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

Puzzles 2

Puzzles 3

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jd2718, on January 10, 2009 at 6:23 pm said: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)

Jonathan

Jason Dyer, on January 10, 2009 at 9:27 pm said:I will see how that looks. Still plenty of room to noodle.

Math Stories : The factoring puzzle, on January 12, 2009 at 12:02 am said:[…] Jason put up a very cool puzzle that is equivalent to solving a quadratic equation. […]

Mr. K, on January 12, 2009 at 12:07 am said: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.

A Puzzle Equivalent to Factoring a Cubic « The Number Warrior, on January 12, 2009 at 9:17 am said:[…] A Puzzle Equivalent to Factoring a Quadratic […]

Mr. K, on January 13, 2009 at 11:08 pm said: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.

Jason Dyer, on January 14, 2009 at 8:54 am said: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.

Carnival of Mathematics 47, where no, well… « JD2718, on January 17, 2009 at 1:56 pm said:[…] 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 […]

Factoring by Stealth « On Another Dimension, on January 19, 2009 at 8:59 am said:[…] 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 […]

mike, on August 30, 2009 at 1:22 am said:can.iget…the.answers

Jason Dyer, on August 30, 2009 at 10:40 am said:First two puzzles: top 2-3 middle 8-9 bottom 4-3

Last puzzle: top 6 bottom 12-18

joshg, on February 5, 2010 at 6:25 pm said:Did you ever end up making a puzzle worksheet that leads into factoring quadratics?

joshg, on February 6, 2010 at 3:50 pm said:Ok, well if you haven’t, I’m making one right now.

Jason Dyer, on February 6, 2010 at 8:39 pm said:I had started but hadn’t finished, so it’s ok whatever you have planned.

Factoring graph puzzles « josh g.’s notes, on February 9, 2010 at 8:01 pm said:[…] 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 […]

Josh Giesbrecht takes on factoring puzzles « The Number Warrior, on February 11, 2010 at 8:51 am said:[…] 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 […]

cvs268, on March 7, 2010 at 11:53 pm said:A simpler number puzzle:-

Can U make 64 using ONLY Two 4’s??…

http://2600hertz.wordpress.com/2010/03/08/four-4s/

Visual Algebra? « The Number Warrior, on June 2, 2011 at 9:53 am said:[…] the same vein as my puzzle equivalent to solving a quadratic, solid lines mean multiply, dotted lines mean […]

A Puzzle Equivalent to Factoring a Quadratic « Running with a compass, on June 2, 2011 at 12:51 pm said:[…] 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 […]

Puzzles with Circles & Algebra « Mathy McMatherson, on May 13, 2012 at 8:01 pm said:[…] 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 […]

Anonymous, on March 3, 2016 at 6:48 am said: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.

Wendy Menard, on October 14, 2016 at 5:38 pm said:Did you ever get an answer key? My students were stumped by the 8/40 one as well.