A Puzzle Equivalent to Factoring a Quadratic

Solid lines mean multiply. Dashes mean add.

Place numbers in the empty circles so all the calculations work out correctly.


Solving this puzzle is equivalent to factoring 6x^2 + 17x + 12.

EDIT: Here’s a version based on jd2718’s comment below.


The whole point of my expressing this as a puzzle is it can be generalized into other puzzles which have nothing to do with factoring, for example:


I was planning a worksheet that started out with only 3 circles and then worked its way up to more complex puzzles. (ADD: See files below.)

It’s also possible to express factoring a cubic this way! It’s a messy diagram but I’ll see if I can get it up this week. (ADD: Post is here.)

ADDED Jan. 31 2012: These were part of a presentation I gave at a conference. The puzzles in “Puzzles 3” are all equivalent to factoring a quadratic.

Instructions for puzzles
Puzzles 2
Puzzles 3

21 Responses

  1. I’m wondering if you can bend the lines a bit? Or maybe it’s me. My mind doesn’t like the “T” shapes and was wishing the two circles would point at their joint product (or sum in the center)


  2. I will see how that looks. Still plenty of room to noodle.

  3. […] Jason put up a very cool puzzle that is equivalent to solving a quadratic equation. […]

  4. I love this idea.

    I have no idea how it’ll fly in my classroom, but I expect to give it a whirl when I get there.

    I wrote up a post with my own variation and more thoughts on how it might go in the classroom.

  5. […] A Puzzle Equivalent to Factoring a Quadratic […]

  6. I’ve been pondering whether I like the non-factoring versions as an approach to teaching how to solve the factoring problem, or if I prefer just working that particular version of the puzzle. I see benefits to both, and I suspect the latter may be more boring for the kids, definitely do less to teach general problem solving skill, but might just reinforce the factoring concepts a bit better.

    BTW, I like how that last problem is an example of the distributive property. It’s quite possible this may have multiple applications.

  7. Another interesting part about the puzzle approach is it is stealthy — you could get students factoring before they even realize what it is! This might be good for a student who has tried factoring before and failed and has a negative opinion of it.

  8. […] the Number Warrior, turns factoring quadratics into a graphic puzzle. Rolfe introduces his sons to binary. Check out this interesting graph of arithmetic knowledge and […]

  9. […] 19, 2009 Factoring by Stealth Posted by Muhammad Alkarouri under mathematics   This is simply brilliant. Humans have always been good at visualisation, and puzzles always generate […]

  10. can.iget…the.answers

  11. Did you ever end up making a puzzle worksheet that leads into factoring quadratics?

  12. […] warm up opener I used for my first Math 12 class last night.  I shamelessly stole Jason Dyer’s idea and turned it into a three-page set of […]

  13. […] Comments Factoring graph puzzles « josh g.’s notes on A Puzzle Equivalent to Factoring a QuadraticFive intuitive approaches to teaching the infinitely small « The Number Warrior on Infinite […]

  14. A simpler number puzzle:-

    Can U make 64 using ONLY Two 4’s??…


  15. […] the same vein as my puzzle equivalent to solving a quadratic, solid lines mean multiply, dotted lines mean […]

  16. […] response to The Number Warrior’s post, I would like to propose an alternative, since the choice to do the sum or product inheres more to […]

  17. […] was at Jason Dyer‘s talk at the Tucson MEAD conference in January where he introduced me to these puzzles. The gist is: dotted lines indicate that circles should add to the desired result, while solid […]

  18. Could you send me the answers to the puzzles 2 problems. I can’t figure out the 3rd problem. It has 8 and 40 in it.

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