Saturday, February 23, 2013

Relation Stations

This semester I desperately wanted to improve how I taught linear relations to Grade 9 students. I had tried some interesting activities in the past, but lost patience and ended up drilling them with notation and algorithms. I wanted to find a way to show the students that equations were just explanations of patterns. I began compiling different linear patterns and dug in for the long haul. 

I stumbled upon a collection of abandoned, square tiles and decided to use them to put students in the center of the pattern making.
I began the lesson by dividing the class into groups of three. Each group was given a handful of tiles and a sheet with three blank T-Charts on it. To begin the activity, I placed the following pattern on the board:

Students were asked to model the pattern for the next three stages and record how many blocks it took to build each instance. As they were filling out their charts, I went around to each group and asked them to explain the pattern. Most began by saying, "you just add four more each time". I kept prying at this explanation. I asked them to be more specific.

"You start with a center, and add one on each arm"
"Every time you make the next step, you need to add a tile to each arm"
"The arms have to be as long as the step you are on"

I left after I was sure the groups had made the connection between the four added tiles and increase of four in the table.

Two more patterns were shown, and I repeated the process. 

After the third time, students started to be very concise in their wording. They began to tell me what I wanted to hear. It was after the third round that I asked:

"What do all of these patterns have in common?"

They quickly decided that each pattern started with a base and then added a constant amount each time. I (happily) wrote that observation on the board, and handed out a sheet 11"x17" blank paper to each group. I wanted students to create linear relations with the two newly-defined parts before I formalized their existence as the constant and rate of change

I explained to the class that each group had three minutes to create their own pattern like the three we had seen so far. The only rules were:
  1. You must use your blocks to model the first three stages of your pattern.
  2. You cannot copy or repeat a pattern we have already seen.
  3. Your pattern must have a starting point and a constant change.
I circulated and heard some great conversations about how they were going to create their patterns. Students would name portions of the pattern "top", "bottom", or "middle" and then determine how much it was going to expand each time. This guaranteed that each stage would have a constant change. A wide variety of patterns emerged at a wide variety of skill levels.

My students have used stations before, so they were very familiar with the concept of circulating and completing a T-Chart for each pattern. They had twenty minutes to copy down the first three stages of the pattern, fill out the T-Chart, and decide what the constant change was from every station. They handed in their work at the end of class.

The nice part about the activity was that I could hand back their work the next day when we worked on building equations from the relations. Students had seven great examples to practice building relations. The paper was large enough that the group could easily share. I noticed that more students understood the make-up of equations once they had built their own pattern. They had experienced (and deduced) the existence of a solid piece around which a constant change was occurring. 

We went on to discuss other examples of linear relations. Ticket sales at a concert, a taxi cab fare, and Pokemon attack strength. Students got so good at finding the rate of change that filling in the charts was a breeze. It provided a great bridge to graphing and using the relation as an input/output machine. 

It's a very scary thing for a teacher to give up control. Last year, I stood in front of the class and delivered a very good (Read: Twenty-five minute) explanation about how to fill out charts based on patterns. When it came to introducing equations, they got lost and I barreled forward. There is something intangible that occurs when you provide a student with a well-structured task. I have seen the benefits countless times, but still get anxious before I loosen my grip on the class.

This activity now provides an anchor for the learning. I can always refer to the two critical parts of a relation and students will have experienced them first-hand and on their terms


Tuesday, February 19, 2013

YouTube Relations

My goal this semester was to continue to improve my use of formative assessment (largely through the use of whiteboarding) and expand the role of Project-Based Learning in my classroom. Up to this point, I have developed a wide-scale PBL framework for an applied stream of math we have in the province called Workplace and Apprenticeship Math. Those specific topics lend themselves very well to the methodology; they are a natural fit for PBL. I am still looking for ways to branch the intangibles from PBL into a more abstract strand of mathematics--one that includes relations, exponents, functions, trig, etc. 

I decided to build a small project for the end of each unit of study in my Grade 9 class. I chose this goal because:

  1. I currently have a half semester of Grade 9s, so I'm only responsible for four topics.
  2. The classes will switch at the midway point. This gives me opportunity to make little adjustments to improve the projects quicker. 
In the midst this goal, I have been asked to sit on an informal committee to review how technology can be embraced in our building. I am no technology expert, but am always willing to try something with upside. 

History lesson over--let's move onto the task.

My students will complete this mini-project at the end of the unit on Linear Relations. They are not given any time in class to complete the project, but half a day is devoted to me introducing the students to the necessary technology. (YouTube, Paint, and 

Students are to use the miracle of self-publishing to find an appropriate video on YouTube. The video must be longer than ten seconds but shorter than thirty. They will then choose variables, create a constant scale, and graph the relation portrayed on the screen. My guess is the grading will involve plenty of sports bloopers and "Fail" compilations. 

