17 January 2018

Equations Exercises

I was looking at some of the old maths textbooks at my school and noticed than even as recently as the 1980s, textbook exercises contained a lot more practice questions than modern textbooks. Below is an example comparing the same topic in a 1980s textbook ('Negative Numbers and Graphs' by Heylings) and a current GCSE textbook ('Edexcel GCSE Maths Higher' published by Oxford University Press). The exercises cover the same skill but the first exercise is double the length of the second. I guess modern textbooks have to fit an entire GCSE course into a single book, restricting the amount of practice questions they can include.
From 'Negative Numbers and Graphs', first published 1984

From 'Edexcel GCSE Maths Higher', first published 2015

Last month I wrote a post about a 1950s algebra textbook called 'A Classbook of Algebra'. The questions in this textbook are generally more challenging than questions in most modern textbooks. In response to this post a number of very generous volunteers stepped forward offering to type up some of the exercises so that teachers can use them in lessons. I am very grateful for the time and effort that has gone into this. In this post I have provided links to all the exercises typed up so far which relate to the skill of solving equations. I have an additional eighteen exercises on other topics which I will collate over the next few weeks.

These exercises are all rather long and the idea is not necessarily to use them in their entirety. Because they have all been typed up in in Word, teachers will easily be able to edit the exercises or cut and paste particular questions to use as examples in class.

Most exercises include answers. I will edit this post if anything is updated. For each exercise listed below I have included a small extract so you can preview the type of questions covered.

1. Very Easy Equations (four exercises) - with thanks to Caroline Beale (@cbealemaths)

2. Easy Equations I - with thanks to Claire Willis (@MissWillisMaths)

3. Easy Equations II - with thanks to Caroline Beale (@cbealemaths)

4. Equations Involving Fractions - with thanks to Michael Allan (@mrallanmaths)

5. Equations Involving Brackets - with thanks to Jane Appleton (@JaneAppleton24)

6. Equations Involving Directed Numbers - with thanks to Caroline Beale (@cbealemaths)

7. Easy Literal Equations - with thanks to Sandra Douglas (@mathsbox1)

8. Miscellaneous Simultaneous Equations - with thanks to Fee Wilson (@fionajw)
9. Miscellaneous Equations - with thanks to Dan Rodriguez-Clark (@InteractMaths). 

I hope this is useful. Thank you again to everyone who has worked on this. And of course full credit to Sidney Trustram, the original author of these exercises which, 70 years on, are still making us think. Look out for my next posts in which I'll share exercises on simplifying algebra, expanding, factorising, writing algebraically and working with directed numbers.

3 January 2018

Lost Vocabulary

While reading the Victorian maths textbook 'Elementary Algebra for Schools' I spotted quite a few words and phrases which are rarely used in modern secondary schools. I'm not saying that this vocabulary has disappeared from the field of mathematics, but I doubt you will hear these terms in GCSE lessons.

Let's take a quick look at some interesting words and how they were used in the 1800s.

Meaning (in this context): the number one.
  • "When the coefficient is unity it is usually omitted. Thus we do not write 1a, but simply a".
  • "If the product of two quantities be equal to unity, each is said to be the reciprocal of each other".
  • "Subtract 3x2 - 5x + 1 from unity, and add 5x2 - 6x to the result".
  • "We proceed now to the resolution into factors of trinomial expressions when the coefficient of the highest power is not unity".

Resolve into Factors
Meaning: factorise (UK) or factor (US) - see Colin Beveridge's post 'Factorise or Factor'.
  • "Resolve into factors x2 - ax + 5x - 5a".
  • "The beginner should be careful not to begin cancelling until he has expressed both numerator and denominator in the most convenient form, by resolution into factors where necessary". 
  • "Resolve 4a2(x3+18ab2) - (32a5+9b2a3 into four factors". 

Meaning: made smaller or less.
  • "The sum of -3x, -5x, -7x, -x is -16x. For a sum of money diminished successively by £3, £5, £7 and £1 is diminished altogether by £16".
  • "Divide 105 into two parts, one of which diminished by 20 shall be equal to the other diminished by 15".

