Probability Introduction

I started the topic of Probability with my Year 10 students yesterday and I was so happy with how the lesson ran and how engaged the students were, so I thought I’d share.

These kids didn’t see probability at all last year, so even though they’re a really strong group of mathematicians, I wasn’t confident with how much they remembered.

We started of by drawing mind maps of everything they could think of to do with probability – individually at first and then as a class on the whiteboard. Most of them were pretty blank. We ended up with words like chance, predictability, certain, impossible, likely, unlikely, 50/50. A few students also wrote that probability can be written as fractions, decimals and percentages so we had a chat about the probability number line.

Rather than start off with relative frequency calculations, I really wanted to get these kids thinking about the broader concepts in probability. I read out the statements below and as a class we talked about whether or not we agreed with it and, more importantly, why.

  • There are 26 letters in the English alphabet. Therefore the chance that someone’s name starts with the letter X is 1 in 26.
  • There are only 2 possible outcomes in a game of tennis – winning or losing. Therefore I have an even chance of wining a game.
  • I have flipped a coin 4 times and each time it has landed on tails. Therefore it is almost certain the next toss will be a tail.

These started some really great conversations about whether the outcomes were equally likely, whether events were independent of previous outcomes, etc. Now that their minds were warmed up and thinking about probability, I handed out these sheets for students to complete in pairs. I didn’t work through the whole activity as it’s described – instead I printed “Are They Correct?” and “Card Set: True, False or Unsure?” back to back and had students write a sentence about each statement.

After a while we came back as a class and spoke about a few of the statements that had caused a bit of debate or that identified an important concept.

To finish off the lesson we had a look at this problem from nrich. I had students stand around the edge of the room with a piece of paper and a pen. I asked them to write down a number between 1 and 225 and not let anyone else see it. Then I asked them what they thought the probability was that at least two people had chosen the same number. The general consensus was that since they had 225 numbers to choose from it was pretty unlikely. Then I had them read out their numbers one by one. The first time we did this there was laughter and playful mocking of the students who had the same numbers and it was largely written off as a coincidence. We repeated this about 5 more times with new numbers and every time there was a minimum of 2 people with the same number. It was beautiful watching my student’s faces as they became really curious about what was happening.

Next lesson we’re going to dive into the maths that explains why they were so likely to pick the same numbers.

Why Algebra?

It’s the start of the new year, and I’m getting to meet all of my new classes. I’m really excited to be teaching a year 11 General Maths class for the first time, and even more excited to find out that I only have 14 students!

Our first topic is Algebraic Manipulation. Sounds like a whole bundle of fun to a group of kids who just told me that more than half of them don’t like maths, and at least 2/3rds of them think they’re really bad at it. Since I knew that at some point most students in the class would probably groan “wwhyyyy do we even have to learn this?” I decided to pre-empt this and have the discussion before we even started.

I started by asking them “So what is algebra?” expecting to get responses about using the alphabet in maths (there were a few along these lines), but I was surprised when students started to tell me that algebra was all about patterns. A good start!

I told them that algebra is a tool (and I think a few agreed with that statement on its own) – a mathematical tool that’s used for two things: describing patterns and relationships, and proving ideas. We had a look at a few formulas that they know already and talked about the relationship that each one was representing. Then I showed then a magic trick.

Pick a number between 1 and 10
Double it
Multiply it by 5
Divide by your original number
Subtract 7
The number you’re thinking of is 3!

And from that ‘trick’ one of the best classroom discussions I’ve ever seen ensued:

Student 1: Why did it have to be a number between 1 and 10? Would it work with other numbers?
Me: Give it a go with other numbers and see if it works!
Everyone picked a new number – bigger than 10 – and were amazed when the magic still worked
Student 2: What if you pick a negative number?
Me: Try it out, see what happens!
There was even more excitement when these numbers worked too
Student 3: What about if it’s not a whole number? What if I pick, like, a half?
Me: What do you think? Will it work or not?
A few nods and shakes of the head. When these worked as well I posed the question:

Will this always work? How do you know?

After a short silence a student tentatively offered the answer “algebra?”
Me: Yes! So we want to pick a number, but we don’t know the value of that number yet. How do we represent it?
Ss: x!
Me: Do we have to use x?
Ss: Nope, you can use any letter
Me: Good. But it doesn’t have to be a letter – it can be any symbol – I could draw a picture of a fish if I really wanted to

Then we went through the process using x (and fish) and showed that our original number divides out, so it’s completely irrelevant.

One student picked up on the fact that we didn’t need to subtract 7 – we could have finished the trick with one less step, but we decided that it makes the trick look more impressive it there’s an extra step in there.

If this lesson is any indication of the year to come, it’s going to be awesome!