Difference of two squares inquiry
Mathematical inquiry processes: Verify; extend the pattern; generalise and prove. Conceptual field of inquiry: Difference of square numbers; rearrangement of algebraic expressions.
Daniel Walker 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 in all such cases. After the initial phase, the inquiry has the potential to develop in different directions. The following lines of inquiry have occurred in classrooms:
Prove the general case, (n + 1)2 - n2 = (n + 1) + n
Explore for cases when the difference between a and b is greater than one and prove;
Make the powers greater than two (see the difference of two cubes prompt);
Express, under the teacher's guidance, the difference of two squares as (a + b)(a - b) and expand the brackets to show (a + b)(a - b) = a2 - b2.
On the second line of inquiry listed above, 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.
At the time of devising the prompt, Daniel Walker was a teacher of mathematics at North Bridge House Canonbury (London, UK).
South Australian unit of work
The Department for Education in South Australia has included the difference of two squares inquiry prompt in a unit of work for year 10 students. The use of the prompt not only covers content in the Number and Algebra strand of the F-10 curriculum, but also involves students in critical and creative thinking - one of the general capabilities in the curriculum. Critical and creative thinking is broken into four inter-related elements (see the diagram). By the end of year 10, students are expected to reach the level 6 descriptors in a learning continuum. The inquiry could involve students in the following processes from the four elements:
pose questions to critically analyse abstract ideas;
create and connect complex ideas using imagery;
speculate on creative options to modify ideas when circumstances change;
give reasons to support their thinking;
identify, plan and justify transference of knowledge to new contexts; and
use logical and abstract thinking to inform a course of action.
The designers of the unit of work have included regulatory statements (see the table) based on the cards to support students in making decisions about the actions they undertake and the aims they set themselves during the inquiry.
Evaluating classroom inquiry
Daniel Walker reports 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 a2 - 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 a2 - 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 5.62 - 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.
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 one (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 consecutive 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.
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 (Wales). 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." Another student 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.”
Tony Fudger, the KS3 Co-ordinator in the mathematics department at Sir John Colfox Academy (Bridport, UK), used the prompt with his year 11 class. He describes the course of the studnets' open inquiry, which started with wide-ranging questions and went on to involve rich and deep mathematical ideas:
"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."
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."
Reporting on inquiry
These are the questions and comments about the prompt from a year 10 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 on the guided poster (see examples below).
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,
(n + 2)² - n² = n² + 4n + 4 - n² = 4n + 4 = 2[(n + 2) + n]
(n + k)² - n² = n² + 2kn + k² - n² = 2kn + k² = k[(n + k) + n]