Here’s my attempt at investigating operations with square roots (surds) when you’re not prescribed by a GCSE/IGCSE syllabus. This is only a general outline of the lessons we’ve had so far but it’d be great to hear how other people go about teaching surds.
To start off, we hadn’t spent much time discussing irrational numbers but I thought the question below would push us into a discussion/debate about them and then things would start flowing:
(An irrational number) x (An irrational number) = (A rational number). Discuss
Being a Year 10 class (Grade 9), they were aware that pi and √2 are irrational (though one student still believed pi could be written as a fraction – see previous post). We wrote down some definitions for rational and irrational numbers and after some thought, calculator work and discussion, the students were quite comfortable that it is possible (i.e. √2 ×√2 = 2). Even though the calculator confirmed it, some of them were not comfortable with this result
“How can a number which carries on forever with no pattern multiplied by another number which carries on forever with no pattern equal a rational number???”
So I showed the equation like this: (√2)²=2 and one of the students realised that if you square root 2, then square it, you get back to 2 (At the time, he said it was like taking 2 from 10 and then adding 2 again to make 10 – they’re inverse operations).
At that point, I showed the slide below and allowed them to battle through it for a while:
To start off the next lesson, I put the following slide on the board:
After a few minutes of independent thinking, half of the students had no clue what the picture represented (√2 ×√2 = 2) so I asked the other half that did understand to explain. Once that had been established, I shared one of the Habits of a Mathematician with them:
Mathematics can be represented in many different ways and different forms can often help us to understand how and why things work.
(Note that no specific learning objective was shared, just a habit). They then went about doing a similar thing to show geometrically how √5 ×√5 = 5 which wasn’t so difficult so I asked those who had finished to tell me why I went straight from √2 to √5? Why didn’t I ask them to show √3 ×√3 = 3?
After showing the result, I was happy to now show them a rule and let them thrash it out:
√a × √b = √(a×b)
I was really impressed at this point at how well this was analysed. One student made a = 5 and b = 5 and realised that it was correct for that case and therefore for every case where a = b. Then the students quite quickly used other numbers with a calculator to confirm that this worked for a ≠ b (no one mentioned negative values and I decided at this moment not to bring it up). Admittedly, I was slightly disappointed that no one tried to show this using a geometrical representation but I guess I can’t complain – it is easier to verify with a calculator. (Writing this post, I see now that I could have been more insistent on proof but then I think I would have been in danger of having too much discussion).
I then gave them a few questions to do on this with the endpoint being that if we can combine square roots by multiplying, could we “break them apart”?. I gave these 4 questions for them to break apart in all of the ways they could:
√20 = √10√2 = √20√1 = √4√5 = 2√5
√45 = √9√5 = 3√5 = √15√3
√23 = (Important to stick one in that can’t be broken apart)
√30 = √3√10 = √15√2 = √6√5 (and important to stick one in that can be broken apart but neither of the roots will contain a square number)
Question: Why do they always ask students to simplify roots in this manner – are the answers actually simpler? How is 2√5 simpler than √20? Neither I nor my students feel this is the case.
After they’d made up some of their own questions which could be broken apart into at least one square number – we moved onto adding square roots:
Does √2 +√2 = √4?
They had an intuitive guess first, at which over half felt it wasn’t true but some felt it might be. Then they were easily able to verify on calculators that this isn’t correct. So I issued an 8 minute challenge:
1. Can you find out what √2 + √2 is equal to as a square root?
2. Can you use Pythagoras’ Theorem to show with a picture representation what √2 + √2 is equal to as a square root?
Everyone answered the first one – no one answered the second one (in fact many were trying to represent it in a similar way to √2 × √2). I was stuck at this point, give those interested a chance to have a go for homework or show a representation and allow them to thrash out the meaning. I went with the latter (without the equations at the bottom or the √2’s and √8 on the diagonal lengths):
Once that was done, I asked them to have a go at √5 + √5 and √7 + √7.
Most verified on the calculator (again, not using a visual representation) that √5 + √5 = √20 and √7 + √7 = √28
A number of students simultaneously conjectured that √a + √a = √(4a) and so I asked one of them to explain what they meant using more examples on the whiteboard.
To help them explain this, I put this on the board: √3 + √3 = 2√3 which helped one student take it further into this: √3 + √3 = 2√3 = √4√3 = √(4×3)
Hence √a + √a = 2√a = √4√a = √(4a)
I then gave a sheet out with more square root addition, but in the questions, the numbers inside the square roots weren’t the same E.g. √8 + √2, √90 + √40 etc.
They were pretty livid at this point: I could tell that they were internally shouting: “Stop making us think so hard about everything – just give us something easy to get on with for a while.” Next lesson we’ll be doing a review/reflection on what we’ve covered so far so it should be less strenuous.
I was really happy though that it wasn’t long before a few students started to figure this out by changing one of the roots so the “base roots” are the same:
√8 + √2 = √2 + √2 + √2 = 3√2 = √9√2 = √18
There’s so much more to investigate and discover! Unfortunately there will come a point when it’ll be time to move on but I’ve been massively enthused and excited about this unit – there’s just so many deep aspects to it and so many questions you can pose!