Meaning: as a consequence of which.
  • "Whence the result follows".
  • "Whence x = 1 is the only solution"

Meaning: multiplying an expression by itself.
Meaning: the operation of finding the root of an expression.

Antecedent and Consequent
Meaning: the first and second term of a ratio respectively.

  • "The ratio a:b is equal to the ratio ma:mb; that is, the value of a ratio remains unaltered if the antecedent and the consequent are multiplied or divided by the same quantity".
  • "A ratio is said to be a ratio of greater inequality, of less inequality, or of equality, according as the antecedent is greater than, less than, or equal to the consequent. 
[There are loads of interesting terms in the ratio chapter - including commensurable and subduplicate.]

Meaning: The act of transferring something to a different place.
[This hasn't disappeared, but the use of the phrase 'by transposition' or 'transposing' in worked examples is something we don't see in schools anymore].

There are many more words and phrases in this book which seem to have disappeared from the daily mathematical vocabulary of secondary schools - far too many to list here. Whence I will save the rest for another post. I hope you find all this as interesting as I do. No doubt someone will now contact me to say that they use all these words in their classroom on a regular basis...!

1 January 2018

5 Maths Gems #81

Happy New Year! Welcome to my 81st gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Number Properties
In Gems 79 I shared this lovely puzzle from Chris Smith (@aap03102):

Since then I've seen two great resources based on a similar idea. The first is the interactive 'Consecutive Number Types' puzzles from Jonathan Hall (@StudyMaths):

And the second is this free worksheet 'Consecutive Chains' from MathsPad.
I think these would work really well at Key Stage 2, 3 or 4 for exploring number properties.

2. Online Textbooks 
After I blogged about a 1950s textbook last month, I gathered together a group of volunteers to type up old algebra exercises into Word and work out the answers. Thank you so much to everyone who has contributed. The project is ongoing and I will blog about it soon. In the meantime, you can see our progress here. Also, thank you to @EmporiumMaths for sharing some 1950s algebra questions from past London O level papers here.

Huge thanks to @NetNym for sharing a beautiful Victorian maths textbook 'Elementary Algebra for Schools' that has been fully digitised. This really is lovely and well worth looking through (I'm a bit addicted to it!). The zoom function works well so the exercises can be used in class if desired - answers are included, and the whole book can be downloaded as a PDF.
I think that we have a lot to learn from the explanations and examples in this textbook. I've started writing a post about this which I hope to publish next week.

Meanwhile, a teacher in Australia (@adaprojectnet) has created a website adaproject.net which is an open online textbook for all to use, covering topics from Key Stage 1 to 5. It's worth having a look at this project which is growing all the time. 
3. Primary Resources
It's great to see John Corbett (@Corbettmaths) publishing lots of new primary 5-a-day content alongside his very popular GCSE 5-a-day collection. Each day there are five KS2 SATs style questions at four different difficult levels.

Also for primaries, Dr Frost (@DrFrostMaths) has now added Primary Maths Challenge questions to his website with the help of @Mathematical_A. You can browse by topic or by paper.
Primary Maths Challenge questions
4. A Level Resources
StudyWell (@_StudyWell) has published a couple of practice papers for the new A level - these are free for a limited time.

Tom Bennison (@DrBennison) shared a new Christmas Calculated Colouring for A level. If you didn't use this at Christmas, the set of questions may be helpful for Year 12 revision later in the year. Tom also published a longer Christmas Calculated Colouring in 2015.
5. Place Value
Thanks to Jonathan Hall (@StudyMaths) for sharing a new interactive place value chart. This is really helpful for demonstrating the effect of multiplying and dividing by ten. You can easily duplicate rows which saves you writing the same digits on the board multiple times.

You can read the latest eNews from the Mathematical Association here.

I'm excited that I've now booked a place at BCME - the biggest maths teacher conference of 2018. BCME conferences only happen once every four years - if you're not sure what BCME is then read my post about it. The price goes up after 31st January so book quickly!

I also hope to see lots of you at #mathsconf14 in Kettering on Saturday 10th March, and for drinks the night before. I had a wonderful time at all the maths events I attended in 2017 - if you've not been to a maths conference before, why not have a go in 2018? All events are listed here.

