# Sharing Math from mathematicians with the Common Core

Last fall Lior Patcher wrote a fantastic blog post about unsolved problems in math and the Common Core math standards:

Lior Pachter’s “Unsolved Problems with the Common Core

This piece made a big impact on how I think about math that is worth sharing with kids.

The general idea of sharing math with kids has been on my mind since I saw Numberphile’s interview with Ed Frenkel:

The line about 50 seconds in to the video, in particular, really resonated with me – “How do we make people realize that mathematics is this incredible archipelago of knowledge?”

In the last several weeks a few other experiences have had me thinking more about sharing math with kids. First, I ran five “Family Math” nights for the kids and parents at my younger son’s elementary school. It was amazing to see how excited and enthusiastic everyone was about topics from pure math. The 2nd and 3rd graders excitement over Larry Guth’s “no rectangles” problem (to be described in more detail below) was incredible.

Second, Cathy O’Neil’s piece How do we make math enrichment less elitist which discusses Peg Tyre’s article in the Atlantic The Math Revolution made me think even more about how to share math with kids – all kids.

Finally, just yesterday at the gym I was re-listening to Marcus de Sautoy’s wonderful series podcasts: “A Brief History of Mathematics.” One part of the section on Hardy and Ramanugan basically stopped me in my tracks – de Sautoy claims that Ramanujan’s discussion of the sum 1 + 2 + 3 + . . . = -1/12 was one of the ideas that caught Hardy’s attention when Ramanujan first wrote to Hardy. That sum was the subject of an interesting (and quite controversial!) video from Numberphile.

Hearing that story sort of tipped me over the edge and made me want to write about sharing fun math ideas with kids. These ideas don’t require anything too fancy or \$1,000 math clubs or anything like that – just an internet connection and some pencil and paper. Finally, with a hat tip to Lior Patcher, I point out how the ideas fit into the Common Core math standards for grades K – 8. All of the information about the Common Core math standards comes from this page: The Common Core Math Standards

For Kindergarten – mathematical coloring sheets:

I first saw the idea of mathematical coloring sheets from the online math magazine Math Munch. They published some coloring sheets made by math artist Dearing Wang. I used Wang’s coloring sheets with my kids here:

If you like great math for kids, check out Math Munch

Then, a Google Plus post from mathematician Richard Green got me thinking more about how you could use advanced mathematical ideas to get kids talking about shapes:

Using a Richard Green Google Plus Post to talk about Geometry with my son

Finally, the amazing book Patterns of the Universe was written by mathematicians Alex Bellos and Edmund Harriss!

So, I think coloring is a great way to get young kids talking about shapes and patterns. The relevant pieces of the Common Core math standards for kindegarden are:

(i) From the introduction:

(2) Students describe their physical world using geometric ideas (e.g., shape, orientation, spatial relations) and vocabulary. They identify, name, and describe basic two-dimensional shapes, such as squares, triangles, circles, rectangles, and hexagons, presented in a variety of ways (e.g., with different sizes and orientations), as well as three-dimensional shapes such as cubes, cones, cylinders, and spheres. They use basic shapes and spatial reasoning to model objects in their environment and to construct more complex shapes.

(ii) From the Standards themselves:

Identify and describe shapes.

CCSS.Math.Content.K.G.A.1

Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.

For 1st Graders – A simple version of the Game of Nim:

Last November I had an interesting twitter conversation with Tracy Johnston Zager that came up when she was talking about a simple version of the game of Nim with elementary school kids. (The version of the game is explained in the project linked below.) The kids Zager was working with were asking lots of different questions about the game, and one set of questions was sort of surprising – does it matter what pieces you use to play the game?

