# Breathing New Life Into Sets

Ms. Walter, my junior high math teacher, sure knew how to get my attention on the first day of class. She told us we would be studying all about sex. Well no, let me restate:  She said we’d learn about sets, but my seventh-grade ears heard otherwise. It didn’t take long for my confusion to clear, and, needless to say, I was disappointed.

My junior high experience aside, I think sets generally get a bad rap in elementary mathematics. Proposing that young learners study sets invariably raises the specter of the New Math movement’s misguided emphasis on set theory in the 1960s.

I propose giving sets a fresh look by considering  a new Sketchpad model that I designed as part of the Dynamic Number project. Start by downloading the Set Puzzles activity. Below is a sample puzzle.

The goal of the puzzle is to arrange the circles and the six numbers (1, 2, 4, 8, 16, and 32) so that three conditions are met simultaneously: The sum of the numbers in the green circle is 21, the sum of the numbers in the blue circle is 26, and the sum of the numbers in the red circle is 14. The numbers can be dragged into the circles, and the circles can be moved as well. Dragging the point that sits on the circumference of each circle changes the circle’s size.

Students find it straightforward to satisfy the first condition of the puzzle. Dragging 1, 4, and 16 into the green circle fills it with numbers whose sum is 21. Creating a sum of 26 in the blue circle is more puzzling. The only way to form 26 is to add 2, 8, and 16, but the 16 already resides in the green circle. How can it perform double duty?

Most students, after pondering this dilemma for a minute or two, have an “aha!” moment: What if the green and blue circles overlap each other, with the 16 sitting in their intersection? The picture below shows how this looks.

Now, all that remains is to satisfy the final condition of the problem by making the numbers in the red circle sum to 14.  The only way to form 14 is to add 2, 4, and 8. But the way those three numbers are currently arranged makes it impossible to enclose them without including the 16 as well. Students decide that they need to adjust their model to allow the 2, 4, and 8 to sit alone in the red circle while still keeping the first two conditions true (green = 21, blue = 26). The picture below shows the final result, with all three conditions of the puzzle simultaneously true.

I’m especially pleased with the way Set Puzzles require young learners to juggle and process multiple pieces of information. More often than not, the placement of the circles and numbers must be adjusted one or more times to ensure that the sum of each set of numbers matches the required value.

And the well of Set Puzzles runs deep. Each new challenge generated randomly by Sketchpad uses the same six numbers—1, 2, 4, 8, 16, and 32—as the elements for the three sets. These powers of two can, amazingly, form any sum from 1 to 63.

The video below gives some more details about the puzzles and shows how students can make custom challenges for each other. Check it out!

## About Daniel Scher

Daniel Scher, Ph.D., is a senior scientist at KCP Technologies, where he co-directs the NSF-funded Dynamic Number project. He has developed Sketchpad activities across the entire mathematics curriculum, from elementary school through college. He received his Ph.D. in Mathematics Education at New York University.
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### 2 Responses to Breathing New Life Into Sets

1. Rebecca says:

So my version of Sketchpad will not run this, so I have a question. How do you determine the sums to use for each circle. Does the sum just need to be a value between 1 and 63?

• Andres Marti says:

Yes, you can create any number between 1 and 63 using sums of the first six powers of 2 (1, 2, 4, 8, 16, 32).

Since you mention having an older version of Sketchpad, we strongly encourage you to upgrade to Sketchpad 5. The prior version is a decade old. You can download a free preview at our website: