# Puzzling Perseverance

School mathematics has a bad reputation for being intellectually unattainable and mind-numbingly boring for many students.  Proclaiming the falsity of these beliefs is usually not enough to convince kids (or people in general) of their untruth.  Students need to experience their own success in mathematics and be given the opportunity to engage in curiosity-sparking mathematics.  For me, one of the very best moments in a classroom is when a self-proclaimed math hater fully engages in a challenge and is motivated to work hard to arrive at a solution.

Enter January 2nd and 3rd.  Students are back for a two-day week which they view as punishment and a rude-awakening from a restful winter break.  To boot, the Governor Dayton announced today at about 11 am that all Minnesota schools will close Monday, January 6th due to impending dangerously cold weather.  You can imagine where the motivation level was in school today.

As the CEO of room 114, I decided to make an executive decision and do a puzzle from Nrich (shocking, I know) in my probability and statistics class.  Technically, the students could use the mean or median to help solve the problem, so I wasn’t veering too far off of what I had previously planned.

The Consecutive Seven puzzle starts like this:

Initially, one student began by explaining to me that she took one number from the beginning of the set, one from the middle and one from the end.  Then she figured the other consecutive sums needed to be above and below that number.  (Spoiler alert:  These numbers actually end up being the seven consecutive sums, so I was very interested in her explanation of how she arrived at those particular answers.  )

It’s worth noting that this student’s first words to me at the beginning of the trimester term were, “I hate math and I hate sitting in the front.”  So you can imagine my excitement when she dove in head first into this particular task, happily and correctly.

Adding to my excitement about the class’s progress, another girl (who was equally enthusiastic about math at the beginning of the term) was the first one to arrive at a correct solution.  And although she probably wouldn’t admit it, she was thrilled when I took a picture of her work.  And I am more than thrilled to display it here:

If you were wondering about how math-love girl #1 fared in completing the task, she persevered and impressed her skeptical cohorts:

This phenomenon fascinates and excites me that students, when confronted with a puzzle, highly engaged and motivated throughout the lesson.  Dan Meyer summarized this idea nicely on his blog recently:

“The “real world” isn’t a guarantee of student engagement. Place your bet, instead, on cultivating a student’s capacity to puzzle and unpuzzle herself. Whether she ends up a poet or a software engineer (and who knows, really) she’ll be well-served by that capacity as an adult and engaged in its pursuit as a child.”

And who knows.  Maybe one of the girls featured above will become a puzzling poet.

# Full Circle Reflection

It’s almost the end of the trimester already which made today my last official “teaching” day with my Algebra class.  I’ve used a lot of the Math Forum’s Problems of the Week in this class.  Since this is a college algebra class, I use the POWS more as problems of the day rather than the week.  As a member, I have access to the library of problems, which I scour quite frequently to find just the right problem to fit the topic at hand.

I used a POW in which the first four terms of a patterned sequence of A’s and B’s are shown.  The students are asked to create an expression to represent the number of B’s in the nth term and then create an expression to represent the ratio of B’s to the total number of letters in the nth term.  What I like about this task in particular is that it isn’t a completely obvious fraction-ladened, asymptote-wielding, makes-a-student-want-to-cry rational function.  The students are able to work through most of the problem forgetting that this is in fact THAT type of function.  In fact, since they weren’t immediately scared off with a 1/x or the like, it seemed easier for them to make connections from their solutions to the graph and equation of the function.

What was particularly fantastic about this problem was that the growth of these students in the problem solving process was so evident.  It was clear as I circulated the room that over the course of a trimester, these students’ goals as mathematicians were evolving:  from “fast and correct” to “patient and curious.”

For example, when asked to find the term that results in 35% B’s, I had many students make a table with # of A’s, B’s, Total letters, and ratio of B’s to Total letters.  At the beginning of the trimester, these kids would accept their correct answer, but then reject their method of arriving at the answer because it was not as quick as those able to recall an equation or procedural method.  Now, after 13 weeks, these same kids are able to look at their table and appreciate the extra questions they can now address about this pattern scenario.  Additionally, some students were willing to attempt multiple methods in arriving at the answer.    It was a pretty profound moment for them as problem solvers and me as an algebra teacher.  I don’t know who was more proud, them or me.

