22 September 2018

Vectors for Enlargement

I'm going to do a series of short posts on approaches or methods that teachers might not have seen before. I'm very aware that there will be many people who already use these approaches, but I figure that not everyone will have seen them so they are worth sharing.

Today, vectors for enlargements. I first saw this on Twitter years ago but I can't remember who tweeted it (apologies!). Later I saw my former colleague Lizzie Stokes (@misstokesmaths) using this approach in her teaching.

How I used to teach enlargement
I'll illustrate my previous method with a negative enlargement question:
I'd tell my students to draw a line to any vertex from the centre of enlargement. Then I'd tell them to count how many squares they've moved to get to the first vertex. I always modelled this slowly on the board. I showed the process of counting carefully ('count the corners, not the spaces!') because I find they often miscount.

Then I circled the scale factor and said that because the scale factor is negative two, to find the image they have to double the distance from the centre of enlargement. But because it's negative, they have to go in the opposite direction from the centre. So in this case, we started by going two units right and one unit up so now we have to go four units left and two units down. I emphasised that we're going the opposite direction because it's a negative enlargement. We then do the same thing for each of the other vertices in turn. At the end, all the corresponding vertices should be joined with straight lines going through the centre of enlargement. If not, you've miscounted!

I'd show another example with a different scale factor, then get them to practise a lot of these on printed worksheets. It was normally a relatively quiet lesson as they all had to do a lot of counting! I'd go round berating them for using pen when I'd told them to use pencil ten thousand times.

In hindsight I realise that this method could be better. It works fine for super smart students (most things do!), but others struggled with it. One problem is that you have to hold lots of details in your head while you work though the question. Another problem is that the method is different to the method for positive enlargement - there's an extra thing to remember.

How I now teach enlargement
They start by labelling each vertex with a letter and then finding the vector that takes them from the centre of enlargement to each vertex. They are already fluent in using vectors from our work on translation so this should be straightforward.

They write down the vectors rather than trying to hold the information in their head like they did before. Then all they have to do is multiply each of their vectors by the scale factor, and this gives them the vector for each of the vertices of the image. So they know exactly where to draw the image - it's all calculated and recorded before they start drawing.
I've used 'Year 11' vector notation in the picture above for clarity, though it's not essential at this stage.

This method is identical regardless of whether we are enlarging with a positive or negative integer or a fraction. So suddenly negative enlargements are no harder than positive enlargements (assuming students know how to multiply negatives). A clear and consistent method all round - I don't know why I didn't always do it like this.

Any others?
I hope that was helpful if you'd not seen it before!

I've blogged a number of times before about alternative methods for various topics (see my posts tagged 'methods'). If there are any approaches or methods that you use that you think other teachers might not use, I'd love to hear them! Please comment below or email me or tweet me.

16 September 2018

5 Maths Gems #94

Welcome to my 94th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Resources Padlet
There's a huge number of maths resource websites available and it can be hard to keep track of them all. Thanks to Hannah (@missradders) for sharing a Padlet of resource websites which is well organised and really easy to use. There may be some websites here that you've not seen before so do explore. It's a good idea to make this your homepage at work for quick access, or if you're unable to do that then perhaps put a link on your desktop.
To see recommended resources for specific topics, check out my resource libraries.

2. Puzzles
If you're a fan of maths puzzles and have a Twitter account, I recommend following Catriona Shearer (@Cshearer41) who has been sharing lots of really lovely problems for you and your students. Here's an example:

"The square, circle and triangle are stacked inside a larger square. What’s the area of the circle?"
There are loads more delightfully challenging and beautifully hand-drawn problems like this on Catriona's Twitter feed.

3. A Level Resources
There has been a flurry of new A level resources lately. CrashMaths (@crashMATHS_CM) has published some really helpful resources including a proof worksheet and some practice exam papers. Also check out their ten 'AS Maths Key Skills Check' sheets which can be used to test your Year 13s on Year 12 content.
A level teachers might also be interested in this excellent Scheme of Work for A level Maths from the Oxfordshire Mathematics Community.

Stuart Price (@sxpmaths) has published a new website of A Level Maths Question Packs full of great sets of questions organised by topic.

Naikermaths (@naikermaths) has published AS Practice Papers that can be used for homeworks or internal tests.

Desmos.com (@Desmos) has released a new activity: Functions and Their Derivatives.

Don't forget that I've updated my A level resource libraries for Year 12 Pure, Year 13 Pure, Statistics and Mechanics.

