Squares and cubes inquiry
Mathematical inquiry processes: Verify; identify and extend patterns; reason. Conceptual field of inquiry: Square and cube numbers.
John Petry, a teacher of mathematics at Parliament Hill School, Camden, devised the prompt from Polya's discussion of induction and mathematical induction in How to Solve It (pp. 114-121). The prompt invites students to identify and extend a pattern using the sum of consecutive cubes. Students can follow different lines of inquiry:
Verify the prompt is true;
Extend the prompt to the sum of the cubes of the first four, five, six (and so on) positive integers;
Change the prompt by starting at a different point - for example,
23 = 8 = 32 – 1
23 + 33 = 35 = 62 – 1
23 + 33 + 43 = 99 = 102 – 1
Change the prompt by increasing the difference between the numbers - for example,
13 = 1 = 12
13 + 33 = 28
13 + 33 + 53 = 153
Generalise the pattern using algebra and prove (see the section in Polya).
In his original prompt, John used the first two lines of the pattern (below). This has the advantage of requiring students to extrapolate from less information, giving them the satisfaction of inferring and developing more of the pattern themselves. The prompt with three lines might be more suitable for students who would find such extrapolation too challenging.
John reports on inquiring into the prompt with a year 7 class below. He concluded that, "All in all the inquiry was very positive. We just had a parents' evening with this year group, and a lot of the students wanted to talk about what they had found with their parents despite it being a couple of weeks ago."
Starting a unit with inquiry
John Petry reports on the inquiry he carried out with his year 7 class and explains the advantages of starting a unit with an inquiry prompt:
I introduced the prompt to the class and these are their questions and comments (above). They verified that both statements were true, although I found they had issues continuing any patterns they saw. A couple of suggestions are more creative around changing the prompt to see if it works. To help the class extend the pattern, I wrote out on the board:
___3 + ___3 + ___3 = ___2
The students wanted to substitute some numbers into the equation and check whether it was ever true. This led to three different lines of inquiry that students decided to pursue for themselves in the second lesson:
Starting with 33 + 33 + 33 = 92, some spotted that if they had four cubes on the left then it worked with 4 and 16, and they followed that pattern and explained it.
A second line of inquiry involved ___4 + ___4 + ___4 = ___3. Most of those that followed this line of inquiry soon dropped it as they decided that the powers of 4 would "become too big, too quickly."
Nearly all the class by the end were trialling consecutive numbers, with some starting at one and others using different starting points.
Three or four continued the pattern that leads to triangular numbers without prompting by me. Having done this I showed their work under the camera at the front, and this led to nearly all the class changing tack onto that. By the end those pupils were happy to come to the board to explain their working. At this point a different student recognised the triangular number pattern and shared that with the class.
I like to introduce inquiry at the start of a unit. I’ve tried it both ways round recently and have achieved more in getting the students interested in their own inquiries, then providing the tools for them to understand further. It also provides a nice over-arching theme to a unit of lessons which we can refer back to, even if we aren’t still working directly on the prompt.
Extracts from the students' inquiries
After the year 7 inquiry, John considered making changes to the original prompt. One alternative (left) would be to leave blanks in the third line in order to suggest the continuation of the pattern. This has the advantage of directing the students towards the standard result, but might close off other interesting continuations. One example of a different continuation that has arisen in a classroom is 13 + 23 + 13 = 10, 23 + 33 + 23 = 43 and so on, although in this case the first line should be 03 + 13 + 03 = 1. Other adaptations involve using Pascal's triangle to get 13 = 1, 13 + 13 = 2, 13 + 23 + 13 = 10 and so on. Another option (left below) is to leave blanks throughout the prompt, which would encourage lines of inquiry that involved other number patterns.