List the first 6 square numbers.
What do you notice about the difference between them?
Express the difference between the square numbers as a sequence in terms of n?
Express algebraically the difference between the nth square number and the (n+1)th square number?
Use you algebra skills to show that they are the same thing.
No 2 in a series of posts based on Thinking Mathematically (1985) by Mason, Burton, Stacey
This is the sort of exercise I can envisage taking a number of different paths depending on what my students do with it which is exciting. The book walks through a generalisation by looking for the lowest 4-digit palindromic number, 1001 and then noting that subsequent palindromic numbers can be found by adding 110. Since 1001 and 110 are multiples of 11, then all numbers in this series are multiples of 11. However, this series misses out other palindromic numbers, e.g. 7557 so we need to refine it further.
I am intrigued to see if this is indeed a path my students would follow or if we would discover something else in these numbers. Depending on the class, I might start by asking “how many 4 digit palindromic numbers are there?” Before getting into the general, I would see this as an opportunity for purposeful practice of long division if that was something that my students require. Some students might need a fair amount of direction to reach a proof, but I would aim to make sure that all students left this lesson with an appreciation of that proof even if I had to lead them through it.
Sometimes it’s worth taking a risk and changing course of a lesson halfway through. And sometimes it pays off.
Today’s lesson was supposed to be about algebraic proof and we started with these nice questions from Don Steward. I thought they looked like good practice for multiplying out double brackets at the same time as introducing algebraic proof.
By the way question 3 is particularly tricky.
Following on from the previous lesson, lots of them started by trying values for n. Great to then have the discussion on what makes a proof vs. an example.
We got into a good discussion on Question 5 and I wanted to know if they were familiar with the difference of two squares. This is where the lesson changed course completely. None of them could tell me what it was called but I got the sense that they had seen it before. So next, I wrote these questions on the board:
- n² – 9
- 4n² – 25
- 81 – n²
- 100 – 81
They raced through these because they had all spotted the short cut. So next, I put this up, again from Don Steward:
Which followed on nicely from the final of the 4 questions I had put on the board. Many of them initially struggled to see the link immediately or see the pattern in the numbers. But with a bit of time and just the right amount of help (i.e. not much actual help, just encouragement!) they started to find others. I heard them forming statements like: “I need to number that multiply to give 2016, then I need another two numbers that add to give one of those number and subtract to give the other.”
About 5 minutes before it was time to pack away there was a great buzz in the room as 3 students found some other numbers and then the race was on. Lots of solutions started coming, but nobody got all 12 (including me!).
I went away and built a spreadsheet to investigate further, but I still didn’t find all 12. Can you help???