4. Global Maths Textbooks
If you're interested in how maths is taught in other countries then you should follow @literallyjustq for fascinating tweets and blog posts. It's so interesting to peek into maths textbooks from all over the world. Here's an example of an exercise in a primary textbook from Timor-Leste:
And here are some fraction questions from a Grade 6 maths textbook from Kyrgyzstan:
And some order of operations practice from the UK in 1976:
Follow @literallyjustq for lots more like this.

5. Multiplication
Thanks to Ashley Booth (@MrBoothY6) for sharing a multiplication knowledge organiser which sets out the reasoning behind related calculations.
Ashley suggested using 'what else do I know? activities for students to explore related calculations.
Clare Sealy (@ClareSealy) suggested that the knowledge organiser could also be used as a matching exercise.

I am so pleased to announce some more exciting additions to my upcoming #latemaths event. The ThinkMaths team (Matt Parker, Katie Steckles and Zoe Griffiths) will be there, as well as members of the Chalkdust team. The wonderful Ben Sparks will be speaking, and we will be launching Ed Southall and Vincent Pantaloni's new book More Geometry Snacks. It's going to be fantastic. Read Ed's post for more on this, and book your ticket now!
In case you missed it, my recent blog posts were Summer Updates, in which I ran through all the new stuff you'll find on resourceaholic.com and Year 7 Maths Activities in which I featured some lovely activities that you could use in a first lesson, or any lesson, with Year 7.

Here are some other updates that you might have missed:
  • How do you factorise harder quadratics? I've blogged about the history of the two most popular methods (inspection and grouping) on La Salle's website ahead of my #mathsconf17 workshop.
  • MathsPad subscribers will have seen lots of great new resources in their September update, including a lovely nets of a cube interactive tool.
  • Mark McCourt published the third instalment of his epic series of posts on Teaching for Mastery. It's a must read for maths teachers.
  • If you are a primary teacher do check out the new Topics in Depth packs by Nikki Martin that have been published for Year 3, 4 and 5 maths topics. Also, don't forget to sign up for the upcoming Primary Maths Challenge!  
  • If you're coming to #mathsconf17 and staying overnight on the Friday then lots of us are staying in the Premier Inn Birmingham New Street. I tweeted this back in August but you might have missed it. Watch this space for news on the location of pre-conference drinks. For more upcoming events, see my 2018/19 conferences page.

Don't forget you can subscribe to my blog posts by email or follow my Facebook page for regular updates.

I'll leave you with this awesome "Mathematicians for All, Equality, Unity..." poster, created by Kate Poirier(@realKatePoirier) and tweeted by Christopher Drupieski (@christopherdrup).

31 August 2018

Summer Updates

I know that many of you have been off having wonderful adventures over summer so may have missed some of my updates. I've summarised them all here so you can start the new school year knowing exactly what's changed on resourceaholic.com.

Resource Libraries
My big job over summer was to totally rewrite my A level resource libraries as they were still aligned to the legacy specification. I have now published the following new libraries:

Year 12 Pure Maths

Year 13 Pure Maths

Statistics (Year 12 and 13)

Mechanics (Year 12 and 13)

They're all accessible from the top menu. I hope they are helpful to A level teachers. Thanks to Ian Tomkin who kindly volunteered to help me rewrite my mechanics page (I don't teach mechanics so I needed help with this!).

I also did some work on my Key Stage 3 and 4 resource libraries (algebra, number, shape and data). Where there were gaps, I added lessons from bossmaths.com. If you haven't seen Boss Maths before (I blogged about it in Gems 89) then do check it out. You might like lessons such as 'Conventions for labelling the sides and angles of triangles' and 'Drawing diagrams from a written description'.

I also made a start on adding resources from variationtheory.com and mathsvenns.com to my resource libraries, though this is still work in progress. And I know that some of my longer topic listings need to be shortened and improved - again, I will get to it (perhaps next summer!).

I updated my maths conferences page for 2018/19. Please let me know if I've missed anything. I will update it throughout the year when events are announced. Have a look now and see if there's anything you want to attend. There are so many great events to choose from!

I will be speaking at #mathsconf17 in Birmingham on 13th October. Here's a description of my workshop:

If you're coming to #mathsconf17 and looking for somewhere to stay on the Friday night a few of us have booked the Premier Inn near New Street Station.

