# Dancing Unknowns: You Haven’t Seen Simultaneous Equations Like These!

When it comes to simultaneous equations, I like to push the bounds of conventional pedagogical wisdom. In an earlier post, I offered a puzzle in which elementary-age students solve for four unknowns given eight equations. Now, I’d like to present a puzzle that might sound even more audacious: Solving for ten unknowns. Oh, and did I mention that the unknowns are not sitting in place but dancing around the screen? More on that below.

Here’s how the static version of the puzzle works: Each of the ten letters in the Web Sketchpad model above has been assigned a random, secret value between 1 and 10 (It’s possible that two or more letters may share the same value.) Your job is to determine these values.  To do so, you’ll use the circle. To move the circle, you can drag its interior. You can also drag the point sitting on the circumference to change the circle’s size.

Whenever two or more letters sit inside the circle, its interior turns green, and you’ll be told the sum of those letters. So, for example, when G and I are in the circle’s interior, Sketchpad reports that G + I equals 16.

To play again with new secret values, press New Puzzle. When your students are ready to try a different challenge, press the arrow in the bottom-right corner of the model. There are still ten letters whose values must be determined, but hey–the letters are moving (dancing) across the screen! The rules are still the same–surround two or more letters with the circle to determine their sum–but the movement of the letters makes the puzzle more challenging.

What strategies do your students use to uncover the values of all 10 letters?

When I tested this activity with a group of fourth graders, Lee solved the static version of the puzzle by dragging the circle so that it covered all ten letters. He then adjusted the circle so that it covered 9 letters, thus allowing him to determine the value of the extra letter. He continued in this fashion, scaling back the number of letters sitting in the circle from 9 to 8 to 7, and so on, all the way to just 2 letters.

When Lee moved on to the “dancing” version of the puzzle, he began with the same strategy of covering all ten letters with the circle and then scaling back one letter at a time. But because the letters did not stay put, he found it difficult, if not impossible, to remove one letter at a time from the interior of the circle. After some consideration, Lee realized he would need to adopt a new strategy: He began by encircling 2 letters, noting their value, and then adjusting the circle to include a third letter.  He continued in this manner, picking other pairs of letters and adding a remaining letter to the pair. It was fascinating to watch Lee modify his original strategy to this new one, motivated by the switch from static to dynamic letters.

This activity first appeared as part of the NSF-funded Dynamic Number project. You can download the original Sketchpad model and teacher notes here.

An annotated list of all our elementary-themed blog posts is here.

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