Sharing the Eigenvectors from Eigenvalues paper with my son

Yesterday I saw a neat tweet from Natalie Wolchover:

I was excited about the result when I first read Wolcover’s original article, but even more excited about the new paper as, by incredibly lucky coincidence, I’m covering eigenvalues and eigenvectors with my older son right now!

The paper gives a simple example of the “eigenvectors from eigenvalues” formula using this matrix:

Yesterday I had my son compute the eigenvalues and eigenvectors for this matrix, which is a nice exercise for someone who learned about those ideas two days ago! Today we tried to use the formula from the paper.

We began by looking at the formula and discussing the 3×3 matrix:

Next I had him work through the standard calculation for one of the eigenvectors:

Before moving on to the final formula, we needed to get some eigenvalues for one of the special submatrices in the formula. Unfortunately we had a little calculation goof that took a minute to find, but we eventually got the right answers:

Finally, we worked through one example of calculating the value for a component of one of the eigenvectors. This part probably could have been done a bit better by us, but live math isn’t always perfect!

I think this new paper is an incredible lucky break for anyone teaching linear algebra now or in the future. It really isn’t that often that a new math paper has a result that is accessible to young students. It was really fun to share these ideas with my son tonight!

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: 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. The Breakthrough Prize lectures Lucky moment for me this morning when I happened to be looking at my screen just as Steven Strogatz posted the links to a few Breakthrough Prize lectures: Michael Harris’s article in Slate a few weeks ago drew my attention to the Breakthrough prize in math: Michael Harris writes about the Breakthrough Prizes in math In particular, this statement from the article got me thinking: “Tao—the only math laureate with any social media presence (29,000-plus followers on Google Plus)—was a guest on The Colbert Report a few days after the ceremony. He is articulate, attractive, and the only one of the five who has done work that can be made accessible to Colbert’s audience in a six-minute segment. But Colbert framed Tao as a genius (which he assuredly is), not as someone who can get them jumping up and down in the aisles.” Though I’m sure that Harris is correct, it seemed to be quite a shame that the work of these mathematicians wasn’t really accessible to the general public. One of the reasons that I thought this was a shame is selfish – I want to learn from these great mathematicians. I want other people to be able to learn from them, too. Terry’s Tao’s public lecture at the Museum of Math is a great example of how you can learn from these world class mathematicians. That lecture is filled with beautiful math that the general public (and kids especially) can understand. I’ve done three projects with my kids already from that lecture: 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 I’d love to see more public lectures like Terry Tao’s lecture at MoMath and use them to show kids more about what mathematicians do. Well . . . what I didn’t know until I saw Strogatz’s post this morning was that each of the Breakthrough Prize winners gave a public lecture as part of the prize. Yes!! I haven’t had time to view all of them, but the lectures by Terry Tao and Jacob Lurie are absolutely tremendous. I’m not sure that they prove Harris wrong, since they aren’t really going into detail about their work, but . . . . If you want to get a better understanding of what math is, watch these lectures. If you want to help other people, including kids, understand what mathematicians do, these lectures are a great starting point: Jacob Lurie’s Breakthrough Prize talk: Terry Tao’s Breakthrough Prize talk: For good measure, here are the other talks. I have not viewed these ones yet, but I am excited to watch them, too. It is so great to see lectures like these ones online. I’m really happy that the Breakthrough Prizes are helping to connect these amazing research mathematicians to the general public. That connection has to be a great step forward for math. The remaining three lectures are below: Maxim Kontsevich’s Breakthrough Prize talk: Richard Taylor’s Breakthrough Prize talk: Simon Donaldson’s Breakthrough Prize talk: 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! Terry Tao’s MoMath Talk Part 2: Clocks and Mars Last week I wrote about finding Terry Tao’s incredible public lecture delivered at the Museum of Math and how that lecture provides many great examples you can use to talk about math with kids: Terry Tao’s MoMath Lecture Part 1: The Earth and the Moon for ease, the direct link to the Terry Tao lecture is here: Today I wanted to use a second example from that lecture for a little math talk with the boys. This topic comes from approximately 42:30 into the video when Tao discusses Copernicus’s calculation of how long it took Mars to orbit the sun. This calculation is an incredible scientific achievement, especially when you consider that telescopes hadn’t even been invented yet! In the lecture Tao describes the remarkable story behind the calculation, but does not go into the details of the calculation itself. To be clear, that’s not a criticism – the point of his lecture was to tell the story not to dive into the details. Exploring the details of this particular calculation is a great topic to discuss with kids, though. The only background material required is some basic knowledge about fractions. We began this morning by watching the (approximately) 5 minute portion of the talk in which