I was interested to hear what my kids would think of this question, so I played the game with them and asked them if the game would change if we used different pieces:

A question from Tracy Johnston Zager that caught my eye

I really like the idea of using this simplified version of Nim with kids because there’s lots of nice arithmetic and problem solving involved in the game itself, and I did use it for a 20 minute project during the 1st grade Family Math night last week:

Plans for the K-1 Family Math nights

It was interesting to see that the 1st grade Common Core math standards specifically address the point the kids were asking about – the difference between defining and non-defining attributes. Here are the sections from the 1st grade standards that support using this game with 1st graders:

(i) From the Introduction:

(1) Students develop strategies for adding and subtracting whole numbers based on their prior work with small numbers. They use a variety of models, including discrete objects and length-based models (e.g., cubes connected to form lengths), to model add-to, take-from, put-together, take-apart, and compare situations to develop meaning for the operations of addition and subtraction, and to develop strategies to solve arithmetic problems with these operations. Students understand connections between counting and addition and subtraction (e.g., adding two is the same as counting on two). They use properties of addition to add whole numbers and to create and use increasingly sophisticated strategies based on these properties (e.g., “making tens”) to solve addition and subtraction problems within 20. By comparing a variety of solution strategies, children build their understanding of the relationship between addition and subtraction.

(ii) From the Standards themselves:

Reason with shapes and their attributes.

CCSS.Math.Content.1.G.A.1

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

For 2nd Graders – Larry Guth’s “No Rectangles” problem

I probably can’t fully explain how happy I was to see the kids at both the 2nd and 3rd grade Family Math nights have fun playing around with this problem. The problem is pretty easy to state:

Suppose you have an NxN square grid – what is the maximum number of squares you can color in without 4 squares that were colored in forming the corners of a rectangle (with horizontal and vertical sides)?

Here’s how the game went when I played it with my own kids:

Larry Guth’s “No Rectangles” problem

Obviously you aren’t going to talk about 10×10 grids with young kids, but the problem using 3×3 and 4×4 grids captivated the younger kids at Family Math night. It was actually hard to wrap up both the 2nd and 3rd grade evenings because the kids were just running up with lots and lots of different patterns that they thought were maximal solutions.

The 2nd grade Common Core math standards that support playing around with the “no rectangles” game are:

(i) From the Introduction:

(4) Students describe and analyze shapes by examining their sides and angles. Students investigate, describe, and reason about decomposing and combining shapes to make other shapes. Through building, drawing, and analyzing two- and three-dimensional shapes, students develop a foundation for understanding area, volume, congruence, similarity, and symmetry in later grades.

(ii) From the Standards themselves:

Reason with shapes and their attributes.

CCSS.Math.Content.2.G.A.2

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

For 3rd Graders – How many times can you fold a piece of paper in half?

This paper folding project that James Tanton did at MIT led to our very first Family Math project:

Toilet Paper used to break paper-folding record at MIT

The project is fun for kids because it is pretty surprising how few times you can fold a piece of paper in half. It is also a great opportunity to talk with kids about both fractions and exponential growth.

The 3rd grade Common Core math standards that support using this paper folding idea with kids are:

(i) From the introduction:

(2) Students develop an understanding of fractions, beginning with unit fractions. Students view fractions in general as being built out of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole. Students understand that the size of a fractional part is relative to the size of the whole. For example, 1/2 of the paint in a small bucket could be less paint than 1/3 of the paint in a larger bucket, but 1/3 of a ribbon is longer than 1/5 of the same ribbon because when the ribbon is divided into 3 equal parts, the parts are longer than when the ribbon is divided into 5 equal parts. Students are able to use fractions to represent numbers equal to, less than, and greater than one. They solve problems that involve comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators.

(ii) From the Standards themselves:

Develop understanding of fractions as numbers.

CCSS.Math.Content.3.NF.A.1

Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

For 4th Graders – Numberphile’s “Pebbling a Chessboard” game

My original introduction to Numberphile’s math videos came from this presentation of the “Pebbling a Chessboard” game by the mathematician Zvezdelina Stankova:

Our project using this video is here:

Numberphile’s Pebbling the Chessboard game and Mr. Honner’s Square

This is a fun game for kids to explore, and the connection with fractions that Stankova explains in the Numberphile video is probably going to be really surprising for kids to see. It isn’t necessary for the kids to have a complete understanding of why 1 + 1/2 + 1/4 + 1/8 + . . . = 2 (in fact, it would be absurd to expect 4th graders to completely understand this idea) but they can certainly have an intuitive feel for why it is true.

