# Click & Reveal

This week I faced a frustration which I’m sure can’t have been for the first time: I had a Word document to use as a “fill in the gaps” style handout, but I had already filled the gaps. I was looking for a way to display the document using the classroom projector, but to easily toggle the gaps between filled and blank.

This situation must surely arise up and down the country quite frequently. I put the call out on Twitter and asked a few colleagues in school as well. It turns out there isn’t a straightforward solution (that I’ve come across so far).

The shortlist seemed to be:

• Use interactive whiteboard software with white boxes covering gaps (downside: I need to create a lot of covering boxes and I don’t finish up with a nicely printable Word document. I also don’t have an IWB…)
• Use Powerpoint and animations (downside: in addition to the above, without clever VBA coding, you can only reveal the gaps in the order in which you animated them)
• Change the colour of the text in Word (downside: need to select all of the text beforehand to make it white, then go through and slowly change the colour of different pieces to reveal them)
• Change the colour in Word using a style – this is the best so far, and the purpose of styles. If every gap is formatted to have the whole style then it’s relatively easy to change that style from white to blue for example. This could even be automated using some VBA and a keyboard shortcut. Thanks to @SamHartburn for help cracking this approach! (Downside: it’s an all-or-nothing reveal and hard to adapt to revealing one gap at a time.)

## There must be another way

I’ve done a fair amount of programming in years gone by, and learned my fair share of HTML, CSS and Javascript. So in a free lesson this morning, I sat down to hack something together and, within an hour, I had a fully working prototype!

Here’s a (silent) video screencast showing how my mocked up page responds:

## The Sorcery

Essentially, each gap (which is typically either a span or div) is given the class reveal. As the page loads, a Javascript onclick function is assigned to every document element with class reveal. The Javascript simply changes the CSS for that element: toggling the colour and toggling the mouse pointer style.

I also found some sample code online that styles the page to mimic an A4 page (in appearance and dimensions) and I include a Google font (PT Sans) as the closest match to the one I always use with my classes (Myriad Pro) – all my handouts have this same style!

So from now on, maybe I’ll switch to HTML over Word documents. We’ll see. Creating equations becomes tedious, and I have yet to fully get the print CSS correct for printing a blank version of the document. That will have to wait until I have another spare hour.. probably half term!

# FP3 Vectors – Developing Deeper Understanding (Part 2)

This part picks up from Part 1 and looks at the ‘shortest distance’ scenarios in FP3 Vectors. I’m always stunned by the textbooks and formula books that promote such formulae as:

$\text{distance} = \left| \dfrac{(\mathbf{a}-\mathbf{c})\cdot(\mathbf{b}\times\mathbf{d})}{|(\mathbf{b}\times\mathbf{d} )|} \right|$

and

$\text{distance} = \dfrac{|n_1\alpha+n_2\beta+n_3\gamma+d|}{\sqrt{n_1^2+n_2^2+n_3^2}}\,.$

The former of these is for the distance between two skew lines, and the latter for the perpendicular distance from a point to a plane. Not only are these a hideous mass of letters but, even more concerning to me, they are both disguising the same mathematical principle. Let’s take a clearer look at things.

## Understanding Components

Here’s my starter question before we consider the ‘shortest distance’ scenarios:

This can take a little time but because the given statement is true, students can invest some time and thought into why it is so. After some short discussion, we are happy that dot and cross product can be used to find components. Moreover, it is only the ‘vertical’ component that we will need: sometimes found with dot product, sometimes with cross.

## Similar Situations

Mathematics to me is all about identifying similarities. The formulae quoted at the start of this post are totally anathema to me. Consider these (beautiful…) diagrams:

In every one of these situations, the ‘formula’ for finding the shortest distance is simply

$\overrightarrow{AB}\cdot\hat{\mathbf{n}}$

Much less to remember, much more understanding developed.

## Two Other Cases

In the case of parallel lines or perpendicular distance from a point to a line, this approach breaks down: we don’t have a convenient unit normal vector. This is where the cross product alternative for finding a component comes in.

## Many Ways

Of course there are many ways to solve these problems. I simply like the above approach because it follows from a clear understanding of components and the geometry of the objects involved.

I wouldn’t often openly criticise a textbook but, for comparison, here is a worked example from the Edexcel/Pearson book [FP3, p128]:

Of course it works. But. Yuck.

