I thought I’d share a first lesson on solving quadratic equations that I developed for year 10 recently. I think many would agree that starting with what the students already know about solving linear equations is an aspect of this learning journey which often gets side-stepped. I therefore tried to make this integral to the learning experience. Looking at the component parts of this lesson, each one is completely obvious to any experienced maths teacher. The reason therefore that I chose to blog about it is because I felt that the learning sequence and journey in this particular case was a quality way to promote real understanding of solving equations. More technical aspects, solving in different ways and real life application will all come later in the unit.
Independent, quick and doable. All three aspects in this case were connected to the key parts of the lesson. Even though it pained me to do it, I taught double bracket factorisation before introducing solving equations this year. Usually I start by asking students to try to solve a quadratic and then introduce the factorisation as a means of necessity. This time however, I felt that to really hit home the concept behind why factorisation is used when solving quadratics, the students needed to be skilled at factorising before understanding why it is important (would be interested to know what other people do in this case).
Starter Q1 was important to remind them of balancing an equation and also to remind them of what the word solve actually means. Normally I would say that solving an equation gives you a value/s which, when substituted into the equation, make it work (e.g. satisfy it). In this lesson, I made the students discuss in more detailed terms what this means and we rested on, “The value/s that, when substituted into the equation, make the left hand side of the equation equal to the right hand side.” We plugged in a few numbers to make this clear, one of them being a decimal and the other being a negative whole number.
It’s obvious that starter Q2 helped to remind them of double bracket factorisation. Starter Q3 however is interesting in that I wanted this concept to be completely obvious and simple to the students before I introduced it as integral to the strategy of solving quadratics. Students said it was easy and I was glad that it was explicitly mentioned at the start.
Main (part 1):
[The second question animated to appear after discussion of the first question]
The first question was easily solved by balancing although students forgot that there would also be a negative solution. It was very important to the lesson to show how balancing equations to solve them is a very general and wide-reaching tool for a mathematician.
The second question is then designed to show that balancing would not work; something the students were quickly able to identify. At that point I took it back to our definition of solving to ask how we might solve this particular quadratic. That is the reason behind the form of the quadratic; I wanted the student to be able to substitute a number into both sides. Students recommended trial and error to which I agreed but only if they use technology to maximise efficiency of the process. One student mentioned at this point that he’d already found the solution (x=2) to which I said that he could move straight onto the next task (I’ll come back to this). Luckily, Year 10 have TI-Nspire Graphic Display Calculators and so we utilised the spread sheet view. Excel would be just as good if you have quick and easy access to it.
I took the students through the use of the calculator with this example. Essentially we wanted to try lots of different input values for x and check whether the LHS of the equation would equal the RHS. (i.e. when is x^2+10 equal to 7x)
Of course the interesting moment was when the student who had found the first solution (x=2) was amazed to see that this equation has another solution (x=5). I responded to his amazement with: “Just because you know a lot about linear equations it does not mean that you know a lot about quadratics. Alarms bells should have been raised when you saw that x^2+8=24 has two solutions. Mathematics requires both intuition and rigour.” His response was both perfectly timed and very ironic: “So all quadratic equations have two solutions?” I told him that he’d be able to answer that questions himself as time went on when he could apply more rigour to his conjecture. No one at this point picked up on the fact that we had only used positive integers as inputs but I was happy to leave unsaid for now.
Students had a go themselves with the following questions and then I wrote their solutions on the board:
Of course some students were confused as to why Q2 didn’t fit the mould and only appeared to have one solution. Fortunately I got a “Ahhhh” moment when one student realised that the second solution may not be one of the input values we’d used in the calculator. That’s when I told them that there was a second solution, and in the interest of the pace of the lesson, I told them it (x=-1).
No one managed to connect the starter questions to the bottom question on this slide and so when we realised that the spread sheet method was unreliable, I went back to the starter questions to highlight Q2 and Q3 before stating that these integral techniques would provide us with a completely new way to look at solving equations. So I put up the next slide and facilitated the class through how factorising the left hand side allowed us to know with certainty that one of the brackets, or both of the brackets must be equal to zero.
[The second question on the slide was animated to show up after we’d discussed the first].
It was completely obvious to them that the only way this method would work is if one side of the equation was equal to zero. I put this completely down to the fact that they had internalised both the skill of factorisation and the concept of multiplying two numbers to make zero, before I introduced the strategy. Therefore it was much easier for them to connect the dots and fully understand why factorisation works.
Here’s the objectives that I showed at the end of the lesson: