Multiplying and dividing by powers of 10

I had my academic year nine class for the first time today and had a lot of fun. I had been warned about a few students, but they were arms deep in the trenches having a good go.

I took an experimental approach today with the class examining how to multiply and divide powers of 10. The idea was to create algorithms in student terms for solving simple equations.

We started with simple examples using whole numbers
5 x 10 = ...

"When multiplying by 10, 100, 1000... count the zeroes and put them on the end of the number you are multiplying."

25 ÷ 10 = ...

This time students considered the position of the decimal point:

"Count the zeroes in the number after the division symbol [divisor] and move the decimal point right of the other number [the dividend] that many digits"

We then looked at the case

2.5 x 10 =

and discovered our first algorithm didn't work as by our first algorithm 2.5 x 10 = 2.50

This lead us to a similar algorithm as for our division case:

"Count the zeroes in the multiplier and move the decimal point left of the other number [the factor] that many digits"

Using the whole number cases gave students an additional method of multiplying powers of ten than the messy loops moving decimal places method. The idea of this lesson was not to deny them mathematical language - but to give them an opportunity to explore a mathematical concept before formal language was introduced. It was a lot of fun for me and engaged them during the lesson.

We then looked at a few cases where the multiplier and divisor were not powers of 10. This exposed that students had difficulty with long division and long multiplication and were over dependent on calculators - which has the possibility of causing issues in non-calculator sections in upper school. We'll now go own to examining factor trees and ease into indices.

We also looked at 250 ÷ 10 where we converted the expression to a fraction and cancelled the zeroes - although we didn't consider why this works and will need to revisit it later.