# FP3 Vectors – Developing Deeper Understanding (Part 1)

I love teaching vectors in FP3 because I see such simplicity behind all of the geometric situations. In Edexcel’s module the chapter essentially breaks down into three sections:

• Learning how to calculate a cross product, culminating in the triple product giving the volume of a parallelepiped.
• Learning that the equation of a line (which was introduced in C4) can be written in several forms; and learning about the equation of a plane, again in several forms.
• Finding points of intersection of lines and planes; angles between lines and planes; and shortest distances between points, lines and planes.

This post picks up on a lesson at the second stage of this journey. Part 2 looks at a lesson snapshot from stage three.

## Productive Struggle

My understanding of the phrase productive struggle is to give students a task which they can’t launch straight into solving (with their current knowledge) but that they can chip away at and explore.

Each year, I look forward to this lesson:

Occasionally near the beginning, I might need to clarify precisely what the question means but I very rarely give any indication about how to proceed. And I give my students a lot of time. Originally I thought I give them around 15 minutes but this week, my class had at least 20 before we shared some ideas on the board. I asked 4 particular students to write up what they had tried:

Each of these contributions had something interesting that I wanted to bring to the whole class discussion. The writing in black identified a geometric meaning to the problem. The student writing in blue worked from the principle that the cross product of a vector with itself is zero (ie the zero vector). The student writing in red took an algebraic approach and realised that the three equations do not have a unique solution, but she was able to identify a valid solution. The student in green worked along similar lines but was able to list multiple solutions and identify a pattern behind them. (The annotations in red and blue came later.)

Eventually we could have a full discussion about why $(\mathbf{r}-\mathbf{a})\times \mathbf{d} = \mathbf{0}$ represents a line – from both an algebraic and a geometric perspective. Students are always quite surprised to see the shortest demonstration:

Of course, a similar lesson could be built around $\mathbf{r}\cdot \mathbf{n} = c$ but I’m not usually mean enough to make the students suffer twice in the same week!

# Playing by the Rules

This post has been motivated by a discussion that’s surfaced on Twitter in recent days of which this is a taste:

Is it wrong to have a game that doesn’t abide by the standard order of operations? Is it likely to teach children misconceptions that will need to be unpicked in maths class?

## Why we have rules and conventions…

Consider the game of chess. If two people decide to play a game of chess, they both know exactly where they stand: white begins, the possible moves are clearly defined, turns alternate, the objective is clear – there really is no room for disagreement. But there are alternatives that can be played: suicide chess where you try to lose your pieces as quickly as possible (I prefer that to chess as I have a short attention span); or any of the many variants such as those listed on Wikipedia.

To me mathematics is in many ways like a board game. We have the pieces (usually set up in terms of definitions) and we have moves (the logic of deduction, theorems and the use of operators and functions etc). We need to agree on a system that is consistent within itself (as far as possible – I don’t want to get into set-based paradoxes or Gödel’s theorems).

In A level Maths and Further Maths, we teach the dot and cross product for vectors. Why do we have two different ways to multiply vectors? Why are they calculated in the way they do? Why don’t we just multiply component-wise? My usual explanation is that it is down to mathematicians to define multiplication and to do so in a way that has longevity/purpose. There is an essence of natural selection in the history of maths: some ideas just don’t have enough in them to last more than a cursory exploration. I’m curious to look back into the history of matrices sometime and see how that multiplication process developed.

## Order of Operations

You might disagree, but I believe the order of operations (BIDMAS, PEMDAS, PIMDAS, and many other variations) exists so that around the globe we all interpret expressions in a consistent manner. Parentheses (and brackets and braces) are a useful marker for prioritising the evaluation of different parts of an expression. But why didn’t we all agree to “left-to-right” evaluation instead? That could still be consistent around the world if we wanted it to be. Simple four-function calculators still work that way (and, in fact, we do use that when an expression is simply say a mix of addition and subtraction). Or why not reverse Polish notation which handles operations in a manner that programmers would call a stack?

Let’s compare them by finding the perimeter of this rectangle:

• Standard Order of Operations: $2\times(3+7)$
• Left-to-right evaluation: $3+7\times 2$
• Reverse Polish: $3\,\,7\,+\,2\,\times$

## Survival of the Fittest

I think it’s clear why RPN didn’t survive in the mainstream: although it’s easiest for a computer program to evaluate, it places a huge demand on the user to enter the input correctly. (As a distraction, there’s a cute discussion on Stackexchange about how to handle the quadratic formula in RPN!)