I'll leave you with this lovely graph, shared by @simongerman600, showing what people really mean when they use vague terminology describing the likelihood of an event. There are other cool graphs here. I think this one would make an excellent discussion point when teaching probability.

Have a fantastic term maths teachers! And all the best for 2018.

20 December 2017

New GCSE: Ratio

I've just marked my Year 11 mocks and noticed that ratio continues to be an area of difficulty. This isn't a surprise - we realised a few years ago that ratio questions on the new GCSE are much harder than they used to be. Here's an example of a straightforward ratio question - this is what ratio used to look like at GCSE:

The ratio of boys to girls at a school is 5:7
There are 600 children at the school.
How many boys are there at the school?

But GCSE questions are more challenging now. We're seeing a lot of 'ratio change' questions - this example, from Access Maths, was recently tweeted by Gayle Head (@maths_head):
How would you solve it? Here's an algebraic approach:
I've used equivalent fractions in my workings here - I think this approach really helps my students get the 5 and the 3 in the right place in the equation.

I've seen other teachers use totally different algebraic methods for questions like this (there's a nice example here).

I don't always use algebra for ratio problems - I also teach my students the 'scaling' method. This involves taking the initial ratio and finding a series of equivalent ratios until you find one that works (essentially finding x by trial and error). In this example, we start with 7:3 and multiply by two to get 14:6. Now if Alice started with 14 sweets and Olivia started with 6 sweets, then Alice giving three to Olivia would result in a ratio of 11:9, which is not what we want. So we try again. Our next equivalent ratio is 21:9, and again if we take three from Alice and give them to Olivia we'd get 18:12, which again is not equivalent to 5:3. So we keep going. Our next try is 28:12 which works.

Scaling is sometimes really time consuming. In this example we could have skipped straight to the correct answer if we'd realised that the total number of sweets has to split into 10 parts (ie 7:3) and also into 8 parts (ie 5:3). If we try the lowest common multiple of 10 and 8 then we get the answer really quickly.

We'd use the lowest common multiple if we were going to use bar modelling too.

On the subject of bar modelling - many of us have always drawn bars and boxes on the board when teaching ratio - bar modelling is most definitely not new! However, it is all the rage at the moment and increasingly used in both primaries and secondaries. Over the next few years we'll start seeing many Year 7s coming through to us using bar modelling as a standard approach for many problems, so I guess it is sensible for all secondary teachers to at least know how it works.

If you're not familiar with bar modelling, Kris Boulton's TES article and William Emeny's blog post are a good place to start for the basics. Your local Maths Hub may run a bar modelling course. If you know the basics but struggle with the harder questions, check out this Twitter thread to see a bar model in action for a trickier ratio problem. It's also worth watching this excellent video on solving harder ratio problems using bar models from Colin Hegarty.

Bar modelling certainly seems to be a clear and accessible approach for simple ratio questions. I would say though that for some trickier ratio questions it's not always quite as intuitive and obvious as expert bar modellers suggest it is.

Let's have a look at methods for another 'ratio change' question - this one is from Don Steward:
The algebra method I described above works very quickly. Using equivalent fractions and setting up an equation gives us 8(5x + 2) = 7(6x). The whole solution takes only a few lines of working.

Scaling works too:
5:6 gives 7:6 when Jan gains 2 marbles 
10:12 gives 12:12 when Jan gains 2 marbles 
15:18 gives 17:18 when Jan gains 2 marbles 
20: 24 gives 22:24 when Jan gains 2 marbles 
25: 30 gives 27:30 when Jan gains 2 marbles 
30: 36 gives 32:36 when Jan gains 2 marbles 
35: 42 gives 37:42 when Jan gains 2 marbles 
40: 48 gives 42:48 when Jan gains 2 marbles - this simplifies to 7:8 

Again, we can get there more quickly if we think about multiples. We start with a multiple of 11 and end with a multiple of 15, but we gained two marbles along the way. So we can list all multiples of 11 and all multiples of 15 and find a pair which are two apart.

11, 22, 33, 44, 55, 66, 77, 88
15, 30, 45, 60, 75, 90
So we started with 88 marbles.

