Teaching rounding to significant figures is a topic I have never felt that I have done particularly well. In the past I have used explanations like "3 significant figures means 3 non-zero digits". I have never felt completely happy with that, nor the way I have taught it in the past.

This year, as part of the White Rose Year 7 scheme, I had to teach rounding to 1 significant figure. I approached it differently to in the past, and it went really well. So I thought I would share (for me to refer back to next year, if nothing else!)

I started by confirming that all students could round to the nearest 10, 100, one, tenth, etc. We had been working on this in previous lessons, but just to make sure this was secure.

I then spent some time focusing on what a significant figure is. I settled on any digit after, and including, the first non-zero digit in a number. I used this excellent task from the Variation Theory website to help demonstrate what counted as a significant figure.

To check their understanding further, I asked them to show me a number with 5 significant figures (in Zoom chat as we are still remote teaching), then extended the idea as below. This was an idea from the excellent book Thinkers from the ATM.

This really got them thinking in more depth about what counted as a significant figure. 

I followed this up with another task from variationtheory.com as below.

Happy that they could all identify the number of significant figures, and, more importantly, identify a given significant figure, I moved on to rounding to 1 significant figure.

I used the WR resources as inspiration here, and developed them.

The explanation of the steps was, I felt, much clearer than I have given in the past. And student success suggested this to be the case too. Only 1 significant figure is in the scheme of work at this point, but I did stretch some with ideas of 2 or 3 significant figures, and the explanation holds up (identify the second significant figure,...)

They then did the WR worksheet in pairs, to great success.

I pulled them back together to look at this question

Finally we used a more-same-less grid

This week I had a breakthrough on how I could teach transforming functions to my IB AA SL students, which as with many of the best ideas, happened almost completely by accident!

The lesson was on combining different transformations to draw complicated functions. The end point for the lesson was questions like this where you have a function f(x) and have to draw something like g(x) given below (f(x) was defined earlier in the example, and is shown in the graph).

But from the previous lesson I knew that several students were still having issues with vertical and horizontal stretches, especially when the factors are negative, so I wanted to practice these first. I decided to use my IB Key Skills question generator to create 3 questions of increasing difficulty. We started with the ones shown below.

This is nothing new. I often do this when I know there are problems for some students. It gives us a chance to discuss as a class the approach to different questions. 

But what I realised whilst doing this was that I could ask students to annotate on the screen to show their answers. I can't believe it has taken me this long to think of this, but it was equivalent to getting students to the front to draw on the board! Anyway, after kicking myself for not thinking of this earlier, I realised I could do a lot more with it.

For starters, I had  three students working at a time, and I chose students to work on the level of difficulty that I thought was appropriate for them based on the previous lesson. I asked the other students to work out what their answers would be.

When a student had finished their question I asked the rest of the class to use the stamps built in to the Annotate function of Zoom, to either tick, cross or ? each answer. This gave me a feeling of what the class thought (and because I could see the names as they annotated, who was right or wrong). The ? was good too, as it allowed students to show they still weren't sure. For those that disagreed with an answer I asked them to explain why.

After we had all three answers done, I pushed the answer button to show the answer. And what worked REALLY well, was when I then scrolled down, the annotations don't move. Usually this is a pain, but in this instance it was perfect, as the graphs they had drawn slid on top of the answers, clearly showing if they were right or not (all of them were by this point).

This short recording gives an idea of the whole process. I made it after the lesson, so it looks like I am annotating, but in class it was students names that appeared.

Of course, the benefit of randomly generated questions is that then I could create 3 more instantly and get 3 different students to have a go (this time choosing those who I knew had struggled last lesson, and had intentionally avoided in round 1). I only needed to do this twice, but I could keep going if I needed.

Then with a quick change of settings we got these questions.

Which I colour coded as below

Then I sent them off to try two Your Turn questions in pairs, suggesting they use the annotate function to communicate with each other as they worked through the problem.

I have previously blogged about some of the activities I use to help students to understand what differentiation tells us (that is what the derivative is), but today I had a great lesson on introducing the actual process of differentiation.​

After exploring the idea of the derivative, I explained that differentiation is an algebraic way to find the function, rather than a graphical way.

I started by using a set of examples and asked students to use the Reflect-Expect-Check idea from Craig Barton. I showed them the first couple, then I asked them to reflect on what had changed in the question, expect what the answer would be and then check when I wrote the correct answer. I also made the different parts a lot more explicit than I normally do, as you can see below. The full set of questions is on pages 2 and 3 of this document.

