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Difference of two squares inquiry

Daniel Walker (a secondary school maths teacher) devised the prompt for his year 10 class. An inquiry might start with students exploring the difference of squares of other consecutive numbers and verifying that a2 - b2 = a + b. After the initial phase, the inquiry has the potential to develop in different directions. The following changes could be made to the prompt:
  • Make the difference between a and b greater than one;
  • Make the powers greater than two;
  • Express, under the teacher's guidance, the difference of two squares as (a + b)(a - b);
  • Expand the brackets to show (a + b)(a - b) = a2 - b2;
  • Represent the numerical and algebraic forms visually (see Daniel's power point).
On the last point, students, even those experienced in inquiry, rarely suggest an alternative mathematical representation spontaneously (see Moving between forms of representation). For this reason, the teacher might introduce the idea of a visual representation in order to deepen the students' understanding of the prompt.
Evaluating classroom inquiry
Daniel Walker reports below on using the prompt with his year 10 middle set.
I decided to keep the inquiry simple and did the following with the class:
1. Put the initial prompt on the board and let the pupils discuss.
2. Type their questions up on the screen, then get them to order the questions in order of priority or level of difficulty.
3. Let the pupils investigate for about 25 minutes, whilst I circulated and nudged them in the right direction if necessary.

4. Finish with a discussion of their findings.
I felt the lesson went well, in so far as many pupils contributed towards the list of questions, made conclusions on one or more of the different lines of inquiry and learnt about how to better investigate problems. However, the pupils were not that successful at working systematically enough to find the 'big' rule that a- b2 = (a - b)(a + b). Their mindset was that they were simply testing whether the given 'rule' worked for other numbers or powers, not looking for other rules. When I do this lesson again I will probably tell the class at the start that there are other rules to be found! 
Whilst pupils quickly realised that the initial prompt only used squares and consecutive numbers, it took a few hints from me for them to acknowledge that next steps would be to investigate the effect of changing only one of these elements and look for patterns across a set of results. For example, after trying once with cubes, once with powers of 4, etc., some pupils simply declared "the rule doesn't work." It didn't occur to them that other sets (for example, where a - b = 2) might have their own rule. It also didn't occur to them that it would be worth calculating a set of results in order to see if there were patterns. Only one pupil (out of 15 in the group) managed to work systematically enough to realise that a- b2 = (a - b)(a + b), although she expressed this is in words: "you multiply a plus b by the difference between them." I got the class to express this algebraically during the final discussion.
There were interesting lines of inquiry suggested and explored - one pupil offered the question "Does the rule work with decimals such as 5.6
2 - 4.62?" and was happy to feedback to the group that this was true. Another pupil suggested swapping the addition and subtraction for multiplication and division, but quickly found it 'didn't work'.
At the end of the final discussion I showed the class the visual representations and explained that sometimes this kind of approach can yield insights. However, I remain unsure about how best to use these - no pupils offered questions relating the initial prompt to shapes and I didn't want to 'interfere' with the process of them coming up with questions. Overall, I'm very happy to have tried a lesson like this and fully intend to do more in the future.

Daniel Walker is a teacher of mathematics at North Bridge House Canonbury (London, UK).

Promethean flipchart    download
Smartboard notebook   download
Guided poster for presenting students' inquiry

An alternative prompt
4² – 3² = 16 – 9 = 7 = 4 + 3
Students first notice that the difference between the squares of consecutive numbers equals the sum of the same consecutive numbers. They can verify this works for other consecutive numbers, before deriving and proving the general case:
(n + 1)² - n² = n² + 2n + 1 - n² = 2n + 1 = (n + 1) + n 
Classes have gone on to explore what happens when the numbers are not consecutive. For example, 
(+ 2)² - n² = n² + 4n + 4 - n² = 4n + 4 = 2[(n + 2) + n]  
In general,
(+ k)² - n² = n² + 2kn + k² n² = 2kn + k² = k[(n + k) + n]
Exploration and proof
Zeb Friedman
used the inquiry for a demonstration lesson she was teaching to a high attaining year 10 class at St John's College, Cardiff. She describes how the inquiry developed: 
“I was teaching a one-off lesson to a class and in a school I didn't know. The students did not have any previous experience of inquiry-based learning. The lesson was 50 minutes long. I followed the Inquiry Maths format and it was seamless! I used the set of six regulatory cards. The students ran with the idea and many of them commented on how much they'd enjoyed having some control over their next steps. I was stunned by the speed of the students’ progress. They did their write-ups in the next lesson with their normal class teacher who is keen to try out more inquiries with them.” 
The students' write-ups (some of which can be viewed here
) show different ways of attempting to prove the general case. One student asserts that "this works for any numbers which are consecutive," while another bases a similar assertion on empirical reasoning: "The proof is that I have used multiple examples for each example and the evidence has not changed." Other students, in their use of algebra, are moving towards 
a mathematical proof. One, for example, shows that x2 - (x -1)2 = (x + 1) + x leads to 2x + 1 = 2x + 1. Overall, Zeb reports that the inquiry process was “amazing.”

Zeb Friedman is a maths consultant and conference presenter based in Cardiff. Her website contains a wealth of ideas on teaching approaches and resources for lessons in secondary schools. You can follow Zeb on twitter @zebfriedman.

Reporting on inquiry
These are the questions and comments about the prompt from a year 10 (grade 9) class. The students had experience of mathematical inquiry and most chose one question to explore independently. The teacher initiated a discussion about finding the product of two linear expressions after one pair of students offered the 'formula' using algebraic terms. At the start of the second hour of inquiry, the teacher introduced a diagrammatic representation of the equation in the prompt and also one of the general form. Students then continued their inquiry and wrote up their findings in the guided poster (see examples below).
Open inquiry
Click on the picture to view the different inquiry pathways.
Tony Fudger used the prompt with his year 11 class. He explains how the inquiry developed: 
"I set the prompt in the middle of the board, and gave students a chance to think about how to investigate (I needed to make a couple of suggestions to get things going). The black text around the centre are the different investigation paths my students came up with. They then worked in pairs on whichever parts they wanted to. Those who chose an 'easier' investigation soon had to move onto harder things as the easier bit didn't take much time! 
"We brought things back to the front a couple of times and as students discovered things I added findings to the board. Most of the blue and red text is students' findings with a couple of bits (such as the proof) a result of me helping a little bit towards the end." The students have created an open inquiry that starts with wide-ranging questions and observations and leads into rich and deep mathematical ideas.
Tony went on to comment on the inquiry maths website: "I've been teaching 12 years and most of the resources I find are variations on the same thing. For me, this really is something different and I've used quite a few of the prompts in lessons. When the inquiries lead to an unexpected discovery, it is a big buzz to see the students' reactions."

Tony Fudger is the KS3 Co-ordinator in the mathematics department at Sir John Colfox Academy (Bridport, UK).