I think the reason the ‘standard order’ generally wins out over left-to-right evaluation is the flexibility it allows, without introducing ambiguity. Following the standard order, commutativity of operations can be used fully:

$2\times(3+7) = 2\times(7+3) = (7+3)\times 2 = (3+7)\times 2$

whereas in left-to-right evaluation, only these two would evaluate the perimeter correctly:

$3+7\times 2 = 7+3\times 2$

and once parentheses are introduced to LTR evaluation, it just feels like a restricted version of the standard order.

## Right or Wrong?

No, just different. Of course, however, young children could receive mixed messages. Are these best handled by only exposing them to the standard order of operations, or  is it better to open up the discussion a little and see why we have the standard order?

I’ll leave all the die-hard BIDMAS fans with this ‘brain training’ from the Daily Mail:

“Multiply by itself twice”… is that $x\times x \times x$ or $(x^2)^2$? 🙂

# A2 Trigonometry (Parts 2 & 3)

Yes, I’m beginning with Part 2. At some point I will try and write about my approach to sec, cosec, cot and the inverse trig functions but following on from some recent Twitter comments, it seems many of us are around the sin(A+B) and Rsin(x+a) mark right now.

My approach this year has made a much bigger deal about the motivation for these formulae and techniques. Here’s a brief run-down of the stages we went through.

# Part 2: sin(A+B)

## Stage 1 – Motivation

A number of statements written on the board. Students to discuss in pairs if each is true or false. Typically something like this:

 1.) $\sqrt{a+b}=\sqrt{a}+\sqrt{b}$ 2.) $(a+b)^2=a^2+b^2$ 3.) $\ln(a+b)=\ln(a)+\ln(b)$ 4.) $e^{a+b}=e^a + e^b$ 5.) $\frac{5}{a+b}=\frac5a+\frac5b$ 6.) $5(a+b)=5a+5b$ 7.) $\sin(a+b)=\sin(a) + \sin(b)$ 8.) $(a+b)^{-1}=a^{-1}+b^{-1}$

Purpose: this gives the opportunity to air a few of the typical misconceptions that always arise again near the beginning of C3 (but hopefully not so much later in the course…) But it also emphasises that most functions cannot simply be ‘expanded’ (mathematically speaking, they are not linear).

Discussion: hopefully students will correctly identify true/false. It’s nice to make improvements to several of them, such as:

• 2.) $(a+b)^2=a^2+2ab+b^2$
• 3.) $\ln(a)+\ln(b)=\ln(ab)$
• 4.) $e^{a+b}=e^a e^b$

Most students are wise enough to not trust number 7, and thus we have some initial motivation for wondering how to ‘expand’ $sin(a+b)$.

## Stage 2 – A Proof

I’ve seen various diagrams and constructions (Cut the Knot has several pages dedicated to them) but my current favourite is this one:

The students are to label all the remaining sides in the diagram (I usually model how to proceed by using say $\cos\beta$ as an example. The worksheet I use has all the necessary labels underneath the diagram to guide them, too. Finally, the sheet poses:

Note, unless I have particularly high-flying students, I don’t make much of a fuss about the construction only really working if $\alpha+\beta<90^\circ$.

## Stage 3 – Examples and Non-Example

Of course, it’s tradition to find an exact value for $\sin 75^\circ$ at this stage, or perhaps $\latex \cos 15^\circ$, but some calculators now provide these values directly which is a shame.

I would typically demonstrate some equation that requires one or both of the new identities, but then I think a non-example is also crucial here. Something like:

Solve $\sin(x+\frac{\pi}{3}) = \frac12$.

Just because we can expand, doesn’t mean we always should!

That’s more than enough for one lesson. The next would begin with this slide

and then move on to the double-angle formulae.

# Part 3: Harmonic Form

For this lesson, I place perhaps an even greater emphasis on motivation. There are two initial activities that I use.

## Stage 0 – The Settler

I make quite a big deal about some things being “just a C2 question” – trying to make the point that C3 isn’t such a big leap as they might think, and that they really should be able to solve C2 questions by now…

Clearly the middle equation causes the problem and they’ll be clamouring to know how to solve it!