Here are some samples of their work:

# Vegan Teacher Crazy about Cheeseburgers

A year and a half ago, I made the best dietary decision of my life and decided to try a vegan diet for 30 days.  Fast forward to now, I love the vegan lifestyle and I’d never go back to a diet filled with animal products.  I know too much.  But that’s a story for another post.

A couple of weeks ago, I logged into Robert Kaplinsky’s presentation on Global Math Department.  He started off with a visual, which is usually good to draw listeners into the presentation.  However, this visual was a cheeseburger.  And he went through more and more visuals, and the cheeseburgers kept getting bigger and bigger until finally I’m face to screen with 100×100 cheeseburger from In N’ Out burger.  I try very hard not to be one of those ‘enlightened and superior’ vegans who constantly judge the dietary choices of others, but these burger pictures were not how I envisioned spending my Tuesday evening.  His methodology had my attention however.

After explaining his problem solving process and distributing his problem solving template, he threw this photo into the mix and asked,

“How much would that 100×100 cost?

Now I was hooked and needed to figure out how much that 100 x 100 cost.  I didn’t care if it was a cheeseburger or a truckload of kale.  The wizardry of Robert Kaplinsky drew this vegan teacher into the problem solving process and made me care how much this monstrosity of a cheeseburger cost.  Brilliant.

Then Robert Kaplinsky threw down the dynamite:

That’s right.  The actual receipt of this 100×100 cheeseburger.  A boatload of kudos to Mr. Kaplinsky for presenting something that was simple, with some great mathematics to go with it.

I’m glad this weeks ExploreMTBos mission was LISTEN and learn.  This was a great presentation, a great lesson, and a great resource.  I’m glad I took the time to listen to Robert Kaplinsky’s presentation, even if it wasn’t so appetizing on the outside.

# Oh, UNIfix cubes! I get it!

I’ve done a lot of professing my new found love for Visual Patterns lately, and today will be no exception.  If I haven’t convinced you of the flexibility and differentiation available in these seemingly simple patterns, let me have one more stab at it.

Today, my College Algebra class looked at pattern # 28.

I took out the unifix cubes for those who wanted to actually have the three dimensional shape in front of them.  This was helpful for some, however, I realized the limitations of the cubes…the fact that they only will “fix” to one other cube (hence the name UNIfix). This may not be mind blowing information to many of you, but I just put those two things together in my brain today.  Because of the one fixture, they were hard to take apart in usable “chunks” without the whole figure falling apart.

Anyway, back to the pattern. I wanted to workout ahead of time all of the possibilities that students would come up with so that I could more effectively use the 5 Practices of Orchestrating a Mathematical Discussion and anticipate their responses.  I’ll tell you what, I played around with those expressions so many times, and thought for sure I had came up with at least the majority of responses I would encounter. They were all quadratic. Then, out of left field, the students threw me for a loop. The majority of students came up with n^3 – (n-1)^3!

Now, you might be thinking, duh! It’s 3 dimensional AND a portion of a cube.  However, my algebraically trained brain started with quadratic expressions and stuck with them since I saw from the difference of differences table that this pattern was in fact, quadratic.  Yes, the n^3 terms get cancelled out when the expression is simplified and the simplified expression becomes quadratic but this opened up a whole new avenue of discussion with my class. We were now able to talk about the misconceptions of expanding something like (n – 1)^3, because if they found the expression for the nth step another way, they could use that as a check for simplifying their answer.

What was eye opening for some of the students that chose an algebraic method (such as using a table of differences and then setting up a system of equations) was that the “c” value in ax^2 + bx + c was hard to conceptualize.  It was very difficult for students to grasp that the first term and the non-existent “zero term” had the same number of cubes.

