Using AoPS’s “Intro. to Number Theory” to build number sense

I love the books from Art of Problem Solving.  Not just because of the subjects that the books cover, but also because the books give me a great chance to secretly review other subjects.

The Introduction to Number Theory book is amazing all by itself.  What makes me love it, though, is the opportunity it gives my younger son to build number sense.

Here’s a pretty typical example from this morning.  The problem (from the review section in Chapter 4) asks you to list the common divisors of 84 and 132.   I assume that the point of the problem is for the student to find the greatest common divisor, but instead my son lists all of the divisors and gets some great arithmetic practice.

Here’s the first part of the problem:

and here’s the 2nd half.

It is fun to watch him build up his number sense. There will be plenty of time to talk about greatest common divisor later!

A nice algebra / geometry problem with lines

My son is working his way through Art of Problem Solving’s Introduction to Algebra book and was stuck on one of the challenge problems in the chapter about lines (chapter 8). We talked through the problem this morning.

Here’s the problem and his solution. He was able to find one of the points that satisfied the conditions of the problem, but he was surprised to learn that there was a second point:

Having learned about the second solution, I asked him to try to find it:

After he found the second point I wanted to show him that there was a little more to this problem that you might think. Here’s how we can get to a solution by using a little algebra and the Pythagorean theorem:

Rather than crunching through all of this algebra, I plugged the equation into Wolfram Alpha. He was pretty surprised to see what the general solution looked like 🙂

So, a fun problem and an pretty challenging one, too. I was happy to see that you could extend the “surprising” fact that there were actually two solutions not just one to a new “surprising” fact that there were actually infinitely many!

An escalator problem

My older son is working his way through Art of Problem Solving’s Introduction to Algebra book. He works through the problems in each section of the book and we discuss the ones that are giving him trouble. Today he had some difficulty with a problem about an escalator.

The problem is a twist on one of my favorite math contest problems from back when I was in high school – Problem 10 from the 1987 AIME:

Screen Shot 2016-04-10 at 9.00.18 PM

Here’s our discussion of the version of the of the problem in Chapter 7 of the AoPS’s Introduction to Algebra:

 

At the end of the last video my son was thinking about trying to find the speed of the sister running up the escalator. Here we turn our attention to a different quantity – how many steps does that sister take?

The main idea in this part was to try to gather together then information that we know in this problem. For example, we know the speed of the two escalators (because he assumed a speed), we know how many steps the escalator has, and we know how long it takes for the sister running up the escalator to get to the top.

 

To wrap up the project, I asked my son to summarize the solution. We had jumped around a little bit to get to the solution, so I wanted him to do this summary just to make sure that he’d understood the ideas that we’d discussed during the solution:

 

Two books that I secretly used to build number sense

[sorry this is written quickly and not really edited – I wrote it while my son was at a school event, so limited time.]

Saw this really great post today from Geoff Krall (via a Tracy Johnston Zager retweet):

It made me think of the two books that I’ve gone back to every now and then to help build a little number sense –

  

I guess I’ve used one of the books so much that it is no longer recognizable!! Here’s where you can buy them:

Art of Problem Solving’s Introduction to Number Theory

Art of Problem Solving’s Introduction to Counting & Probability

For me the point of these books was never about learning number theory or probability, though I’m sure the boys picked up a few things here and there. Instead the point was finding ways to build number sense by talking through either (i) some interesting properties of numbers, and (ii) some neat counting problems.

Here’s one example project from number theory:

Using Divisibility Rules to Build Number Sense

The last part of this project uses a really neat divisibility rule for 7 idea that I found on Tanya Khovanova’s website.

