Wednesday, January 8, 2014

Using Pictures to Prepare for MFM 2P Board Wide Summative (OntarioOCDSB)

In our grade 10 applied math course in the OCDSB students are required to write a board wide Summative. It is different. It requires the students to look at pictures of a scenario ( for example a bunch of pictures related to a massive snowfall ), ask questions about the pictures ( for example What is the volume of the snowman? What is the slope of the toboggan hill? ), once given some data about the pictures they are to then answer some of their questions. Questions are asked individually, then shared in groups, then shared as a class. Once all students have lots of questions the students then work individually to answer some of their questions - translation - demonstrate what they have learned in the course in this context.

In an attempt to prepare my students for this ( it takes place January 14th and January 16th in class ) I did this activity prior to the holiday break.

Initial Set Up
Snowballing for Questions
5 stations / groups were set up with a theme at each group and a colour associated with each group. Groups were to write down questions in their colour on chart paper. Groups rotated around the room (snowballed) taking their colored markers with them. Eventually groups had visited all 5 stations / themes.

Here are the pictures I provided and the chart paper with all the questions.

A couple of observations at this point. This took a period. Groups left their questions and the pictures behind when they rotated. So new groups that arrived could read all the previous groups questions. By the time the 4th and 5th groups arrived to each station / theme, most questions had been asked and the behaviour wasn't great. Also there were lots of silly and inappropriate questions. I wonder how many of these questions might have been escalated into conflict between student and teacher in a different format?
In retrospect, I think I would generate the questions over 5 days at the start of those 5 periods. Also I would not allow groups to see other groups questions. Live and learn.

Snowballing to Categorize the Questions into the Curriculum of our Course
At the start of the second period groups then started at their original station / theme and categorized the questions into the curriculum headings for the course. Groups then snowballed to all 5 stations / themes until they had categorized the questions into the curriculum headings. This forced groups to discard irrelevant or inappropriate questions.

Here are a few samples (Of course the sheets were colour coded based on the colour of their group):

Snowballing to Pick Best Three Questions
Groups then started at their original station / theme and came to a concensus on the best three questions that were categorized. Groups snowballed around the room until all groups had 15 best questions (3 per station / theme). Every person had to have the three best questions written down on a sheet of paper for each station / theme. Here are some sample questions that came out of the different groups. (curriculum topic in italics).

Carnival theme
When is it better to go to Overwijk's carnival? McLaurin 's Carnival? (Linear relations and intersections of lines)
How far does the bean bag go? (Quadratics)
What is the maximum height of the bean bag toss? (Quadratics)
How much space does the Skee Ball machine take up? (Volume)
What is the angle of incline of the beanbag boards? Skee ball machine? (Trigonometry)

Snow theme
What is the volume of the snowman? (Volume)
What is the volume of snow in the backyard? (Volume)
What is the maximum height of the snowball throw? (Quadratics)
How far does the snowball land from where it was thrown? (Quadratics)
What is the height of the doorway in the igloo? (Quadratics)
What is the length of the outside of the igloo? (Sum of Squares)
What is the width of the igloo? ( Sum of Squares)
What is the angle of incline of the side of the igloo? (Trigonometry)
What is the surface area of the igloo? (Surface Area)

Bridges theme
What is the height of each pillar? (Similar Triangles)
What is the Height of the parabolic bridge? (Quadratics)
What is the length of the wire? (Sum of Squares or Trigonometry)
What is the angle between the wire and the bridge? (Trigonometry)
What is the angle between the two wires? (Trigonometry)

Swimming Pool theme
What is the Volume of the pool? (Volume)
What is the surface area of the thatch hut? (Sum of Squares and Surface Area)
What is the height of the pool ladder? (Sum of Squares or Trigonometry)
What is the angle of the poll ladder? (Trigonometry)
What is the maximum height of the dive? (Quadratics)

Skateboarding theme
What is the length  (height) of the ramp? (Sum of Squares or Trigonometry or Similar Triangles)
What is the volume of the ramp? (Volume)
What is the depth of the parabola? (Quadratics)
What is the height of each board along the ramp? (Similar Triangles)
How high does the skateboarder jump? (Quadratics)
What is the difference in height of the skateboarder and the skateboard? ( Quadratics)

A Day Break
The class then took a day break from this activity. This gave me time to create the pictures with the appropriate data for each group for each theme so that the groups could answer their best three questions per theme. For some photos I did not add data because there were people in the pictures that the students could use to create a scale.

This should give you the idea:

Answering the Best Three Questions per Theme

This took the better of one and a half periods. Groups started at their original theme and we rotated around the room (snowballed) to answer their best three questions per theme. All groups members had the same three questions because they decided on them on day 2. Students were asked to work individually on each theme and then collaborated with the others in their group to come up with answers. At the end of the period and half all students had five sheets of paper with three questions and three answers on each (one for each theme)

A couple of observations:

1) Since this basically covered most of the curriculum in our course, as I observed students working on solutions, it was obvious where students had weaknesses.

2) The collaboration and engagement during this time was great. Not sure why? Because they made the questions?

3) Some of the questions were difficult. This caused some frustration.

Picking Three Artifacts

In the other half of the period I asked students to cut out their best solution, a solution that was OK , and a solution that needed help.
Once they had these cut out they were to comment on some process expectations by filling in the following form - one for each of the three solutions. They then stapled the form to the solution.

        EXIT CARD                                                      OK
                ¨      Did you connect math to the problem?

  ¨      Can you apply any of the math            covered in class to this problem?

  ¨     Did you translate the problem into math?

  ¨  Have you use as many ways as you can (e.g. table of values, graphs, equations, diagrams, words, numbers)?

  ¨  Do you have all the tools you need? (e.g. graphing calculator, grid paper, measuring tools, formula sheets, etc.)

  ¨  Do you have a plan?

  ¨  Does your math make sense?

  ¨  Have you read over your work?

  ¨  Does your answer make sense?

  ¨  Can you explain how you know your answer makes sense?

  ¨  Is there another way you could have approached this problem?

  ¨  Could someone else understand your work?

  ¨  Are you able to clearly explain your work to someone else?

  ¨  Have you used units correctly?

  ¨  Have you used symbols correctly?

  ¨  Have you used math terms you learned in class?

  ¨  Did you write concluding statements, highlighting your answer?


On the back of their best work I asked them to rank the curriculum from 1 - what they were best at to 6 -  what they were weakest at. The six curriculum areas were Sum of Squares,  Surface area / Volume of 3D figures, Similar Triangles, Trigonometry, Linear relations, Quadratic Relations.

Doing Other Groups Questions
When the students came in for the fifth day (that's right - a whole week's worth of stuff) I had this posted on the board.

This was their ranking from the day before. I then threw all the pictures with all the data from all the groups on some desks in the middle of the room. Their assignment on this day was to find a question that addressed each of the 6 curriculum areas ( it had to be from a different group) and solve it. I asked them to do it in reverse order of how they ranked the curriculum from the day before i.e. they started with what they were weakest at and finished with what they were best at.

I will report back after the summative to tell you if this was worth the effort and time.

Thoughts? Questions?


  1. Not surprised by the students having trouble with quadratics. A little surprised by how many ranked linear relations a 5 or 6. Very cool.

    1. This could be due to the way in which I teach linear relations. I cycle the curriculum through activities and don't really emphasize y=mx+b. The students need to create an equation that works for a particular context. I think this makes it more difficult for them. Trying to talk less.

  2. I posted this on my blog this week. Can you also post the files with the pictures for each station?

  3. I really like the form that you had your students evaluate their solutions with. Thank you for sharing.