Expected value and Dice

Saw this post on twitter tonight:

The exercise for the students is a neat one involving biased and unbiased dice.  If you read the Twitter thread a little further you’ll see suggestions about running Chi squared stats, too.  All great stats examples.

In 2003 and 2004 I was involved in an television game show called “Play for a Billion.”  On the show 1,000 contestants tried to guess a 6 digit number from a number that I had selected.   If anyone did successfully guess the number I had selected they would have won one billion dollars.

I did an interview for the show, but it didn’t make it on air.  A first cut of that interview is below.  I thought it might be to use for a stats example because I picked my six digit number using dice.  To determine if it would be ok to use the dice I had to run through a bunch of stats that were pretty similar to what the exercise above is asking the students to do.

This project was one of the most fun math-related projects that I’ve worked on in my career.

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

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

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

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

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

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

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

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

Mr. Honner’s 13-14-15 Triangle and a surprising unsolved problem

Last week Patrick Honner wrote about an old New York State Regents exam question involving a 13-14-15 triangle:

Yesterday I was pleasantly surprised to see that the Art of Problem Solving Introduction to Geometry book led off the section on Heron’s formula by looking at a 13-14-15 triangle.  Fun little coincidence:

Finding the area of the 13-14-15 triangle is a nice exercise in both algebra and geometry.  We start in on the algebra first:

Having set up two equations relating our two variables, we now tackle the algebra of solving those equations.  I like using examples like this to build up a little algebra sense.  The first instinct is often to multiply out all of the terms and see what happens, but here we have a few other algebraic techniques which provide an interesting (and informative) alternate path.

At the end of this video we discuss the surprising geometric solution to this problem, too.

We just found out that the 13-14-15 triangle can break into two right triangles with integer side lengths.  It turns out that constructing a box with integer lengths for the sides and for the diagonals is a famous (and unsolved) problem.  It seemed like talking about that problem was a natural extension of the discussion we had on the 13-14-15 triangle, so we took a short little diversion and looked at the “Euler brick.”

Finally, we went to the kitchen table to discuss the surprising property of the 13-14-15 triangle that Patrick Honner pointed out in his blog post.   The problem from an old NY State Regents exam asked students to find the angle between the sides of length 14 and 15 in the 13-14-15 triangle.  The question is designed  to test a student’s knowledge of the law of cosines, but, as Mr. Honner points out in his blog post, there are geometric solutions that do not require the law of cosines to answer this multiple choice question.

One of the geometric solution is so clever that I had to try it out for myself.   First I used a rule and compass to construct the triangles that Mr. Honner uses in his blog post:

Compass Triangles

An amazingly close fit – just as the picture in the blog suggested!

Following this construction, I used our 3D printer to make physical copies of these triangles so you could see how they fit together.  We used these objects to walk through Mr. Honner’s solution:

So, a fun project with the 13-14-15 triangle and an unsolved problem.  Love finding a neat blog post online just as the same concepts are coming up in one of the books we are following.  Can’t wait for the next time it happens!


Grothendieck, Heron, and Brahmagupta

My son is taking the AMC 8 today so I thought we’d have sort of a light morning with math.  The section of the book we were meant to cover today was Heron’s formula for the area of a triangle.  Not exactly a “light” subject if you delve into the proof!  Instead, I took today as an opportunity to talk a little math history.

Quite a bit after Heron, in the 600’s actually, the Indian mathematician Brahmagupta found an amazing generalization of Heron’s formula.  Brahmagupta’s formula calculates the area of a cyclic quadrilateral and the triangle is a special case when one of the sides has length zero.

Though I know very little of the mathematics that Grothendieck actually studied, I took the example of Brahmagupta finding Heron’s formula as a special case of his formula as an example of solving a problem via generalization.   In the pieces I’ve read about Grothendieck in the last few days, his ability to find the right generalization of a given problem seemed to be one of his great gifts that the writers focused on.  The analogy with Heron’s formula and Brahmagupta’s formula  appeared to my best shot at mentioning Grothendieck’s contribution to math to my son.

The first video introduced Heron’s formula and discussed a little bit of history.

The next part of our talk introduced Brahmagupta’s formula and talked about why the triangle is (possibly!) a special case of this formula.  For it to be a special case, we just need to show that you can circumscribe a circle about any triangle.

In the next video we try to see if it is possible to always circumscribe a circle about a triangle.  This led to a discussion of a slightly easier problem – how do you find the set of points that is an equal distance from two given points?  Solving that problem leads to the idea that it may, in fact, always be possible to draw a circle that hits all three verticies of a triangle.

Finally, we jump over to the kitchen table to try to construct the circumcircle of a given triangle.  It is a neat construction and it is always a nice surprise when you get the three perpendicular bisectors to intersect in a single point!

So, a fun morning showing a little math history and a couple of really amazing geometry formulas.   Nice to have a light day every not and then.

Adding in binary with Duplo blocks

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

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

Our project from this morning went pretty well:

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

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

Inequalities and Mr. Honner’s triangles

Earlier today Patrick Honner posted an interesting piece about a famous triangle:

His post sparked a perhaps odd connection in my mind with a blog post about solving inequalities that Bob Lochel had posted earlier in the week:

The connection had to do with how you describe a triangle using inequalities.  I thought that creating 3D prints of the triangles in Patrick Honner’s blog would be an interesting exercise in inequalities, so I gave it a shot.

First, though, I was pretty surprised by the 2nd picture in the triangle blog, so I used a ruler and compass to draw the triangles (yes, a ruler – I measured out lengths of 13 cm, 14 cm, and 15 cm.  Sorry construction purists!!)

Compass Triangles

Amazing how close the 13-13-13 equilateral triangle is to fitting perfectly into the 13-14-15 triangle – wow!

Next I went to Mathematica to see how tough it would be to describe a 3D triangle.  This is only the third shape hat I’ve created on Mathematica to 3D print, so I am by no means an expert.  I wouldn’t be surprised at all to learn that there is an easier way to do what I did.  Actually, I’d be stunned if there wasn’t, but I wanted to use inequalities specifically for this project even if there was a better way.

Though my code is clumsy, it wasn’t too difficult to create the inequalities that would describe the 3D triangles:


The code above was for the equilateral triangle.  It assumes that the three verticies of the triangle are at points (0,0), (13,0), and (6.5, 13  \sqrt{3} / 2) in the x-y plane.  The 4 variables after the a,b,c, and d are just giving me the slope and y intercept of the 3rd side of the triangle (that I used “xint” for the y intercept says more about the quality of the code than any other words I could speak .. . . ).

The next line defines the triangle and gets to the part about inequalities.  We’ll define this triangle by all of the points satisfying:

(i) the y coordinate is greater than 0.  (y = 0 describes the line segment at the bottom of the triangle.)

(ii) the point is underneath the line y = (a/b) * x.  (This formula describes the line segment on the left side of the triangle.)

(iiI) the point is under the line y = mx + xint.  (This formula describes the line segment on the right hand side of the triangle.)

(iv) and finally, to make the triangle three dimensional, I’ll take the z coordinate be between -0.25 and 0.25.

That’s it for the simple equilateral triangle.  The 13-14-15 triangle and the third triangle I made – the 13-14-2 triangle – are a tiny bit harder since they require a little trigonometry to get the third coordinate (I assumed (0,0) and (13,0) were points in all three triangles).  If you are working through this same exercise with students, I suggest also using (0,0) and (13,0) for two of the points in all three triangles.  The reason is that you have to switch up one of the inequalities to describe the 13-14-2 triangle, and thinking through that switch of inequalities is a nice exercise for students.

Printed Triangles

So, a fun connection between two blog posts and a neat printing exercise which doubles as an interesting inequality exercise.  Made for a fun Sunday afternoon recreational math adventure.  Even got to show my 8 year old how to make an equilateral triangle using a ruler and compass!

As usual, always happy to see and play around with the math ideas that people share on Twitter.

Which reminds me, I’d previously written a post about Mr. Honner’s square, so I guess this post was just a matter of time!!


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?!?

Complete this sentence: Math is ______!

