Here’s a big number:

Try different single-digit divisors. No remainders.

This is an example of purposeful practice – exposing the wonder of mathematics whilst providing a reason to practise lots of of bus stop division.

You might want to start by asking pupils to come up with their own dividend “in the tens of millions” and try different divisors. (Here for a quick primer on the mathematical language.) Inevitably they will end up with remainders, which they may or may not carry into decimal places. Then let show them this “magic” number.

Questions to ask:

- What divisors does this work for and why? (Purposeful practice)
- What other dividends could I make like this? (Purposeful practice + reasoning)
- What smaller dividends could I make like this? (reasoning)
- What is the smallest dividend I could make that all numbers 1-9 will divide into without remainders? (reasoning)

Whilst I would want everyone in the class to understand the reasoning through a whole-class discussion, you may have some learners who need the practice on bus stop long division and spend most of their time doing this. Those that are confident with this technique can spend their time exploring deeper into the structure of the number.

Whilst we are on the subject of “Bus Stop”, maybe this technique actually has nothing to do with standing in line waiting for a bus:

##### 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.

## Ideas for better maths teaching