90 = 2 x 45, but I’m actually pretty happy to see that he didn’t think about this problem in that way.

135 = . . . . long pause.  Long, long pause, but he’s thinking so I don’t interrupt.  Suddenly he writes that 135 = 9 x 15 = 3 x 3 x 3 x 5.   I like the long think about factoring 135.  Hopefully that thinking is helping to build up a little number sense.

Next 180 = 10 x 18 = 2 x 5 x 2 x 9 = 2 x 5 x 2 x 3 x 3.

Now for the fun:

225 = long pause.

“Well, I know that 300 equals 15 x 20 and that 90 equals 15 x 6, so I know that 210 is 15 times something.”

Long pause.  Long pause and then he writes that 225 = 14 * 15.

“Are you sure?”


“Ok, l’ll tell you this – I don’t know what 14 x 15 is, but I know that it isn’t 225.  How could I know that?”

“Pause . . . . 14 is even and 14 x 15 has to be even. ”

“Interesting –  why don’t you multiply out 14 x 15 and see what it is.”

“[working it out]  210.”

“Good.  Remember that you said that 300 was 15 x 20 and that 90 was 15  x  6.  Do you see how to get to 210?”

“Yes, just subtract.”

“Great, now lets look at 225 again. ”

“225 = 15 x 15 = 3 x 5 x 3 x 5.”


Now, on to part (c) – what is the relationship between the factors above?  The goal here, I think,  is to notice that all of the numbers we factored in parts (a) and (b) could be written as 1 x 45, 2 x 45, 3 x 45, 4 x 45, and 5 x 45, but the way the numbers (or factors, I guess) were written on the board did not make that relationship obvious.    He thought about the question for a while and noticed that all of the numbers on the board had two 3’s and a 5 as factors.  It was neat to see him come to that conclusion and then eventually notice that what was left over was 1, 2, 3, 4, and 5.

So, a nice arithmetic review and a neat way to learn about factors and multiples all in one innocent litte problem.  He seems to really enjoy writing out the factor trees for various numbers – easy to forget how fun it is to learn ways to represent numbers that you’ve never seen before.  I also think that exercises like this are a great way to build number sense – so much thinking about multiplication in this problem.

As I said above, I’m a little sad to agree with the idea that number theory isn’t going to play much of a rule in a normal school math curriculum any time soon.  Maybe not every single kid is going to find exercises like this to be exciting, but I think that lots of kids will.   I’m sure enjoying walking through this book with my son.  Sort of sad to think that it is going to be my last time through it 😦