Not all tricky ratio questions are in the form of these 'ratio change' problems. We also now get GCSE questions like this:

If the ratio a:b is 4:7, write a in terms of b

Though this may seem obvious to many of us (a is the smaller of the two, so a is four sevenths of b), writing the ratios as equivalent fractions can help students get their numbers the right way round (providing they are confident in rearranging equations).

Another type of question is this:

If the ratio a:b is 2:5 and the ratio b:c is 3:10, what is the ratio a:c?

Again, fractions might help.

A quicker alternative is scaling here. We can write a:b:c as one ratio if we get the b parts to match.

a: b can be written as 6:15
b:c can be written as 15:50
So a:b:c is 6:15:50
This shows that the ratio a:c is 6:50, which simplifies to 3:25

And here's another type of question:

Punch is made my mixing orange juice and cranberry juice in the ratio 7:2. Mark has 30 litres of orange juice and 8 litres of cranberry juice. What is the maximum amount of punch that Mark can make?

Again, I think that scaling is probably the quickest approach here. Multiplying the punch ratio by 4 gives us 28:8. This is the most punch we can make because we're using all the cranberry juice. So in total we're making 36 litres of punch.

In summary, there are a variety of approaches for solving trickier ratio problems - most can be solved efficiently using algebra, scaling or bar modelling. If you're teaching this topic for the first time at GCSE it's worth spending some time looking at the various methods. I think our students will need a lot of practice of numerous different types of ratio question to prepare for their GCSE.

Finally, here are some resources that you might find helpful:

There are loads more ratio resources in my number resources library.

I hope this post will be helpful when you teach ratio at GCSE. Do tweet me to let me know what methods you use if I haven't mentioned them here.

18 December 2017

Jo's Blog Posts in 2017

I wrote 55 blog posts in 2017 - just over one per week - which is far fewer than the 78 posts I published in 2016. This wasn't intentional. My new job is pretty full on and since September it's been hard to find the time to write. This is frustrating because I love blogging!

Every December I like to reflect on what I've written about over the course of the year. Today I'm featuring a selection of the posts I particularly enjoyed writing - just in case you missed them.

Dan Walker
I loved writing a post about Dan Walker's Resources back in February because I got to spend time looking through all his lovely PowerPoints. It ended up being one of my favourite posts of the year. If you're not familiar with these fantastic resources then have a look now!
Challenging Algebra
My recent post 'Algebraic Fluency - 50s Style' generated a lot of interest and enthusiasm. A number of teachers emailed me to say that this post led to discussions in their maths department about how we can increase the level of challenge in the algebra questions we give to our GCSE students. This post also initiated a collaborative project to get some of the exercises typed up - I will blog about this over Christmas.
National Challenges and Competitions
When I joined the teaching profession I was surprised that there appeared to be no central source of information - many things were communicated by word of mouth or through local authority networks, and some schools knew very little about what was available nationally to support teachers and students. Over the years I have attempted to fix this by publishing 'listings' - conference dates, school tripsin-school enrichment, and so on. The most recent addition to this catalogue of information is my post about maths challenges and competitions. I hope that by publishing this post, more students will be given access to these fantastic opportunities.
Source: Primary Maths Challenge
What's Happening In My Classroom
I like to write about things that I do in my classroom - it's a good opportunity for me to reflect and improve. There used to be lots of maths bloggers who did this, but it's increasingly rare. I'd love to read more about what goes on in other schools. My post about Papers Society and Structured Revision Lessons gave teachers an insight into what I was doing with my Year 11s in the run up to their GCSE exams, and my post The Folder Experiment... Revisited described why my students now use folders instead of exercises books. In my post Working Well: C1 and C2 I wrote about some things I was trying with my Year 12s including vertical binomial expansions and the grid method for division.
Cover Work
In Cover Work I shared a standard template for a cover lesson in maths - one that I ended up using many times in setting work for a supply teacher over the last few weeks. I hope that other teachers have found this template useful too.

