Adding and subtracting directed numbers

Download the PowerPoint here.

OK. So this is a big one.

Firstly, I have found teaching negatives difficult in the past. Teaching the rules is a bit useless. ‘Two negatives make a positive’ often leads students to doing things like -5 – 9 = 4 (or similar). I avoid the rules like the plague.

I guess pattern spotting helps.

Here I have used a Craig Barton idea. The idea is that you present each set of examples, asking students to predict the result. You then wait during the … (the gap of understanding) and ask them to predict the result of the next one. Hopefully they see through pattern spotting what the answer is.

I’ve got some questions to support this.

I have tried to write questions that help develop student’s strategies like grouping.

Notice at this point it’s still all addition.

Increasingly I have found, however, that multiple representations are helpful. I used to think that I didn’t want to complicate things in student’s minds, and that showing differing methods would confuse them.

I’m now of the opinion that switching between multiple representations is really, really useful. Students who can do this are much more successful and I’ve tried to be more explicit in adding in differing representations into my work.

This is an idea I took from the excellent Pondering Planning website.

Notice I’ve done this AFTER the questions. I think sometimes it’s helpful to get students being able to solve the problems first and then supporting understanding later.

Even more representations!

I like double sided counters for negatives. They fall apart when multiplying or dividing by a negative, but they’re a useful representation for supporting addition and subtraction.

Hopefully, having differing representations helps students do some worded questions. I think this is probably a nice time to break for a lesson.

When we come to the second lesson, we could start by trying some arithmogons. I like arithmogons. They feel ‘puzzly’. The numbers in the box is equal to the sum of the circles either side. They test a bit of fluency.

Time to move on to subtraction using a similar frame. Including more patterns, questions and representations.

These questions tend to come up a lot in SATs.

I even chucked in a crossnumber.

How about some negative arithmogons? You subtract along the direction of the arrow.

Maybe a little complex. I’ve written a sheet of them, though.

And of course, JMC questions often bring out a little understanding.

This is a pretty massive PowerPoint. 41 Slides. Probably 4 lessons. I don’t think I devoted enough time to doing negatives, before, though. This is pretty exhaustive.

Add a comment if you’ve got any suggestions for improvements.

Calculating bearings

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Bearings is a topic that I’ve taught really badly in the past. It’s due to a number of factors (teaching measuring, use of worksheets) but mainly I think due to a lack of thinking on my part about the different processes and skills contained within bearings work.

I often lumped all of bearings together, maybe throwing a few calculation questions around as they’re not seen too much at GCSE. This was clearly a mistake.

In my measuring and constructing bearings lesson I tried to really atomise the concepts involved in bearings as much I could and sequence the lesson in a much more thoughtful way.

I’ve tried to do the same here with my calculating bearings lesson. It mainly covers question types like the following.

But I tried to think about what challenges pupils could have. Pupils could have different angles given. They could have questions that require them to remember that a bearing is a three-figure value. There could be more than one bearing. My attempt to look at these question types is below.

Looking at these, there’s a few things I like and a few things I don’t.

I’ve managed to build in a bit of variation so that students have to think about the angle they’re trying to find. Maybe there could have been some more straightforward questions to build confidence first, though. I also tink it’s a little crowded and messy from a visual standpoint, but I found it very hard to make it look any nicer whilst still being readable.

I took a lot of inspiration from this excellent solvemymaths sheet.

I also added some worded questions.

And this

I love the question ‘what can you work out?’ when it comes to angles. It really does make such a huge difference in how student approach their work.

Index Notation

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So … I can’t claim that I’ve done much original work here.

Most of this is stolen from Jo Morgan’s indices in depth presentation. Sorry Jo!

Normally I make a lot of stuff from scratch, but Jo’s work here is so good that I can’t really improve on it. If you get to see Jo do an ‘in depth’ presentation, go. The time and effort behind them is phenomenal.

PowerPoint contains a section on writing things in index and expanded form.

As well as some questions people might find a bit old fashioned. Also the first time I’ve put a ‘reveal all’ button on a page. Can’t believe I’ve not thought to do that before.

I like these little discussion slides. I think they’re fun.

Some UKMT questions, which I stole from this book. It’s a lovely little book.

I also added a little problem solvey bit where we talk about powers of two and binary.

Multiplying to come next (it’s basically copied and pasted from Jo again. Sorry Jo!)

Applications of Pythagoras

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I find telling pupils the Pythagoras rules and then getting them to solve ‘problems’ isn’t enough. Often, the unstructured problem solving can be overwhelming for students.

Opinions vary. You might think that I’ve structured this waaay too much, and removed too much thinking from the students.

It would be interesting to hear people’s point of view on this.

Starts with a ‘do Pythagoras but with a little context’ and leads into a question that’s a little more unstructured.

Moves onto using Pythagoras to find the area of a triangle.

Maybe I’m over explaining things, but in the past I have not made enough things explicit and I’ve just assumed my students will work it out. I think I’d rather be more explicit.

Next, we pick from a complex diagram.

Now points. Which I don’t actually think is too complicated.

I’d love for people to feedback on this lesson. Is there too much spoon feeding going on here? How can I reduce the spoon feeding whilst also making skills explicit?

Ratio problems

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Just a load of interesting (ish) ratio problems to work though.

I used bar modelling. I am increasingly into bar modelling for KS3, although I’m still not massively convinced when it comes to negatives.

I call these ‘reverse’ ratio problems. I don’t know if that’s the correct terminology.

I have been watching a lot of Queer Eye recently.

I like bar modelling for questions like this.


Some questions adapted from Don Steward and White Rose Maths Hub’s Barvember 2017.

No posts for two weeks now. I’m on holiday. When I get back, Pythagoras and Trig!