My favorite – watching problem solving ideas develop

This is sort of an accidental “my favorite” but I spent 30 minutes with my older son today and found myself thinking about how much I love watching problem solving ideas develop in kids.

“Problem Solving” is notoriously hard to define – and since I’m in a happy mood, I’m not going to try to define it 🙂  It is, at least in my mind, though, a skill that you can watch develop over time.

Today my son worked through four old AMC 10 problems that had given him difficulty the first time through them.  We had not looked at or reviewed these problems since he worked through these tests, so, although he had seen the problems before, he’s not previously been able to solve them.  This afternoon with just a few nudges here and there he was able to work through all of them.  Along the way are some pretty nice examples of what a kid looks like doing math.

All of the problems can be found on Art of Problem Solving’s site here:

The 2009 AMC 10 A hosted on Art of Problem Solving’s website

and here:

The 2008 AMC 10 b hosted on Art of Problem Solving’s website

The first problem was #14 from the 2009 AMC 10 A – it is a problem about absolute values:

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There are a lot of plus and minus signs to keep track of in this problem, and he does a nice job of approaching the problem in a pretty systematic way to help keep track of all of those signs.


The second problem is #17 from the 2009 AMC 10 a – it is a problem about 3-4-5 right triangles

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There are a lot triangles similar to a 3-4-5 triangle in this problem and keeping track of the sides is made a little extra difficult by how the various triangles scale. In one of the triangles the longer leg has length 3 and in another triangle the shorter leg has length 4. That trickiness does trip him up, but luckily he does catch the mistake (because his original answer wasn’t one of the 5 choices).


The next problem was #14 from the 2008 AMC 10 b – it is a problem about rotations:

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This problem gave him a lot of difficulty. There’s a little bit of geometry to keep track of and also you have to keep track of a few plus and minus signs at the end. His solution here is a good example of working through a few initial misconceptions to arrive at the correct solution:


The last problem was #19 from the 2008 AMC 10b.


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This problem looks like a super challenging 3d geometry problem – it is really easy to have a reaction similar to – “well, I really don’t know anything about water in cylinders.”

What I loved about his work in this problem is that he figures out what he can do and then figures out how to use that information to solve the problem. At one point he says something like “wouldn’t it be great if these were 30-60-90 triangles?” Loved that!


So, the work today really did make me happy. I love having the opportunity to work on math with my kids, and I love watching their problem solving skills slowly develop over time.

Some book suggestions for a 14 year old who loves math

I saw this tweet from Aperiodical retweeted by Steven Strogatz this morning:

Thinking about book recommendations for a 14 year old interested in math was sort of fun, so I pulled a few books that we’ve used for projects (plus one or two more) off of our bookshelf.  Here are some that I think a kid interested in math would probably find interesting:

A wonderful mathematical coloring book from Alex Bellos and Edmund Harriss, we’ve done three projects from it already!

Fibonacci, Zome, and Patterns of the Universe

Patterns of the Universe Part 2

I found out about this book through one of Evelyn Lamb’s blog posts. It gives you a fascinating look at the various aspects of Egyptian mathematics – we did three projects from it that are all linked here:

Count Like an Egyptian Part 3

I assume this is the Matt Parker book that was referenced in the Aperiodical tweet. I just found it last week at the same time I found Patterns in the Universe:

It looks like a great way to introduce a kid to some fun ideas in math.

This book is one that you’d have to use more selectively with a younger kid because lots of the examples assume knowledge of Calculus. However not all do and there are some fantastic ideas that really show the power of mathematical thinking.

Zome Geometry combined with a Zometool building set opens up the world of 3d geometry to a kid in ways that are almost impossible to describe. I wish there was a cost-effective way to get the Zometool sets in the hands of every kid.

Here’s our latest Zometool project and there are probably 40 more on the blog.

Can you believe that a dodecahderon folds into a cube?

Patty Paper Geometry, like Zome Geometry, is an eye opener. I’ve never seen an approach to geometry (2d geometry in this case) like the one outlined in Serra’s book. Essentially an approach that simply involves tracing and folding figures allows idea after idea from geometry to fall right into your lap.