September Lessons
I don't like all my posts to be about resources - sometimes I have opinions to share too! - but I know that maths teachers benefit most when I share resources that save them time. In my post Bridging the Gap... Revisited I wrote about the transition from GCSE to A level and shared a simple algebra test for assessing students at the start of Year 12. In Planning for September: Year 12 I shared some ideas and resources for the first week of term.

New GCSE Topics
To conclude my series of posts looking at new GCSE topics, I wrote New GCSE: Functions in which I explained the new GCSE specification for functions and shared some helpful resources for this topic. With over 6,000 visits so far it's the most viewed post I wrote in 2017. I guess people use it when they're planning their functions lessons. This makes me happy.
Surprisingly, my most viewed post of all time is still - after three years - my post about methods for finding a Highest Common Factor and Lowest Common Multiple, which has had almost 30,000 views. It's worth a read if you're teaching this topic - Venns are not your only option here.
I hope you found my blog posts helpful in 2017. Do check out my archive for the full collection of posts and my list of maths gems for all the latest resources and updates.

13 December 2017

New this Christmas

I have a page of seasonal resources which includes Christmas-themed maths activities. I think it's really important to do mathematics in every maths lesson throughout the year, but if half the class are out at carol service rehearsals or a lesson is cut short by an end of term assembly, then I see no harm in putting on a bit of Christmas background music while students get stuck into some festive algebra. I love a bit of Wham with my equations.

There are some classics on my seasonal resources page, such as Chris Smith's much-loved relay, but also quite a lot of new additions this year. I thought it might be helpful to list a few of these new resources here so you don't miss anything.

Topic Based
Dave (@d_e_humpty) has produced a lovely set of Christmas shape transformation activities which include translations, reflections and rotations.
Grant (@AccessMaths) has been hard at work producing Christmas resources including a Bauble Puzzle and a Higher and Foundation Christmas Tree Algebra Puzzle. 
MyMaths also has a new collection of Christmas resources including Christmas Algebra which might be suitable for Key Stage 2. 
Mr Bayle (@mrbaylemaths) shared a quick Christmas data collection activity where you play Jingle Bell Rock and students keep a tally of the word count for 'Jingle', 'Bell' and 'Rock'. 
Students at @Maths_CCB had a go, and made festive pie charts of the results.
Danielle (@PixiMaths) has helpfully shared a mixed-topic maths quiz that she made for her Year 9s.

It's nice to veer off-syllabus for a lesson or two. La Salle have a selection of Christmas enrichment resources including How to make an origami Santa and Christmas Colours which relates to the Four Colour Theorem.
Hexaflexagons are a personal favourite of mine. I became addicted last year after watching Vi Harts's outstanding hexaflexagon videos (if you haven't watched these yet, please do!).

I ran a couple of hexaflexagon lessons in the last week of the summer term. I showed the Vi Hart videos then handed out some templates for colouring. The videos got my students suitably excited about making hexaflexagons but some of them spent absolutely ages colouring, and then for the last 15 minutes of the lesson I had 30 hands up wanting help with the folding and flexing! Hard work. Most students ended up leaving the lesson with both a working hexaflexagon and cool mathematical stories to share at home.
Thanks to Jo Tomalin for sharing a festive hexaflexagon design.

Holiday Work
If your Year 11s have mocks in January then you might want to set them some Christmas holiday homework. Mel (@Just_Maths) has created lovely holiday GCSE homeworks for both Foundation and Higher tier. She has also shared a similar holiday homework for Key Stage 2.
Form Time
If you have any extended form time on the last day of term and your school hasn't provided any resources to keep your students entertained, try Mel's (@Just_Maths) excellent Christmas Pub Quiz.

Alternatively, maths teacher Graham Coleman (@colmanweb) has updated his awesome website Guess the Tunes which now includes Guess the Lyrics and Guess the Faces.

Cards and Gifts 
Maths Ed (@MathsEdIdeas) has shared a Christmaths card for schools to distribute to students and families, encouraging shared maths-play over the holidays. Editable files can be downloaded here and printed onto A4 to fold to A5. 
Finally, do check out my post The Top 5 Christmas Presents for Maths Teachers if you still have some Christmas shopping to do.

Enjoy the last week of term! The end is in sight.

Teachers Wine Glass from notonthehighstreet.com