Algebraic Fluency - 50s Style

Whilst visiting the Mathematical Association in Leicester on Saturday, I picked up a couple of old maths textbooks from the 1950s. 'A Classbook of Algebra' is my favourite.

In it I found some wonderful exercises for developing algebraic fluency, including this set of questions on 'Miscellaneous Factors':

The instruction 'factorise where possible' adds a delightful extra level of intrigue and challenge to this exercise. I love these questions...

Factorise c - 3 + 2x(3 - c)

Factorise c² - (c - d)²

Factorise ef - 1 + e - f

Factorise -14yz + y² + 49z²

Factorise 81 - 9a  + 0.25a²

When I get a chance I'll type them all up. It does make you wonder why the vast majority of textbooks and worksheets on factorising these days simply have questions like this:

Factorise x² +7x

Factorise x² + 5x + 6

Factorise x² - 9

and maybe this kind of thing for stretch:

Factorise 3x² + 10x + 8

Factorise 2x² - 18

Is that really the best we can do? We could at least change the order of the terms! No wonder so many of our students lack algebraic fluency when they get to A level.

If you're thinking that this factorising exercise from the 1950s is a bit tricky, you will be relieved to hear that in the same book there are a number of exercises breaking down the factorising skills into stages, leading up to the 'Miscellaneous Factors' activity which brings it all together. For example there is a whole section on 'Factors by Grouping' which explains how to factorise expressions like ax + 2x + 3a + 6. I'm not sure this skill is widely taught these days.

I have plenty of students who would love to get stuck into these questions. Back in 2015 I wrote the post 'Stretching Practice' which is about where to find good practice resources for high attainers. There's a decent selection of resources available, but nothing quite at the same level as the exercises from 1950s textbooks. I think that the closest online resources for developing algebraic fluency are some of Don Steward's tasks, like this excellent difference of two squares exercise:

I wish I could show you every exercise from 'A Classbook of Algebra'. There are so many great questions. I'll just share a few more examples...

Under the title 'Easy Brackets' we find questions that many teachers would now use as extension work in a lesson on expanding single brackets:

And under 'Miscellaneous Easy Brackets' the questions look very different to what we now consider to be 'easy':

 

When we teach 'collecting like terms' I doubt many of us use exercises like this one on 'Addition and Subtraction of Polynomials':

Check out the last question: 'To the excess of 6 - x + x² over 5 + x² - 3x add x² -3x + 4'!

Do you want to check that your students really know how to expand brackets and solve equations? Try some of these:

Here we have double brackets too:

I wrote about multiplying negatives in this post, but the resources I featured didn't include algebra like this exercise does:

Finally (though I could go on all day!), here's something for your indices lesson:

If only I had time to type all these up!

I hope you enjoyed looking at exercises from the 1950s as much as I did.