Finding the surface area formula for step n was even more awesome, because it was in this portion of the pattern that I was able to see real growth in my students’ willingness to attempt a more conceptual method.  There are certain students whose default method is to set up a system of equations using the table of values for the pattern steps. These students are noticing more that they encounter errors much more often than those who have a conceptual understanding of how the pattern is built.  I found this time around, less students relied on the algebraic method (about 7/35) whereas last week, probably 15/35 of them were starting algebraically.  As we are covering more and more concepts in this course, the students are realizing that they do not remember specifics about formulas and procedures from their previous algebra courses.  They remember “learning” the topics, but they usually can’t quite nail down the specifics of each method.  I really feel that we are making some good headway toward solidifying their conceptual understanding of the algebra as I see more and more students break away from the procedural methods toward a more conceptual one.

We talked about this pattern for an entire 60 minute class period.  You know it’s a good day when kids look at the clock and say, “whoa! Class is over already?”

# A Visual Patterns Trifecta

This is my third (and most exciting) post about my new found love for Visual Patterns.  My enthusiasm stems from a growing appreciation of how these patterns can be used in such a wide range of grade-levels, including advanced algebra.  The use in an elementary or lower-level secondary classroom is easy to see.  However, the teacher and student need to dig a bit deeper into the make-up of these patterns in order to generalize them.

For example, here is Pattern #8.  Kudos to Fawn Nguyen on this one.

It’s not immediately apparent what step 4 should be.  But even more so, the quadratic nature of this pattern is not necessarily simple to comprehend.  From yesterday’s pattern #5, the students had a method for finding the number of penguins in the nth step by converting the penguins into a table and creating a system of equations.  I didn’t want to encourage this method, as it is very procedural and tedious.  However, it was a good place for students who liked to work in a more algebraic way to feel successful.

Also, the table allowed them to explore what the difference of differences really told them.  I had a student, let’s call her Kay, ask “I wonder what the constant difference of differences represents in our equation for the nth step.”  She came up with a conjecture by comparing it to our problem from pattern #5.  Kay concluded that the “a” value in the equation ax^2 + bx + c = y is half of the constant difference of differences.  I challenged Kay to continue to examine these values in future problems to see if her conjecture holds true.  I had another student, Em, wonder if that meant that the “a” value in a cubic function is equal to one third of the difference of difference of differences.  This she will investigate as well.  What is very exciting about these questions is that they were non-existent 5 weeks ago.  It wasn’t that the students didn’t WANT to be mathematically curious, they just didn’t know HOW.  It was a huge thrill for me as a teacher to see these kids move from looking at a math problem with a single solution to being able to ask new questions.  A nod to Christopher Danielson for helping me realize that learning is having new questions to ask.

###### Back to the problem at hand:  How many penguins are in step n?  A few of the students were able to get the answer without using a table.  These were mostly the students who like to do things in their head.  The ones who want to fully process the problem in their brain, but not write any of it down.  [Side note:  these are usually the ones who are brilliant with numbers but get lower grades in traditional math classes because they don’t want to “show their work.”] Anyway, I wanted to challenge those who used the table method and set up a system of equations to relate their model back to the picture.  Spoiler alert!  The answer is 1/2n^2 + 1/2n + 1, but I wanted my students to be able to relate that back to the picture.  What do the individual pieces of the expression represent in penguins? This way, the students were able to make that connection of a picture or pattern that didn’t seem quadratic to begin with and flesh out its quadratic properties.

When the students figured this out, it was a magical moment.  I had to capture it:

Another cool experience with this problem:  the same evening that I did this problem, our school hosted parent-teacher conferences.  One of my students came into conferences with her parents and her three little sisters, ranging in age from about 5 to 10.  One of the little sisters sat down and wanted to be part of the conference.  I pulled up the visual pattern and asked her how many penguins would be in the next step.  It was a validation of my initial thoughts of how open and accessible these problems are to all levels of mathematics.  Here was an 8? year-old looking at the same pattern that her 17 year-old sister explored earlier that day.  And it was mathematically applicable to them both.  Beautiful.

