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:
Anyway, the 6th grade standards that support talking about this Numberphile video with kids are:
(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.
For 8th Graders – Terry Tao’s “Cosmic Distance Ladder” lecture
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.
Mike's Math Page 