Introducing Trigonometry

Introducing Trig - NZmaths

This week I will be introducing Trigonometry to Year 9 (UK Key Stage 3, age 13-14). I always like to begin trigonometry with students actually measuring lengths in triangles, I believe they get more of a feel for the meaning of the ratios of the sides of a triangle if they have actually measured the length of the sides and calculated the ratios themselves.

I decided what I need is some accurate drawings of triangles of various dimensions that they could work on. It took a few seconds (the third entry in the search results for introducing trigonometry) to discover not only the drawings I wanted but a perfect recording sheet! From NZmaths (New Zealand Maths) – a site I have mentioned before for its excellent resources comes Introducing Trig. As well as the resources, teachers’ notes are provided. (Scroll to the end, past the teachers’ notes  for the resources)

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So after an introduction including a reminder of Pythagoras they met last year, we’ll just need to be clear on the terms opposite, adjacent and hypotenuse, how to complete the recording sheet and the measuring can begin! This is a very able group of students and I suspect several of them to be telling me for example by the end of the lesson that sin 60° is the same as cos 30°.

Checking a few more links in the results of the search I see the excellent Math Open Reference site which I have referred to on several occasions. I also see that I am in very good company in my desire to get the students measuring themselves and using the nzmath resource – see Dan Pearcy’s post.

Something else I like to do when discussing trigonometry is to discuss all the possible types of problems that can come up because whether they are disguised as buildings / trees / ladders or whatever there are still only a limited number of problem types, eg find the angle given the opposite and hypotenuse. The students can work out how many problems there are.

I’ll update this post with how we got on!

Desmos – tangents to curve

At the moment I am introducing calculus to my sixth form classes and I wanted to use Desmos to illustrate drawing a tangent to a curve. A quick search found thisCopying the page to my own account I then modified the page to show a simple quadratic (f(x) = x2).

Desmos tangent

Desmos – tangent to a curve

Something I have been meaning to do for a while is try the folder feature on Desmos, a rather neat way to create a tidy looking set of items.

Desmos page notes

Desmos page notes

Show / hide folder contents

Show / hide folder contents

Folders are easy to create:

Desmos Folders

Desmos - add folder item

Desmos – add folder item

Having created a folder, press enter and the next item will be placed in the folder. If you want to add items later, ‘nudge’ your item into the contents of the folder.

I recommend that students use Desmos to help them understand any questions they do; finding the equation of a tangent to a curve is a good example, Desmos can be used very quickly to illustrate a correct (or not!) solution.

For checking calculus examples students can use WolframAlpha, examples are included in the fourth slideshow here).



Update ….and along came @Desmos! (select either picture for graph page)
Desmos tangent tweet

Tangent to curve - Desmos improved version

Tangent to curve – fabulous version by Desmos!

Thank you @Desmos! (Think I’ll put in a few requests!)

This week…

…such a busy one – so I’ll be brief (but can’t break that new year resolution of January 2011, the only one I have ever kept, to write a post each week!)

So this week – I have been using my name cards – I’m learning names fast! My classes all know that I think WolframAlpha and Desmos are rather useful and we have had some great discussions on the learning behaviours I’ll be noting about them all on ClassCharts.

I must recommend the free samples from Bring On the Maths, the Core 3 activity – Logarithmic equations worked really well with my Year 13 class and next week I’ll use the C4 Binomial Expansion resource when we are talking about the validity of a given expansion. I have used the Trigonometric Ratios resource before – and will again; there are several other great resources in that list of samples – explore!

Now to hand over to a master (and his son), I love this TED talk from Hans and Ola Rosling. Can you do better than the chimps? Why we are all so statistically ignorant! (More on The Ignorance Project)


Diagnostic Questions

Logs & ExponentialsPlanning some lessons for the week I realised that I will be telling all my students about Craig Barton’s and Simon Woodhead’s wonderful Diagnostic Questions website so thought I would write this week’s post on that (again!). By some amazing coincidence as I started typing – into my inbox comes notification of a new post from Craig! He has been writing lots of lovely new questions and we can look forward to more (thank you so much Craig!).

Diagnostic Questions can make a great start (or any time!) to a sixth form lesson. I will be teaching Year 13 (UK age 17-18) this week about logs and exponentials. The Post 16 – Pure section has a growing and great selection of questions. Checking Logarithms and Exponentials I think those will do nicely as part of my lesson! As a registered user I was able to create a quiz consisting of all these questions. By creating a quiz one can order the questions exactly as required and also very easily create a PowerPoint slideshow for offline use, alternatively or as well – simply download the pdf version.

