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!

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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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8 Comments

Lynn Charland on May 7, 2016 at 9:58 pm

I want to try this next year. I have never done 3 act math. Wish me luck.
Reply
- Kyle Pearce on May 9, 2016 at 10:03 am
  
  Awesome! Give it a go! Heck, might be a good way to get into reviewing course material as the end of the school year comes up. Let me know if I can help!
  Reply
  - Lynn Charland on May 10, 2016 at 12:11 pm
    
    Thanks Kyle, Do you have notes or some littérature on how to do a 3 act math lesson?
    
    2016-05-09 10:03 GMT-04:00 Disqus :
    Reply
    - Kyle Pearce on May 10, 2016 at 2:25 pm
      
      Hi Lynn,
      
      I just recently posted a blog about what a 3 act task is and why they are useful here:
      https://tapintoteenminds.com/what-makes-3-act-math-tasks-so-effective/
      
      However, the original creator of the idea is Dan Meyer. His post is a must read:
      http://blog.mrmeyer.com/2011/the-three-acts-of-a-mathematical-story/
      
      Let me know how it goes!
      Reply
Scott Phillips on June 14, 2020 at 6:47 pm

Kyle and Justin,

Thanks for sharing this 3 Act Task. How much time in total do you think this takes? I’m planning on doing at least one 3 Act Task a week this school year as a warm up, but also want to make sure we can get in the grove of not taking up too much class time. Thanks for any information you’d be able to provide!
Reply
- Kyle Pearce on June 15, 2020 at 12:38 pm
  
  Hi Scott,
  
  So happy to hear you’re looking forward to using this task and maybe some other 3 act tasks!
  
  Firstly, I’d nudge you to consider using 3 act tasks as the “guts” of your problem based lesson rather than a “warm-up”. They aren’t an “add on” to a direct instruction lesson, but rather as a means to spark curiosity and help to lead to fuel sense making in students through reasoning…
  
  As for time – it always depends on the learners… grade level… student readiness… etc. but I tend to try to make a task the core of the lesson then build some of the big ideas via consolidation.
  
  Check out makemathmoments.com/framework for more details on this…
  
  Hope this helps!
  Reply
Greg on December 14, 2020 at 1:54 am

A darker background would help the toothpicks stand out.
Reply
- Kyle Pearce on December 14, 2020 at 10:31 am
  
  Hi Greg – so true. I’m a *little* better now at planning that sort of thing. May need to go back and reshoot at some point 🙂
  Reply