The 4th grade Common Core math standards that support using this project with kids are:

(i) From the Introduction:

(2) Students develop understanding of fraction equivalence and operations with fractions. They recognize that two different fractions can be equal (e.g., 15/9 = 5/3), and they develop methods for generating and recognizing equivalent fractions. Students extend previous understandings about how fractions are built from unit fractions, composing fractions from unit fractions, decomposing fractions into unit fractions, and using the meaning of fractions and the meaning of multiplication to multiply a fraction by a whole number.

(3) Students describe, analyze, compare, and classify two-dimensional shapes. Through building, drawing, and analyzing two-dimensional shapes, students deepen their understanding of properties of two-dimensional objects and the use of them to solve problems involving symmetry.

(ii) From the standards themselves:

Build fractions from unit fractions.

CCSS.Math.Content.4.NF.B.3

Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

For 5th Graders – The Collatz Conjecture

This’ll be my one overlap with Patcher’s post – he uses the Collatz Conjecture for his 3rd grade example.

The Collatz Conjecture looks at the following procedure:

Start with any positive integer. If the integer is even divide it by two, and if it is odd multiply it by 3 and then add 1. Now, repeat the process until you end up with the number 1.

The question is – do you always end up at 1? No one knows the answer to this question – the problem is unsolved!

We’ve looked at this problem (and variations of the problem) a few times:

The Collatz Conjecture and John Conway’s Amusical Variation

There aren’t many unsolved problems in math that young kids can understand, so that alone makes this problem fun to share with kids. The extra opportunity kids get to get in a little arithmetic practice also makes this a nice activity.

The 6th grade Common Core math standards that support sharing this problem with kids are:

(i) From the Introduction:

(2) Students develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of operations. They finalize fluency with multi-digit addition, subtraction, multiplication, and division. They apply their understandings of models for decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths. They develop fluency in these computations, and make reasonable estimates of their results. Students use the relationship between decimals and fractions, as well as the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number), to understand and explain why the procedures for multiplying and dividing finite decimals make sense. They compute products and quotients of decimals to hundredths efficiently and accurately.

(ii) From the Standards themselves:

Analyze patterns and relationships.

CCSS.Math.Content.5.OA.B.3

Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

For 6th Graders – Numberphile’s “-1/12” video

I know there is are many different opinions of Numberphile’s video about the series 1 + 2 + 3 + . . ., but I loved it:

I’ve also loved talking about this series with my kids – using the idea of “algebraic intimidation” from Jordan Ellenberg’s How not to be Wrong:

Jordan Ellenberg’s “Algebraic Intimidation”

As I mentioned in the beginning of this post, it was Marcus de Sautoy’s description of Hardy and Ramanujan discussing this exact series that gave me the idea to write this post. I think this is a wonderful example to use with kids because it shows them, among other things, that you shouldn’t feel bullied by a bunch of math. That’s why I asked me kids at the end of the project if they believed what we just did.

I also think that you can share this idea with kids, along with some historical context, and they can see some really beautiful ideas in math. I personally tracked down a copy of Hardy’s Divergent Series book to try to understand what was going on after seeing the video. Some other nice background is in this Numberphile video with Ed Frenkel:

(i) From the Introduction:

(3) Students understand the use of variables in mathematical expressions. They write expressions and equations that correspond to given situations, evaluate expressions, and use expressions and formulas to solve problems. Students understand that expressions in different forms can be equivalent, and they use the properties of operations to rewrite expressions in equivalent forms. Students know that the solutions of an equation are the values of the variables that make the equation true. Students use properties of operations and the idea of maintaining the equality of both sides of an equation to solve simple one-step equations. Students construct and analyze tables, such as tables of quantities that are in equivalent ratios, and they use equations (such as 3x = y) to describe relationships between quantities.

(ii) From the Standards

Reason about and solve one-variable equations and inequalities.