Tao describes how Copernicus calculated the time it took for Mars to Orbit the sun. Following that we went to the whiteboard to talk about what we learned, and to head down the path of understanding the calculation in detail. The starting point I chose for understanding the calculation is asking questions about the angles formed by the hands of clocks. I will say at the start that it was a little harder for my kids than I was expecting. The discussion and the explanations below are not at all flawless and have several false starts. As I’ve said many times, that’s what learning math (and, in this case, a little physics) looks like. Watching the films of this discussion prior to publishing this post has reinforced my feeling that Tao’s lecture is a great spring board to talking math with kids. Having looked at a few examples of when the angles between the hour hand and minute hand of a clock would be zero, in the next part of the talk we began to try to drill down on the math. The starting point for the discussion here was the observation by my older son that the minute hand moves 12x faster than the hour hand. In this video we try to write down some expressions that describe how fast the two hands of the clocks are moving: The next step was writing down an equation that told us how far the hour and minute hands would move in “t” minutes. In retrospect I wish I would have made a different choice in the approach here since jumping directly to the algebra made a simple idea a little harder than it needed to be. If I could do it again I’d probably cover the ideas in this video nearly in reverse (and I’m annoyed with myself for getting frequency and period reversed, too. Can’t get everything right . . . .) However, even with the little bit of extra time that introducing the algebra at the wrong moment led to, the discussion here did get us to an equation that looked a lot like the equation Terry Tao had written down in his presentation slides. At the end of the last video we got to an equation that helps us understand when the hands of a clock are exactly on top of each other – now we solve it! Solving this equation is a great exercise for kids who have a little familiarity with fractions. We sort of stumble out of the gates with the solution, but once we get on the right track we actually get to the end in sort of a neat way. With all of this background out of the way we can return to the equation that Terry Tao had in his presentation. We being this part by briefly talking about difference between our clock equation and the equation that Copernicus solved.. After that introduction we solve the equation and determine how long it takes for Mars to orbit the Sun! I’m really excited about using more examples from Terry Tao’s lecture to talk math with kids. There are so many great things about this lecture – for instance the incredible historical information and the great opportunity to see Terry Tao speak on an accessible topic – but for me the new examples the talk contains for talking about some basic school math with kids is the best thing about this public lecture. Who would have thought that calculating the orbit of Mars just boiled down to simple fractions?!? Using Terry Tao’s MoMath public lecture to show math to kids Recently I became aware that the Museum of Math in NYC has videos of more than 50 lectures given at the Museum in the last year (or maybe few years). The lectures discuss the use of math in an astonishing variety of fields ranging from pure theory to every day life (a recent lecture was about math and cooking, for example, though I don’t know if that one is online yet). The collection of lectures is here (click on the “Math Encounters” link at the top of this video to see the list): One of the lectures that caught my eye was Terry Tao’s “The Cosmic Distance Ladder.” That talk is here: It caught my eye for a couple of different reasons. First, Tao is one of the top mathematicians in the world and it is a rare treat to see him speak. An even rarer treat to see him deliver a lecture designed (quite successfully) to be accessible to the public. Having watched this talk several times now, I think many parts of it provide fun and exciting examples for kids of how math has been used to advance science. I hope to use many pieces of his lecture to show my own kids some important math and science. I began with two fun, and frankly amazing, examples today: (1) The approximation of the radius of the Earth by Eratosthenes (beginning around 17:20 in the video of the lecture), and (2) The approximation of the radius of the orbit of the Moon by Aristarchus (beginning around 25:45) Both of these are obviously remarkable scientific achievements, and Tao’s lecture does a wonderful job of explaining the ideas behind the discoveries. The lecture doesn’t dive too deep into the calculation of these results, though. That was not the point of the lecture – not at all – so my point isn’t even remotely a criticism. Rather I took it as an exciting opportunity to use the videos to teach. Exploring the calculations in the lecture a little more carefully seems to me to be a great way to use this lecture to help kids learn a bit of math, a bit of physics, and a bit of history. Only some basic geometry is needed to understand the calculations. Diving in a little deeper into the math this morning with the boys was really fun. First off is the approximation of the radius of the Earth: Second is the approximation of the radius of the orbit of the Moon: The calculations we do are slight simplifications (as is noted in the original lecture), but I think the important mathematical ideas are here. Discussing the limitations of these calculations and ways to improve them could be a fun student project. I’m really happy to have stumbled