A Visual Comeback

Please excuse me while I geek out for a few minutes about Visual Patterns.  My love affair with this versatile website has made the transition from autumn to winter as I engage in select patterns with my Algebra classes.  I didn’t start using these until a unit on quadratics last trimester, so I was very pleased that a linear pattern could create just as much conversation and mathematical excitement.

For example, this is a replica of pattern #114 that we looked at in class today:

Lego 114

The equation y = 3x + 4 was not terribly difficult for these kids to decipher. But the fun began, as usual, when I asked them to relate their equation back to the figure.  Here are some of their findings:

1.  Students used the idea of slope and recognized that the slope is the change in the number of squares divided by the change in the step.  The y-intercept is the value when the “zero” step is determined.

2.  There are always 4 squares in the corner and each “branch” off of that square has a length of x.

3.  SImilarly, there is one square in the corner and each branch from that one square has a length of x+1

4.  There are always x “sets” of three squares, and four squares left over.

5.  The arithmetic sequence formula works nicely here, common difference of 3 and first term of 7.

The final observation deserves its own paragraph, as I was completely blown away by the thought process.  The student noticed that if we made each step in the pattern a square, then the formula would be (x+2)^2.  He then noticed that the portions that were missing were two sections, each consisting of a triangular number.  Recalling the formula we worked out last week (by accident) for the triangular numbers, (.5x^2 + .5x) he took (x+2)^2 -2(.5x^2+.5x) and simplified it.  The result is, you guessed it, 3x + 4.  Below is a photo of this amazing insight:

Pattern 114 Triangle

What I like most about these visual patterns this time around is that it helps the kids get comfortable having a mathematical conversation.  Students build on each other’s thinking and discover new insights by listening to their classmates.  This was difficult to do last trimester with a similar group of kids.  I think that by starting with a linear patterns, rather than quadratic, the students have acclimated themselves to different ways of approaching the patterns.

Curiosity Driven Mathematics

In my very first years of teaching, I used to have students ask me, in that age-old, cliche teenage fashion, “When are we ever going to use this?”  I vividly remember my response being, “Maybe never.  But there are plenty of other things we do in life, like play video games, that have no real-world application. That doesn’t seem to bother us too much.”

In fact, if every moment of our lives needed to apply to the bigger picture, the REAL-world, when would we do anything for pure enjoyment? or challenge?  or even spite?  I know kids are capable of this because some of them spend hours upon hours a day engaging not only with a video game but also collaborating with other people through their game system.

And furthermore, where do we think this resentment for learning math really comes from?  I have a guess…probably adults who have realized that through the course of their lives, being able to solve a polynomial equation algebraically is not all that useful! News flash, math teachers:  Our secret is out! 

There are many kids across all levels of achievement that will not engage in the learning process simply because the state mandates it or the teacher swears by its real-world relevance.  Students (and arguably people in general) are motivated by immediate consequences and results and cannot easily connect that the algebra they are learning today will be the key to success in the future.  They do not care that if they don’t nail down lines, they’ll never have a prayer understanding quadratics.  If they are bored to death by linear functions, I can’t imagine that they have even an inkling of desire to comprehend the inner workings of a parabola.  

What does resonate with learners is the satisfaction of completing a difficult task, puzzling through a complicated scenario, or engaging in something for pure enjoyment.  Kids are naturally problem-solving balls of curiosity.   There are ways to provoke curiosity and interest while simultaneously engaging in rich mathematics.  I think many teachers assume that in mathematics, especially Algebra, curiosity and deep understanding need to be mutually exclusive, and I’m positive that mindset is dead wrong.  For example, show this card trick to any group of kids, and you’d be hard-pressed to find a group who isn’t trying to figure out how it works.  I also think you’d be hard-pressed to find the real-world relevance to a card trick.  It’s still no less amazing, as well as algebraic.  



A Top Down Approach?

A new trimester is upon us in St. Francis, MN which means a new group of advanced algebra students as well as two classes full of squirrely 9th graders.  I’m amazed that these sets of students can have things in common and a lesson for one class can serve as a bell-ringer for another.  I have said in the past that my favorite activities are the ones that can be used across multiple ability levels and this task is no exception.

This week, in advanced algebra, we’ve been working on problems that allow the students to connect specific patterns and examples to general formulae.  I feel that this trimester, I have done a much better job of sequencing the class problems in a way that has help build student confidence in the problem solving process.  As I’ve done in the past, I chose some nrich problems that have a low barrier to entry and a high ceiling.  These problems feel like number play:  Pair ProductsAlways a Multiple, Think of Two Numbers, and Calendar Capers.  Although I’ve had the occasional moan from students who prefer their math to be in lecture/practice format, I’ve seen much more willingness to engage in the problem-solving process this time around.

One particularly memorable day, we used a Math Forum problem called Baffling Brother in which a brother is attempting to amaze his younger sister by having her choose a number, perform some operations on the number and then telling her the result.  I’m disappointed that I didn’t think at the time to have the students act out this scenario.  That could have been spectacular!

These being upper level students, I always encourage them to attempt the “extra” for these problems.  On this task, they needed to come up with a number puzzle of their own that resulted in an answer of 7 each time.  I told them that I would be giving these number puzzles to my 9th grade classes to amaze.