## Stage 1 – The Motivation

I then give the students a sheet that begins like this:

There are a couple more parts further down, but you get the idea. Given time to work in pairs and discuss what they are able to do, the students realise that filling in the column for $f(x)$ is relatively straightforward (and essentially C2 work, but it gives the opportunity to get a bit of help too) whereas the column for $g(x)$ is nigh on impossible.

It’s great to see what they then try: some will plug a few values in their calculator, using a bit of trial and error to find the domain of g. Others will try and guess what the sum (or rather, difference) of a sine and cosine graph might look like. Basically though, they’re stumped.

We use Desmos to do the big reveal. Type in f(x) to check all their answers to the left column. Then type in g(x) and. Wait. What? It’s the same function? *general surprise*

## Stage 2 – Backwards and Forwards

Some then realise that they could ‘expand’ f(x) using the cos(A+B) identity and we do this on the board. The stage is then set for, effectively, the reverse of this process.

## Stage 3 – So, it’s not a big deal

The whole point of harmonic form is to make very difficult questions (the right column) very easy (the left column)! They love it.

## Stage 4 – Practice

All of the above, perhaps with the inclusion of just another example or question is usually plenty for the first (hour-long) lesson.

I don’t have much more to add, except this idea which occurred to me for no particular reason. I make use of the classic Solomon worksheets relatively often (maybe on per week, ish) but this is the first time I’ve annotated one before issuing it to students:

I’ll try and get a higher resolution pic at some point, but the idea was to highlight the purpose (or distinguishing feature) of each question [done in red ink] and then recommend what to complete, such as “just do one of these” [done in green ink]. I’m sure the big piece of paper and colour-copying played its part in engaging the students, but they responded to it very well: for the able ones, it gave them clear guidance about what was worthwhile to complete; for the weaker ones, they had the detail of the question emphasised and felt more secure in tackling them.

# The Organised(?) Teacher

For some reason, my colleagues are often surprised when I say that I’m disorganised. I think people expect mathematicians to be organised as a consequence of the logic of the subject. It doesn’t take a mathematician (or perhaps it does) to see the fault in this. I can think through a problem, come up with a strategy and follow it through sequentially to a conclusion. But if you tell me the problem when I’m on my way to a lesson and then ask me about it later in the day, I’ll more than likely say: “Ah yes, I forgot about that. What was it again?” I’m forgetful, I procrastinate, I’m easily-distracted and I’m disorganised when it comes to managing tasks. But memory, focus, organisation and record-keeping are crucial for teachers, so I thought I’d take some time to share my strategies (or trade secrets, perhaps).

## The Year Plan

I know we have a scheme of work. But it’s in a shared network folder which isn’t easy to access outside of school. If I made an electronic copy it wouldn’t get updated. If I printed it, it would use too much paper and I would leave it in the wrong place. Each year, I now produce a “student year plan” for the courses I teach, which briefly summarise the topics on a week-by-week basis. Here’s an example:

Every student has a copy to put in their folder so they never have to ask which topic is next, or which textbook to bring. I have a copy pinned above my desk so even if by Wednesday I have forgotten what we’re doing that week, it’s right in front of me. It’s also saved on Google Drive so I can access it from anywhere.

Do we stick to it? Well, no. Of course there is a natural ebb and flow to teaching. Right now, we’re ahead of the game with M2 so Statics will shift from November to October. That doesn’t mean I’ll re-print the whole thing – we’ll all just annotate our own copies. Come November, I might try and start teaching Statics again but I’ve told the students to remind me not to…

## The Week Plan

I’ve never been able to plan a whole week’s lessons in advance. Every day I like to see how the class have found that day’s material and what they would benefit from as a next step. However, we do have some weekly structure set by the school: students are tested every week and they are graded every week. Thus to maintain a routine, I allocate Thursday as test day for all my groups so they can have their tests returned and discussed on Fridays. Grades are then entered on our system by Monday morning.

A corner of one of my whiteboards is always set aside to record the topics that will be covered in that week’s test for each group. This reminds the students permanently that tests are on Thursdays and stops them giving me a headache asking “What’s this week’s test going to be on?” three times a day. But again, it’s there for me as much as for the students: sometimes I have to look there on a Wednesday afternoon as I ask myself “what do I need to put on this week’s test?”.