CCSS.Math.Content.6.EE.B.5

Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

CCSS.Math.Content.6.EE.B.6

Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

CCSS.Math.Content.6.EE.B.7

Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

For 7th Graders – the Surreal Numbers

This tweet from Jordan Ellenberg from last August started me down a path that ended last week with an hour long discussion of the Surreal Numbers with the 4th and 5th graders at my younger son’s school:

I wrote about the night (and the prep work) here:

Sharing the Surreal Numbers with Kids

Just as with Larry Guth’s “No Rectangles” problem, I was thrilled to see the kids (and parents) getting so excited about the surreal numbers. I think kids will really enjoy solving the little puzzles that come up – finding the value of the “blue / red” stack, for example. The usual ideas about “infinity” and the “infinitesimal” are tremendously fun, too.

The 7th grade Common Core math standards that support sharing the Surreal Numbers with kids are:

(i) From the Introduction:

(2) Students develop a unified understanding of number, recognizing fractions, decimals (that have a finite or a repeating decimal representation), and percents as different representations of rational numbers. Students extend addition, subtraction, multiplication, and division to all rational numbers, maintaining the properties of operations and the relationships between addition and subtraction, and multiplication and division. By applying these properties, and by viewing negative numbers in terms of everyday contexts (e.g., amounts owed or temperatures below zero), students explain and interpret the rules for adding, subtracting, multiplying, and dividing with negative numbers. They use the arithmetic of rational numbers as they formulate expressions and equations in one variable and use these equations to solve problems.

(ii) From the Standards themselves:

Apply and extend previous understandings of operations with fractions.

CCSS.Math.Content.7.NS.A.1

Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

CCSS.Math.Content.7.NS.A.1.a

Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.

CCSS.Math.Content.7.NS.A.1.b

Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.

CCSS.Math.Content.7.NS.A.1.c

Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

CCSS.Math.Content.7.NS.A.1.d

Apply properties of operations as strategies to add and subtract rational numbers.

Terry Tao’s “Cosmic Distance Ladder” public lecture at MoMath is an absolute gem:

In the first 45 minutes you learn how some basic ideas from geometry helped Greek scientists  find good approximations to (i) the radius of the earth, (ii) the size of the moon, (iii) the distance to the moon, (iv) the size of the Sun, and (v) the distance to the Sun. All without any technology at all – just math ideas 🙂

This lecture is a fascinating history of science and an incredible opportunity for kids to see a lecture from one of the most respected mathematicians alive today. On top of that, you get some fantastic examples of how elementary geometry helped us understand “the real world.”

The 8th grade standards that support sharing Tao’s lecture with kids are:

(i) From the introduction:

(3) Students use ideas about distance and angles, how they behave under translations, rotations, reflections, and dilations, and ideas about congruence and similarity to describe and analyze two-dimensional figures and to solve problems. Students show that the sum of the angles in a triangle is the angle formed by a straight line, and that various configurations of lines give rise to similar triangles because of the angles created when a transversal cuts parallel lines. Students understand the statement of the Pythagorean Theorem and its converse, and can explain why the Pythagorean Theorem holds, for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points on the coordinate plane, to find lengths, and to analyze polygons. Students complete their work on volume by solving problems involving cones, cylinders, and spheres.

(ii) From the Standards themselves:

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

CCSS.Math.Content.8.G.C.9

Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

In addition to the projects above, we’ve looked at ideas from mathematicians like Katie Steckles , Barry Mazur , Jacob Lurie , Laura Taalman , and too many more to name, I guess!!

One thing that makes me really happy is that so many mathematicians are sharing great math ideas on line. I’m excited to do a bit more thinking about how to share more and more of those ideas with kids.

# Numberphile’s “Freaky Dot Patterns” video

Saw this neat tweet from Patrick Honner earlier in the week:

The video itself will blow you away:

I shared it with Dan Anderson yesterday who made a couple of computer versions of patterns from the video:

Sorry the video quality isn’t so great, but it was fun talking through these patterns with the kids. We started with the square pattern:

Then we moved on to the triangle pattern. It was surprisingly difficult for the kids to understand how to describe the rotations, but eventually they figured it out. I think I’ll revisit a bit more about rotations for a Family Math Project this weekend.

The patterns in the Numberphile video make a great project to talk through with kids. I can’t wait to try a few more ideas from their video.

