Wednesday, May 16, 2012

Using Real-Time Graphs

I have a class of grade nine students this semester that are part of a stretch program. This essentially means that they get 160 hours to complete a 120 hour course. The class is designed to accommodate the transition from elementary school (Grades 1-8) into high school (Grades 9-12) for those students who feel uncomfortable with their math ability.

It also affords me a few extra days here and there to stress certain topics. One of my foci this semester has been pattern modeling. Essentially, we work with various patterns and develop generic rules to describe their behaviour. Linear relations will be our finish line, but I am making sure to provide ample concreteness before abstracting into notations.

This week we began the topic of coordinate graphs; we did this without formal variables. We didn't graph functions, relations, or even tables of values--we graphed life.

The first time students saw a grid was on Monday morning. As they filtered into the room, I had a blank GeoGebra file open. Unannounced to them, I plotted each student as they came into the room. If a student left, the graph decreased. If a pack entered, I plotted them quickly. It didn't take long for some to take notice and begin to make sense of the situation.

Soon questions began to fly:

"Which dot am I?"
"What are you drawing?"
"What is that clump of dots?"

Very preliminary. I kept my mouth shut and continued. If the students were going to figure this out, they were going to do it as a collective. Some began hypothesizing that the graph must somehow be a graph of grades. A piece of me died inside when I realized that some students saw this as the only valid function of a graph, but I remained silent.

The bell rang, and the national anthem began. At our school, every student stops and stands until the anthem is complete. This means that students who are late for period one must remain in the hallway until the anthem is completed. This provided a great talking point because during the anthem, the graph stopped.

Once it had finished, more dots appeared. One student then shouted out:

"You're graphing us!"

I finally answered:

"What do you mean, graphing you?"


"You are using that grid to keep track of who is in the room."

This was a pretty good explanation, but I countered:

"How am I supposed to know which dot is who?"


"That's not important. The graph just tracks how many people are in the room now."

Two things impressed me about this comment. First, he used the word graph. I liked that after they kept calling "points", "dots". Some part of my OCD math brain found that incredibly offensive. Second, it opened up the class for some interpretation questions. By now, the graph looked like this:

Please excuse the awful quality
I addressed the class and began to identify certain points as people. I said that point A (the first one) was a certain student. From that, the class deduced that the graph could not represent grades because this student would not be at the bottom. One girl pointed out that he always came early. From there, the class was reasonably sure that the graph represented time.

I asked what two points almost directly on top of one another meant. They correctly deduced that it represented two people entering together.

At that moment, another student walked in. I plotted them. The class took notice.

We talked about what happened when the points lowered. Someone must have left. Why was there a pause? Students knew that it was the time every student was idle during the anthem.

Things were going very well, and I was about to move on, when a student bolted out of his desk and out into the hall. I lowered the graph. He then poked his head in, and jumped through the door. I plotted him again. He left; I plotted. He entered; I plotted.

He was proving to himself, quite ingeniously, that their interpretation was correct. As the class went on, I made sure to plot every late student. If a student had to go to the bathroom, they plotted themselves, and explained how.

The next day, students entered with a fresh file on the board. They were much quicker to get to deducing. When I spoke to them, the graph looked like this:

It was a graph that compared the number of boys to the number of girls. A girl counted as a positive on the y-axis, while a boy counted as a negative. It took them a while to decipher.

During our process, the same student tried his door trick. He discovered that the graph was effected. Soon after, a group of girls came late. This messed things up because the graph was supposed to fall when someone entered the room.

A girl then got up and tried the door trick. The class soon deduced the graph. I then could ask questions:

"What do these three points in a straight line mean?"
"How many people are in the class?"
"Are there more girls or boys in the room?"
"Am I included on the graph?"

The real-time strategy helped students gain an orientation of the coordinate plane. We have since graphed real-world relations, used Dan Meyer's Graphing Stories, created our own graphs, and began a study on the shapes of linear relations.

I have also graphed "Number of Students in Desks" vs. "Time" and "Number of Students at Desks" vs. "Number of Students at Tables" in real time. Their power is not in their complexity. In fact, they present quite simple graphs. It comes from their relevance; they place the students as active pieces in the lesson. It gives them the unique role of being both problem creators and solvers.


Wednesday, May 9, 2012

Soft Drink Project Part 5: The Show

This is the finale of a series of blog posts detailing a student posed project. To get the full picture, begin reading at part one:

As the project drew to a close, students began to place a valuation on their work. Very seldom did the topic of grades come up during the process, but even students know they are playing a game. They asked me how I would be grading, and I told them we would be using our self/peer/teacher model as always.

Even after the entire process, students were still musing on the possibilities of their work.

Two tweets sent to me from students
One student asked if she could put her work on her blog. Her innocence surprised me. She felt like she had to ask permission to put her own work on a personal blog. Up to this point, she has seen learning as the teacher's possession. She had never encountered anything that she would want to share. I told her that I was the one who needed to ask permission for her brilliant work. The public audience was growing.

Administrators poked their heads in throughout the process. Input from grocery store managers (in the form of a student's boss) and other business experts contributed to the authenticity. In the end, the audience helped, but the students didn't need outside approval to feel truly proud of the work they had done.

A teacher wandered into my class today and commented that they loved the atmosphere. After I explained their new project, she asked if it was the same class who designed the cool pop boxes. Apparently the students carried their work into other classes.

A display case was designed, and most designs made it in. It now stands as a proof of the intriguing mixture of mathematical imagination, rigor, and creativity.

The finalized display case

A closer look

One last time
What began as an innocent opportunity to pose questions about a curious situation bloomed into excellent mathematics. The sheer volume of work that went into the different designs was several times more than I could have ever gotten out of the same students with a dictated assignment. It showed me that an interesting starting point, a little bit of student control, and a willingness to learn alongside can create unbelievably powerful learning.

Now my efforts turn to replicating the feeling as we move into a new unit, but such is the life of a teacher.


Sunday, May 6, 2012

Soft Drink Project Part 4: The Math

This is the fourth in a series of posts detailing a student-posed math project. To get the full picture, please read the previous posts beginning with:

This post is designed to dampen the fear of math teachers. I know, because I was very afraid that the project had missed the mark until students moved into this phase. For some reason, teachers feel like they have more ability to complete a list of outcomes if they dictate the exact way, pace, and form that the learning will take. My division states they want to create life-long learners; in this model, the only lifelong learners are teachers because they must continue to do all the learning for their students day after day.

I gave a presentation on the blended project-based and problem-based system that I have adapted for my classroom this year. During the session, I claimed that great tasks involved student decisions, voice, and innovation. My goal was to show that creating an ecology where students feel comfortable expanding their problems will end up covering far more curricular topics that you would expect.

Maybe in my little world I thought this is what teachers needed to hear. I know I needed to be convinced until I saw it first hand.

The pop box designs were all unique. I honestly didn't see that coming even though the project began with such a high volume of student influence. I thought that a "best way" would be democratically deciphered, and several groups would then move on to design it to exact specifications.

Not a chance.

On the whole, groups encountered the same mathematics. There were deviations from the status quo, but they occurred in extreme circumstances. The lesson for me (and all teachers interested) is that we can still tailor a student-driven class around a content-driven curricula.