I find the easiest way to create quizzes is to use the Instant Quiz Facility which I have written about before; I thought it would be worth putting all the instructions together so created the following slideshow showing how I created the quiz on logs and exponentials. To create a new quiz I make sure that the Instant Quiz has no questions currently in it so have got into the habit of clearing it out once I have created a new quiz. The instructions for doing so are included here.

Using the option to download images I created the following PowerPoint:
Logarithms & Exponentials diagnostic questions 

For completeness this slideshow gives instructions for downloading a quiz you have created and quickly creatng a PowerPoint slideshow for offline use.

To see the pdf version choose this file: Logs & Exponentials Diagnostic Questions

To view the quiz online then follow this link.

Know Thy Impact – John Hattie

Hattie - Visible Learning

So – back to school again and I thought I would make a final and rather important update to Resolutions for (Mathematics) Teachers. Reading John Hattie’s Visible Learning for Teachers is such an important reminder that we should really be looking at the impact of all we do on our students. We might think a particular method or resource is amazing, but do we think so because we have considered very carefully how it will help our students learn?  
For a summary of the book, read this from The Main Idea.

The five dimensions of Expert Teachers Hattie identified were based on a review of the literature.
In summary:

  1. Expert teachers identify the most important ways to represent the subjects they teach
  2. Expert teachers create an optimal classroom climate for learning
  3. Expert teachers monitor learning and provide feedback
  4. Expert teachers believe all students can reach the success criteria
  5. Expert teachers influence a wide range of student outcomes not solely limited to test scores

Dimensions 4 and 5 remind me of Carol Dweck, I wrote recently that these points she made struck a chord with me:  for teachers to develop a growth mindset in their students they need to develop their own growth mindset; do we ever judge our students too quickly? Also, such a useful reminder that we may sometimes worry too much about ‘teaching to the test’ when we just need to remember that ‘The outcomes are natural byproducts of engaging in good practice’.

A thought provoking interview with John Hattie can be found here in an issue of  ‘In Conversation’ (Spring 2013). The interview is structured round the eight “mind frames” discussed by Hattie in Visible Learning for Teachers.

I have sometimes listened to audio books as I do like to hear authors read their own work, I believe it helps understanding. You can hear John Hattie himself on the principles discussed in Visible Learning in these two videos: Visible Learning Part 1: Disasters and below average methods  and Visible Learning Part 2: effective methods. If you are in a hurry you might want to skip straight to the last part of the second video! For anyone who can’t get enough of Hattie, he was interviewed recently as part of Radio 4’s series The Educators, you can listen here.

For further reading of current ideas, see Tom Sherrington’s excellent collection: Contemporary educational ideas all my staff should know about.

If you are about to return to school (or have already done so) then I wish you and your students a great year.

Resources for Students


One of my personal resolutions for the coming year is to carry on with my practice of using resources that students can then refer to or use at home if they wish. Mathematics notes and calculators are a good example of such resources.

To consider an example, early this term with the Further Mathematicians I will be studying matrices and I will let them know the sources of any resources I use in lessons. I use a blog to provide the details of my students’ homework so I can simply add the links to their homework page. Sometimes where there are several useful resources I think maybe of interest to a wider audience I also add a post to Mathematics for Students, see for example, Polar Coordinates. In fact I think I will do that more this year.

To return to matrices, some useful resources include the following:

On the AQA website the Teaching and learning resources page for A Level Further Maths includes three online textbooks under the Resources for students heading. For example if I want a worked example of finding the inverse of a 3×3 matrix then we can look at  Chapter 5 of AQA’s Further Pure 4 text. This also has an exercise with the answers at the back if they want additional examples.

The Math Centre

The Math Centre

More sources of notes and examples include Chapter 9 on Matrices and Transformations from the CIMT Further Pure Mathematics A Level material, Just the Mathsthe Math Centre and The HELM Project. If you have not come across the HELM Project before, the project was designed to support the mathematical education of engineering students and includes an extensive collection of notes which include clear worked examples. You can see on the list that a very small number of titles (that you are unlikely to want A Level) are ‘not ready yet'; for the sake of completeness I discovered the complete set hosted by the Open University. To access the Open University resources you will need to create an account (easy and free), this will also give you access to the numerous free online courses.