Our grading system is excellent: students are graded in every subject, every week, for both achievement and effort. This is all recorded online and students, tutors, parents and teachers can all see the grades each week. And I get a relatively friendly nudge at 10:30am if this hasn’t been done…

## Planner or Markbook?

And planners? They just don’t seem to work for me. I’ve tried the standard teacher’s planner product. I’ve tried creating my own custom version, printed at the beginning of term. I’ve tried an old-fashioned simple markbook. All the same problems come up: it’s never in the place I need it and I fail abysmally at keeping it up to date. It becomes another chore in the week to retrospectively fill in all that week’s gaps, by which point 90% of the information is redundant. I’ve even tried the app iDoceo: a “powerful teaching assistant for teachers”, which some people swear by. I’ve tried it twice or three times now. I get as far as adding in students and blocking out term dates/holidays and adding our timetable structure. Then I don’t have my iPad with me or I don’t have an internet connection and it all goes out the window before the first full week of teaching. I’ve tried Moodle too: you can record test scores, attendance and share documents with the class. This time, sadly, it was the clunkiness of the gradebook that let me down and, although I could access it anywhere through a browser, the sheer number of clicks needed to get through to any useful information also put me off.

## New Year, New Plan

This year, I’ve tried to rationalise everything: I need convenient access and I need to store some key student data. It turns out Google Sheets might be the solution that works for me: quick to access, available anywhere and I’ve learned about cell comments.

I spent quite a while tinkering and experimenting with the layout and merged cells, but I like the simplicity of what I’ve set up: attendance is taken care of with the 5 small boxes each week; and underneath these are the student’s test score and weekly grade. At the top I can also quickly see the main topic, and with conditional formatting I get the ‘at a glance’ view of strengths and weaknesses. I have one spreadsheet per class and one tab per half term to reduce scrolling. As for cell comments, this is what I always felt was lacking but I Googled for them out of desperation and they do exist! If you notice, for example, the cell saying ABSENT. It has an orange mark in the upper right corner. This indicates I’ve added a comment, and by hovering the mouse on that cell it will pop-up giving more details about the absence.

## Other Records

I should have a record of teaching and for me this is covered by 3 things: the year plan, the scheme of work and the Google Slides presentations that I build up over the course of a topic. See for example these ones on M2 Moments or C3 Trigonometry II – depending on when you view them there may be more slides. Of course, I can also easily share these with students via the same direct links. If I make any alterations, I don’t have to re-send the whole presentation or re-upload it to a VLE.

I should have a record of prep. I think I’m in a grey area right now, if I’m honest. As you may well have read from my posts here and here, I’m running prep in a different way from my colleagues. The students themselves have the full record of all the prep they’ve completed, along with my feedback. They’re all tackling different combinations of questions, so I could never hope to record that myself. If they lose their orange book, this record is gone. But they have still done the prep and received feedback, and I think that’s the most important part of the system.

## All The Other Jobs

As you might be able to imagine, I am also hopeless at keeping on top of any kind of “to do” list. The problem isn’t recording jobs to be done, it’s getting in the habit of checking that list and acting on it. Again, I’ve tried apps, post-its, and even the old-fashioned note on the back of my hand. (I currently have ‘Q6’ written there in red, but I haven’t the faintest clue why. I’m sure it’ll come back to me…)

The habit of checking is always the downfall to the systems I try. For my boarding duties it’s simple: I have the rota stuck on my fridge door. I only need to check it when I’m at home, and I always see it when I’m at home.

This term, I’m trying out a system called the Bullet Journal. I’ve bought a Leuchtturm notebook, so I’m literally invested in the system, and I took the time to mark out calendar pages and the beginnings of my to-do list. The essence is that tasks are listed, then either completed or rescheduled. Each day you add to and/or cross tasks out from the list. It turns out I’m not alone as @dazmck, @MathematicQuinn and @MrDraperMaths are also using this system. However, it does rely on my keeping this notebook at my side throughout the day. I chose an orange one – my favourite colour and hard to lose. Only time will tell if this helps!

## Basically, I Cheat

The secret to my organisation is really relying on other people. I give the students full information so they know what’s happening on a weekly basis. All teachers love the starter task of “Tell me what we learned last lesson”. It’s rare that I have to rely on that to know what I’m doing, but it has happened… I keep minimal records in an easy-to-access online place and I drop in to see my HoD when I have that feeling that I should be doing something, just to check if I really should be doing something.

What works for you?

