Early algebra ideas about binomial expansion, Stephanie's interview five of seven, Clip 4 of 10: Investigating the "doubling pattern" for unifix towers of increasing heights.

In the fourth clip in a series of ten from the fifth of seven interviews, 8th grader Stephanie continues her exploration of Early Algebraic Ideas about Binomial Expansion. Using combinatorics notation, researcher Carolyn Maher records as Stephanie reviews the total number of towers 4-tall selecting from red and yellow cubes. In each of the five cases for n equal height 4 and r equal the number of red cubes from zero to 4, Stephanie indicates the number of towers and then adds the numbers from the subtotals to find the total. She then revisits the totals for towers one, two and three cubes high and notes that the totals double for each additional cube. Maher asks Stephanie to begin with the two, one-tall towers and to generate towers two-cubes tall by adding either a red or a yellow cube above the original base. From these four, Stephanie then generates the 8 towers that are 3 cubes tall and justifies her solution by referring to the tree diagram that she has drawn. She posits that, on the basis of her findings, this doubling pattern will continue. The problems as presented to Stephanie: Based on your combinatorics notation, what is the total number of Unifix Towers that are four-cubes tall? What are the total numbers for towers that are: one-cube tall, two-cubes tall and three-cubes tall? What would you predict as the total for towers 5-cubes tall? If there are exactly two towers one-cube tall, how can you generate towers that are 2-cubes tall? Three cubes tall? Why does this pattern work? Do you think that it would continue for towers 4-cubes tall and taller?Transcript and student work are also available.Robert B. Davis Institute for Learning. (1996). Early algebra ideas about binomial expansion, Stephanie's interview five of seven: Clip 4 of 10, Investigating the "doubling pattern" for unifix towers of increasing heights [video]. Retrieved fro