Review: “Using Friday Puzzlers to Discover Arithmetic Sequences”

The author teaches eighth-grade mathematics and writes about using related puzzles to lead students to discover the formula for the sum of an arithmetic sequence. The puzzles are presented each Friday, and students look forward to the experience each week.

The guidelines are presented before the first puzzler:

  1. Each puzzler can be completed within a class period.
  2. Your current mathematics knowledge is all you need to solve the puzzle.
  3. You will never be given a puzzle that is unsolvable.
  4. Often by first thinking about it, a technique will emerge that allow you to complete the puzzle in less time than if you plod ahead without thought.

Yolles uses the following five core Puzzlers:

  1. Gauss: A Child Prodigy. This is the story of Gauss’s 3rd-grade teacher assigning the class “find the sum of the numbers from 1 to 100” and having Gauss almost instantly produce the answer.
  2. Handshakes All Around. Ten friends attend a party where each person shakes everyone else’s hand, exactly once. How many handshakes occur?
  3. Lighting the Chanukah Menorah. Over the course of eight days, how many candles are lit in all on a menorah?
  4. The Twelve Days of Christmas. How many gifts were sent on Day 12?
  5. Clock Face Puzzle. Can you locate one straight line to split a clock face in two so that the sums of the numbers on the two parts are equal?

Students work on each problem for an entire class period, earning stickers if they solve the problem. The author says that few succeed at first, but the proportion of students solving each problem grows through the year. Puzzlers are not provided each Friday; the students take a few weeks between sessions.

The author provides samples of student work; her students show surprising creativity in their approaches to these problems of counting. She also describes Socratic conversations which attempt to bring together the results of multiple weeks’ work.

By the end of the Friday Puzzler sequence, she says, “all students have earned an ‘I got the Friday Puzzler!’ sticker,” and the class has discovered the formula S = (n/2)(a + l), where n is the number of terms in the sequence, a is the first term, and l is the last term.

I found the article fascinating and inspiring, and I look forward to using it in my own classroom.

References

1. Yolles, A. (2003, November). Using Friday Puzzlers to Discover Arithmetic Sequences. Mathematics Teaching in the Middle School, 9(3), 180-185.

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