One other project mentioned in this blog is learning to do arithmetic in binary using duplo blocks. Here’s one of our addition projects which I thought was a great way for kids to see numbers in a slightly different light:

We were even able to use some of the ideas we learned in binary to reinforce some ideas about decimals:

Writing 1/3 in binary

Here are a few number sense examples inspired by ideas in the Introduction to Counting and Probability book:

Counting Arrangements around a Table

The hockey stick theorem and some fun geometry in Pascal’s Triangle

The discussions that we’ve had over the years about Pascal’s triangle sometimes let the kids find Pascal’s triangle in surprising places – :

Talking about Pólya’s Urn with kids – inspired by Jim Propp’s blog post

Again, the point of using these books for me wasn’t to teach number theory or probability, but rather just to find some fun problems that would hopefully help to build some number sense. The idea from Geoff Krall’s post that really reinforced this idea for me was this one:

Also, we’re not talking about shutting everything else down classroom-wise, lest you’re worried about losing precious class time. While coverage is overrated, let’s put that aside for now, shall we? We’re talking 10-20 minute activities and discussion here, maybe a couple times a week.

The nice thing about these two books from Art of Problem Solving is that they are full of neat problems that you can use for 10-20 minutes here and there for a little non-standard (and hopefully fun!) number sense building 🙂

Why I love the Art of Problem Solving books – part 2

Had two really wonderful experiences talking through problems in our math books tonight. My older son is studying in Art of Problem Solving’s Introduction to Geometry book and my younger son is studying in their Prealgebra book.

The current topic for my older son is 3D geometry and the specific topic today was pyramids. An exercise that gave my son a little trouble today is an absolutely brilliant problem about pyramids. Here’s our walk through it tonight:

The current topic for my younger son is a little introductory geometry and the specific topic is circles. The problem we talked through tonight was the last example in the circle section – I love that this problem is just right at the edge of his reach right now. His ideas are really wonderful and open the door for some really fun follow up conversations tomorrow morning:

I’m sure there are other great math books for kids out there, but I love being able to crack open these books and see wonderful example after wonderful example after wonderful example. I love that the variation in problem difficulty, too. The problems always seem to keep the kids engaged and excited about math. It is so much fun to have the opportunity to go through these books with them.

The Power of a Point

My older son and I are studying the Power of a Point chapter in our Introduction to Geometry book right now. Yesterday I saw a fun problem on Art of Problem Solving’s website and thought it would make for a neat discussion with my son.

The problem is Intermediate Problem #1 here:

See Intermediate Problem #1 here

Just after going through the project I saw this tweet from Patrick Honner:

I mentioned to him that we’d just looked at a fun problem which led to this exchange – I hadn’t noticed that the problem could be interpreted in two different ways:

With that introduction, here’s our talk about this problem.

My son is initially surprised that you can even talk about the perimeter of the triangle since you don’t know the lengths of any of the sides. He notices a few other geometric ideas, too, and eventually uses the some of the ideas about the power of a point to conclude / remember that two tangents to a circle from the same point have equal length.

This is the key idea in the problem, though we don’t head directly to the solution after discovering this idea.

So, having walked up to inches away from the solution in the last video, now we finish it. It is pretty surprising that you can say anything about the perimeter of a triangle when you seem to know almost nothing about the triangle. That’s one of the nice surprises of this problem.

After we finish up with his solution, we wrap up by showing another little geometric trick that gives you an idea of what the perimeter might be. Fun little problem:

A nice problem from our Geometry book and a neat Julie Rehmeyer interview.

I’ve written previously about why I like the books from Art of Problem Solving:

Why I like the Art of Problem Solving Books

and I got another nice reason to like them today with an excellent problem in our geometry book’s section about parallelograms. I’ve written two other posts recently about parallelograms in the last week, too – we had quite a lot of fun in this section 🙂

Patrick Honner’s Challenging Parallelogram Problem

Cleaning up an Oversight in our talk about Patrick Honner’s Parallelogram Problem

This section of our Geometry book tilted a little more theoretical than some of the previous sections. This tilt led to some good conversations about techniques of proof and also equivalence. The chapter ends with a statement that these three basic properties of a quadrilateral are equivalent:

(1) The opposite sides are equal,
(2) The opposite angles are equal, and
(3) The diagonals bisect each other.

If any one of the above statements is true, the quadrilateral is a parallelogram. It is pretty tough work to get through all of the examples and proofs that lead to this statement.

The exercises at the end of this section end with a real gem of a challenge problem:

Problem 8.3.7 The diagonals of a convex quadrilateral ABCD meet at E. Prove that the centers of the circumcircles of triangles ABE, BCE, CDE, and DAE are the vertices of a parallelogram.