on this collection of lectures at the Museum of Math, and am super excited to spend some time over the next few months trying to figure out fun ways to used them to help kids see interesting examples of how math is used (and has been used) in the world. Fun math that I saw this week At the beginning of 2014 Numberphile published an incredible interview with Ed Frenkel with the provocative title: “Why do people hate mathematics?”” There is a particularly interesting exchange around the 5:00 mark: Numberphile: “Let’s apportion some blame. Let’s blame someone. Sounds to me like you are blaming high school teachers . . . . back in our school days they were making us paint fences instead of showing us Picasso.” Frenkel: “Well, if I really were to assign blame, I would assign the blame to myself and to my colleagues – professional mathematicians. We don’t do nearly enough of exposing these ideas to the public in an accessible way. Often times we aren’t willing to come up with good metaphors and analogies. That exchange has really stayed with me. Later in the year, actually about a month ago, David Coffey wrote a nice piece in which he answered a similar question – Whose fault is it that you aren’t good at math?: Whose Fault is it that you aren’t good at math? I like his answer a lot -> you aren’t good at math because you didn’t have the experiences that you needed to be good at math. I wrote a little too long of a response to Coffey’s piece here where I suggested some places where students (and, well, everyone) might find fun math experiences: Responding to Cafid Coffey’s Challenge Despite writing way too much last time, I want to write more because I’ve seen so many great math experiences just in the last few days. (1) Let’s start with this incredible public lecture from Terry Tao at New York’s Museum of Mathematics: This lecture seems to be almost exactly what Frenkel was taking about in the piece of his interview that I quoted above. Tao shows some wonderful ideas of mathematics and how those ideas helped us understand how to measure distance in the universe. A beautiful and accessible lecture from one of the world’s top mathematicians. It has been online since the beginning of June and as of today doesn’t even have 700 views!! Ugh, though to be fair I try to look out for stuff like this and didn’t even see it until yesterday. I hope more people find out about this lecture and are able to watch it. (2) Speaking of the Museum of Math, this weekend they finished their portion of the MegaMenger project: http://www.megamenger.com/ . My family went down to participate in the build and it was really fun to see all of the kids helping out and playing around in the Museum. Hopefully there will be more projects like this one that can show kids that math is more than their 20 question algebra assignment. Here’s a picture of my son standing inside of the finished product: (3) Fawn Nguyen’s digit puzzle. Last week Fawn Nguyen shared a wonderful digit problem that she did with her class. A few days later the online math world was buzzing left and right about her problem. On Sunday morning 5 different people had shared it on my twitter feed. So fun and I’m so happy for Fawn’s incredible work is being recognized by an ever-growing audience. But don’t take my word that this is an “utterly kick-ass” exercise. Try it out, too: (4) Continued Fractions Continued fractions have a special place in my heart because my high school math teacher, Mr. Waterman, loved them. He taught us out of the book pictured below (sorry I only have the picture side by side with Geometry Revisited, but trust me, that’s a great book as well!): Amazingly I saw two ideas in the last week where continued fractions either have or could have played a role. It really is a beautiful subject and it is a shame that it no longer has much of a place in high school or college math programs. In any case, here’s a really neat blog post from Sam Shah showing how he incorporated continued fractions into a lesson in his class: Substitution and Continued Fractions and here’s a fun tweet from Steven Strogatz from this morning showing a problem in which continued fractions could help students make a fun connection: Since the picture from Gilbert Strang’s book doesn’t come through, let me expand a little on the continued fraction connection. On page 36 of the book Strang mentions the pattern he’s showing comes from a connection that $2\pi$ has to the fractions 44/7, 25/4, and 19/3. Those three fractions just so happen to be the first three “convergents” of the continued fraction for $2 \pi$. The next one is 333/53 which might be fun to look for in Strang’s pattern. Anyway, it has been great to see all of this fun math online (and in person) during the last week. Hopefully there will be many more weeks like this one to come. Terry Tao shares a math circle problem I saw Patrick Honner tweet about Terry Tao’s math circle post last night: Hopefully the link in the tweet to the original post works, but in case it doesn’t Tao’s post on google+ is here: https://plus.google.com/u/0/114134834346472219368/posts/CR1ZoNe9ojQ%20 and here’s the problem itself: “Three farmers were selling chickens at the local market. One farmer had 10 chickens to sell, another had 16 chickens to sell, and the last had 26 chickens to sell. In order not to compete with each other, they agreed to all sell their chickens at the same price. But by lunchtime, they decided that sales were not going so well, and they all decided to lower their prices to the same lower price point. By the end of the day, they had sold all their chickens. It turned out that they all collected the same amount of money,$35, from the day’s chicken sales.  What was the price of the chickens before lunchtime and after lunchtime?”

