Difference of two cubes inquiry

The prompt

Mathematical inquiry processes: Verify; extend the pattern; generalise and prove. Conceptual field of inquiry: Difference of cube numbers; rearrangement of algebraic expressions.

Emmy Bennett designed this prompt as an extension to the difference of two squares inquiry (see the report below). Students can prove the general case for the difference of two cubes (see below).   

There is also a pattern for the difference of two consecutive numbers to the power four. For example,  44 - 3= 43 + 33 + (4 + 3) x 4 x 3.

Challenge through inquiry

The picture above shows the course of a dynamic and exciting inquiry by a year 9 class at Priory School, Edgbaston (UK). The inquiry lasted two one-hour lessons. Emmy Bennett, the class teacher, reports on how a variety of lines of inquiry developed:  

My first attempt at an inquiry lesson was with a year 9 class. We had just finished an algebra topic and I wanted to give them something to really challenge them. I started with the prompt and let them discuss it for around five minutes. We then shared ideas which I wrote up on the board. I gave each pair the six regulatory cards and asked them to pick where to go with the inquiry. Most wanted to Find more examples or Decide the aim of the inquiry. As a class we decided to try and prove the prompt algebraically.

During this first lesson some pupils were changing the prompt and looking at what happened when the difference between a and b was more than 1 (i.e. a2 - b2= a + b). Others, who were trying to prove algebraically for a difference of one, spent quite a while trying to write the prompt algebraically, which can be seen at the top of the image above. Other pupils were exploring what happens when square numbers are added and others wondered what would happen with cube numbers.

After the first lesson pupils were really excited to continue with the prompt into a second lesson. At the start of this lesson I showed some pupils how to expand double brackets as they had not done this before. 

One group of six students were working on proving the prompt is true for any two numbers, and not necessarily with a difference of one. (They had been able to show this quite quickly once they were shown expansion of double brackets.) They did achieve this by the end of the lesson and were working on proving the difference of cube numbers. I introduced a related prompt to the class as some wanted to explore cube numbers:

The class then split into another two groups. One group was looking at the patterns made by the differences between squares. Pupils realised it was a linear sequence and this led them on to finding the nth term. When exploring cube numbers, pupils found the differences created a quadratic sequence. I had not covered this with them previously but all spotted the 'second difference' of the sequence. Finally one group was trying to draw the second prompt. This was pretty tricky and I have tried to recreate it below.

I was really happy with both of these lessons. All pupils were engaged throughout the whole two hours. I will definitely be trying more prompts with this class in the future as it really gave them a chance to think mathematically and challenge themselves.