6 December 2017

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 + x- 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.

1. Hi Jo, thank you very much for this insight, amazing how we are reluctant to challenge our students nowadays. I am sure this will find its way to many lessons. It is useful to have photos, we teachers can still use it.
Erna

2. This is really wonderful Jo! Many thanks! Will try and find out which font it is indeed!

3. Thank you so much...being of a certain age I remember when I was at school we got stacks and stacks of homework each day...the emphasis was on practice and consolidation rather than a mad rush to cover every maths topic under the sun as is the case with current GCSE syllabus...

4. Very interesting post, Jo. Thank you so much.

5. Fantastic! Thanks a lot.

6. As one who was learningat the end of the 40s and beginning of the 50's (in melbourne aus.)the pages bring back many memories. I cant remember the name of the books but they all looked like this one.Tons of pages of excersises but we did get to a high level of expertize in algebraic manipulation.I suppose school work was what there was to do. No tv in aussie until 1956,extremely puritan and films were very much censored.

7. Did you check how many of these are handled reasonably by CAS tools like Maxima, Mathematica, or Maple?

8. Have you seen the series of textbooks by Heylings? They are similarly excellent; loads of questions, no pictures of bunnies!

9. Scan them in! Converts to electronic straight away :)

10. Dear Jo,
Thanks so much for these they are fantastic. Shared it with my grade 7 class and they seemed excited to work out these problems. Would you be interested in a class generated set of answers? Im asking them to show their working out as well so hopefully will be a good guide, what do you think?

1. Hi. That would be excellent! Some teachers have already started working out the answers, but we still have lots of gaps: https://docs.google.com/spreadsheets/d/1CFTEkJwqLegd9_kxTnLmSERlLCBsiWSivV5835dGILA/edit#gid=0

11. Thank you for all the comments, I'm so glad people have enjoyed looking at these questions.

Some wonderful volunteers have started typing these exercises up into Word format here: Word versions

And thanks to @NetNym for sharing this lovely Victorian textbook that has been fully digitised: Elementary Algebra for Schools