Placing Toothpicks

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

Patterning, Proportional/Direct Variation Linear Relations

Placing Toothpicks 3 Act Math Task - Patterning, Proportional Direct Variation Linear Relations

Patterning is something that comes up early in the Ontario Elementary Math Curriculum during the primary grades often connecting directly to proportional reasoning, and then evolves into direct variation linear relations in the grade 9 academic and applied courses. In both the academic and applied grade 9 courses I teach, I try to integrate proportional reasoning early and often – slowly scaffolding students towards one-step equations and the characteristics of linear relations.

Today, I’m going to share out the following 3 act math task that Justin Levack and I filmed last year keeping the grade 9 academic curriculum in mind. In particular, this task can be connected to:

  • NA2.02 – I can solve problems requiring the manipulation of expressions arising from applications of percent, ratio, rate, and proportion.
  • NA2.07 – I can solve first-degree equations, including equations with fractional coefficients, using a variety of tools and strategies. (minus fractional coefficients)
  • 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).
  • LR2.04 – I can compare the properties of direct variation and partial variation in applications, and identify the initial value when described in words, represented as a table, a graph, or an equation.
  • …and many more!

Let’s get started!

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 Padlet wall:

Can’t see the padlet wall? Click here.

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.

The questions 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 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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New to Using 3 Act Math Tasks?

Download the 2-page printable 3 Act Math Tip Sheet to ensure that you have the best start to your journey using 3 Act math Tasks to spark curiosity and fuel sense making in your math classroom!



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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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