Tag Archives: geometry

Top 5 Things I Love About Pascal’s Triangle

This week, my 9th grade Prob/Stat class has been working through the Binomial Theorem. With many rich patterns and connections to explore, the unit begins with the class jigsaw, where students examine seemingly disconnected math concepts, and discover unexpected relationships.  This year we have been using Wolfram|Alpha to validate our work, and discover more patterns.

Sharing the many patterns in Pascal’s Triangle is a real boost for me. This year, few students had seen the triangle before (which is a shame, as they are 9th graders!), so this was all new to them.  If math is the science of patterns, then this is the center of the universe…would love to build a fun elective around it.  Here are my 5 favorite Pascal’s Triangle ideas to share:


Start at any of the 1’s on the outside, slide your finger along the diagonal, going deeper into the triangle. Stop at any point, and instead of continuing along the diagonal, divert to the connected number in the next row. This number will be equal to the sum of the numbers along your traced path.

Here, 462 = 1 + 5 + 15 + 35 + 70 + 126 + 210


Can you verify this one?


And think about the proofs we can develop together. Move proofs from geometry and into combinatorics.


Many students quickly identify what is happening along the diagonals:

  • Along the sides, there are 1’s
  • Along the next diagonal, we have consecutive numbers (the counting numbers)

But is that all?  Nope…more cool stuff!

  • The next diagonal reveals the TRIANGULAR NUMBERS: 1, 3, 6, 10, 15… (those numbers which make cute equilateral triangles if we make them with dots.
  • Next, we have the TETRAHEDRAL NUMBERS: 1, 4, 10, 20, 35… (making cool pyramids with marbles!)
  • And we can even locate the FIBONACCI NUMBERS, if we hunt enough!


Take your triangle, and color in all the multiples of 2.  You’ll find some cool patterns emerge.  How about multiples of 3?


This fun applet colors up to 256 rows. Try some prime numbers for wild, unexpected results. The coloring party continues when you look at coloring remainders.


All of the entries in any row tell us the number of possible combinations in a binomial experiment. While coin flipping is not all that sexy, I have used these facts to play Plinko with my classes, which I have used in STEM talks before. Today in class, students created their own graphs of flipping results, using randint on their graphing calculators.

And…Galton Boards!!! Which I told my students I would offer 1,000,000 bonus points if they built for me.


A few days before this unit, I had an animated gif of Sierpinski’s Triangle projected on my board as students walked in. Many students identified it from their Geometry class, and most were enthralled by the seemingly infinite self-similarity.  So when a few days later I go rambling on about Pascal, dive into coloring patterns, and reveal the first 265 rows of colored even numbers……wait…what’s this?????

It’s our old friend Sierpinski out to play. And if you haven’t before, develop this cool visual by playing the Chaos Game on the rich Cut The Knot site.


Encouraging Persistence Through Contest Problems, Part 2

In my last post, we looked at an AMC-12 problem of moderate difficulty, but with an premise that could be understood by many.  This time, we’ll take a look at a problem which delves into more abstract concepts, and explore how technology can allow students to consider solutions.  The following problem was question #23 from this year’s AMC-12.

Let S be the square one of whose diagonals has endpoints (0.1, 0.7) and (-0.1, -0.7).  A point v = (x,y) is chosen uniformly at random over all real numbers x and y such that x is between 0 and 2012, inclusive, and y is between 0 and 2012, inclusive.  Let T(v) be a translated copy of S centered at v.  What is the probability that the square region determined by T(v) contains exactly two points with integer coordinates in its interior?

In this year’s contest, where over 72,000 students participated, this question was answered correctly by only 4.5% of students, and was left blank by 81.6%.

Tomorrow morning, write this question in its entirety on a side board, and observe student reactions.  How many students begin to sketch the square described in the first sentence?  How many ask questions about some of the sophisticated language?  How many shrug and turn away?  This problem presents a number of chance for students to summarize given information, summarize “scary” language, and consider possibilities.

The first sentence describes a task which all geometry students, regardless of phase or level, can pursue.  Give students the task of telling you everything they can about that square, and share out their ideas.  Let’s look at it piece-by-piece.  First, we have a diagonal with defined endpoints:


Can we find the other diagonal? This is a great opportunity to look at perpendicular bisectors, and consider slope. After the other diagonal is found, we can see the given square:


Is there anything else we might need to know about this square? Can we find its side lengths? How can we find its dimensions? Another nice connection, to our old friend the Pythagorean Theorem, emerges…


How convenient for us! We have a unit square, where all sides have length 1. Even if we don’t consider the rest or the problem, think about how rich of a discussion we have already had!

Now for the scary part….all that spooky language. But is it so bad really? What is the question really asking us to do? Challenge students to re-write the premise of this problem, so that it can be explained easily to a friend or neighbor. Here’s what we are being asked to do:

  • Pick any random point in the first quadrant, but don’t go above 2012 for x or y.  We’ll call it point v.
  • Take the square we just made, and copy it, so that v is the center.
  • How likely is it that the new square contains two points with “integer coordinates”?  This is great time to introduce the term “lattice point”.

