A unique sculpture greeted students as they entered class today:

There’s a lot of math goodness happening in this picture, but I don’t want to steer conversation in any particular direction right off. Time for some Noticing and Wondering! Students shared their thoughts on the back board:

Most of our class time today will be spent completing a jigsaw activity which guides students through many of the rich connections between Pascal’s Triangle, Combinations and the Binomial Theorem. Knowing that I would eventually talk about Pascal’s Triangle (one of my favorite shares of the year!), I was hoping to see if we could generate ideas on triangular and tetrahedral numbers organically. This visual opener did the trick. And while I ran out of time today for my Triangle chat, it’s in my pocket for tomorrow!

After sharing this experience on Twitter, Annie Fetter (the queen of noticing and wondering) chimed in with her ideas:

@bobloch Ooooh! Any one wondering about optimal packing??

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:

THE HOCKEY STICK THEOREM:

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.

COOL DIAGONAL PATTERNS

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!)

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.

COIN FLIPPING FUN

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.

IT RELATES TO GEOMETRY AND CHAOS

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.

Is there a rule in math which encapsulates more great stuff than the binomial theorem? Increasing powers of x, decreasing powers of y, a bunch of terms to look at…and hey, what’s this….combinations, you say? I’m in. But please don’t remove all the fun discovery moments there are to be had here. Here’s a jigsaw activity which was handed down by some of my “senior” and now-retired colleagues, and is now infused with my need to get kids moving around and discovering things. I hope you enjoy it.

The students will need some background on a few things here. In particular, students should already be fluent with multiplying binomials, and have some familiarity with combinations. Copy the first 4 pages of the binomial theorem Jigsaw Activity and have them ready to go. Now it’s time for some movement.

Arrange classroom desks into groups of 4. If your class roster number is not a multiple of 4, then you can have some 5’s, and we will deal with them soon.

In each group, have the students number themselves from 1 to 4. It is important that each group have one of each number, and that students know their number. If you have any groups of 5, then allow for two “1’s” in a group. This will be helpful, as #1’s actually have the trickiest job in this task (but don’t tell them that yet!).

Next, hand out the packet of tasks. The packet looks intimidating, but each student is only responsible for completing one page. All students who are “1’s” are responsible for completing Task 1. 2’s will complete Task 2. 3’s = Task 3. And 4’s have Task 4. On your command, students will break from the group, and move to a new location and meet with all oftheir similarly-numbered classmates to complete the task together. After all students have completed their task, they will re-join their group and share their findings. It is important that students understand the need to complete their task thoroughly and thoughtfully, as their group success depends upon it!

Now, send all the numbered students to meet with their group and complete their task:

SUMMARY OF THE TASKS:

In Task 1, students are asked to expand (x+y)^4, starting with (x+y)^2 and working their way up. This task usually takes the longest for groups to complete, and requires attention to detail. In the end, groups are asked to list the co-efficients of the terms they get.

In Task 2, students are asked to list the sample space for 4 tossed coins. The task is not difficult, but requires attention to detail in making sure all possibilities are provided. Groups are then asked to list number of ways to get 0 heads, 1 head, 2 heads, etc., in the 4 coins. Note, groups often give probabilities here….we want the COUNTS!

In Task 3, students complete a number of rows in Pascal’s Triangle. The task in not tricky, but requires attention to detail in filling out the small boxes. The group is then asked to list the numbers in row 4 of the triangle.

In Task 4, students are asked to compute a number of combinations by formula. I usually try to have 4-function calculators at the ready for this group, and want them to really focus on the formulas. In the end, the group is asked to list the combinations of 4 items taken 0, 1, 2 3 and 4 at a time.

The tasks often take about 20-30 minutes for groups to complete and check. Bringing groups back together and having them share their findings with their teammmates often takes another 20-30 minutes, so this may need to be done over 2 days.

The big reveal occurs as groups begin to realize that all 4 tasks have the same “answers”:

1-4-6-4-1

We’re now ready to start exploring the amazing connections between 4 ideas (binomials, coins, Pascal’s Triangle, combinations) which seemed quite different math-wise, but have some strong connections. Now the ideas come fast and furious as we explore the connections. Page 5 of the packet provides some guiding questions, but don’t feel tied to the linearity of my questions.

Let’s think about (x+y)^5…..

How many terms will it have? 6

How will the powers of x and y behave?

How can we find the coefficients? Pascal’s Traingle!

How else can we find the coefficients? Combinations!

How about coin flipping:

If we flipped 6 coins, how many items would be in the sample space? 2^6

How many of these possibilities will have all heads? 1

How can we find out the number of ways to get exactly 2 heads? Pascal’s Triangle

Eventually, this will become the basis for “assembling” the Binomial Theorem. You’ll be surprised at how much of it your students will be able to piece together after this activity. Your job is to just help out with some symbols and some common language.

Can we just have a conversation about Marla Gibbs somehow ending up in the Breaking Bad universe?
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