# Amazing math from mathematicians to share with kids

About two years ago I saw this Numberphile interview with Ed Frenkel:

One of the ideas that Frenkel mentions in the interview is that professional mathematicians haven’t done a good job sharing math with the general public. Although I’m not really the kind of professional mathematician Frenkel was talking about, I took his words to heart and have been on the lookout for math to share – especially with kids.

It turns out that there are some fantastic ideas that are out there for kids to see. Some surprising fun I had sharing Larry Guth’s “no rectangles” problem with kids earlier this week (see below) made me want to share some of the ideas I’ve found in the last couple of years, so here are a few examples:

(1) One of the most incredible lectures that you’ll ever see is Terry Tao’s “Cosmic Distance Ladder” lecture at the Museum of Mathematics in New York City:

I used Tao’s video for three projects with my kids – but there are probably 20 math projects for kids you could get out of it.

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

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

Terry Tao’s MoMath lecture part 3 – the speed of light and paralax

(2) The Museum of Math’s public lectures are a great source beyond Tao’s lecture.

Here’s a project based on Bryna Kra’s lecture:

Angry Birds and Snap Cubes – Using Bryna Kra’s MoMath public lecture to talk math with kids

Eric Demaine’s lecture was part of our Fold and Cut theorem project:

Fold and Cut part 3

and I can’t say enough good thinks about Laura Taalman’s work – she’s inspired dozens of our projects.  Just search for her name on the blog:

(3) and Speaking of Fold and Cut . . .

Katie Steckles and Numberphile put together an incredible video about the Fold and Cut theorem. I used the video this week for project with 2nd and 3rd graders at my younger son’s school earlier this week.  Steckles’s presentation is so incredible – this is the kind of math that really inspires kids:

We used it for three projects (including the Eric Demaine one above):

Our One Cut Project

The Fold and Cut Theorem is Awesome!

In prepping for the grades 2 and 3 projects I also totally coincidentally ran across a “fold and punch” exercise that is a great activity to try with kids before trying out fold and cut:

(4) Another great success with the 2nd and 3rd graders was Larry Guth’s “no rectangles” problem. I had a great time playing around with this problem with my kids, but nothing prepared me for how enthusiastic the kids in the two programs were about this problem.

Larry Guth’s “No Rectangles” problem

After the 3rd grade night, Patrick Honner sent me this picture that I used to wrap things up with the 2nd graders.

(5) The Surreal Numbers

I’d seen John Conway’s surreal numbers previously via an amazing Jim Propp blog post:

and I wanted to revisit them after finally reading Donald Knuth’s book:

Revisiting the Surreal Numbers

Infinity + 1 and other Surreal Numbers

Playing with the surreal numbers via checker stacks is an incredibly engaging way for kids to learn about mathematical thinking.

(6) Speaking of John Conway –

In the 2014 edition of the Best Writing in Mathematics Conway had an article about variations on the Collatz conjecture. It was a fascinating article that even gave us the idea to translate some of the math into music.

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

I’ve also talked with the boys about the standard version of the Collatz conjecture:

It is a great way to introduce kids to an unsolved problem in math while also sneaking in a little bit of arithmetic practice!

(7) Occasional contest math problems

I happened to run across another MoMath lecture yesterday – this one by Po-Shen Loh. He was talking about “Massive Numbers.” I thought maybe he’d be talking about the book “Really Big Numbers” by Richard Evan Schwartz:

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

or maybe Graham’s Number:

An attempt to explain Graham’s number to kids

The last 4 digits of Graham’s number

His presentation is fascinating and I even talked through the first version of the problem with my younger son:

Show that any positive integer n has a (positive) multiple which has only the digits 1 and 0 when represented in base 10.

A challenging arithmetic / number theory problem

(8) Building off of popular books by mathematicians as well as public lectures

I was surprised at how much great math writing and speaking there has been for the general public in the last couple of years.