As we went through we stopped at various points, talked about expectations, talked about the "obviousness" of the answer to y=3x (the derivative is the gradient, which is 3) and y=7 (the derivative is the gradient, which is 0) and that these fitted in with the patterns they had already spotted.

One thing I did differently in writing this sequence compared to normal is starting with the general case and showing y=x^2 as a specific case within this.

After this we then did loads of practice, but where I would normally do this via mini-whiteboards in class, since we are remote teaching, I had to find a technological solution. For most things this year, typing in the chat in Zoom has been enough, but I wanted to see the full written derivative from students.

Desmos comes to the rescue. I set up this assignment called My Whiteboards (copied one of the Desmos templates and added a few extra of my own). Then I paced them to the second slide so they were typing Maths. I decided to do this as they need to practice writing in Maths Type before they do their coursework next year. I then projected questions through sharing screen, and students wrote the answer in the desmos, deleting each time to write the next one.

This way I could see their answers as they wrote them, give immediate feedback and see who was participating and who was clearly unsure at any stage.

I used my website to generate the questions, starting at

And ramping up to

Slowly increasing the complexity by adding fractional coefficients, then multiple terms, then negative powers. We ran out of time to get to fractional powers, but will bring those in next lesson.

Overall, all students were successfully differentiating functions like the one shown by the end of the lesson, so I am happy with the progress they made. Unfortunately we only have one more lesson before the end of our school year, so will probably have to review a lot when we go back in March.

Due to COVID-19, we are unable to run our IB and IGCSE exams this November. Local restrictions make it impossible. It has been tough for students and teachers to have 2 years worth of work count for nothing. Our students will not get an IGCSE set this year (as an international school, no systems in place like there were for the UK). The IB students will have grades awarded purely on coursework. 

The announcement that there would be no IB exams this year was made the day before I was due to finish teaching the syllabus, and my IB HL class were keen to finish that last little bit of vectors. But we still had two weeks left of term left, and with no exams, the usual rush of exam papers was pointless.

So between me and the other HL teacher we decided to offer two separate options: she taught the calculus option (we did a different option, but many students were interested in this) and I did a series of lessons on random mathematical diversions. 

Here I will share those diversions, along with the resources, in case anybody ever feels like using them.

Taxicab numbers
We started by looking at this problem.

I blogged about this problem in the past here when a student brought it to me. I ran the session basically as an open problem, with students in breakout rooms in Zoom, and me popping between them.

Benford's Law
Next we took a look at Benford's Law. 

I asked students to think about a set of data. They could choose anything, but I gave some ideas like

• Populations
• Number of goals scored in a season
• Average playing time per player
• House numbers on a street
• Cost of items in a weekly shop

Once they all had an idea of their data set, I asked them to think about the first digits of each data point in the set, and to decide what the probability distribution would be for them. That is, what is the probability if you choose a random point in your data set, the first digit is a 1 (or 2,3,…)

We had a brief discussion about this, with the first answer being the expected 11% each as they are all equally likely. One student suggested that they would be clumped around a number (probably the mean) value.

After a brief discussion, I told them to go away and find the data set they had thought about in the first place.

We entered them all into this Google Sheet.

And then I added the data sets one by one to Autograph.

Obviously, with any activity like this you are open to it failing dramatically, but below you will see all the data sets plotted together, and Benford's Law falls out beautifully. There is even an excellent non-example.

The non-example was of weights of NBA players, and we discussed why this did not fit the pattern (weights will all fall within the range 70 - 120 kg approximately, so the first digits will be 7, 8, 9, 1, and the mean is in the 80s, so more 8s as this is a normal distribution.

But what about the other data sets? Why do they follow this same pattern?

I finished the session by explaining Benford's Law, the percentages it predicts and the formula, and how it can be used to spot people who have made up a data set.

Exploding Dots
Exploding Dots was a part of the Global Maths Project a couple of years back, set up by James Tanton. It is definitely worth checking out the website.

However, I prefer to teach it a bit more actively, and so created a version that I can present to students, with questions for them to do along the way. You can find a blank PDF of this here. 

It starts from the very beginning of school level Maths, with counting (in different bases), followed by the four operations. It introduces the idea of zero pairs to perform subtractions, and then builds up to unknown bases: that is polynomials. Within an hour you can take a group from counting to performing polynomial divisions.

