My classes have been working on:

- S2: expanding single brackets and stem and leaf diagrams
- S3: factorising quadratics
- N5: vectors
- AH: inverse matrices, transformations and Euclidean Algorithm

**Progress of the week**

My S3 class have been working on their retrieval starters for three weeks now. You can read about them here. After a bit of apprehension from some of the class, they are all finally buying into the process. During each starter, lots of questions are being asked and pupils who wouldn’t have normally bothered attempting the starter are now doing it. Pupils are noting down the solutions to questions that they didn’t get correct so that they have something to refer to the next day. The highlight came on Friday when the majority of the class scored over 70% in mini test 3. Lots of smiles and I have noticed an increase of confidence. Positives signs for the rest of S3.

**Treat of the week**

Ok, so technically this happened on Saturday but I’m still counting it as part of the week. I attended Muckle Grampian in Aberdeen on Saturday. What is Muckle Grampian? It’s a teach meet where teachers meet to share what is going on in our classrooms. It was a fabulous day. I met some wonderful teachers who love to discuss teaching and left at the end of the day buzzing with enthusiasm. This is the kind of event I would love to see happening in schools on a regular occasion but I’m not sure if it ever will. The trouble when having this type of event within a school is that teachers inevitably end up talking about the day-to-day teaching problems rather than sharing practice.

**Lessons of the week**

My S3s are studying factorising at the moment and I decided to go at it from a slightly different direction. Rather than giving notes I jumped straight into examples. I asked the class to expand these brackets and see what they notice:

Without sharing what they had noticed, I then set them the task of writing this expression in brackets.

Most of the class had the correct answer but a few had yet to spot the connection between the constant term and the coefficient of x. I then asked the class to speak to their partner to share the connection they had noticed. The next ten minutes or so were spent factorising trinomials with positive coefficients. In the next lesson we explored what would happen if we introduced negative coefficients. The conversation around factorising something like x² – 8x – 20 went a bit like this:

Think about or write down two numbers that multiply to make 20 (note: not -20)

Which pair could be used to make an 8? (note: not -8)

Ok, so now we have chosen 2 and 10, how would we use 2 and 10 to make -8?

So our answer is (x + 2)(x – 10)

Previously I would have focussed on writing a list of numbers that multiply to make -20 then choosing which added to negative 8. By removing the focus from the negatives, it became more of a puzzle to solve rather than negative number calculations.

At one point I mentioned the word “algebra”. The pupils stopped and said “Is this algebra?” I hadn’t specifically mentioned it (although I thought it was obvious) and they just thought of factorising as a little puzzle involving multiplying and adding. I actually think by not making a big deal of it, not completing notes formally and treating as a puzzle meant the class were not intimidated by it. I am hoping that this more informal approach will lead to more confidence when factorising and completing it by inspection rather than a formal procedure.

**Weird notation of the week**

I have been teaching for a long time and have never seen a pupil write down a vector addition in this way.

Three equals signs!! The funny thing was, just after I said to the pupil I had never seen this before, I turned to another pupil who had done exactly the same thing.

]]>My classes have been studying the following:

- S2 – expanding single brackets
- S3 – factorising expressions
- National 5 – vectors
- Advanced higher – matrices

**Struggle of the week**

This week my N5 classes were learning about vectors. This is always an interesting topic as those who study physics already have a pretty strong awareness of vectors and how they work. We were looking at drawing vectors, which went well. The trouble started when drawing addition or subtraction of vectors. It seemed to really divide the room (in both my N5 classes). It seems that the pupils either “got it” and raced through the selection of questions or they didn’t have a clue. I found this tricky to deal with as I couldn’t get to grips with understanding what is was they weren’t understanding. The issue was not reversing for a negative vector but seemed to be where to attach one vector to another. I tried labelling the ends of the vector with start and finish but it didn’t seem to help. I’m hoping that I will be able to figure out another way to explain it to help those who were really struggling.

Topic of the week

I love algebra and I love teaching algebra. I actually changed the order of our course slightly so I could teach algebra before teaching more graphs!!. My S2 class had learned about expanding a single bracket last week and I wanted to spend this week reinforcing the skill as well as practising simplification and use of negative numbers. Here is the task that the class worked on.

This task is from Don Steward and can be found here. Although initially I thought it would require factorising, we discussed as a class that the expression would be a multiple of a number if all terms were a multiple of the number. I particularly liked Q17 – 20 which provided a real challenge for the few that got to them.

**CPD of the weekend**

I spent the weekend visiting family in the central belt of Scotland. This meant a 3 and a half hour journey there and back again. I love being able to use this time to listen to the latest podcast from Craig Barton and this weekend I was in for a treat. There were two I hadn’t listened to so I had one for each part of my trip. The first one I listened to was with Naveen Rizvi. This was about scripted lessons and so much more. I had heard Naveen talk about atomizing at a maths conference earlier this year so I was fascinated to hear more. I love the passion she has for maths and for making resources. I haven’t yet had time to process my thoughts but I have a few ideas buzzing around.

