I had two aims for this lesson - a mathematical aim and a wider 'social' aim. They were:
- To be able to evaluate negative and fractional indices
- To work together in pairs and be able to talk about this work in whole-class discussions.
Students were told that they would be asked to work together in pairs, but that either student may be called upon to discuss the findings of the pair. This came from noticing that some students weren't participating as I would have liked. I wanted students to be able to participate without the pressure of responses being just their own, and to have time to discuss and think about their ideas before sharing with the class.
Having just evaluated a couple of simple powers, a student asked "What happens if we have a fractional power like 2/3?" I wrote the question on the board as asked students to work together to describe what this might mean.
I wrote the two statements down on the board, and asked the rest of the class: "What do we think about this? Have a minute to think about which statement you agree with."
At this point, I monitored the students discussing the two statements. I noticed one of the more reserved students writing the following in his book:
Around 30 seconds later I stopped the students working and asked which statement the students agreed with. To my pleasure, the pair who had made the statements decided to go first and revise their answer in public, giving the correct answer - cube root then squared.
I want my students to be able to revise their answers in public, and praised these students specifically for doing so - then added public revision as an additional lesson aim on the board.
I then asked students to explain why the power 2/3 means cube root and squared. Many students gave informal reasons, such as, "Because the 3 is on the bottom and the 2 on the top." I explained that this is how we might think about it, but that this doesn't really explain why.
I then asked the shy student to tell everyone what he told me. He replied: "I just wrote it down," and offered no further response. I paused for a moment of two, then replied, "Well, could you tell me what you wrote?"
He then explained his method to rest of the class. I praised him specifically and publicly for exhibiting the two main skills I was looking for in the lesson.
After some work on adding and subtracting algebraic fractions, I introduced the following equation with the intention that students could apply what they had just learned:
However, I had noticed that Amir's pair had followed a different approach and got a different answer. I asked him to write his method on the board next to the other method. We now had the following:
I was amazed when he wrote every step of his method systematically on the board, and made sure I praised him specifically for this. I then asked: "What do we think about this? Which approach do you agree with, which do you disagree with?"
The majority of the students agreed with the method on the right-hand side. I asked them why they disagreed with Amir's method. They found this difficult to answer, and a debate broke out. Initially, students were shouting that it was wrong. I reminded them that no answer is ever fully wrong, that there is always logic and reasoning behind what people do - and that they were both currently conjectures that we either agreed and disagreed with until proven otherwise.
At this point the discussion became much more mathematical in nature, as students attempted to disprove Amir's conjecture. I sat back and let the students make sense of it with each other, with Amir defending his method to the rest of the class.
However, they found Amir's conjecture very difficult to disagree with; there is much of his conjecture that is based in sound mathematical logic! He described how he multiplied both sides of the equation by 4 (using the balance method), in order to 'get rid of' the /4 in the first fraction, and so on... The rest of the class started to see the elegance of the solution.
After some time, we as a group identified that he had not multiplied all the terms in the equation by 4 (and 3), which was a crucial requirement of the balance method. An important lesson learnt.
Amir then revised his second line to include the fraction 4x/3. Some students suggested that multiplying x/3 by 4 should give 4x/12 (rather than 4x/3); we addressed this common misunderstanding.
We then worked on some other questions. At the end of the lesson, Amir came to me with his book. He had been working on adapting his method on other examples. Under his new method, he showed how you can multiply both sides of the equation by 12 in the first step. When I described this approach as a work of genius he smiled.