Placing Toothpicks Sequel

How many toothpicks are in the 6th figure? ... the 11th figure?

Ontario Alignment By Grade
Ontario Alignment By Overall Expectation
CCSS Alignment By Grade
CCSS Alignment By Standard

Patterning, Identifying Linear and Non-Linear Relations

Placing Toothpicks Sequel 3 Act Math Task - Identifying Linear and Non-Linear Relations

Here’s an extension to the Placing Toothpicks task I posted recently. While the previous task hit on quite a few learning goals from the Grade 9 Academic and Applied courses in Ontario, this task will focus on identifying linear and non-linear relations in those same courses. It should be noted that this task could also be used in grade 10 courses for learning goals related to Quadratics. On my radar right now would be the following grade 9 academic expectations:

  • LR2.02 – I can construct tables of values, scatter plots, and lines or curves of best fit as appropriate, using a variety of tools for linearly related and non-linearly related data collected from a variety of sources.
  • LR2.03 – I can identify, through investigation, some properties of linear relations and apply these properties to determine whether a relation is linear or non-linear (by rate of change/initial value when described in words, by first differences in a table, straight/curved graph, degree of terms in equation).
  • AG1.01 – I can determine, through investigation, the characteristics that distinguish the equation of a linear relation (straight line) from the equations of non-linear relations (curves).

Act 1 – What’s the Question?

Show the students the video below:

Can’t see the video? Click here.

At this point, I’d be asking my students what questions they have. You can write these questions on the board or if the kids have devices, use something collaborative like a Google Doc or Padlet wall.

Act 2 – Revealing More Information

After some questions are shared out, you can show students the following video:

Can’t see the video? Click here.

As we did in the original Placing Toothpicks task, the question(s) I want students to think about are:

How many toothpicks are in the 6th term? … the 11th term?

I would likely have students figure this out on their own, using any strategy and then consolidate the task. The following math task template might be a good option to consolidate student thinking:

Placing Toothpicks Sequel 3 Act Math Task Template

Act 3 – Reveal the Solutions

You can now let students see their solution in action!

Act 3, Scene 1 – Figure 6 Solution

Can’t see the video? Click here.

Act 3, Scene 2 – Figure 11 Solution

Can’t see the video? Click here.

Have you tried this task? How can we make it better? Share your thoughts in the comments below!

Download The Task and Resources

Grab all the videos, images and resources by clicking download below:


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About Kyle Pearce

I’m Kyle Pearce and I am a former high school math teacher. I’m now the K-12 Mathematics Consultant with the Greater Essex County District School Board, where I uncover creative ways to spark curiosity and fuel sense making in mathematics. Read more.


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