The second one was with David Didau. This was incredibly interesting and thought provoking. I find it quite difficult to read research papers and pick out key information. I tend to get bogged down by the technical language so I really enjoyed hearing David and Craig talking about current issues but in a way I could actually understand. Again, I still need some processing time and certainly have a lot to consider.

If you haven’t been listening but would like to then you can find the podcast here.

]]>I really like this method as it shows very clearly why 12(x + 1) = 12x + 12 and leads nicely into using a grid for expanding brackets.

So why did I want to change to a different method?

Whilst exploring Open Middle I came across this problem.

I then thought about exploring expanding brackets using the distributive property of numbers. I tried to think about how I could get my pupils to play around with some numbers and consider how to expand brackets. The trouble was that I did it in a bit of a hurry and this is what I came up with:

To start with it worked really well. There was good discussion around the room and they came up with two things:

What I had failed to include was the link to brackets. We then discussed the link as a whole class and then looked at the relationship to algebra. What I wonder is how to create a better task so that the pupils investigate the relationship rather than me show them. I have had a few thoughts and will show them below.

- change the questions to include the brackets

- include the opportunity for pupils to create their own to see if they understand the rule
- make the link to algebra within the original set of problems

I would really appreciate some feedback so that I can create a better task for pupils the next time I am introducing expanding single brackets.

]]>My classes this week were working on:

- S2 – calculating bearings and simplifying expressions

- S3 – foreign currency

- N5 – trigonometric identities and percentages

- AH – no lessons this week since I only see them on Mon and Fri

**Worksheet of the week**

I used this worksheet from www.maths4everyone.com.

I really like the way it is structured, building in difficulty and not too many questions. Most of the class managed well with it, using previous knowledge of angles to calculate the bearing. I’m not sure I’ve ever spent much time on this previously but will definitely be doing so in the future as it provided great angles review and showed me who understood the concept of a bearing (use north line, measure clockwise, writing in 3 figures)

**Question of the week**

As a review of simplifying expressions, I set this task for my S2 class. I took the questions from www.startingpointsmaths.blogspot.com but added a couple extra.

Having the answers and trying to create the question gave a good structure and it meant nobody was making the error of adding x’s and y’s together. This task also allowed a review of negative numbers and I was pleased to see some of the class using the number line that is up in my classroom.

**Resource of the week**

Following our recap of simplifying the S2s worked on this task from www.mathspad.co.uk.

Often matching tasks require a matching pair to be found. I really appreciated the added dimension of finding three equivalent expressions. Despite having spent time on the recap task described above there will still a few in my class who wrote that x + x + x + 4 + 4 + 4 = 15x. I need to keep emphasising the need to collect like terms.

**Task of the week**

The S3 class were having a quick look at foreign currency. The basics of converting currencies is pretty straightforward so I wanted to find more challenging tasks. I set the class this question taken from a N5 Lifeskills paper.

Once they had completed it they were asked to create their own word problem involving at least three different currencies and provide a fully worked solution. As an extension to this I asked them to make a non-calculator question.

**Re-used task of the week**

In S3 we spend a lot of time on percentages, essentially covering all of the N5 content. This means that in N5 we don’t need to reteach it but do need to review it. So, rather than creating a new task, I re-used a sorting task from the S3 course. The task involved deciding whether a question was appreciation/depreciation or reverse percentage.

]]>

- S2: bearings – measuring, writing and drawing
- S3: volume of prisms
- N5: solving trigonometric equations
- Advanced Higher: Gaussian Elimination

**New plan of the week**

Last week I wrote about my plan for building up the retention of skills and knowledge with my S3 class. This week it began. The class seemed keen to buy into the process. I think the fact that they had all struggled with the assessment meant they knew something had to be changed. The topics I chose to start with were rounding, solving equations and circumference/area of circle. After three starters on Monday, Wednesday and Thursday, the pupils sat a mini assessment on Friday which we then peer marked. The majority of the class were successful in answering the basic questions and some managed the extension questions. This coming week there will two new topics introduced. These will be mixed up with the topics from last week with the aim of seeing if the class can retain the knowledge from last week. Hopefully as we continue with this we will all see an increase in retention of knowledge.

**Feedback of the week**

My N5 and AH classes returned from study leave and I knew that I needed to give feedback from their prelims. Over the years I have tried many different ways to do this but never really found a way that I thought was really beneficial. One thing I was definitely sure of was that I didn’t want to stand up at the front of the room and spend two periods going over all the questions. I know that the best way to learn maths is to do maths so I didn’t want to waste time. I had my classes complete a prelim analysis form which took about 20 minutes then we moved on. I am not overly convinced that having a question analysis really works as the pupil might only focus on those particular areas and not think about the whole course. However, I wanted them to know where their errors were coming from. Was it level C questions? Was it A/B questions? Often pupils think that the reason they fail N5 maths is that they “can’t do quadratics”. However, there are many areas which they lose marks on but don’t seem to worry about. From this form, I am encouraging my pupils to identify the basic questions where they lost marks. If they can improve in these areas then their marks will improve. I haven’t got a copy of the form to put on here just now but will update tomorrow.