This problem seems pretty intimidating when you first read it. In fact, when I read the problem for the first time it was certainly not obvious to me why the statement would be true. What do the centers of those circumcircles have to do with each other?

We started with a simple picture

Problem Picture

and then discussed how you would find the centers of the circumscribed circles. My son remembered that the center of the circumscribed circle of a triangle is the intersection of the perpendicular bisectors of the sides. This observation led to the following picture:

Perpendicular Bisectors

You get the idea from the picture that you may indeed have a parallelogram (though, less so from the picture we had on the whiteboard – ha!). Even with the idea that you may have a parallelogram, there’s no doubt that this new picture is pretty intimidating.

But, intimidating or not, it led to two really great discussions. In the morning we focused on definition (2) above and found why the opposite angles in quadrilateral XYZW above are equal. Along the way we learned about the angles made by pairs of intersecting perpendicular bisectors in a triangle.

In the evening we came at the problem a different way by understanding why the opposite sides of quadrilateral XYZW are parallel.

So, I was happy to have the opportunity to talk through this problem with my son. I think it is really important to learn how to approach problems that seem intimidating (or problems where you have no idea at all where to start). I had no idea how to solve this problem when I first read it and I really hope that our two talks through this problem helped my son see how to make progress in this type of situation.

The math lesson I’m trying to get through to my son is one that Julie Rehmeyer talks about in this interview:

Julie Rehmeyer’s “Inspired by Math” interview

What’s always stuck with me from this interview is the story that begins around 31:30 and in particular the part beginning around 34:40 about proving that 0 + 0 = 0.

She comes to meet her adviser, Phil, and is stuck on problem about proving that 0 + 0 = 0. Phil has not worked with the axiomatic system that Julie is studying and does not immediately know how the proof will go. He tells her, though, that he knows that if sits down and thinks about it for a bit that he will figure it out. He knows this because he is a mathematician and he’s learned how to work through proofs like this one. He tells her that if she sticks with math for a while that she’ll develop that same set of skills, too.

That’s a better summary than I could ever give about what I think working through the Art of Problem Solving books helps me do for my kids.

Problem Solving part 2 – an old AIME problem

[note: I’m up in Boston today and decided to take a trip to the MIT library to check out a book that Jordan Ellenberg mentions in his book “How not to be Wrong.” I didn’t know the MIT libraries were closed until 11:00 am today (they used to be open 24 hours!!), so I wrote this post fairly quickly in the hour I had to kill waiting for the library to open!. Sorry if it seems rushed – it sort of was!]

I few months ago I wrote about Tim Gowers live blogging his solution to one of the IMO problems:

Problem Solving and Tim Gowers’s live blogging an IMO problem

I thought that live blogging problem solving was a good idea because I think that kids (and everyone) needs to see that most solutions you get to problems aren’t the super perfect “official solutions” and don’t come to mind immediately. In that post I talked through my solution to a number theory problem that David Radcliffe had put on Twitter. Today’s blog is my second attempt at live blogging a problem. I saw today’s problem in Art of Problem Solving’s Precalculus book in the chapter 8 challenge problems.

It also happens to be problem 15 from the 1991 AIME – see the bottom of the page here:

The 1991 AIME Problems hosted by Art of Problem Solving

Here is the problem:

For a positive integer n define S_n to be the minimum value of the sum:

\sum_{k = 1}^{n} \sqrt{ (2k - 1)^2 + a_{k}^{2} }

where a_1, a_2, \ldots, a_n are positive real numbers whose sum is 17. There is a unique positive integer n for which S_n is also an integer, find this n.

 

The “live blogging” of my solution is below:

Background:

I had a nice (and short) discussion on twitter with Patrick Honner about Art of Problem Solving’s Precalculus book. I personally like this book because of chapter 8 – the geometry of complex numbers. This chapter, and in particular section 8.5, has content that I’ve not seen anywhere else.

Following that discussion, I grabbed my copy of the book to take a fresh look at chapter 8. I ended up back in the challenge problems and the problem I’m writing about today caught my eye.