# Purposeful Prep, Part II

This post follows on from Purposeful Prep and Manageable Marking that I wrote last week. In that post I explained briefly the idea that I’m trialling with all of my classes: maths homework every night, self-checked by the students, handed in for comments and feedback, perhaps leading to a one-to-one discussion as well.

## Aims – My point of view

My main aims were to ensure that students were working on mathematics much more frequently outside of lessons (i.e., every evening) and that they were ultimately completing a greater quantity of work that is directly relevant to them. Here is an enlarged image of the expectations sheet, much easier to read than last week’s (highlighting added by the student):

I’m also trying to keep my marking load well under control. By my nature, I’m relatively disorganised and hopeless at settling down to work on jobs. Face-to-face, I have all the time in the world for my students. Sat at my desk, I could win a gold medal for procrastination.

Ultimately, I decided that the best value I can add for my students is to provide specific feeback, advice and corrections on questions that they have identified as problematic. I could never keep on top of every class’s work, when every student is potentially tackling different questions and topics!

I frequently chat very openly with my students about how we organise our week, about the nature of class tasks and work, and, now, about how homework can be most effective. Some people see marking as a significant part of a teacher’s role, but I have to disagree. The marking isn’t anywhere near as important as expert feedback. (Expert in any sense: efficiency of method, clarity of presentation, accuracy of algebraic work, precision of written responses, providing a structured strategy when students are stuck etc.)

## The Routine

“Orange books, please” is now my opening sentence in every lesson. They’re all handed in – I count them explicitly as they reach me. If no homework was done, I note it in their book. If they failed to check their work, I comment on that – then return the book and expect the student to check the work during the lesson so that I can review it again.

At some stage in every lesson, the students are busy working on tasks and I have the time to look over each book, carefully taking in:

• the quantity of work they have done
• the source of the work and it’s approximate level of difficulty
• the topic of the work, and if it is similar to past work or something new/more advanced
• the presentation of the work: legibility, clarity of algebraic work etc
• the problems that the student has highlighted (and often they will have written in corrections in a different colour, which I can review for correctness)
• the wording of written responses, especially in statistics (many of my students are EAL and, moreover, I don’t expect students to be able to mark such questions simply from comments in a mark scheme for example)

This sounds like a lot but the whole process is incredibly efficient. I can add my annotations, closing comments, recommendations for possible future work etc within a few minutes. Where necessary, I can also chat one-to-one with the student to explain a point or, if necessary, fix a time for them to come back later in the afternoon.

Indeed, much of what I write is common across many books and so today I took some time preparing stickers that I can use (somewhat sparingly..  I don’t want to go sticker crazy):

These are just a first iteration and, as with everything I do, I will ask students for their honest thoughts on using them. It is likely that I will adapt some after another week of seeing what works. Of course, I will still be writing comments too!

## The Guilt

After mentioning this trial approach with another colleague in the department, she has also adopted it (taking care to use red books instead of orange, as we have a number of students in common!) We have both seen a wonderful shift in balance: the students are completing more independent work than ever before, and our ‘marking’ (for want of a better label) is taking less time than ever before. It all happens within lesson time.

It is a strange feeling to be free during a free lesson, let alone not having to take a pile of marking home for the weekend. We are almost creating odd jobs to do just so that we don’t appear idle. (Hence printing sheets of coloured sticks today, for example!) Of course, there are term reports, leavers’ references, forecast grades etc all to be done, so we’re not that idle really.

## Sustainability in the Future

This approach really does seem ideal for this summer term: there is no new subject knowledge to be taught this term as the students prepare for AS and A2 exams. As I mentioned in a tweet earlier this evening, it is possible that everyone of my students completes different classwork and different homework from all the others. But, equally, I can set a common homework task if I believe they would all benefit from it.

The big question on my mind now is to what extent I can continue this approach next year, during the Autumn and Spring terms when we are teaching the subject. I have plenty of time to mull that one over!

### A Note on Preparing the Stickers

The stickers I’m using come on A4 sheets with 65 per sheet. I ordered them online a long time ago, but have never made great use of them. Avery provide template documents for all manner of sticker sheets, so I downloaded one of those to help with the layout.

Within each sticker, I created a two-column table. The left column contains an emoji (which I copy and paste from iemoji.com) and the right column contains the text. Print them on a colour printer and voila.

I’ve shared my sheet on Dropbox – feel free to play with it.