Jordan Ellenberg’s “How not to be Wrong” inspired several projects – probably my favorite was using his idea of “algebraic intimidation” to talk about the famous 1 + 2 + 3 + . . . = -1/12 video by Numberphile. :

Jordan Ellenberg’s Algebraic Intimidation

Jacob Lurie’s Breakthrough Prize public lecture inspired two projects about a year apart from each other:

Using Jacob Lurie’s Breakthrough Prize Lecture to Inspire Kids

Using Jacob Lurie’s Breakthrough Prize talk with kids

And, Ed Frenkel, who got me thinking about sharing advanced math with kids in the first place has inspired a few projects, too:

Fine Ed Frenkel – you convinced me

Ed Frenkel, the square root of 2, and i

and one of my all time favorites:

A list Ed Frenkel will love

(9) Finally, it would be impossible to write a post like this one without mentioning the work that Evelyn Lamb is doing writing math articles for the general public. I’ve lost count of how many projects she’s inspired, but it is probably well over 20. I’m especially grateful for her talk about topology which have generated really fun conversations with the boys. For example:

Using Evelyn Lamb’s Infinite Earring with kids

Evelyn Lamb’s fun torus tweet

and

Henry Segerman’s Flat Torus

which arose after Lamb pointed out this video:

So, I’m really happy that mathematicians are sharing so many amazing ideas. I think this is the sort of math promotion that Frenkel had in mind. Hopefully it continues for many years to come 🙂

# 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 🙂

# Relations from i to geometry

Saw this video from Numberphile a few weeks ago:

The boys and I had some fun talking through a few of the examples. For instance, we saw that 5 is not a prime in the Gaussian integers because 5 = (2 + i)(2 – i).

We also saw that the set of integers combined with integer multiples of $\sqrt{2}$ has some unexpected factorization properties, for example, like 2 = (2 – $\sqrt{2}$)*(2 + $\sqrt{2}$), and 14 = 2 * 7 = (4 – $\sqrt{2}$)(4 + $\sqrt{2}$). We also played with the $\sqrt{-5}$ example in the video.

I spent the last few days kicking around a few ideas about how to explore these ideas in other ways with the boys. Part of the struggle was figuring out how to translate advanced math ideas like the polynomial ring Z($x$] / ($x^2 + 1$) into something that the boys could understand.

I couldn’t crack the code for that, but thinking about this idea led to a fun coincidence when I saw this problem from Matt Enlow today:

I won’t give away the precise solution to the problem, but my approach used the idea of a polynomial relation. A simple relation like $x^5 = 2x^3 + x$ (but not that exact relation) arose from the conditions in the problem and that relation allowed me to solve Enlow’s problem without actually having to solve for $x$.

It is neat to see similar algebraic ideas arising in totally different contests.

# Graham’s number and Skewes’ Number

Saw another great video from Numberphile today:

One thing that got my attention in the video was the comparison to Graham’s Number. Some of their previous videos about Graham’s number had inspired a few projects with the boys. For instance:

The last 4 digits of Graham’s number

The fun (and sometimes frustrating) thing about talking about Graham’s number is that it is so large that it is nearly impossible to describe. In fact, the title of this Evelyn Lame piece on Graham’s number sums it up perfectly: Graham’s numbers is too big for me to tell you how bit it is!

Skewes’number doesn’t have this little problem – you can even write the number 🙂

Skewes’ number = $10^{10^{10^{34}}}$

After seeing the video I thought these two questions might be interesting to kids learning about powers (or logarithms):

Question 1: Which is larger, Skewes’ number or this tower of 5 powers of 3 -> $3^{3^{3^{3^3}}}$

Question 2: Which is larger, Skewes’ number or this tower of 6 powers of 3 -> $3^{3^{3^{3^{3^3}}}}$

The reason I thought it would be interesting to compare to power of 3 is because of how Graham’s number is constructed.

More questions like these can be found in Richard Evan Schwartz’s book “Really Big Numbers.” A few projects that we did from that book are here (with the 3rd one being similar to the two questions above):

A few porjects for kids from Ricahrd Evan Schwartz’s “Really Big Numbers”

Oh, and by luck my older son came home from school as I was finishing this post, so I tried out question 1 with him:

# Fold and cut part 3

We decided to do one more fold and cut project tonight. Our first two are here:

Fold and Cut project #1

Fold and cut project #2

Tonight we decided to try to fold and cut some of the letters that Katie Steckles demonstrated at the end of the Numberphile video. My older son chose the letter ‘B’ and my younger son chose the letter ‘L’. We didn’t quite get the ‘L’ right, but overall both letters were a great challenge.