I initially was going to get the students just to write out the video's URL and hand in a hard copy of the graph, but then decided that the entire class would benefit from seeing the relations. I needed a place where the graph and video could be fully functional in one place. I decided to use

LinoIt is an excellent place for online collaboration. I have used it as a simple class website in the past. Parents could access the URL and see homework assignments, student work, links to school sites, and class announcements. The boards can be private or public. My suggestion is to make the board public (anyone can post) but not to broadcast the URL. That way, students do not need to make an account. This speeds up the posting process. 

The students download a standard blank graph (directly from the LinoIt board) and are asked to graph their relation accurately. When they are done, they can easily post the video, the graph, and their name to the board. All the instructions are given to them in a handout. The handout as well as graph template can be downloaded here.

When completed, each student's homework will look something like this:

The tools are connective and intuitive. The project is simple and creative. It opens up an opportunity for students to begin to see the world through numerate eyes. 

The assessment is three tiered:
  • Students assess themselves based on accuracy, challenge, and creativity.
  • Students rate peer's graphs based on the same three criteria. (As we show them to the class).
  • I will assess each graph on the previous three plus a completion grade. 
My hope is that small, unit-ending projects such as this will begin to include some of the intangible benefits that PBL has brought to my other classes. I want to leave ample opportunity for student choice, autonomy, and innovation. Adding technology only opens those three avenues further. 


Sunday, February 3, 2013

Sorting Set(s)

Set Theory, Counting Methods, and Probability are probably my three favourite topics to teach. For the first time under our new curricular framework, I got to teach these topics to a group of seniors. I decided to build up large themes and understandings through introductory tasks; my goal was to create an "unflippable" entry point where students could work together to complete tasks and filter out necessary details such as rules, notation, etc. I began our study of Set Theory with this task.

The students were introduced to the idea of what a set is. They also were given some elementary verbiage. I wanted them to become comfortable using words like set, subset, and disjoint throughout the task. I did not introduce them to the idea of intersection and union--those were to be formalized through the task.

The day before I rolled out the task, I showed them some Venn diagrams. Many of them had seen them before and we had fun deciphering their meaning. Never did I use the words intersection or union. Students naturally asked about overlap and exclusion though. I just went with the flow and brought up those questions in the classes that followed. 

When the students came into class, I asked them to get into groups of 3. Each group was given a large piece of paper (from the SRC room) with a blank Venn diagram written on it. The diagram had three sets all interlocking--this is the most familiar look to students and I wanted to begin within their familiarity. I stood at the front and introduced them to the cards from the game of Set

For those of you who have never seen it, it is a game with cards that have four attributes--Shape, Number, Shade, and Colour. Each attribute has three options. After a short introduction, I handed out 10 cards to each group. 
Single, Green, Hollow, Diamond
Double, Purple, Shaded, Squiggly
Triple, Red, Solid, Oval
Soon students wondered what they were supposed to do. I made sure to take my time circulating with the cards because I wanted to see how students would begin to organize them. Most primitive structures were based on number and colour. After two minutes of stalling, I began to circulate again writing an attribute inside each set on the Venn diagram. I was sure not to double up within one category (i.e. Red and Purple) in the first round. Students were asked to sort their cards based on the new information.

The circulation allowed time for each group to finish. When I returned to a station, we pointed out some trends in the sorting. Some students didn't like the fact that many of their cards didn't fit into a set; that led to discussions of the universal set. Other students provided a very concise argument for why they were correct. I took mental notes every time the words and, or, & not were used. 

After discussing with all groups I asked them to take a picture with their phone and flip their paper over to a blank side. I then circulated and gave them three new attributes. This time I was sure to double up one of them (i.e. Hollow and Shaded). They needed to draw their diagram and then place the cards. Not surprisingly, many groups traced the pattern through the paper and began. 

On my circulation, all but one group needed to discuss disjoint sets. How can the sets be arranged so that only possible sets exist? What is the difference between an empty set and impossible set? All good conversations to have. groups began to re-draw their diagrams. I asked the more advanced group if they could find any subsets among their cards. That question took them a lot of thinking.

After round two, groups had two pictures of their work:

For round three, I projected some groups' work and we began naming the different sections of the diagram. The words and, or, & not came up again and pieces of notation were introduced. At this point, the students had gained a deep understanding of the sorting process and notation was just a formality. 

The activity anchored the whole unit. When discussions of the principle of inclusion and exclusion came up, we could go back to the archived images and discuss options. It provided me (and the class) with a deep starting point that grew with the topics being addressed. 

A student came in the next day and confessed that she actually had fun during the activity. After realizing what she just said, she looked at her friend who reassured her that she also understood the concepts for the first time. The whole conversation was one of those "jump-for-joy-it-actually-worked" moments. If you were to look at their binders, all the necessary notes are there; we covered all the definitions and rigor necessary to move on. We accomplished all of this with the added bonus of student productivity and engagement.