Lines and Squares

A sculpture of San Francisco by Liz Hickok, made of Jell-O and gelatin squares and other shapes.

This week’s challenge was suggested by Jack Dieckmann, a math teaching expert at Stanford University. Jack is a core member of the popular YouCubed, a Stanford-based online resource for K-12 math teachers founded by math teaching superstar Jo Boaler and her colleague Cathy Williams. YouCubed’s mission is to help ignite students’ natural enthusiasm for math, which may have been crushed by worksheets, drills and homework.

Homework! That’s something I thought was a good thing. At least a little homework. How else can you really learn? But studies are mixed on the value of doing schoolwork at home, with opponents claiming it produces little, if any, gains in learning at the cost of family time. (For more from this perspective, check out The Homework Myth (Ch. 2 here) or Race to Nowhere.).

I visited Jack in his office recently and he shared the following problem, which he thought might be fun to try out on Numberplay. I agreed completely. Let’s try —

Here are 10 straight lines and 17 squares.


Here are 9 straight lines and 20 squares.


Find the smallest number of lines needed to make exactly 100 squares.

Once you’ve done this, investigate further.

How many different ways can you make a particular number of squares?

That’s it for the challenge. Jack describes his own experience sharing this exploratory treasure:

”I’ve given Lines and Boxes to a range of learners including both students and math teachers. With middle school students, one of the biggest insights that comes from working on the challenge is that organizing your findings (exploratory work, scratch work, drawings and diagrams) makes it easier to study the relationship between the number of squares and the number of lines and to make progress on the task. Students are challenged to develop their own way to organize their work as a part of the authoring their own mathematics.”

Thank you, Jack!

This problem was part of the Points of Departure series, a set of mathematical investigation starting points provided by the Association of Teachers of Mathematics in Britain.

Check back on Friday for solutions and commentary by Jack Dieckmann.

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