Really Big Numbers is a book for kids with some “really big” ideas hiding in the background. We used it for a neat project here:

A few project for kids from Richard Evan Schwartz’s “Really Big Numbers”

and Jim Propp did a nice review of the book here:

Jim Propp on “Really Big Numbers”

Keith Devlin’s book isn’t aimed at kids, but I think a kid interested in math will find it fascinating. It walks you through some of the most challenging unsolved (at least at the time of publication) problems in math today and is a great introduction to the ideas that mathematicians think are important.

Like “Street Fighting Mathematics”, not all of this book is going to be accessible to a 14 year old. Parts of it are, though, and those parts plus the incredible pictures might be incredibly inspirational to a kid who is interested in math. Here’s one project we based on Farriss’s ideas:

Frank Farriss’s Patterns

Tanton’s book is really hard to find, but if you do stumble on it you’ll find tons of clever math ideas, questions, and projects that should delight a kid interested in math. If you can’t find it – don’t worry too much, Tanton is an incredibly active writer and sharer of math. Just follow him on twitter – he’s inspired tons of our projects!

Our projects inspired by James Tanton

“Bridges to Infinity” was a gift from my high school math teacher when I was 15. It was my first introduction to math that was outside of traditional school math / math contest math. It was an amazing thing to read back in 1987 – I had no idea that the world this book describes even existed.

Pickover’s book is full of amazing ideas from 100’s of different areas of math. Each comes with a picture and a short, one-page explanation. Great fun to just flip through and if something catches your eye just hop on the internet to find out more. We’ve done many projects based on my kids asking questions about something they saw in this book. For example:

Banach Tarski, Hilbert curves, and Infinite sets


Counting Geometric Properties in 4 and 6 dimensions

These last three pics come from some fun books by Ivan Moscovich and Theoni Pappas. The books by these two authors should be on the shelf of any kids who are interested in math – they are absolutely wonderful.

Christopher Danielson’s square problem

Saw an interesting problem on twitter last night:

with this one extra clarification on the shape:

Seems like a good problem for a kid learning geometry, so I tried it out with my older son tonight. First we looked at the two twitter posts to make sure that he understood the problem and I asked him to come up with a plan for how to solve it:


Next we went to the white board to work out the solution. He did a good job following through on this plan, which was nice. One thing that I thought might give him a little difficult was that the 4th root of 3 appears in his solution. However, he did not simplify the numbers in the problem the way I expected him to, so the 4th root of 3 didn’t appear 🙂


Definitely a fun little problem. One alternate problem that I skipped because I thought it would be too difficult was finding the area of a different square hiding in Danielson’s picture. The “center” of the 4 pointed star is also a square – what area does it have if the kites have area 1?

Fibonacci, Zome, and Patterns of the Universe

The kids wanted to do an other project using Patterns of the Universe today:



Flipping through the book last night I found a neat page featuring golden rectangles, so I turned that into a little half coloring / half zometool project for this morning.

We started by talking about the picture in the book. The kids were interested in the spiral pattern and my older son remembered a few of the properties of “golden” rectangles:


Next we built a golden spiral on the living room floor using our zometool set. We actually did a similar project about a year ago, so the ideas weren’t totally new:

Fibonacci Spirals and Pentagons with our Zometool Set

I liked hearing their description of how the Fibonacci sequence appear in the shape – it was a nice way to get talking and thinking about math ideas:


While I cleaned up the Zometool pieces the kids colored in Patterns of the Universe. Here are their descriptions of their colorings. Here’s the one from my younger son:


and here’s my older son:


So, a fun little project for kids connecting the Fibonacci numbers to some geometric patterns. It is neat to hear how the kids think about these connections. I really love how the coloring ideas in Patterns of the Universe lead so naturally to fun projects!

Comparing Zager and Zimba

Saw a couple of interesting reads in the last couple of weeks, and they don’t have much overlap.

