Placing Toothpicks Part 4

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

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More Patterning and Partial Variation Linear Relations

Placing Toothpicks Part 4 - Partial Variation Linear Relations

Yet another task in the Placing Toothpicks Series (Placing Toothpicks, Placing Toothpicks Sequel, Placing Toothpicks Part 3) I posted recently. The first task was proportional, followed by a quadratic in the second and a partial variation linear relationship in the third. This task is going to give my students another go at partial variation linear relations. As were the learning goals from the Part 3 Task, here’s the grade 9 academic expectations we can make connections to:

  • 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).
  • 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.

Act 1 & 2- Revealing The Task

Show the students the video below:

Can’t see the video? Click here.

As we did in the previous toothpick tasks, the question(s) I want students to think about are:

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

As usual, I like having students figure this out on their own, using any strategy and then consolidate the task. However, the following math task template might be a good option to consolidate student thinking:

Placing Toothpicks Part 4 Math Task Template Screenshot

You’ll notice that I tried to make the PDF file interactive as I was going to be absent the day my students were going to do this task. How I did it was by adding hyperlinks in the PDF that will allow students to jump straight to the Act 1 video:

Placing Toothpicks Part 4 - Watch This Video Hyperlink

After students have come to their conclusions, they can watch the Act 3 video by clicking the link embedded in the math task template:

Placing Toothpicks Part 4 - Watch This Act 3 Video Hyperlink

The task template is structurally organized to ask students to do specific things like create a table, graph and answer some questions on the back. I’m not super thrilled with using this approach as of late, but these math task templates are really the only structure from my former, more traditional, teaching approach. I would be lying if I said I don’t worry about my moving to a less structured teaching style, but I feel deep inside that it is the right thing to do.

If you want to grab the template, click below:

Grab It!

Act 3 – Reveal the Solutions

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!





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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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