I have done this with other classes before, and it went down well with this class too. By the end of the first double period we were answering questions like the one below (admittedly, they do this in the course, but we did the more traditional long division).

​We took this further to polynomial division that creates infinite polynomials, and the students wondered what would happen in other situations.

It was so popular, that we did a second double period on Exploding Dots, and this was all new stuff that I hadn't done with classes before.

We looked at decimals and fractions in the exploding dots model, which allowed us to look at fractions in different bases.

Then we looked at fractional bases, and explored what they might look like.

I love the exploding dots model. The students all commented on how visual it was, an how easy it was to understand what is going on. I really need to make more of this in 'normal' teaching, and not just as an enrichment activity.

Bayes Theorem
We study this in the course, but a couple of students asked to look at it in more detail. I was fairly lazy with this one, relying on some excellent resources available online.

First I sent students to this page to read the examples. We discussed the importance of the size of the population, and then did a few of the questions at the bottom of the page.

Then we watched this excellent video from 3Blue1Brown which visualises the whole thing beautifully.

Chinese Postman
I taught D1 once before leaving the UK, but it has been a while since I have done any decision maths. I thought this was a wonderful opportunity to take a look at the Chinese postman problem. I based the lesson on the plan from the Standards Unit, and turned it into a Desmos Activity. It was a very discursive session, so I paced them through the activity to start with, and also talked about the ideas whilst demonstrating and collating their ideas on a whiteboard.

Continued Fractions
Based on a couple of articles from nrich, I put together an activity on continued fractions. We started with evaluating them (like the one below).

We quickly moved on to look at infinite continued fractions

And generalised this

We took a quick look at how we can write any rational number as a continued fraction by using reciprocals.

And then how we can use continued fractions of surds to get pretty good rational approximations for them

 Finally we took a look at some of the continued fraction representations of other irrational numbers such as pi and e.

It was fun to get to explore some different aspects of mathematics with the class. I really must try to build it into my teaching more often, and not as enrichment, but as an integrated part of teaching Mathematics. If you have any go to activities like this, I would love to hear about them.

The second book by Craig Barton (well, ignoring the non Maths teaching ones) is everything a sequel should be: it builds upon the greatness of the first, but has its own tale to tell. It gets into the nitty gritty of the story, focusing on one of the smaller parts of the first. And yes, it is a bit controversial.

I am not going to write a summary here. I thought I was, but that would not be able to do any justice to the book. If you are a Maths teacher, you should read the book. Craig is open throughout about not trying to tell you what to do, but rather telling what he does, why he does it, and provoking you to think about how you could adapt those things to work for you (if indeed you find them valuable in the first place). But even if you disagree with everything Craig has to say, then you will still learn a lot from reading the book. If nothing else, if you do all the sequences of questions he provides, you will be giving your subject knowledge a good servicing!

Here I am going to share a few of my main takeaways, and what I want to incorporate into the book.

In my teaching ​

In the very last chapter of the book, Craig gives some advice on "Making it work", and the first thing is to choose one thing you want to try. Well, I am going to ignore him on that! Well, not completely. I am going to choose one thing that is new, and 3 things that I currently do but want to adapt after reading the book.

Reflect, Expect, Check, Explain - this is the new one. I have been using some sequences of questions from Craig's www.variationtheory.com website, as well as making some of my own, but, honestly, they have really just been a set of questions. They have allowed me to direct student attention to certain things, but I have not been systematic enough in my approach to develop their mathematical thinking in the way Craig describes. 

Since starting the book I have been adding some elements, in particular the reflect stage, but I want to make more of this. So I am going to try the full structure, and use the Prompt Questions that Craig suggests (available on his website: http://mrbartonmaths.com/booklinks/). I think I will need to use a template to help them structure the process too. My plan is to try this with my first year IB class, though I need to think carefully about what topic to do this with. We have some recap of indices and logarithms coming up, so that seems like a good fit. I will blog again on this when I give it a go. I have been using some sets of questions with them (such as the one below on Binomial Expansion), and have made short references to the ideas of reflecting on what has changed, but will need to be more explicit about this.

