Last Saturday (24th June) La Salle Education held a national teacher’s conference #MathsConf19. It was a great event where I enjoyed attending talks and speaking to teachers about Maths in Action enrichment days.
I particularly enjoyed a talk on Making Mathematicians by Kate Milnes, a teacher of maths for over twelve years, at #MathsConf19. Kate argued that in most jobs we are required to solve problems, we start with a scenario, ask our self questions and then pursue a line of enquiry. She then shared a lot of ideas about how teachers can set up their maths lessons to develop these problem-solving skills in students.
Two ideas that Kate kept returning to were conjectures and generalisations, these excited me because of my background in pure maths. Kate used the 1089 trick to illustrate her ideas:
- Take any three-digit number where the first and last digits differ by 2 or more.
- Reverse the digits, and subtract the smaller from the larger one.
- Add to this result the number produced by reversing its digits
The resulting number, subject to a few conditions, is always 1089. To begin with a class of students has a go at the arithmetic with their favourite three digit numbers. They’re then encouraged to conjecture whether it works for all three digit numbers. Then they prove it.
Next we generalise, the class are asked to do steps 1,2 and 3 above with a four digit number and it turns out (again, subject to a few conditions) that the resulting number is always 10890. The pupils prove that. Then the students can conjecture what will happen if we use five digit numbers. It turns out that it’s not 108900. But that’s OK because not all conjectures turn out to be true.
We can generalise again, what happens if we follow steps 1,2, and 3 with a three digit number base seven? Usually you get 1056. The students can conjecture what happens in base eight, and then base . They can have a go at proving it too!
This investigation is really fantastic. Never mind the students, I had a whale of a time conjecturing, calculating and proving during Kate’s session! You can generalise this task so many different ways, each time encouraging students to ask themselves questions, and pursue a line of enquiry.
I often hear the question “what’s the point” when it come to maths, pure maths in particular. Sometimes my answers feel a bit hollow but the next time I’m asked that question I will bring up 1089.