Blog Posts
In case you missed it, I wrote Gems 92 right at the start of the holidays. It included loads of lovely stuff, particularly on ratio. In early August I also wrote Gems 93 which featured lots of great tasks and resources.

In conjunction with Craig Barton's latest podcast, Slice of Advice: First Lessons, yesterday I published a post on Year 7 Maths Activities in which I shared six tasks which might work well in a first lesson of the year.

Do have a listen to Craig's podcast for loads for great advice on first lessons.

For those of you who are planning your first Year 12 lesson, check out my post from last year 'Planning for September: Year 12' in which I shared some lesson slides that you might find helpful.

I launched my fourth maths social event and have already sold a quarter of the tickets. #Latemaths takes place in London during half term on Saturday 27th October. It features a book launch and loads of mathematical fun so do book tickets now before they sell out. Bring your colleagues! All the details are at latemaths.weebly.com.

If you enjoy putting up a couple of new posters at the start of a new school year, or if you work in a school that insists on fresh corridor displays for Open Evening, you can find my updated displays page here.
I'm currently loving @jaegetsreal' s lovely 'First 1000 Digits of Pi' display which really brightens up dingy school corridors and is a great talking point for passing students. Importantly, it's also very quick and easy to put up!

And finally...
I wrote the MA August e-News which you can read here for the latest news from The Mathematical Association. I also wrote a magazine article about teaching indices and an accompanying resource which (hopefully) will be published at some point in the next couple of months.

Before anyone tells me off for working over summer instead of relaxing, I can assure you that I did a huge amount of relaxing! Because I'm starting a new job, I had no school work to do over summer for the first time in ten years. I spent the first two weeks of the summer break away on a wonderful family holiday, staying in a cottage near Hay-on-Wye. I had a lovely time in the remaining four weeks - highlights included an amazing night out with friends at my first ever Secret Cinema event, enjoying two awesome escape rooms with lovely colleagues, watching numerous films with my husband (A Quiet Place is so good!), and enjoying lots of drawing, jigsaws and days out with my two daughters. I also got the chance to catch up with some friends from Twitter, thanks to Dr Frost.

Megan, Adam, me, Stuart, Chris, Nikki, Colin, Jamie and Daniel.
Photo taken by @CantabKitty at Jamie Frost's summer drinks.

My youngest daughter is starting reception next week and I'm going full-time after six years of working part-time... So I guess it's nearly time to switch the alarm clock back on. Like most people, I'm feeling very sad that summer is over but I'm sure that within a week or two we'll all be back in the swing of things.

Good luck with new school year everyone!

30 August 2018

Year 7 Maths Activities

Craig Barton has just released his podcast Slice of Advice: First Lessons. It's packed full of good advice so do have a listen. In my contribution to this podcast, I promised that I would share some activities for first lessons (or indeed, any lesson) with Year 7. Here we are looking at activities that are accessible without being patronising, and give students the opportunity to show us what they can do (mathematically speaking) as we circulate and meet them for the first time. There are hundreds of great activities that would work well. This post features just six examples. The pictures shown are only extracts - please click on the links for the full resource.

1. In-Betweens from Colin Foster
This is an enjoyable and highly accessible activity with a good stretch task. It will start to give you a good idea of how much your new Year 7s know about place value.
2. Rainbow Squares from Math Pickle
The idea is to find pairs of numbers that add up to square numbers (children were taught square numbers at Key Stage 2).  There is a high level of challenge as the task progresses. If anyone struggles to get started here they could be given a list of square numbers.
3. Addition Pyramids 
In Gems 84 I wrote about the time I saw this classic activity in an interview lesson. It's really simple and engaging. It looks like it may have been based on this Nrich task.
Read this post to see a similar task in action.

4.  Consecutive Chains from MathsPad
I love this - it's fun for everyone! Children first meet square and cube number in Year 5 and should be familiar with primes, factors and multiples. I think this task will be accessible to some Year 7s (perhaps after a reminder of number properties) but not all.
5. Loops by Colin Foster
This pattern spotting task is slightly more challenging, but still accessible. This is good if you want your students to start secondary school with maths that looks quite different to anything they've done before.

There are loads more lesson activities in this set of numeracy activities from Colin Foster. I particularly like 'Musical Composers' and 'Number Triangles'.

6. Number Properties Challenge from Stephen Bodman
Give students three or four random digits and they have to generate numbers with specific properties - such as 'biggest number', 'smallest odd number', 'number closest to 3000' and so on. You can download the resource from TES and it could become a regular feature of your Year 7 maths lessons.


After the first lesson, I'd get straight on with teaching the content of the scheme of work. We must of course be mindful of what maths our students have been taught at primary school, but take into account that they are likely to be a bit rusty after the long summer break (this applies to every year group!). If you have one main feeder school it's worth finding out if there are any particular methods (eg bar modelling) that your students will be familiar with.

All secondary teachers, but particularly Year 7 teachers, should know the content of the primary maths curriculum. This summary by Michael Tidd is very helpful, and perhaps worth looking at during a maths department meeting at the start of the new school year. It's also a good idea to look through the SATs papers that your new Year 7s took at the end of Year 6. To help you translate their SATs scaled score data - in 2018, a mark of 61 out of 110 (ie 55%) translated to a scaled score of 100, which is the Government's 'expected standard' pass mark. Of course the numbers don't tell us much, so hopefully our students will start to show us much of their mathematical knowledge and understanding over the first few weeks of Year 7. We must give them the opportunity to do so!

18 August 2018


I am incredibly excited to announce my fourth annual mathematical event.

In 2015 and 2016 I hosted Christmas events for maths teachers. Christmas events are a bit stressful to organise because of the timing, the cost, the risk of snow etc...  So last year I held a summer event at Bletchley Park instead. We had a gloriously hot day and a really wonderful time. 

This year I'm hosting a book launch! This is exciting. I've always wanted to go to a book launch...

Ed Southall and Vincent Pantaloni have written a sequel to their successful puzzle book Geometry Snacks. At my #latemaths event in Central London on Saturday 27th October, you'll get to meet the authors. Plus you'll get a signed copy of More Geometry Snacks, hot off the press.

The evening will be utterly mathematical from start to finish. There will be plenty of entertainment, including a talk from the wonderful Ben Sparks.

This is a great opportunity to have a drink with some of your favourite mathematicians.

For full details, and to book tickets, visit latemaths.weebly.com.

Come on your own or with a partner or with friends - all welcome! 

I hope to see you there.


You can read write-ups of my previous events here:

6 August 2018

5 Maths Gems #93

Welcome to my 93rd gems post. This is where I share some of the latest news, ideas and resources for maths teachers. It's the summer holidays! Many of you will be on a Twitter break right now, so this post will fill you in on some things you might have missed in the last couple of weeks.

It was in the summer holidays - four years ago this week- that I first started writing my gems posts. You can see the full collection here.

1. Triangles
Thanks to John Rowe (@MrJohnRowe) for sharing this right-angled trigonometry activity. Students have the find the length x.
Benjamin Leis‏ (@benjamin_leis) has blogged about his approach to this problem here.

If you like this then you might also enjoy this problem from UEA which requires knowledge of the sine and cosine rules - I featured it in Gems 54. It takes a while to solve

You might also like the angle chase problems I shared in Gems 35. I love angle chases!

Speaking of triangles, thanks to Mark Horley (@mhorley) for sharing @DrPMaths' triangle generator. This helpful tool creates possible triangles with integer sides, area and height. If you make your own resources then this will be helpful to ensure you don't include impossible triangles.
2. Sums and Products
Thanks to Shaun Carter (@theshauncarter) for sharing a new resource which he calls 'Diamond Problems'. He blogged about it here.
The questions are similar to those on this shorter 'Sum Products' worksheet which I always use before my students start factorising quadratics. Of course, as Shaun says in his post, these puzzles are suitable for students of any age even if they're not studying quadratics - they are good practice for working with negative numbers and decimals.

Thanks also to Meredith Purser (@MeredithPurser) for suggesting that students first work out the rule themselves before completing the blank grids.

3. #midweekmaths
The White Rose Maths Secondary Twitter account (@WRMathsSec) is sharing a weekly problem throughout the holidays using the hashtag #midweekmaths
It's worth following @WRMathsSec because they regularly share lovely tasks for students. 
Also check out their new secondary five year plan.

4. Interleaved Homework
I have always set homeworks that directly relate to the topic I'm teaching, but for the last couple of years I have been meaning to change that. Continually revisiting past topics through homework is a great way to help students remember things.

Thanks to David Wees (@davidwees) for sharing 125 interleaved practice assignments - although these are aligned to a specific curriculum, it's so helpful to see the format and approach. This is definitely on my list of things to start doing!
5. Geogebra Whiteboard
I know many of you already use Geogebra, but you might not have seen this beta version of Geogebra Whiteboard. Thanks to Pip (@AccomplishEdu) for sharing this. The interface is brilliant - it's so easy to construct and annotate diagrams. Have a play with it and you'll see what I mean.