Another aspect that I have not been building in that Is important is the idea of Fluency Practice before the intelligent practice. I used to do too much fluency practice, now I am not doing enough for students to get the most out of these sequences of questions. For students to develop the mathematical skills, they need to be more confident with the method they need to apply first,

Atomisation - this is something I have been exploring, in particular with putting together the IGCSE Booklets and IB Lesson Sheets, but the systematic way Craig approaches it grabbed my attention. Going through the small atoms that make up a new idea and ensuring they are all secure first is something I want to look into further, but think that will definitely need to be a collaborative project. I am also thinking about how I could do some of that in the "flipped" model with IB classes.

Example problem pairs - just a minor adaption to my current process, but I need to find a consistent way to get all their attention on the example. I print examples and your turns in the booklet/lesson sheet, so the easiest way seems to be to get them to shut their booklet when I go through the example. Even go as far as put it on the floor if necessary. Then they can open their booklet once I am done to try the Your Turn. I also make scans of the sheets available to students after they are complete, so I might get students to NOT copy the example in class, and then get them to do the example as a homework, where they can check against my version. This would give them another exposure pretty soon after the first, and give them instant feedback on it.

Rule - I have been playing around with Frayer Diagrams for a year or so too, and the Rule sequence is a nice structure to lead into these. So far I have been using them to introduce the definitions, but this has not really been successful. Flipping this and getting students to fill them in themselves after seeing a sequence of examples, non-examples and boundary examples is a much better approach. 

In out department​

I also want to get my department thinking more deeply about the questions we offer students and the experiences they get of thinking mathematically. I am hoping to get some time with the whole department to get them to do some sets of questions over the coming months, and then start building some of our own sets. I think I will start with the Reflect, Expect, Check, Explain cycle.

It is difficult as we are still in lockdown from Covid, but I think we can make it work via breakout rooms in Zoom. Thoughts and plans are coming together…

We have moved to using the White Rose schemes of learning this year. In the current unit on place value, I was surprised to see the inclusion of Range and Median as Small Steps in their guidance. But when I thought about it more, it makes perfect sense. Separate these similar ideas from mean and mode. Both these require students to write a list of numbers in ascending order, which has been covered a couple of steps previously, so they get more practice. We then move on to ordering decimals, so we can come back to range and median in that context, and again later when we hit negative numbers.

But when I was looking for some tasks for students to do to practice these skills beyond the worksheets that White Rose provide, I realised nearly all resources either cover just one, or the whole mixture of averages. So I went ahead and adapted a few resources to fit what we have covered.

The first is a set of questions that I put together to try to get students thinking about what it means when the data set changes and only one of the median or range changes. It is meant to lead them towards the idea that both a measure of position (median) and spread (range) are necessary when looking at data.

The second is a More Less Same grid (check out this website for more).

The third and forth are a pair of Maths Venns tasks.

The final is an Open Middle style problem.

The PowerPoint file that contains all 5 is here.

There has been plenty of debate on twitter the last few days over the effectiveness of teaching live online lessons vs setting work for students to complete in their own time. In other words, whether we should be teaching in a syncronous or asynchronous way in the current school closures.

Mark Enser goes into detail of why he thinks the asynchronous model is a better approach here as a response to some rather antagonistic tweets from a former Schools Minister. Enser accepts that different circumstances will mean that each model will be more effective in different situations. At the end of the post he asks for anyone who has been teaching live online lessons to share how they have made it successful, so that is my plan for this post.

My situation
First I want to state that I know that what I am doing would not work for everybody. Even those in a similar school situation have different home lives. I am not sharing this to say others should follow my lead, but rather that here are some things that have worked for me.

At the start of the lockdown I posted a blog with 5 tips. I stand by those now, though there are certainly things I might add!

As I write this we have been doing online teaching for 7.5 weeks since the schools were closed here in Peru. This started 1.5 weeks into our new school year. We are currently on a week's break before starting the second term of 9 weeks which will be done online too. It is very possible we do not return to school premises until 2021.

I mention this to make it clear that we are in this position for the long term, and so suggestions of just reviewing and mastering content students have already studied are not appropriate for us.

I work in a private school. This brings two things into the mix. Firstly, our students largely have their own devices (year 9 upwards all have laptops in school normally, and years 7 and 8 use them in certain lessons, so most also have their own) and Internet connections are not an issue (well, no more than are usually an issue here). Secondly, our salaries are dependent on parents continuing to pay the fees, and so there has to be a large element of 'pleasing the customer' at this time (more so than usual).

