This week I embarked upon the topic of algebra manipulation with my top set year 9s. The fact that they are year 9s means that they have already met algebra several times before and should have a firm grasp of the basic manipulations such as expanding brackets. For this reason, when I saw expanding double brackets appear on the SOW I automatically started to look for something a bit different, something for them to practise the skill in a different way but also to bring in some thinking/problem solving questions. I found some questions that I decided to try on Median by Don Steward – You can visit the resource here: https://donsteward.blogspot.com/2012/02/expanding-brackets-quadratic.html
The particular question set I used was:
The reasons I decided to use these particular questions were:
- The notation for squaring brackets is used and I was unsure as to whether my year 9s would have seen this before so for some of my students might be a new way to see expanding double brackets
- The element of having a term outside of the bracket is used, which again, when introducing expanding double brackets lower done this element may be omitted so the skill of expanding brackets can be focussed upon
- The answers on the left hand side are related (some are even the same) and so it allows the students to see that the dame quadratic expressions can be manipulated differently to produce different expressions
- The concept of proof is introduced. Recently on an A-Level course that I attended we discussed proof at A-Level and the struggle students apppear to have with it. We discussed whether this is due to proof not really being used throughout KS3/KS4 but something we tend to look at as a single topic in year 11 rather than scattering it throughout KS3/KS4. Ever since then I have thought that if an opportunity arose to be able to use proof lower down then I would take it.
At the start of the lesson we disucssed the key word ‘quadratic’ as non of my students were actually able to verbalise the meaning and so we wrote down a definition and we were all happy with this (never assume that because the students have been using the word means they can verbalise its meaning). We then went on to look at two examples of expanding double brackets (one example was a standard expand these two brackets and the other used the square notation). I highlighted with the square notation the common error that most students make which is to omit the ‘x’ term of the expression as the temptation to treat any bracket squared as the difference of two squares is too great. Once we had done these I simply put these questions on the board and let the students go.
Question 5 prompted the first question, as some students were unsure how to approach this. The 6 squared element seemed to confuse some of them and they weren’t quite sure how to treat it. Once I had been through this with them they were then happy with the concept and continued on with the questions.
Some of my students reached the proof element of the questions which was very interesting. I think this may have been a first time that proof had really been introduced as my students took some different approaches to this. One of my students for question 17 simply substituted into a value for n to see if it worked and once they had found one example they moved onto the next question. I noticed this and so challenged this method ‘how many examples does it take to prove something is true?’ and ‘how many does it take to prove something is not true’. From here I lead into that algebra is used in Maths to prove statements and so was able to show the student what this meant:
I was able to go through some of the common concepts of proof such as having to actually write a statement in proof to communicate what it is that you have actually found. I was able to show how we can compare expressions to show that something is true and introduce some very basic concepts of proof. My students next time they come across proof, which I hope I will be able to interject again this year, should at least have a little knowledge and hopefully start to remember that Mathematically we use algebra to prove.
Reflecing upon this lesson there are a few things that I might now go back and add in. I don’t think I fully utililised the left hand side of the questions, I simply asked the students to complete the questions but never really made them think about the connections between the answers. I could have actually asked the students to predict what they thought the question they were about to complete would produce and encourage them to start making the links between the questions. I could have asked them to a write a sentence after completing two questions with the same answer as to why the questions both led to a same answer even though the question was presented differently. This may have been a missed opportunity to help my students start building connections and go beyond simply completing the questions.
Overall, I would certainly recommend using this resource with a class that is expanding quadratics. There was certainly lots in this activity for the students to get their teeth into and personally for me, any resouce that starts to introduce the concept of proof further down is important and should be maximised.
You can find my power point for the lesson here: expanding double brackets