I've updated my conference listings for 2018/19. There are lots of great events coming up. Next term I intend to go to #mathsconf17 and MathsJam. And I hope to announce my own event soon...

I've been working on my A level resource libraries - they need a total rewrite because of the new A level specifications. It's a huge, time consuming job!

If you're a member of The Mathematical Association and interested in volunteering, please get in touch. I chair the Publicity and Media Committee and am looking for a couple of new members of my committee. I'm also looking for people to help man the MA bookstand at conferences. Please let me know if you can help. A small honorarium will be paid to conference volunteers.

In other news:

I'll leave you with this arithmetic maze from @MathforLove, shared by @MathsEdIdeas. Without passing through the same cell twice, what’s the largest total you can make? This might be a nice activity for Year 7 to have a go at in September.

22 July 2018

5 Maths Gems #92

Welcome to my 92nd gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

The summer holidays are here - hurrah! Time to relax, have fun and let our mental and physical health recover before we go back for fresh starts with new classes in September.

1. Equivalent Ratios
I like this equivalent ratios activity from Richard Tock (@ticktockmaths). Below is an extract showing some of the ratios that must be simplified and grouped. These have been carefully chosen to reveal misconceptions.
Henk Reuling‏ (@HenkReuling) shared an interactive version of a similar activity here.
2. Probability Problems
Thanks to Jason Steele‏ (@steelemaths) for sharing a nice probability task. Here students have to write numbers on the spinners to meet the stated conditions.
Do check out Jason's other resources too - he recently shared a nice set of questions on angles and ratio.
3. Real-Life Graphs
MathsPad has a brilliant set of interactive resources which are great for classroom demonstrations. Their latest interactive tool for subscribers demonstrates real-life graphs based on filling containers with liquid at a constant rate. There's an accompanying worksheet too.
4. Open Middle Angles
Here's another fantastic Open Middle problem from John Rowe (@MrJohnRowe):
5. Symmetry Display
Thanks to David Morse (@Maths4Everyone) for sharing a new alphabet symmetry display. This might be useful for making 'Welcome to the Maths Department' signs.
If you want to brighten up a maths corridor or classroom, you will find a large collection of maths displays here.

Here are a few things you might have missed:
  • My most recent post was about surds and how the teaching of the topic has changed over the years.
  • If you're launching Hegarty Maths in September, you might find Ben Gordon's (@mathsmrgordon) launch PowerPoint helpful.
  • If you're setting up a group or individual intervention with Key Stage 3 students next year, or if you're a private tutor taking on a new tutee, you might find this assessment helpful for identifying topics to focus on.
  • If you're teaching Year 12 in September, check out my entry assessment and first lesson slides.
  • In case you missed it, do check out the lovely blog post "Not As Complicated As It Looks…" by Mr Rowlandson (@Mr_Rowlandson).
  • I had a fantastic time with three of my Year 7 students at the London Rock Wrangle a couple of weeks ago. We didn't win the helicopter ride, but we came sixth out of 43 schools overall which was awesome. If your school doesn't already use Times Tables Rock Stars, check it out.
  • If you have time over summer, do listen to Craig Barton's latest podcast which features 55 individuals answering the question “what have you learned this year?”.
  • I really enjoyed taking part in The Aperiodical's The Big Internet Math Off over the last few weeks. I was honoured to be invited to take part. I wrote two pieces for the competition - one on hexaflexagons and one on Klein bottles. I was knocked out in the second round by Matt Parker! Do look out for the final on 24th July.
  • Thanks to David Faram (@dagsmaths) for showing me the maths education research pages from the University of East Anglia which are full of interesting reads for maths teachers. Have a look at the tasks here and here and the student responses.
  • Congratulations to Jemma Sherwood on the publication of her excellent new book "How to Enhance Your Mathematics Subject Knowledge". 

Finally, a quick shout out to the wonderful Glyn maths teachers who joined me for the best leaving do ever! We did bottomless brunch, an escape room, pedalos in Battersea Park and burgers in Clapham. What a great day! It was my first ever escape room and I loved it so much I now intend to try every escape room in London! Thank you to everyone on Twitter who recommended it.