Here’s an example of what we are looking at.  The original square remains at the origin, but a new one, with center v1, has also been introduced.  Notice that this new square captures only one lattice point.


So, how are we going to find that wicked probability? Using Geometer’s Sketchpad, I created a model of this problem, which students could then use to explore the premise. Contact me if you would like the Sketchpad file to use.  Enjoy my tinkering in the video below:

Encourage your students to break down problems into smaller, digestible pieces, and not be afraid or scary-looking language. The rich class discussions which come from allowing time for questions to stew and worth it!

So, what’s the answer? Below is a link to a great summary video by Richard Rusczyk from the excellent website artofproblemsolving.com. It’s perfect for sharing with your classes after you have explored the problem and discussed ideas: http://youtu.be/9ytTJr6ctwY

Encouraging Persistence Through Contest Problems, part 1

One of the many tasks I perform for my school district is serving as math club sponsor.  While I often attempt to find interesting activities and experiences for the club, many students join math club to participate in the contests we tackle each year, and this month tends to be a particularly busy period for contests.  In-house contests include the Pennsylvania Math League and the American Mathematics Competition exam series.  Today, I am writing my blog post from a lecture hall at Lehigh University, while two floors below hundreds of students are participating in an annual contest organized by Dr. Don Davis, who uses the event to recruit students for his American Regions Math League teams.  Tomorrow, two teams of students I work with will attempt the Moody’s Math Challenge, where students are given 14 hours to complete and open-ended question and submit a solution.

I recall a conversation I had with a math department head I worked with when I first became a teacher, and the conversation centered around why our school didn’t have a math club.  The veteran teacher responded that he didn’t believe in doing math problems as competition, and I suppose that I agree with the essence of his argument: that sitting alone, isolated, doing a series of problems may not be the most enriching of pursuits.  But the conversations that take place surrounding challenging problems can lead in interesting directions which often encourage collaborative thinking and build confidence in approaching “scary-looking” problems.

The problems from the AMC 10 and 12 exams, in particular, lend themselves to discussions of problem-solving approaches.  Each exam is set up with 25 multiple-choice questions which loosely go in sequence from least to most challenging.  Many students I coach can handle the first 10 to 12 questions, and may venture as high as question 20, before sensing that the questions have taken a turn towards the evil…questions with wording and symbols beyond their experiences.

We can use contest questions to encourage not only higher-level thinking from all of our students, but also develop persistence in problem solving.  Consider the following problem, which was #15 on this years AMC-12:

A 3×3 square is partitioned into 9 unit squares.  Each unit square is painted either white or black with each color being equally likely, chosen independently and at random.  The square is then rotated 90 degrees clockwise about its center, and every white square in a position formerly occupied by a black square is painted black.  The colors of all other squares are left unchanged.  What is the probability that the grid is now entirely black?

This is a problem 15 from the contest, which implies that it is bordering on the medium to hard-type of problem.  I like this question for two reasons:

  • It’s a probability question.  I know that these types of problems often appeal to me, as opposed to geometry questions, which often interest me less.
  • It has an accessible premise.  While we may have trouble down the road computing the probability, this problem can be easily de-constructed, simulated and discussed, even by middle-school students.

We can walk through this problem by giving students some 3×3 grids and a black marker.  According to the problem, each square is painted white or black at random.  Have students make their own grids, then make a copy of it, which will then be rotated 90 degrees:


Then, follow the directions to make an altered grid: every white square in a position formerly occupied by a black square becomes black.  All other squares are left unchanged.  In the example below, our grid fails, since the results is not all black.


Where can we head from here?  Depending on the maturity and sophistication of the students, there are a few paths to consider:

  • With younger students, hand out some pre-made grids, where some will become all black after the transformation.  Can students categorize those which become all black?
  • For more sophisticated students, hand out more 3×3 grids and experiment to see if they can develop one or more grids which satisfy the problem.


For some students, the problem may stop here, which is fine.  The experience of having tackled part of a complex problem is a success unto itself.  You can even let the problem stew with students for a few days to discuss with friends and parents before reaching some conclusions.  In this problem, there are 3 dependencies, 3 different aspects of the grid to consider:

  • The center square must be black
  • The corner squares (A,B,C,D, below) must meet certain arrangements
  • The non-corner squares (e,f,g,h, below) must also meet certain arrangements


Can we list arrangements for ABCD which will result in all black?  Certainly black-black-black-black works, but so does white-black-white-black.  Are there others?  This then shifts the nature of the problem from a scary-looking probability question to a more tame (but still semi-scary) counting problem.  I’ll leave the counting to you and your students.

In my next post, we’ll look at a geometry example from this year’s AMC-12.