Obviously we need to keep an eye on the specification when looking at alternative sources of examples but surely that can only be a good thing, particularly for our students who will be off to university in the near future.

Matrices is an example of a topic where it can be very useful to check work with WolframAlpha; I have created a new slideshow of Matrix Examples to add to the WolframAlpha slideshow series so we can easily check any work.
The series is on Mathematics for Students also and a post including the matrices resources discussed here has been added also.

Learning Names

I have written on this before but something coming up again soon for UK teachers and no doubt many readers are already doing this – lots of new names to learn! Something we all need to do at the start of each year, learn the names of our classes as fast as we can! Certainly I think this is worth spending time on and should be a priority, we want our students to know that we know who they are!

Name card

I used name cards last year and will certainly use them again for the coming academic year. These are simple to make from an A4 piece of paper which can be folded in half and then folded in half again. Students can then write their name clearly on one side of the card. The other side of the card visible to the student could be a reminder of anything you want; the above illustration shows the card I used for a lesson observation with a class I was unfamiliar with. For my own students I have the details of when homework will be set and handed in and the address of the blog I use to post details of homework. In case it’s useful this is the Word file for the above example.
name card template

There is plenty of useful advice for learning names, these suggestions might be helpful:

I was interested to see a suggestion to seat your class alphabetically by their first name rather than surname on Classrooms and Staffrooms; that could be worth a try.

TES – Learning Names on New Teachers

Learning Students’ Names from the University of Nebraska includes many suggestions. I might try a variation on suggestion 15 here with younger students, perhaps they could try and think of a mathematical term which begins with the same initial letter as their name, Colleen calculator, Tina triangle….!

10 Techniques for learning Names from Cuesta College.

On the subject of names it is worth mentioning the lists compiled by the Office for National StatisticsIn fact the top names for England and Wales for 2013 were published on August 15th 2014. This pdf  details the key findings from the data and includes Excel files to download various tables. There are clear infographics showing the changes from 2003 to 2013 for girls and boys.

Anna Powell-Smith’s website  England & Wales Baby Names has details of names chosen by parents in England & Wales each year from 1996 to 2013 (based on the ONS data discussed above), using this site makes it easy to see the popularity of a name over time, we could search on Colleen for example (or this link for the US version)!

I think teachers and students can also be users of the baby name statistics because in my experience it goes down very well with students! What’s in a name? is a lesson from the excellent Census at school site; the lesson is suggested for Year 7 (age 11-12) and learners are asked to investigate popular first names and do a survey for their class on the image of first names and to report their results. This involves data collection, presenting data and designing a survey. Another suggested lesson which I have have successfully used myself is Baby Names from Stats4schools. The lesson involves students investigating the popularity of names and asks whether names get more or less popular over time.

Students might be interested to see how their school compares to the ONS data.

Further websites offering Statistics on names:

Entering a name into WolframAlpha shows US Statistics for that name and gives the etymology of the name and notable people with that name.

This Wikipedia entry has the top 10 names for various regions of the world.

And just what you always wanted to know, it seems Max and Bella are the most popular names for US dogs – Popular Dog Names – 2013! In fact Max tops this UK list as well.

Resolutions for Mathematics Teachers

Everywhere I look I see references to Back to School (there were signs in the shops before we even finished term!) and I know many of you are already back or about to begin term so I thought I’d post my updated ‘Resolutions for Mathematics Teachers’ earlier than originally planned.

All links have been checked and updated where necessary; in many cases the posts linked to have also been checked and updated with new additions.
To highlight some updates:
On gritty students, I have updated the blog post with a video where you can hear Carol Dweck talking about teachers and coaches developing a growth mindset in their students in an interview with Basketball School. Carol Dweck made a couple of points that struck me in particular, for teachers to develop a growth mindset in their students they need to develop their own growth mindset; do we ever judge our students too quickly? Also, such a useful reminder that we may sometimes worry too much about ‘teaching to the test’ when we just need to remember that ‘The outcomes are natural byproducts of engaging in good practice’.
Negative numbers
A new reference to coding has been added. I recently came across CodeMonkey (a subject I will return too in later posts) which is a superb site for teaching coding and is easy to use for students of all ages. I have previously written about Scratch which also offers us the chance to get some coding into our Maths lessons. There are many topics where we can use a little coding including polygons and making the monkey walk about in all directions it struck me that this would be great for directed numbers!
I always try to use resources in class that interested students can then use themselves at home, hence the addition of the reference to Calculators and tools. When one of my very good students last year marched into the lesson and announced that she know the text answer was wrong because she had checked it on WolframAlpha I saw the impact of this strategy. She had been working on a problem and was confident in her methods but the text answer had a typo so she turned to WolframAlpha to check. There are so many excellent tools out there for students to explore and check their work, this can encourage their independence in learning.
A reference to ending lessons well has been added as I recently created a new slideshow on the subject.
My post on feedback was updated recently including the need to allow time to respond to feedback.
I learn much from reading various blogs and tweets and about teaching generally not just Mathematics, the reading pages here have been updated to include some favourite blogs on learning and teaching relevant for teachers of any subject. On the subject of reading, have a look at the addition of Nix the Tricks to the Free Books page.
If you have just returned or are about to return to school then I wish you a brilliant year. If you are still on holiday I hope you are having a well deserved rest…but maybe there are a few useful thoughts here for you when your thought return to school!