In terms of my technology I have a work laptop which I am using for the Zoom call, and then have my personal laptop set up beside this so I can do registers, see classkick progress, view the worksheet without having to jump between tabs on the work laptop. This has been an incredibly important part of my work flow solution.

Working at a private international school also has an impact on the number of periods we teach. We have 40 minute lessons and there are 40 periods a week. The maximum teaching load is 28, and most do not have more than 25/26. I also have a post of responsibility so have a lighter timetable.

I have a two and a half year old at home. He has not been able to leave the house for 7 weeks and is going stir crazy because of it. But fortunately my wife stopped working when he was born, and so she is taking the brunt of looking after him. Of course, he doesn't completely understand and finds it hard for me to be at home and not be able to see me, and this does lead to some interruptions.

But my school have been incredibly understanding of home situations from the outset. We are sticking to the normal timetable, and the only requirements on staff have been that they must have at least one live sessions with each class each week, and they should be available during timetabled class periods, but this can be via email/Google Classroom.

So given that many of the practical problems with live teaching are not an issue for me, I have decided (as have our whole Mathematics department) to teach all lessons live through Zoom.

What I am doing
This year I am teaching two year 7 classes, a year 12 IB standard level class and a year 13 IB Higher Level class. I have taken a different approach with the different age groups, but I do a Zoom lesson every period.

With the IB classes I have broadly followed what I do in school normally. As we are working towards and external qualification, there is an element of needing to cover the content, and this is taking a little longer than it normally would. I have cut the retrieval starters down for this reason, doing one question in single periods, and then following the Spacing Concepts I started this year in the double period.

In terms of the rest of the lesson, I am still using the lesson sheets I produce for IB classes. Students either print these or have them open of their screens and write in an exercise book if they have no printer. I do not have a printer at home, so I am writing in an exercise book.

I use my visualiser and screen share with the class my book, and work through examples as I do in class. Sometimes I will bring out a mini whiteboard under the visualiser to answer tangentially questions. Then students do a your turn. Where in class I can wander around to see their work, in Zoom I am making more use of cold calling students to talk through their entire solution, and asking if anybody did it differently. For shorter questions I get them all to type their answer in the chat function on Zoom, which I have set so that only I see their responses.

I am more reliant on them asking for help than I would like, but it seems to have worked well, as the quieter students are asking through the chat.

After some input, they generally work on some independent practice. I am making use of Classkick (I made this guide for our staff) and Desmos activities which both allow me to see student responses, but mainly for IB they have questions to do from the textbook, which have answers they can check. One of the mistakes I made early on was not ensuring they knew where the answers were, but now they are in the habit of checking themselves.

Keeping them on the Zoom call but muted has become the norm here. This was a request from the students who said they were too easily distracted working in their rooms without it. This also enables them to ask questions if they get stuck, and sometimes I will put them in a Breakout Room to discuss it with somebody else from the class.

With my year 7 classes I am taking a different approach. I am uploading a presentation to Classkick, and producing an assignment in Classkick for each lesson (labelled week 7 lesson 3 etc). The first slide is the starter activity which is Numeracy Ninjas. I have found this more important for the younger students as they arrive to the Zoom call in dribs and drabs, and this gives them something to do straight away. One benefit of Classkick is that I set it to mark automatically after 5 minutes.

After this I will usually introduce the idea for today's lesson through an example. I have mostly been doing this by editing the Classkick assignment live. If they are on the page they see my edits appear immediately. I then talk through these on the Zoom call. Whilst doing this I lock the assignment in Classkick so they cannot edit it. Then they do some practice. This will probably involve some your turns first which I check before they can move on to the main exercise (in Classkick they can call me to check their work). The main exercise is from the White Rose Maths resources (we changed our scheme of work to theirs this year), and I have set it to self mark where possible. I then keep a view of the whole classes work and can see their work live. I will check questions that can't be self checked (written answers) and answer questions which they can ask through Zoom or Classkick.

Some concluding thoughts
Some of the benefits of doing lessons live have been:

• Less work out of class as I do not have to mark every piece of work submitted. I can keep on top of this during class time like I would in school.
  
• Students can ask questions when they are stuck or unsure, like they would in class.
  
• I can monitor misconceptions more easily and address them earlier, through classkick and targeted cold calling. Again, more like being in school.
  
• The social aspect for our kids is important. They have not been able to leave their houses for 7 weeks now, so the normality of school and social interactions is really important.
  