In his post Tao asks not for the solution but for the thought process you went through to solve the problem.  I like these “thought process” posts, so I thought I’d give another one a shot.  My first one was here:

http://mikesmathpage.wordpress.com/2014/08/12/problem-solving/%20

For this math circle problem I made the same assumptions that many of the commentators on Tao’s blog were making -> (i) the number of chickens sold by each of the farmers in both the morning and the afternoon was not zero, and (ii) that the both the morning and afternoon prices were an integer number of cents.

The first thing I did was look for a simple example just to get my arms around the problem.  With a morning price of $4 per chicken and an afternoon price of$1 per chicken you can get close to the situation the problem describes.  I didn’t put much thought into those two prices – I just chose those two prices because $4 was above the average price for the 10 chickens and$1 was below the average price of the 26 chickens.

Farmer 1:  8 chickens at $4 plus 2 chickens at$1 = $34 Farmer 2: 6 chickens at$4 plus 10 chickens at $1 =$34

Farmer 3:  3 chickens at $4 plus 23 chickens at$1 = \$35

This short exercise gave me some hope that this initial guess was pretty close to the answer to the problem.

Next I wrote down some equations.   Call the morning price x and the afternoon price y, and let A, B, and C represent the number of chickens sold at the morning price by Farmers 1,2, and 3 respectively.  We have (in cents):

(1) Ax + (10 – A)y = 3500

(2) Bx + (16 – B)y = 3500

(3) Cx + (26 – C)y = 3500

A little equation combining, namely (1) + (2) – (3), yields the equation:

(4) (A + B – C)*(x – y) = 3500

Since both terms on the left hand side are integers the problem now is to find the right factors of 3500.  Fortunately 3500 doesn’t have too many factors, and double fortunately the example from the beginning tells me to expect (A + B – C) to have a value around 11.

There are two factors of 3500 that are on either side of 11, namely 10 and 14.  The values of (x – y) for those two choices are 350 and 250.  I checked 350 first.

Rewriting equation (3) you can get:

(5) C*(x – y) + 26y = 3500.

When (x – y) is 350 this equation becomes 350C + 26y = 3500, or

(6)  26y = 350*(10 – C).

This equation will not have any solutions when both y and C are positive integers since the left hand side is divisible by 13 and the right hand side isn’t.

Bad luck, I suppose, but we do have the second potential solution from above when (A + B – C) = 14 and the price change from morning to afternoon is 250 cents.  As Tao has asked to not give away the solution to the problem, I won’t work through that math but will say you can find a way to make both sides of the equation similar to (6) be divisible by 13 here.  Yay!

Definitely a fun challenge problem.  Unlike the Tim Gowers’s IMO problem that inspired my first “thought process” post, I hope to use this problem for a neat little Family Math project with my kids later this week!

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:

http://www.artofproblemsolving.com/Store/products/intro-numtheory/toc.pdf%5B/embed

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!!).

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?”

“Yes.”

“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.”

“Awesome.”

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 😦