Binomial expansion inquiry

The prompt

Mathematical inquiry processes: Notice properties; verify equivalence; extend the pattern; find counter-examples; reason. Conceptual field of inquiry: Binomial expansion; factorisation; Yang Hui's (Pascal's) Triangle.

The prompt shows two identities, which seem to be the start of an intriguing pattern. In the initial phase of noticing and wondering, students make the following contributions:

  • Are the 'equations' true?

  • Why are there three lines and not two as normal?

  • How do you expand (x + 1)3?

  • There's a pattern - the highest number in a bracket on the right-hand side is one less than the index on the left-hand side.

  • The next two cases will be

(x + 1)4 - (x + 1) = x(x + 1)(x + 2)(x + 3) and

(x + 1)5 - (x + 1) = x(x + 1)(x + 2)(x + 3)(x + 4)

  • The pattern does not work for the first one: (x + 1)1 - (x + 1) = 0 (and not x)

After verifying the truth of the two cases in the prompt (with the teacher explaining how to expand triple brackets if necessary), students are enthusiastic to test their conjectures. However, the pattern breaks down for the next two cases. Far from being discouraged, students are hooked into the inquiry by this stage and want to know why the cases in the prompt factorise so neatly, but the other cases do not. The teacher might split the class into groups, with each group having to expand a new case.

Grid multiplication suffices to expand the brackets in the prompt. The expansion of three brackets - that is, (x + 1)3 - can be approached by multiplying the product of the first two by the third. The next cases are not so straightforward and give the teacher an opportunity to introduce the Binomial Theorem, its links to Yang Hui's (Pascal's) Triangle and its use in expanding binomials.

Once the next cases are fully expanded and the results factorised and rearranged, the class can analyse their mathematical structure. What is it about the two cases in the prompt that mean they can be factorised fully? Why does the pattern break down so quickly?

After the first phase, the teacher could direct students to explore other patterns, such as:

(x + 2)2 - 2(x + 2) = x(x + 2)

(x + 2)3 - 4(x + 2) = x(x + 2)(x + 4) and

(x + 3)2 - 3(x + 3) = x(x + 3)

(x + 3)3 - 9(x + 3) = x(x + 3)(x + 6)

Alternatively, students could suggest their own changes to the prompt (see the next section). Throughout the inquiry, students should be learning when and how to use different methods to expand binomials.

Changing the prompt

The prompt can be changed to develop new lines of inquiry. Students have explored the following changes in classroom inquiry:

  • Reverse the signs: (x - 1)2 + (x - 1) and (x - 1)3 + (x - 1)

  • Change the constant: (x + 2)2 - (x + 2) and (x + 2)3 - (x + 2)

  • Change the coefficient of x: (2x + 1)2 - (2x - 1) and (2x + 1)3 - (2x + 1)

  • Analyse the general case: (ax + b)2 - (ax + b) and (ax + b)3 - (ax + b) and then expand (ax + b)n.

April 2021

Mathematical notes

The mathematical notes include:

  • the use of the Binomial Theorem;

  • an analysis of the structure of different cases and the general form; and

  • the consequences of making changes to the prompt.