• The kids and parents have asked for it. I have sent a couple of emails to parents explaining my process at this time. I have also sent a survey to kids and their parents. My school is doing the same on a whole school level. In all cases, the majority want Zoom classes if possible.

As I said at the start, I am not saying anybody else should go down this route, and I accept that the practical limitations could get in the way of this being a reality for many teachers. But I have found a way to make it work for me, and I feel like I am able to meet the learning objectives for the students this way in a more effective way than through an asynchronous model.

The biggest problem with online teaching (in either model) is checking for student understanding at the time of input. Through Classkick, questions and answers through Zoom chat and cold calling explanations of answers I think I have managed to make a good stab at being able to do this fairly effectively.

This year I have started trying something new with my IB class to promote their retention of key facts, concepts and skills. I have previously blogged about using the Last Lesson, Last Week, Last Unit, Further Back starters but having had our teaching time reduced I now struggle to feel these are a worthwhile use of time every lesson, and instead have moved to weekly quizzes made up of past exam questions. They get the same number of questions but I mark them and we 'waste' less lesson time in transitions.

But I still wanted to do some daily recall (it is a Rosenshine Principle after all!) and with this particular class was a little worried about their knowledge and fluency of key terms and basic skills. I decided to keep a track of the new vocabulary we meet in class, along with key facts and any simple key skills. That is, the things I want them to be fluent in doing. 

On top of this, I wanted a more systematic way to review these things keeping the spacing effect in mind.

To do this I created a spreadsheet!

I input the concept/fact/skill into the first sheet and it automatically copies across into the Review Timeline sheet. Then I enter a 1 in the cell that matches when I first taught the concept to students. So in the first lesson of the first week I taught them the concept Gradient and how to find a gradient (The ones before were taught in the taster sessions last year).

The sheet then automatically populates the rest of the row with when to do the next review. So the following lesson is a 2 which is the second review. After three more lessons, the 3 tells me when to do the next review. A larger gap appears before the 4th, then 5th, 6th, 7th, 8th and 9th review sessions. We teach the course over six 9 week bimesters, with a final bimester of revision before the exams, so I have set it up for those 6 bimesters. Not all topics are going to get the full 9 reviews, but for gradient the final review occurs in Week 8 of the fifth bimester at which point there is a full 9 weeks between each review.

Then for each lesson I look at the lesson we are in (Bimester 1 Week 5 Lesson 2) and look down the column to see which concepts etc I should review. 

With the current remote teaching I am assigning these as the starting activity as a Google Form for students to do as we wait for everyone to arrive in the Zoom class. I then check their answers and return it using the Google Forms features. My plan is to also increase the difficulty of the skills questions as the review stage increases.

When we go back to teaching in a classroom (which seems like it may still be a while off for us here), I am thinking about the best way to do this. It doesn't need to be at the start of the lesson.

If you would like to adapt this for your own teaching there is a template version here. There is a template version for having 1, 2, 3, 4 or 5 lessons a week. But you might need to adapt the headings for your situation. I suggest only adding extra columns at the end, rather than deleting columns in the middle, as this will mess up the formulas.

Obviously, this could be used to schedule a lot more than key concepts etc. Perhaps you could give it to students to help them schedule their revision. Or to schedule when you will set exam questions. Or anything else. But I have found it a very visual way to see the idea of spacing, and it is also useful to help explain what it should look like to students.

We started teaching online on Friday 13 March, after all schools in Peru were closed due to COVID-19. In this post I will share a few things I have learned in the first 2 weeks.

The main difficulty I am having at the moment is:

• How to effectively and efficiently check for student understanding. Zoom chat is good for easy to type answers, but it can be a bit overwhelming for anything other than a short number. A sentence of equation here becomes hard to keep track of. I am trialing classtick this week to see if this will help.

In Part 1 I looked at why I have started to use Booklets in my teaching. In Part 2 I looked at how I go about making a booklet for a unit. In this part I want to have a look at how I use the booklet, once it is made. There are two parts to this: how I use the booklet to plan the actual lessons; and how I use the booklet in class.

In planning
With a completed booklet (either from a previous year, or just done for an upcoming unit), I will set aside two copies for myself. The first is used in planning the lessons.

