Four Points, One Line

As my students have worked through a series of linear graphing challenges this month, I’ve been looking for a way to challenge them to synthesize (and hopefully even extend) what they’ve noticed over the past few weeks.

I think I’ve found my culminating challenge.

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My Goal

My goal is to elicit a variety of equation styles (point-slope, slope-intercept, etc), and my hope is that the restriction (which numbers they may use in the equations) is not only clear enough, but also provides the right dose of structure to encourage students to think more deeply about the relationships between the rate of change, intercepts, non-intercept points, and the parameters in each equation.

To give it a test run before sharing it in my own class, I hereby offer you this:

Your Challenge

How many different equations can you write using only the numbers included in the ordered pairs? Can you get to three? How about five? Maybe even 10? Or more?!

Do the work in Desmos, and drop a line in the comments!

As always, feedback—on the challenge in general, or the restriction in particular—is 100% welcome.


I struggled with the wording in the original challenge. As I shared above, my goal is to draw out from students a variety of equation forms, each one utilizing information revealed by a particular point or pair of points. After some back and forth on Twitter, I settled on this reframing of the task:

Four Points, One Line.003

I’d love to know whether you think that drives more quickly and clearly to the heart of what I want students to focus on (while leaving it open enough that students will feel freedom to tinker and explore).

Comments 5

  1. I was able to find 4 pretry quickly. Minimum level of proficiency might need to be lower than 4, but the challenge for differentation is present. My students definitely need practice synthesizing what they have learned. Thanks, as always, for sharing.

  2. Fun question. You may want to include the restriction that the equations be well-formed (or not!) — as soon as you have one equation, you can get another equation that appears different by multiplying both sides by any algebraic combination of those six numbers; if you are allowed to repeat the numbers used, then you’ve got infinite equations possible.

    I counted up to 276 before I fully saw the infinite possibilities afforded by the assumption that (0-6)/(8-0) and -3/4 are valid different representations of the slope…

    If you do intend for students to be writing fully simplified equations, I’m a touch concerned about what happens with the slope. Since you’re emphasizing where the parameters come from for the standard form / intercept form equations, there’s the potential that a student will see 3/4 and think it has something to do with the (4,3) point. How would you feel about putting fractions in the points so you can include (8/3, 4) and (3, 15/4) instead of (4,3)? I’m not sure if that would really help, though — if a kids going to think (4,3) has something to do with the slope, they will probably think those fraction points have something to do with it, too.

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    Andrew, thanks for playing with the proble, and for all the great feedback/questions. I’ll have to wrestle a bit with what you’ve raised. I’ll be back with more ideas soon. 🙂

  5. My desmos small group activity is to have them graph y = x then play around with changing the slope, then the intercept, then combinations of both (without using those words). They have a sheet they fill out describing the changes, then there are different challenges like trying to get an equation that goes through 2 specific points, or finding an equation perpendicular to a specific line.

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