Here are some examples:

image (1) image

What happened next I could not have predicted and was not an iota shy of completely awesome!  I presented one of these number puzzles to my 9th grade class and stood in the back of the room as I read the steps to them.  When they arrived at their final answers, I had them compare with one another.  I wish I had a camera on the room to capture the amazed look on their faces when they realized they all got an answer of 7.  Icing on the cake:  the advanced algebra students were very satisfied that they were able to amaze 9th graders with problems that they created.  I’ll call that one a win for engaging kids in “boring” old, non-applicable, relevant Algebra.

Motivating with a Math Story

I’ve observed over my career that as a high school teacher, 9th graders are amongst the most challenging yet most rewarding groups of students.  Challenging in the sense that they never stop talking or moving, but rewarding nonetheless because of their naivety and innocence. This combination makes engagement and relevance easier to create day to day.  

For example, today in my 9th grade probability and statistics class, I adapted an IMP activity involving sample size and the ratio of mixed nuts.  I literally had these 9th graders believing that I counted all of the nuts in a container of mixed nuts and compared it to a fictitious “nut ratio” from Planters’ website. It’s worth noting that my intention was to tell a story about mixed nuts, but they seemed to believe that this must be true, so I just played along. They think I’m a little crazy for counting the number of nuts in a can, but they were bound to reach that conclusion at some point, nuts or no nuts.  This little “fib” served me very well today as the students now wanted to figure out if I was short-changed on the number of cashews in my can of mixed nuts and whether I had enough evidence here to sue Planter’s Peanuts.  I don’t plan on making up stories of this nature all trimester, but the fact that changing the character in the problem from Mr. Swenson to Mrs. Schmidt played out in my favor was satisfying.

I felt a little guilty having mislead them, however, so I scoured the internet for anything relating to Planters nut ratio.  I found this interesting post about a similar (albeit smaller) bag of mixed nuts. The entire blog was actually pretty intriguing as its entire purpose is to critique gas station food fare.  I’ll probably show this to my students tomorrow just to see where their brains go with it.  

Pushy vs. Persistent

“Sharing is caring” does have a nice rhymey ring to it. Although lately, I’ve felt a little bit like my version comes off as ‘sharing is pushing and over-feeding’.  I’ve had teachers in my department inquire about problem solving and desire to get kids to invest and engage.  I like sharing what I’ve discovered and what I have found that works, but sometimes I get so excited about sharing resources that I end up like Tommy Boy and his pretty new pet.  I sometimes fail to realize that trying new approaches can be uncomfortable, unpredictable and downright scary and not all teachers want to dive into the change head first as I did.

Here’s a great example: we had final exams in 2-hour blocks right before Thanksgiving break.  To say that the kids get “restless” by the middle of the second day is sugar-coating it.  A new teacher in our department, (let’s call her Sheryl) sent this picture with the caption, “My algebra kids were bored after their final and built this with their textbooks.” booktower

Of course, my brain couldn’t just let that one go and say, “Nice book tower, Sheryl.” Dan Meyer calls this perplexity and modeling this behavior is a key to getting students curious. Instead, my eyes lit up and I thought, “what a great math problem!”  As we looked at this photo, I said, “what do you think kids will notice and wonder about this photo? Do you think you could get them to come up with how many books are in the 10th row or the nth row?”  Of course the question that’s raised, legitimately, is “what do you do when students say ‘there are green books and red books’ or ‘some are faced forward and some are faced backward’?”  This is the part that I believe is scary for a lot of teachers is relinquishing control of the immediate direction of the lesson and not being so certain about how students will respond.  At least when we give them a quadratic to factor, we have a pretty good idea of the limited number of directions they can move to arrive at a singular correct answer.

But what I believe is imperative here is validating and acknowledging those seemingly math-less observations and creating a math opportunity with it.  With the instance of “some are red and some are green,” we can now extend that declaration of color to ideas like percentages, ratios, and so forth.  But by first validating this red/green response, we’ve invited this student to the conversation and made them part of the creation of the problem we are about to solve.  Now they are empowered by the process and more motivated to step in to the problem-solving ring.  Whereas before, this same student might have disengaged completely.

A recent example of this from my own classroom:  We were beginning Dan Meyer’s 3-act task using the Penny Pyramid.  When collecting wonderings, one student asked how many 1996 pennies were in the pyramid. He was born in 1996, and was probably just fascinated by that year, but I didn’t want to dismiss that from the discussion.  I have a bucket of pennies in my room that I use occasionally for probability experiments, and I hoped that this kid could draw from his knowledge about samples to make a reasonable estimation of how many 1996 pennies were in that pyramid.  As it turns out, that students off-hand question turned into a great math discussion about random sampling.

But back to this book tower:  After I’m sure that I’ve thoroughly freaked out this new teacher with my enthusiasm over a book tower, something awesome happens.  This new teacher, races into my room after 1st period on Monday and says, “I did it!  I did the book tower, and it was AWESOME!”  I’ve had some great moments with other teachers, but that one is going to rank pretty high on my list for a long time. At lunch, she was STILL raving about it. She even said that the students were so engaged, that they ran out of time talking about it during class. Maybe there’s hope for Tommy Boy after all.