My first reaction, frankly, was fear. Back in high school I’d not really understood analysis all that well, and did not have a good grasp of theorems that helped with this type of summation problem – the Cauchy Schwarz inequality, for example: The Cauchy-Schwarz Inequality. An “analysis” approach to this problem probably requires an even more general inequality – Hölder’s inequality – though I would not have known that in high school.

Whenever I see this type of problem, even today, I remember the fear of the analysis ideas that I had in high school. Funny how those old high school fears never seem to go away!

But wait – why would the Art of Problem Solving folks put an analysis problem in the chapter about the geometry of complex numbers? There must be a geometric solution here somewhere? Where is it?

So, let’s look at a few terms:

(1) n = 1. The “sum” is just one term \sqrt{1^2 + 17^2}. Certainly not an integer, but also certainly reminiscent of the Pythagorean theorem and a right triangle with legs of length 1 and 17.

(2) n = 2. The sum is now \sqrt{1^2 + a^{2}} + \sqrt{3^2 + (17 - a)^2}. Now there are two right triangles – one with legs 1 and a, and another with legs 3 and (17 - a). The sum of the integer length legs is 4, and the sum of the other legs is 17.

Oh, wait, I see what’s going on. The general case is going to have a bunch of right triangles whose “heights” always adds up to 17 and whose “lengths” will sum up to 1 + 3 + 5 + (2n - 1) = n^2.

See here for the sum of the odd numbers:

 

How cool! The minimum value of the sum we are being asked to consider is going to come when we can make all of the hypotenuses of these triangles form a straight line, and that length will be \sqrt{ (n^2)^2 + 17^2} from a “large” right triangle with length n^2 and height 17.

Sweet – per the statement of the problem, there must only be one right triangle with integer side lengths where one of the side lengths is 17. That shouldn’t be too hard to find.

We know there’s a 3,4,5 triangle, and a 5, 12, 13 triangle, and a 7, 24, 25 triangle – which helps see the pattern fairly quickly. For an odd number, you’ll get a right triangle with integer sides by finding two consecutive numbers that add up to the square of the odd number. From the three examples above: 4 + 5 = 3*3, 12 + 13 = 5*5, and 24 + 25 = 7*7.

So . . . 144 + 145 = 289 = 17*17, so there must be a 17, 144, 145 triangle. By luck 144 is a perfect square and we indeed have a right triangle with integer sides having one leg equal to 17 and the other leg equal to a perfect square.

That means the minimum value of the sum we were looking for is 145, and the value n for that particular triangle is 12.

Fun and super clever problem. Love the connection between geometry and algebra here – especially because the algebra side of this problem still makes me nervous 🙂

Most of the thoughts above were in my head – my “work” for the problem is in the picture below. I’m glad I saw the geometric solution before trying to dive into all of the analysis which, I assume, is way more grungy and way less of a fun solution.

Solution Pic

Why I like the Art of Problem Solving books

I’m working through two of Art of Problem Solving’s math books with the kids this year.  My older son is studying  “Introduction to Geometry” and my younger son is studying their “Introduction to Number Theory” book.   You can buy those books here:

http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:geometry

http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:nt%20

I’ve never taught any elementary math before, but the approach that Richard Rusczyk and his team take to explaining the subjects really resonates with me.  It was actually their Prealgebra book that got me hooked on their approach, and it sure seems the more of their books I work through the more I love what they do.    Though the approach is great, the icing on the cake is the collection of problems.  Just an absolutely outstanding set of problems ranging from introductory to Olympiad level problems to challenge all types of kids looking to learn from their books.

We ran across a really nice challenge problem in the chapter about congruence today that made lots great math conversation.  Both the problem itself and the more advanced theorem from “Geometry Revisited” that it hinted at were super fun to talk through with my son.   Here’s us doing a quick review of the problem itself tonight:

and then here are a couple of theorems on Napoleon triangles from “Geometry Revisited” that the problem practically begs you to talk about  🙂

The Art of Problem Solving problems are so great to begin with and it is sort of doubly fun to be able to use them as stepping stones to show some more advanced geometry.   Next up for us is the section on perimeter and area.  Can’t wait.