I have a few things to work on over summer, but not a huge amount as I really need a rest! I will continue my old textbook research and my topics in depth project. I plan to update my resource libraries - in particular, I need to totally reorganise my A level libraries. I also need to update my conference listings. But before I do any of that, I'm off on holiday with my daughters... first stop, Cadbury World.

I'll leave you with this nice little puzzle from MathsPad's latest standard form resource.
Have a great summer!

15 July 2018

A Look Back at Surds

This year I've written a number of posts about maths textbooks from the last 300 years. Buying and reading historical maths textbooks has fast become my favourite hobby - it's fascinating, and it has done a lot for developing my subject knowledge this year.

Subject knowledge for maths teachers isn't just about being able to do the maths - that's the easy bit - it's also about knowing how to explain concepts clearly, knowing multiple methods and approaches, knowing common misconceptions, knowing history, etymology, narrative and so on. So, in the interest of subject knowledge development, today I bring you some maths from the past. I'm focusing on surds here (because everyone loves surds!), and I hope to bring you similar posts about other topics over the coming months.

In Elementary Algebra for Schools (Hall & Knight, 1885), the chapter entitled 'Elementary Surds' starts with some definitions:

Whether we'd still classify those algebraic terms as surds is debatable, but otherwise the wording of the definition ('when a root cannot be exactly obtained') has been pretty consistent over the years. Here it is again in 'A Shorter Algebra' (Baker & Bourne, 1927):
The book later goes on to refer to 'surdic expressions' - I've never used the word surdic and I'd like to see it back in common usage!
In 'The Essentials of School Algebra' (Mayne, 1961) we have the following definition of surd, which offers more clarity by providing a few examples and non-examples:
All three books explain what is meant by the 'order' of surds. There are exercises on transforming surds of different orders into surds of the same order - something that we don't do in secondary mathematics anymore.

Note also that surds of order two were sometimes referred to as 'quadratic surds'. This is another expression that we seem to have lost from our secondary school vocabulary. We now deal almost exclusively with quadratic surds, in fact I have seen a number of websites and resources claim that only square roots can be called surds, which is just plain wrong.
Contrast the precise vocabulary and detailed definitions in old textbooks with the disappointing one line description we have in a modern day textbook (Edexcel GCSE Maths, OUP, 2016).
I wonder why textbook authors gave up on thorough definitions.

Interestingly, old textbooks claim that the use of surds isn't absolutely necessary because, with extensive number work (ie using the process of evolution to find roots), we can find the value of any surd to a suitable degree of accuracy.

Back in the days when maths was all done by hand, they were generally happy with 'accurate enough' and didn't insist on exactness. In fact in 'A Shorter Algebra' it says,

"Results in surds are only practically useful when expressed as decimals".

It seems that although surds were often used for efficient workings, numerical answers were rarely given in surd form. Here we see that rationalising the denominator is done to make numerical calculations easier, because multiplying by a decimal is easier than dividing by a decimal:
Though we all know and use the phrase 'simplest form' to describe a surd that has been simplified, I'm not sure that many of us use the term 'entire surd' to describe an unsimplifed surd.
Textbooks used to feature exercises where students had to convert simplified surds into entire surds - it's rare to see exercises on this now.

The method for adding surds hasn't changed over the years. We still add 'like surds' or 'similar surds' (perhaps we more often refer to them 'like terms' though),

Adding unlike surds leaves us with a compound surd ('an expression involving two or more surds'):
And the process for rationalising hasn't changed over the last 130 years. We multiply the denominator by a 'rationalising factor'. Where necessary, the rationalising factor will be the conjugate of the denominator. We still refer to the conjugate now - it's good that at least some of the formal language has not been lost.
Of course, standard problems in the 1960s were far harder than they are now.
I don't need to tell you that exercises used to be a lot longer and more challenging than they are in most modern maths classrooms. Here's an example of an exercise on rationalising - this one is from 1885. Note that the first ten questions are just for practising finding the product of two conjugates:

Notice the tricky typesetting of those pesky vinculums!

For comparison, modern textbooks have only half a dozen questions on the same skill.

The books I have been looking at (one from 1885, one from 1927 and one from 1961) continue with square rooting binomial surds and solving 'irrational equations'. I could include a lot more about surds here, but to keep this post to a reasonable length I will leave it there! You get the idea - a lot has changed, all except the mathematics itself.

I've recently bought loads of old textbooks from the early 1900s and am hoping to do a presentation about them at an upcoming conference (possibly #mathsconf17), so if you find this stuff interesting do come along.

Part of my old textbook collection!