Yesterday I saw a super cool student project posted on twitter – click the link in the tweet to see the stop motion video:

and a few follow up pictures, too:

The student’s project gave me an idea for a fun activity that would tie together some ideas from arithmetic, algebra, and geometry.  We started by building a replica of the student’s cube out of our snap cube set and then talking through some of the ideas we’d be looking at today:

After the short introduction we moved to the living room to start in on the project.  I like to emphasize the problem solving strategy of looking at easier problems first.  Keeping with that spirit, to start in on the geometry we first looked at a line.

I’m not sure that I did a good job explaining the connection between the algebra and the geometry right away, but hopefully looking at this easier example helped get going with that connection.  Also, because my son mentioned the relationship between Pascal’s triangle and (x + 1)^n in the last video, I decided to count the type of building block we use at each stage so that we return to that connection at the end:

Now we moved on to talking about turning a 3×3 square into a 5×5 square with the snap cube pieces we have handy.  My younger son picked up on a pattern was able to construct the 5×5 square pretty quickly.  He struggled with the algebra (which isn’t surprising, since we’ve not talked about algebra!) so we spent a bit of extra time on the connection to algebra in this movie.

I got so caught up in the algebra in the last video, that I forgot about the geometry.  Instead of moving on to the three dimensional case right away we returned to the 2 dimensional case to talk about dimensions and geometry.  The specific geometry that I wanted them to see here is how the number building blocks of each type we have at a given stage relates to the number of those blocks in the next stage.    Understanding how to count the number of building blocks at each stage helps understand the connection to Pascal’s triangle (although, I did not emphasize this connection at the end, unfortunately):

With all of this background we are now able to understand the student project that was posted on Twitter yesterday.  We look at (i) how this cube arises from sliding a square in the third dimension, (ii) how to count the number of different building blocks, and (iiI) how to write down an expression for (x + 2)^3 from our construction.  There is so much fun math hiding in this shape!

. . . and whoops – watching the movie just now it turns out that we did not actually write down the expression for (x + 3)^3  Dang!  Oh well, it comes from the numbers on the screen:  (x + 2)^3 = x^3 + 6x^2 + 12x + 8.

Now that we’ve understood a little bit about how to make 3 dimensional cubes from 2 dimensional cubes, we tried the next step – understanding how to make a 4 dimensional cube!  Everything works the same way as before, though it is now much harder to visualize so we have to let the math guide us..   The multiple connections that we’ve talked about in the prior stages help us understand the 4 dimensional shape even without being able to see it.  Amazing!

To wrap it all up we went back to the white board to see the connection between Pascal’s triangle and the number of pieces in each of our constructions.  I wrote the two triangles on the board side by side and my younger son noticed the connection between the two number triangles.  I was hoping one of them would see it, so I was super happy when he noticed the relationship!  We finished up just by talking about how cool all these connections are:

The connection between the algebra, geometry, and arithmetic demonstrated by this student project are really are amazing.  Seeing the project posted on Twitter yesterday helped me see a great way to explore these connections – that’s why I love looking at all of the math people share online!  Probably no better way to end the blog post than with what my younger son said at the end of the last video:

Me:  Complete this sentence:  Math is . . . . .

him:  crazy!

What learning math sometimes looks like part 2

Yesterday we had a really good talk about Pythagorean triples.  My son struggled a little bit with the algebra, but I was happy to see that struggle since I think it represents what learning math really looks like.  That post is here:

What Learning Math Often Looks Like

Today we talked a little more about Pythagorean triples during our normal school time and it seemed like it was time to move on to another topic.  Luckily for me Kate Nowak had shared her thoughts on a great Five Triangles geometry problem yesterday evening:

Because of her post I didn’t have to think too hard about what problem to talk about next.  My son worked for a little while and came up with a solution to the problem that had a small error.  We discussed the error and I wanted to see if he would be able to talk about the error and how to modify his solution on camera.

I was happy to see him talk through that solution and followed up with a new question.  That question gave him a little trouble, but just like yesterday that trouble turned out to be a good example of thinking about math.  As I said in the first blog post, learning math isn’t always a straight line.   I’m really happy to see (and share) these examples because I really think this is what learning and thinking about math looks like:

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

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

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

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

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

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

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

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

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

Pascal’s Triangle and Powers of 11

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

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

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