Tracy Johnston Zager wrote a piece on math apps for kids:

My Criteria for Fact-Based Apps

and Jason Zimba, one of the people behind the Common Core Math curriculum, wrote about how he’s helped his own kids with math here:

Can Parents Help With Math Homework? YES

Giving the timing, I assume the 2nd article was at least in part written to clarify some points in this article from a few weeks ago

Back Off Parents: It’s not your job to teach Common Core Math when helping with homework

where his quote:

“The math instruction on the part of parents should be low. The teacher is there to explain the curriculum,” said Zimba.

got a little more publicity than usual.

What caught my eye in Zimba’s more recent piece was this paragraph:

“Parents can also help at home with skill building and fluency practice—things like memorizing basic math facts. When it comes to skills, practice is essential. It helps students to have someone to flash the cards or pose calculations to them. I have made flashcards that we use at home, and my kids sometimes use digital apps like Math Drills.”


Particularly because Zager’s piece went in nearly the opposite direction when it came to math apps – for example:

“I don’t want to see naked number drills, especially not for 3rd graders. Flashcards embedded in silly or glitzy contexts are still flashcards. I want to see mathematical models like arrays, groups, hundreds charts, and number lines. ”

It certainly appears from the screen shots on the Math Drills app page:

Math Drills on the Itunes web page

that this app wouldn’t meet many (if any) of the criteria that Zager looks for in a math app for kids.


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Anyway, both articles are fascinating reads. It is interesting to me to see influential people in math education having ideas that seem almost almost totally opposite of each other.

Can you believe that a dodecahedron folds into a cube

Last week I saw an incredible post by Simon Gregg:

In Gregg’s post there is an amazing GIF of a dodecahedron unfolding into a cube:

dodecahedron fold

which Gregg found on this other amazing blog post:

The Golden Section, The Golden Triangle, The Regular Pentagon and the Pentagram, The Dodecahedron

After seeing the post I guessed that there would be a way to make the shape from our Zometool set and gave it a shot on Tuesday while the kids were out for the evening:

Playing around with a neat post from Simon Gregg

Today I went through the same exercise with the boys. It took about an hour – because it isn’t obvious, especially for kids, how to make the shapes – but it was so much fun. The project has a great balance between “there’s no way to do this” and “there must be because I just saw it”. Definitely one of the most interesting Zome projects we’ve ever done.

The boys did all of the work on their own. My only suggestion to them was to build the original dodecahedron out of medium blue zome struts. This choice minimizes the number of Zome balls you need later in the project which also minimizes confusing about the shape.

We started by looking at the original gif:


After that the boys decided to build a dodecahedron and along the way the thought that putting a cube on the inside would help:


The shape in the previous video gave them an idea of how the dodecahedron would unfold into a cube. It took probably 20 minutes for them to build the shape representing the folded up doedcahedron. Their approach was to build the cube first and they try to construct the parts on the inside. Understanding all of the rotations and how the various pieces fit together is fairly challenging.


One bit of this shape that the boys found interesting was the shape on the inside, so I had them build that shape next. This part of the build probably took about 15 minutes – even holding the shape in the cube right in front of you, this shape is not super easy to understand:


Finally, I had them connect up the Zome balls inside of the shape to form an icosahedron. It was pretty surprising to me to find this icosahedron hiding inside of this “8 pointed star.” They built that shape and we wrapped up the project:


So, I think this is a fantastic project for kids. The approach that Simon Gregg took with paper is incredible, and if you have a Zometool set you can create the various shapes pretty quickly. I’m still amazed that an dodecahedron can fold up into a cube! Platonic solids are amazing 🙂

A good (and fun) thing that happened today – half a punch!

Through my prep work for Family Math night at my younger son’s school I ran across a fun little activity that had been used in a prior year – Fold and Punch:

Today I did the activity with 4th grader before he went off to school. The full write up is here:

Fold and Punch with my younger son

The really great part of this little morning activity came at the end when he was working on the last challenge – three dots:



It might not be obvious from the picture, but it appears that he’s got too many folds in the paper to solve this puzzle correctly. He started by folding the paper in half down the diagonal so the “punch” is going to produce more than three dots.

