4 Spotting patterns and adjusting algebraic identities

In Activity 2, you worked with your students on developing an image of multiplication and algebraic identities. Your students will now be aware of methods other than memorising formulae and algorithms to work out the products of multiplication and algebraic identities.

The power of understanding algebraic identities in mathematics is not only in being able to work out their products, but also (and perhaps more importantly) in being able to spot them when they are not in an easily recognisable form. To be able to ‘tweak’ expressions so they can be written as variations of algebraic identities is also a very powerful skill to have.

Activity 3 focuses on this. It requires students to actively develop ways to spot patterns and manipulate expressions in the context of algebraic identities.

Activity 3: Spotting patterns

This is an activity about spotting patterns and manipulating mathematical expressions in the context of algebraic identities.

Ask your students to decide whether each of the calculations below is an example of an ‘algebraic identity’. They can find these in their textbook:

  • 5.62 − 0.32 = 31.27
  • (x − 3)(x + 5) = x2 + 2x – 15
  • 118 × 123 = 14514
  • 25/4x2 – y2/9 = (5/2x + y/3)(5/2x – y/3)

Case Study 3: Mrs Agarwal reflects on using Activity 3

I told the students to look up the algebraic activities to remind themselves what these are. After that, the students happily started comparing the given questions with the identities. For the first one, they did identify the correct identity, but Suman and a few others wrote it as 5.62 − 0.32 = (5.62 − 0.32)(5.62 + 0.32). I thought her error might be good to share with other students so they could all learn from her mistake. So I asked her to come to the blackboard and write it down. At once, Ravi asked ‘How can it be that on the right-hand side [RHS] we have the same expression, but then it is multiplied by another?’ Suman immediately saw what she had done and rubbed out the indices on the RHS, leaving the correct answer.

The second one was done easily enough, but for the third, some distributed it as 100 + 18 and 100 + 23. This led to a discussion on whether that was simplified enough or if there another way of making it simpler. I liked that the students were actually thinking about different ways to get to an answer.

Some of the students wanted to write the last one as open five times times times four x minus y times nine close times open five times times times four x plus y times nine close. There was a lot of discussion on what was right and wrong with this suggestion. Then I asked them to take out their textbook, not to ‘do the exercise’, but to now see if they could easily identify the identities that they needed to use.

You may also want to have a look at the key resource ‘Assessing progress and performance [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] ’.

Pause for thought

Good questions to think about after the lesson are:

  • What responses from students were unexpected? Why?
  • What questions did you use to evaluate your students’ understanding?
  • Did you modify the task in any way? If so, what was your reasoning for this?

3 Algebraic identities seen as special cases of multiplication