Two Points and Graduation

We’ve recently been talking with teachers a lot about the following problem from last year’s EQAO test:

The total cost of swimming at a community swimming pool is made up of a membership fee and a cost per swim. At this community centre, Jake pays a total of $100 and swims 40 times. Paula pays a total of $70 and swims 25 times. Which of the following statements is true?

                                                           a) The membership fee is $20.

                                                           b) The membership fee is $30.    

                                                          c) The cost per swim is $2.50.           

                                                          d) The cost per swim is $2.80.

How would you do it?  Or maybe more importantly, how would you expect your students to do it?

We’ve heard a variety of methods:  create a system of equations and solve, write as two points and find the slope and y-intercept, think about the context (i.e. I’m paying $30 more for 15 more swims…), trial and error for each option, etc., etc.

Through talking with teachers we conjectured that students really don’t need to have any knowledge of linear equations at all to do the swimming pool problem.  They just need to have a sense of the context and operations.  We decided to test this conjecture with the Graduation Problem which we found on Dan Meyer’s website, and adapted to the class we were working with.

Students are given a class graduation roster (we got a list of the students currently in grade 9 and made up a “Class of 2016” roster) with the times that two students crossed the stage (see Dan’s activity).  Students know that graduation starts at 1:00, there are some speeches, and then students start crossing the stage one by one.  The overarching questions are:  a) how long are the speeches? and b) how long does it take each student to cross the stage. 

Students were immediately engaged in the problem.  Many had experienced a Grade 8 graduation not that long ago so could relate to the context.  One student shouted out “I wonder what time I would cross the stage at?”  so we knew we had their interest.

Here’s some samples of student work:  (note that these students knew NOTHING about linear equations yet, they were just finishing their first unit of the course on measurement)




Most teams managed to solve the problem but the approaches varied.  Many students eventually chose to find a rate but they described the rate in different ways: 25 students/5 mins., 12 seconds/student, 5 students/1 minute.   We used the same numbers as Dan for the problem and we found this choice of numbers was helpful in allowing students to represent their rates in a way that they were comfortable with.  Students also used different strategies for determining the time of the initial speeches including just guessing initially but then returning to test their guess once they came up with a different approach.

After sharing solutions, Liisa told the class that the roster the students had been given actually was missing about 20 people from their grade (which was true!).  She asked them how they could figure out the length of the graduation if they were added to the list.

One girl raised her hand: “You could make an equation… it’d be like… um… 12 seconds times the number of people + 20 minutes for the speeches”. 

This student essentially had developed an equation of a line given two points (with some minor issues with units….) and she did this without any formal lesson on how to find the equation of a line given two points, in fact, she hadn’t even heard about slopes and y-intercepts yet – she just made sense of the problem and solved it!

Would this be an effective way to introduce the equation of line given two points?  If you started with this problem, could students move to generalizing an approach that would work in any situation?


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