The question I chose for us to investigate (decided ahead of time to meet the curriculum learning goal for that day):

Act 2

Groups were given toothpicks to play with & this image with the number of toothpicks (250) in the jar displayed:

It’s important to give students enough toothpicks to extend the pattern seen in the video, but not as many as 250. My goal is for them to model mathematically, which they won’t do if they can complete the physical model w/ the manipulatives given.

I have also found lately that my students prefer to draw their model on their boards, rather than use the physical manipulatives. Perhaps I should have them work for 2 minutes with the physical manipulatives first before sending them to the boards? Thoughts?

The solutions from each group:You can see that they all used a table of values with second differences to continue the pattern & find an answer. These 2 groups were trying to use some calculations but the top group could not explain to me their reason (within the context of the toothpick triangles) for dividing the total number of toothpicks by 2.
The bottom group was trying some linear proportional reasoning and using the idea of 3 toothpicks per triangle in their calculations. They didn’t make the connection between their constant second difference meaning that linear reasoning wouldn’t work here.

Act 3

The answer:The video version (which we watched) is here.

Even though our curriculum doesn’t require students to determine a quadratic equation from a table of values, it does expect them to graph the data (and I like them to use Desmos to perform a regression & find the equation since Desmos makes it so easy!). Instead of the typical handout I’ve used in the past to consolidate these sorts of activities, I created a Desmos Activity version which you can find (& use) here:This was their homework after the toothpick task.

On day 2, we used this handout to work on describing the steps in some of our solutions:
Student solution:Desmos solution (which I created):

As always, all materials for this activity can be found here.

– Laura Wheeler (Teacher @ Ridgemont High School, OCDSB; Ottawa, ON)

Thanks Laura for your Desmos Activity Builder! It was my first time using the activity builder, but thanks to you, I got to see how it could work in my class (great for engagement)! With a bit of tweaking (my Grade 9 Applied class has no idea of quadratics yet), I wanted them to make the connection between difference between linear vs. non-linear relations and they totally got it! Thanks again.

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Thanks Laura for your Desmos Activity Builder! It was my first time using the activity builder, but thanks to you, I got to see how it could work in my class (great for engagement)! With a bit of tweaking (my Grade 9 Applied class has no idea of quadratics yet), I wanted them to make the connection between difference between linear vs. non-linear relations and they totally got it! Thanks again.

Hi Jacqueline, I’m so happy you found it useful & your class went well! Thanks for letting me know how it went!