Here’s my older son and ‘B’:

and here here’s my younger son with ‘L’:

Feels like we could do projects like these for a long time! For some more amazing one cut shapes, check out Erik Demaine’s MoMath lecture (with the one cut shapes starting around 2:30 in the video below): (h/t to Patrick Honner for this amazing lecture by Demaine, btw)

# The fold and cut theorem is awesome

Yesterday we saw this incredible Numberphile video:

We did a really fun project this morning based on the video, too:

Our One Cut Project

There was much more to do, though – this activity has so many different possibilities for kids! One thing that was on my mind all day was shapes with holes. While my younger son was taking a bath, I decided to try out a simple square in a square shape with my older son:

When my younger son got out of the tub he gave it a try. The cool thing was that he did it a slightly different way!

Just to emphasize that comment near the end: “hey cool . . . the fold and cut theorem is awesome!”

Can’t wait to try more shapes!

# Using Numberphile’s “blob Pythagorean theorem” video in a lesson

Ran across an interesting problem in our Introduction to Geometry book today that reminded me of an old Numberphile video. My son didn’t make the connection right away, but he did have some interesting observations about the problem so I thought it would make a fun little project.

I wanted to do a quick review of the problem first, but an accidental mistake sent us down the wrong path at the start. We sort of start over about 2 minutes in to the video below. Solving problems isn’t always a straight line . . .

Now that we were on the right path, we finished up the calculation and talked about the geometric situation. He recognized that the Pythagorean theorem was hanging around somewhere, and that there might be an analogous theorem for half circles:

After we finished up I had him watch Numberphile’s excellent “blob Pythagorean theorem” video featuring Harvard’s Barry Mazur:

So, a fun morning project (even with the little false start). Nice to see some of the ideas in Numberphile’s videos coming up when we study geometry.

# 10 fun math things from 2014

I’ve been paying attention to math a little more in 2014 than I have in previous years and thought it would be fun to put together a list of fun math-related things I’ll remember from this year:

(10) Dan Anderson’s “My Favorite” post

Dan asks his students to talk about things they would like to learn more about in math class, and the students talked about subjects ranging from topology to diving scoring. I was really happy to see the incredibly wide range of topics that the kids thought would be interesting. Beautiful post by Dan and a fantastic list of topics chosen by his students – this one made a big impression on me:

Dan Anderson’s “My Favorite” post

My initial reaction to Dan’s post is here:

A list Ed Frenkel will love

(9) Laura Taalman’s Makerhome blog:

We bought a 3D printer early in the year and it allowed us to do a bunch of math projects that wouldn’t have occurred to me in a million years. Most of those projects came either directly or indirectly from reading Laura Taalman’s 3D printing blog. As 3D printing becomes cheaper and hopefully more available in schools, Taalman’s blog is going to become the go to resource for math and 3D printing. It is an absolute treasure:

Laura Taalman’s Makerhome blog

An early post of mine about the possibilities of 3D printing in education is here:

Learning from 3D Printing

and one of our later projects is here:

Klein Bottles and Möbius Strips

(8) Numberphile

It has been nearly a year since Numberphile’s fun infinite series video hit the web. I know people had mixed feelings about it, but I loved seeing a math video spark so many discussions:

I’ve used so many of their videos to talk math with my kids, I’m not even sure which of them to pick for examples. Here are two:

Using Numberphile’s “All Triangles are Equilateral” video to talk about constructions

Some fun with Numberphile’s Pythagorean Theorem video

(7) Fields Medals and the Breakthrough Prizes

Erica Klarreich’s coverage of the Fields Medals over at Quanta Magazine was absolutely amazing. Two of her articles are below, but all of them (including the videos) are must reads. Her work her made it possible for anyone to meet the four 2014 Fields medal winners:

Erica Klarreich on Manjul Bhargava

Erica Klarreich on Maryam Mirzakhan

A really cool opportunity to understand the work of one of the Fields Medal winners came when the Mathematical Association of America made an old Manjul Bhargava’s paper available to the public. I had a lot of fun playing around with this paper (that he wrote as an undergraduate, btw). It made me feel sort of connected to math research again:

A fun surprised with Euler’s identity coming from Manjul Bhargava’s generalized factorials

The Breakthrough Prizes in math didn’t seem to get as much attention as the Fields Medals did, which is too bad. The Breakthrough Prize winners each gave a public lecture about math. Jacob Lurie’s lecture was absolutely wonderful and a great opportunity to show kids a little bit of fun math and a little bit about the kinds of problems that mathematicians think about:

Using Jacob Lurie’s Breakthrough Prize talk with kids

I’m glad to see more and more opportunities for the general public to see and appreciate the work of the mathematical community. Speaking of which . . . .

(6) Jordan Ellenberg’s “How Not to be Wrong”

Jordan Ellenberg’s book How not to be Wrong is one of the best books about math for the general public I’ve ever read. I have it on audiobook and have been through it probably 3 times in various trips back and forth to Boston. My kids even enjoy listening to it – “consider the set of all integers plus a pig” always gets a laugh.

One of the more mathy takeaways for me was his discussion of infinite series and what he calls “algebraic intimidation.” Both led to fun (and overlapping) discussions with my kids:

Jordan Ellenberg’s “Algebraic Intimidation”

(5) The Mega Menger Project

The Mega Menger project was a world wide project that involved building a “level 4” Menger sponge out of special business cards. We participated in the project at the Museum of Math in NYC. The kids had such a good time that they asked to go down again the following weekend to help finish the build.

It was nice to see so many kids involved with the build in New York. It also made for another fun opportunity to explore the math behind the project a little more deeply:

The Museum of Math and Mega Menger

(4) People having a little fun with math and math results

For some family fun, check out the new game Prime Climb:

our review is here:

A review of Prime Climb by Math for Love

Also, don’t forget to have a little fun when tweeting about new and important math results. Like Jordan Ellenberg tweeting about the solution of an old Paul Erdos conjecture:

Erica Klarreich’s Quanta Magazine article on the same result was just published yesterday by coincidence:

Erica Klarreich on prime gaps

For me the math laugh of the year was Aperiodical announcing the results of an 8 year search confirming the 44th Mersenne Prime:

(3) Evelyn Lamb’s writing

Evelyn Lamb’s blog is a must read for me. I love the wide range of topics and am pretty jealous of her incredible ability to communicate abstract math ideas with ease. Her coverage of the Heidelberg Laureate Forum was sensational (ahem Breakthrough Prize folks, take note!). This post, in particular, gave me quite a bit to think about:

A Computer Scientist Tells Mathematicians How To Write Proofs

My thoughts on proof in math are here:

Proof in math

Away from her blog, if you want a constant source of fun and interesting math ideas just follow her on Twitter. For instance this tweet:

led to a great little project with the boys:

Irrationality of the Square root of 2

(2) Terry Tao’s public lecture at the Museum of Math

On of the most amazing lectures that I’ve ever seen is Terry Tao’s public lecture at the Museum of Math. I don’t know how it had escaped my attention previously, but I finally ran across it about a month ago. What an incredible – probably unparalleled – opportunity to learn from one of the greatest mathematicians alive today:

Explaining a few bits of his talk in more detail led to three super fun projects with the boys:

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

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

Part 3 of using Terry Tao’s MoMath lecture to talk about math with kids – the speed of light and paralax

(1) Fawn Nguyen’s work

When one of the top mathematicians around is tweeting about projects going on in a 6th grade classroom 2000 miles away, the world is working the right way!

Fawn is producing and sharing some of the most interesting math projects for kids that I have ever seen, and I’m super happy that her work is getting recognized. She’s probably inspired more than 20 projects with the boys, and I can’t wait for the next 20 in 2015. Here are two from this year:

Fawn Nguyen’s Geometry Problem

A 3d Geometry proof without words courtesy of Fawn Nguyen

If you have even a passing interest in fun, exciting, and generally kick-ass math projects for kids – you have to follow Fawn.