Below are some of the mathematical skills my students encountered:
  • Surface Area
    • I would hope so. The initial lesson was designed to have students explore how changing the dimensions of a prism would affect the surface area. This was the starting point of the learning. Students designed several boxes and calculated areas to compare. Students were discussing the formulae for areas of triangles and rectangles--using arguments to justify their design. Triangles, circles, and various polygons were calculated. Even an irregular shape was estimated.
A student estimates the SA of his newly minted shape
Student uses grid shading to create an accurate estimate

  • Measurement
    • The dimensions of every shape had to be labeled. This was particularly interesting for triangles. The students had to decide which dimensions were necessary/important to include to calculate the surface area. The most eluding measurement was the diameter of the can. The bevel along the top and bottom made this an interesting problem. Students recalled their skills from a previous lesson to take the measurement.
Student measures the diameter of two cans
  • Pythagorean Theorem
    • I wasn't expecting to fit this into the project, but it usually manages to squeeze itself into places where triangles are present. Students often forgot that the height of any triangle is needed to calculate its area. (Well at least until they become familiar with the idea of semi perimeter). I would ask them if they had a way to calculate the height and most remembered the theorem. Few knew how to use it. I think students remember the theorem because it represents a particularly traumatic experience in school mathematics. Regardless, students found their height and calculated areas.
  • Error Estimation
    • Many students realized the effect that human error has on mathematics. Their centimetre rulers only contained one digit of accuracy while their tracing and cutting skills contained far less. One group found it particularly distressing that their Pythagorean Theorem calculation differed from their measurement of the height.
    • Student 1 - "Does that mean the theorem is wrong?"
    • Student 2 - "No, it means that our measurements are wrong."
    • Students blind recognition of mathematics is troubling. Nowhere along the line did they ever question the fact that side-squared-plus-side-squared-equals-hypotenuse-squared.
  • Combinatorics
    • This was pretty much exclusive to the group that created the Tetris box design. As they were drawing up possible arrangements, I asked them how many possible arrangements existed. They could then make an objective choice as to which to bring to life. Essentially this was a tiling problem with irregular tiles composed of 4 squares.

  • Algebra
    • The student who designed the triangular prism cans to fit snugly inside the triangular prism box was well on his way to completion when I asked him how big each of his cans were. After a discussion, and a quick Google-ing to see the conversion between cubed centimetres and millilitres, we decided that we should shift the height of the cans so they held exactly 355mL. The area of the base of his can was fixed at 21.35 squared centimetres. (An equilateral triangle with sides of 7cm). He knew he wanted the volume to be 355 cubic centimetres. He recalled our prior tasks with prisms to set up the following equation:
    • Volume = Area of base * height = 355
    • Volume = 21.35 * h = 355
    • He then went on to experience his very first real application of algebra. He made sure to mention to everyone who would listen that his cans contained exactly the same amount of liquid as the regular cans.
The last 355mL triangular prism can is created (with help from my fabulous EA)
Other, specialized, areas of mathematics emerged as well. One group used their knowledge of computer assisted drafting to complete their design; another group discovered that a number can be considered "triangular" just like some numbers are considered "square". I was initially worried that the mathematics would take a back seat to colouring, cutting, and taping, but this could not have been further from the truth.

Throughout the entire process, the mathematics remained front-and-center. One colleague asked if I considered the task to be student-centered. I think that the project's room for student innovation created a task that was mathematics centered. Isn't that the way a classroom should operate?

Tuesday, May 1, 2012

Soft Drink Project Part 3: The Design

This post is the third in a series of posts detailing the happenings of a math project. To better understand the whole story, please start reading at the beginning:

The next few classes after the brainstorming class were a blur. Students would come into class, grab their previous work, and get down to business. It was the best I could do to have supplies waiting for them. I learned quickly that students can become pretty demanding when it came to their learning.

I didn't have any problems granting their requests; none of them seemed unreasonable.

"Mr. Banting, we need some tape"
"Mr Banting, do you have any string?"
"Mr. Banting, what are we going to build these out of?"

I ran down to the main office for supplies two or three times. Every time, I would have a line of students waiting for my arrival. They were all anxious to show me their new developments. A trip to the mall produced sheets of poster board, the SRC room provided large sheets of paper for nets, the Home Economics lab had extra shoelaces...

Students had made it very clear--I was not to stand between them and their vision. Luckily for me, that vision was well steeped in mathematical know-how. The entire process was summed up nicely by one student. In the midst of designing his zero-waste pop can/box design, he looked up at me as I was looking for a set of protractors. He smirked:

"This whole thing has kinda taken off, hey?"