He had a clever solution to that problem – half a punch:



It was fun to see him thinking outside of boundaries of the problem!

Fold and Punch

I’ve been spending a little time over the last week getting ready to run the Family Math nights at my younger son’s school. There are 5 nights – one for each of the grades K-5 (with the grade 4 and 5 night combined for some reason).

The way the nights run is that there are a bunch of intro activities for kids (and parents!) to do as they arrive and then 2 or 3 longer projects over the next hour. Three boxes filled with the intro activities from prior years were handed to me when I agreed to run Family Math night this year – and some of these activities are really great. For instance, there was this amazing coincidence when Anna Weltman tweeted the “H puzzle” this past weekend:

There was also a great activity called “Fold and Punch”:

I’m extra excited about this activity because one of the longer activities that I’m doing with the 2nd and 3rd graders uses Katie Steckles’ amazing video about the Fold and Cut theorem:


Here’s a link that has the 3 projects that we after watching Steckles’s video:

Our 3 fold and cut projects

Today I tried out the fold and punch activity with my 4th grade son. Here’s how it went:


Also, he kept working and finished the remaining 4 patterns when the camera was off. The way he did the last pattern was pretty clever. Hopefully you can see the folds he used, especially the one down the diagonal of the paper:


How did he get only 3 holes with those three folds? Well . . . half a punch 🙂


I can’t wait to see how both the fold and punch and fold and cut activities go with the 2nd and 3rd graders – such fun projects 🙂

First root problems

My son had bit of a struggle with problem #18 from the 2007 AMC 10 b today. Here’s a link to the problem:

and here’s the problem itself:

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Tonight we took a closer look at this problem.  He was able to solve it, but during the solution we had an interesting detour caused by the square root of 2.

Here are his introductory thoughts about the problem as well has his first steps toward the solution:


Next we went on to solve the equation he came up with in the last video:

\sqrt{2} x = x + 2

His idea for how to solve this equation is to square both sides:


After finding the answer to the equation, we went back to the original linear equation and found other ways to solve it. My son’s solution starting around 1:05 and going for maybe 45 seconds was fascinating to me. I hope that after this short little additional talk that he is more comfortable solving linear equations.


So, a challenging problem for sure, but it was nice to stumble on the little algebra trouble, too. Hopefully a little extra practice here and there will help him gain a better understanding of both algebra and geometry.

A post from Simon Gregg showing a dodecahedron folding up into a cube

Saw this cool. Tweet from Simon Gregg today:

Which has this incredible image:

dodecahedron fold
which Gregg found on this other amazing blog post:

The Golden Section, The Golden Triangle, The Regular Pentagon and the Pentagram, The Dodecahedron

I couldn’t stop thinking about the dodecahedron folding up into a cube. We’ve done several similar projects:

A Rhombic dodecahedron folding up into a cube

Cubes in a dodecahedron

5 tetrahedrons in a dodecahedron

But I never knew that a dodecahedron could fold up into a cube – why did nobody tell me this! I figured it would be possible to do it with our Zometool set, so I gave it a shot while the kids were out tonight. I’m definitely going to have the kids try it out this weekend.

Here’s a dodecahedron that shows one of the inscribed cubes. This shape is made with medium blue Zome struts as the edges of the dodecahedron and long blue struts as the edges of the cube:

Once you see the shape above made out of Zome pieces, you can see how it might fold up, so I made the 6 individual pieces of the dodecahedron that were outside of the faces of the cube:

Next I put them together! Here’s what a dodecahedron folded up into a cube looks like:

And here’s what the inside shape looks like, the “pyritohedral dodecahedron” from Gregg’s blog post:

Looking at the inside shape spinning around, I wondered what it would look like if you connected up the Zome balls on the inside. It turns out that those balls connect up to form an icosahedron made out of short blue struts!!

What an amazing thing to learn on twitter today! Can’t wait to try out this project with the boys this weekend.