The first thing I do is complete the booklet how I would expect a student to have theirs completed at the end of the unit. This includes doing each of the examples and your turns, writing out the notes, completing some of the exercises (depending on the skill I will do more or less of these based on how difficult they are), and the practice activities. There are three benefits to going through this process:

1. I can see any mistakes that might have managed to slip past me (the occasional copy and paste for a your turn where I forgot to change something);
2. I get a clearer idea of the progression through the skill, and what students will be thinking about at each stage;
3. When the lesson comes around, I have everything worked out already, so do not have to 'live perform' reducing my cognitive load to focus on other things in class.

This first stage is completed in red pen throughout.

Alongside completing the booklet, I also add notes to myself, usually in green pen. These might be things like:

• Show a diagnostic question here;
• More details of explanations;
• Breaking down the steps for a worked example to make it clear to me in lesson what to do.

These notes are there to help me do the important thinking before class, and once again, help me focus on other things during class.

Finally, in a blue pen, I add extra questions that I want to ask along the way. This could be as part of a worked example to draw attention to a particular part, or they could just be oral questions that I don't want to include in the booklet itself.

This is a process I have worked up to over the last couple of years, and I find that all three stages of the planning process with the booklet help dramatically reduce my cognitive overload during class. This allows me to give my attention to other things, such as focusing on checking for understanding and dealing with misconceptions and being more aware of behaviour issues and acting promptly. The next stage for me is to start thinking about who I am going to ask the questions to. 

This post from Doug Lemov explores a lesson preparation process he observed that follows a similar outline.

In lesson
When the lesson comes around, I am well prepared to get going. I now teach using a visualiser/document camera, and make use of the second booklet that I put aside. 

For the notes section of the booklet, I started by writing notes on my blank copy and getting students to copy this down, but more recently I have moved away from this. Now I go through my explanation (penned in green on my copy), and will then direct students to fill in the notes. 

For example, when looking at the magnitude of a vector (one of the notes examples in Part 2), I would explain with a semi-script: "A vector is a quantity with both direction and magnitude. The direction is given by where the vector points. The magnitude is the size of the vector. It is the length of the line segment that represents the vector. If the vector has a direction parallel to the x or y axis, then the magnitude is easy to count. For other vectors, we can draw a right angled triangle where the vector is the hypotenuse and so we can use Pythagoras' Theorem to calculate the magnitude of the vector. If the vector is given in component form, then we can either draw a diagram to help us, or use the formula |a| = sqrt(x^2 + y^2) for a = (x,y)."

Then I would direct students to the page in the booklet, and ask them to fill in the notes based on what was just said. This acts as an immediate form of retrieval practice for students, and gives them a second exposure to the idea. Whilst students write it down, I will circulate and check what they are writing over their shoulder. After time to do this, I will cold call on a student to read out their notes, and ask all students to check theirs against this. This gives students a third exposure to the notes in a short period of time.

I also use the visualiser for doing the examples. I will ask all students to put their pens down, and look at the board. I will then go through the example completely, using silent teacher as described by Craig Barton. I will then ask students some elaboration questions to get them to explain what I did at each stage, and why. If I think there is a step that the class will not know why it is there, I will narrate it instead. Then I will give students time to copy the example down (exactly as I wrote it), and they can move on to the Your Turn. As they do this, again I circulate the classroom, with my complete booklet, looking at their work and checking it against mine. If it is a very short question that requires little work, I would probably not do this. Then I will cold call a student to give their solution, talking through the whole solution, not just giving the answer. At the end I will ask the rest of the class if they agree. The student I cold call may have given an excellent answer or made a misconception (that I picked up on when circulating).

The other way I use the booklet is when students are doing practice on the big whiteboards. I have found that having students hold their booklet as they do this just gets in the way, so I will project the activity on the board for them to refer to.

As I go through the lesson I will also make further notes in my completed copy (with black pen) as to any issues that arise with the booklet, or anything I want to change for the next year. Perhaps a note to add more examples, or remove some, or how to make the explanation clearer, etc

One thing I have not been doing, but am planning to start doing, is to also take notes of students who struggle with a particular part (I guess a purple pen for that!). This would give me something to refer back to when assessments and quizzes come up. 

After the unit
After the unit is complete, I scan the 'live' copy of the booklet (the one I worked on under the visualiser) and upload it to the class website. This acts as an aid to those who missed any lessons as they can see some bits of notes and also all the examples. This is the reason I don't write the your turns in this booklet, so that students can download a new copy to do the your turns again if they wish, whilst having the examples.

Do you use booklets in your teaching? How do you use them effectively to plan for lessons? How do you use them in class? ​