I bought the caretaker donuts because my room was looking more and more like a garbage dump. Cut cardboard boxes, crumpled papers filled with scratched ideas, and empty pop cans filled the place.
The growing mess behind my desk.

Slowly, but surely, the designs came to life. Each group had a rationale as to why their box provided a solution to a problem.

One group began with the idea of the 6x2 box produced by Coca-Cola. Instead of leaving the cans stacked directly on top of each other, they eliminated a single can and moved the top five cans into the awaiting gaps created by the bottom row. The purpose was to sacrifice one can of product to greatly diminish the amount of wasted space within the box. Also, that move reduced the height of the box, thus reducing the surface area as well. All great thoughts, and an impeccable design that used the principles of both rectangular and triangular prisms.
An 11-can prototype

A couple groups chose a triangular option, but each with their own distinctive changes. One used a base of 4x2 cans on their sides. The triangle was constructed on this base layer. The result was a 20-pack of cans that used far less volume than an inefficient rectangular variety. They also assured me that their model afforded for easy fridge dispensation.
Student tests out their 20-pack
The problem of convenience was addressed by another triangular prism group. This group, the group that was initially arguing at the board, arrived at a compromise where their box would be composed of 4 parts-- all triangular prism 6-packs. They would then fuse them together to make a 24-pack. This model still cut down on volume, but actually increased surface area marginally. They rationalized this trade off. The customer would be able to rip apart the 4 packages and store them as they saw fit. Fridge, closet, bar, camper, cabin, you name it... They were combining math skills with the 21st century skills of marketing, economics, etc.

Another group was not satisfied with the wasted volume in the corners of a triangular prism. They wanted to round the edges to follow the curved contours of the cylindrical cans. They took 3 cans and traced them as the were lying tangent to each other. From there they got their design. This group spent the majority of fabrication time in the computer lab next door working with a computer drafting program. It was the first time the student saw connections between his high school classes. This is a pretty sad, yet common, occurrence.
The modified triangular prism takes shape

A group designed a hexagonal model that was designed to look like a large pop can itself. This was the first design that took the practicality of the task and married it to the aesthetics of product design. To me, it had the best of both worlds. They figured that a hexagon would better match the natural shape of the can--thus reducing volume waste. They were also confident that the six smaller panels would use far less material than the four larger, rectangular ones.
Student estimate a loss in surface area
The hexagonal shape is mapped out
On the more creative side, one group decided to keep the same rectangular box, but fill it with Tetris inspired mini packs. They were more than up-front about the drastic increase in surface area, but thought the idea was cool enough to compensate. They went into designing the various orientations and nets for the pieces. The results were quite cool.
A Tetris piece is netted and constructed
The design process was highlighted by two things:
  1. A visit from my administrators encouraged my kids. They could not wait to show the math off to them. One student followed my vice-principal around asking for advise and assistance. Another student talked how they shared their math work with their family for the first time. A significant audience is crucial for PBL; seeing other people take significant interest in their work was extremely uplifting.
  2. A student went to work one evening and asked his boss about the design. He then tweeted me the results. This doesn't sound that amazing--unless you knew the student. He is a 50% attender who has repeated math multiple times. He just laughed it off when I highlighted the fact that he was actually doing math outside of class.
Once the initial designs were in place, the challenging task of mathematical accuracy began. I designed the task to cover the topics of surface area and volume, but it ballooned into combinatorics, geometry, circle geometry, regular polygons, estimation, triangular numbers, etc.

The important thing to note is I did not force any specific design on a group. Some groups embraced the "go green" approach, while others were engaged in their novel ideas. The group who doubled the surface area, but designed a box they felt was exciting learned just as much as the practical-minded groups who designed a box to reduce both surface area and volume. I simply sat back, provided the opportunity and freedom, and watched as the math fell from the experiment.

And fall it more ways and forms than I could have possibly predicted.