I have read some great posts on feedback, see Alex Quigley’s Improving Written Feedback  and Verbal Feedback Given….. on Shaun Allison’s ‘Class Teaching’. Such a good idea, I couldn’t resist – so I bought myself the stamper and decided to try this with Year 7 – early days yet, but I love the idea so far..

Nought & CrossesA student in my Year 7 class was determined to find the number of winning lines in a game of 3D noughts and crosses which she did successfully. She drew some clear sketches of the different groups of winning lines and after a discussion with me was the recipient of my first stamp! I was very enthusiastic about her written work on this as you can see from her response! I had explained to the class that if I discuss their work with them, then give them a stamp – they have to write down that feedback as Shaun suggests in his post.

verbal feedback v3

It is important for students to make a note of verbal feedback and I have asked students to do this more often recently, so for example if I am giving any verbal feedback to the class after a test or homework I expect them to make a note of any verbal feedback they believe applies to them.

Something we must do of course is allow students time to respond to feedback, ideally I want a conversation in their books! I acknowledge where a student has responded to feedback by awarding a Feedback Response point (see ClassCharts for recording learning behaviours). We need to allow time in class for students to act on any feedback given; another possibility is to give freedom of choice for homework. My notes for my Year 7 class on our homework blog includes the following:

Note that the independent homework gives you the chance to respond to feedback; for example you might want to try to solve some equations and present your solutions very logically and show that you are checking your work. It may be a response to my feedback or your ‘self-feedback’. When you reflect on something we study in class, sometimes you might think ‘I’d like to practise some more examples’. Your independent homework provides that chance. Remember you could choose any topic, your homework offers you the chance to revise work.
Staying with the use of homework, it may be appropriate to give a follow up homework allowing students the opportunity to act on feedback received from the first attempt.
See also: Dollops of Feedback which includes some useful resources on feedback.
The most powerful single modification that enhances achievement is feedback. The simplest prescription for improving education must be “dollops of feedback”.
Hattie, J.A. (1992). Measuring the effects of schooling. Australian Journal of Education (see page 9).

Time to Tidy Up!

Tidy UpIt’s holiday time so time for a bit of tidying up – my house and my blog!

During the holidays I’ll be checking and updating New Academic Year Resolutions for Mathematics Teachers and will publish the updated version at the end of August.

In the meantime I’ll be doing some tidying up and updating in the never ending quest to make useful things easy to find! I’ll post my weekly tidy ups!

This last week I destroyed and recreated the UK Assessment Pages, a completely new page, Assessment without Levels KS3  gives details of the packages of the winning Assessment Innovation Fund. The principles here may also be of interest to non UK readers. On the subject of assessment I rather like the student rubrics I found on the US site, Exemplars, particularly the Jigsaw Rubric. Actually I think some teachers would probably appreciate rubrics like that – plain and simple language!

The GCSE & A Level Reform page has also been checked and updated with new links added, including the chart showing the old and new GCSE grades and subject content.

The Reading pages have also been updated recently including the addition of a new page Blogs – Learning and Teaching, these are on learning and teaching generally and are not Mathematics specific. It’s a short list, these are my personal favourites – so much common sense here by such open-minded authors!

Excel spreadsheet for symbols & charactersFinally for this week, in response to a query I had on last week’s post on symbols, I have updated that post (at the end) with a spreadsheet you can use to see which key on your keyboard will give you each symbol in a symbol font.

If you have just broken up for the holidays or have been on holiday for a while, wishing you a lovely break.

Thank you for reading this academic year.