# Visual patterns with a side of awesome sauce

Regular old Wednesday turned amazing today when I posed pattern #2 to my math recovery class, a remedial math class for kids to recover credit from a previously failed course. It may not need mentioning, but just to be clear, these kids hate math and think they’re no good at it. In pattern #2, the kids need to find how many cubes are in step 43 and the surface area of step 43. Side note:  My kids wondered, why 43, Mrs.Nguyen?

Anyway, finding the surface area was where the magic started to happen. I had 4 or 5 kids out of this class of about 15 get seriously invested in finding out the answer. They were drawing pictures, explaining their thinking to one another, figuring out different ways to think about the problem. It was inspiring and motivating for both them and me.

As if that wasn’t enough to make it a great day, I decided to pose the problem to my College Algebra class as a starter and try my hand at the 5 Practices for Orchestrating Productive Mathematics Discussions. My expectation was that they found the number of cubes and surface area of step ‘n.’ What was gorgeous about this problem was not necessarily the answer, but the numerous ways they came up with to arrive at the nth step. Here are a few:

n + n + (n-1) + (n-1) + n + (n-1) + n + (n-1) + 2

4(n-1) + 4n +2

4(2n – 1) + 2

6 + 8(n-1)

4[n+(n-3)] +10

6(n-1) + 2n + 4

8n – 2

What was even more powerful was, as Ben Blum-Smith calls, an effing game changer.  He’s right, and this was beautiful.  I used the tactic he lays out in his blogpost where students are asked to summarize the ideas of someone else.  I had a few try to slyly summarize their own ideas, but alas, I would have none of it.  As a result, I had more engagement, more involvement, and more buy-in that this problem solving process is helping them to understand the mathematics more deeply.

Here is an exchange between two students (T and C) that is worth highlighting.  T is the student who came up with 6 + 8(n-1) as the surface area for step n:

C: Oh, I see.  T just used the arithmetic sequence formula.  The first term is 6 and it goes up by 8.

T:  Actually, that’s not what I was thinking.  I thought that there were 8 sides of the figure that had ‘n-1’ squares and then 6 squares left over, two on the caps and 4 in the corner.  OH, you’re right, it is the formula.

Then the lights came on.  This girl who had probably only known mathematics and algebra to be a long list of rules, procedures, formulas, and practice was able to experience that developing a conceptual understanding of this pattern help her to create the arithmetic sequence formula.  It was the bottom-up approach that I’d been talking about all trimester where developing conceptual foundations are where real math learning happens.

# Nrich Love Affair: MTBoS challenge #1

I told my husband that if we weren’t already married, I’d run away with nrich in a heartbeat.
That being said, it’s probably no surprise that my favorite problem comes from nrich: consecutive sums and is my response to the explore the mtbos mission 1.
I’ve used this problem a few times, with high level students, low level, and in between as well.

Here’s a poster with a general overview of the problem.  The link from nrich provides starting help as well as teacher notes and a solution.

Some things I love about it:
1. So many points of entry and a low barrier.
2. So many paths. I’ve had so many different conjectures arise from this problem because of the open-ended nature of it and its ease of exploration. The numbers are not intimidating so students are unafraid to explore some of their findings.
3. Multiple extensions. For example, do consecutive differences work similarly? What about consecutive products? Or better yet, the difference of consecutive products!
4. Students organize their work in so many different ways. It’s completely fascinating to see it happen.

When doing this with a lower level class, I usually have them make a list of noticings and/or wonderings. This way, the patterns they learn to communicate what they believe to be true in their head. I may challenge them to generalize a little, especially with odd numbers always being a consecutive sum.

The most exciting thing that happens when I do this problem is the “what ifs” that students can’t help but think up themselves.  For example, What if we took the difference of consecutive numbers?  What if we took the sum of consecutive odd or even numbers?  Consecutive square numbers?  Triangle numbers?  Negative numbers?  It’s pretty amazing to be part of.

If you have tried or try this problem in the future, I’d love to collaborate on it.