**Worksheet of the week**

I love the worksheets produced by Corbett Maths. My S3 class were looking at volume of prisms this week and I used this sheet from Corbett Maths.

This sheet provided those who needed it straight forward “find the volume of” questions but also gave others the “apply” questions which provided a nice bit of problem solving.

**Success of the week**

This question is one that N5 pupils often struggle with. The usual first thought is “how can I do this without a calculator?” I think part of the problem in the past is that I have never given over time in class to it. It is one of these questions where I used to think pupils could just apply knowledge to solve without being explicitly taught. After discussing a few examples at the board, I set the class these questions from www.startingpointsmaths.blogspot.com created by Chris McGrane.

It was perfect. A small collection of questions to allow practise using trigonometric graphs to order values.

]]>Here’s what my classes (that I had this week) were doing:

- S2 – review of decimals and fractions
- S3 – review of trigonometry and circle and assessments

Worksheet of the week

I decided to have a break from our scheduled topics with my S2 class this week to spend a little time reviewing decimals and fractions. I have noticed how some of my class are still struggling to perform these calculations accurately.

I found these sheets on Don Steward’s Practice Makes Perfect website and loved how they provide basic practice questions but also provide extension tasks. This means that everyone in the class is working on tasks that are accessible yet challenging.

As I went around the room to provide help to pupils I noticed an increase confidence with most of the class. I was delighted when a pupil very loudly shouted “I did it” when working on the addition sheet Q13. There were still a couple of pupils who struggled with the concept of place value when adding decimals. I tried using place value cards to help but it didn’t seem to matter. It’s at these moments when I feel that I am not properly trained to teach pupils at these very early levels and need to spend more time liaising with primary teachers about different strategies.

Consideration of the week

My S3 class are the middle of three sets and my goal with this class is to prepare as many of the class as possible to go onto N5 maths in S4. On a day to day basis most of the class manage well with the topics we cover but their retention of skills is very poor. I have tried to build in retrieval practice in the form of daily homework and starters but it does not seem to be helping at the moment. The class had an assessment this week testing all of the content of S3 covered so far. I could see on their faces whilst they were taking the assessment that they were finding it really difficult. It was upsetting for me to see this. Part of me thought – I have failed them as a teacher, not preparing them well enough. The other part of me thought – they have failed themselves by not completing daily homework and not working as hard as they need to. As I reflected on what to do next I realised that assigning blame was not the right way to think and that my thoughts should be on how can we stop this happening again.

I then remembered reading a blog post on Jo Morgan’s Resourceaholic website about Five for Five. You can read the post here. I have decided to adapt this process to use with my S3s. Here are my plans: I see the class for 4 lessons per week so have structured this to fit a week.

Lesson 1: Set 3 questions on the board as a starter. The pupils will get time to complete then we will discuss as a class. The rest of the lesson will be continuing on with new topics.

Lesson 2: Set 3 questions on the board as a starter. These questions will be the same as lesson 1 but with different numbers. The rest of the lesson will be continuing on with new topics.

Lesson 3: Set 3 questions on the board as a starter. These questions will be the same as lesson 1 and 2 but slightly more challenging. The rest of the lesson will be continuing on with new topics.

Lesson 4: The class will have a low stakes assessment based on the starter questions from lessons 1 to 3. These will be marked in class then I will take in their answers so I can see which areas still need work.

The following week will follow the same structure but instead of having 3 new questions, there will be two new questions and the third will be one of the questions from the week before. This means that the class will be tested on the retention of the skills from the previous week.

As the weeks go on, I will continue to review previous skills along with the new content. I am going to focus on the key skills that these pupils will need for N5 maths.

]]>

Here’s what my classes have been doing:

- S2 finding the formula of the nth term of a sequence, linear patterns in a context
- S3 surface area of a triangular prism, solving equations
- S1 angles – calculating missing angles

**Surprise moment of the week**

Whilst working on a problem involving surface area, there was need to solve an equation. Many in the class struggled with this even though we had previously worked on solving equations. As it is such an important skill I decided to pause from our work on volume and surface area and spend a few periods working on equations. After some example problem pairs completed using mini whiteboards, we played the Shape game. This is a game I have played before with a smaller class (you can read about it here under the title Game of the Week). I was unsure how it would work with 24 pupils.

Here is an example of the questions I set:

In order to write their group number in a shape the pupils had to correctly answer all the questions on the card. If they were incorrect I looked at their working and gave a suggestion on how they might correct it. The surprise was how quiet and focussed the class were while working on their questions. They were supporting each other, checking solutions and genuinely working together. Why was this so surprising? Most of the time when I organise group work I spend more time on behaviour issues than helping with maths. It was wonderful to see a whole class fully engaged with the work.

**Task of the week**

This task was completed in pairs by my S2 class. The concept was very simple – organise the cards into groups of 4 (sequence, nth term formula, formula in words, 50th term). I decided to remove some of the answers so that the pupils had some blank cards left over to fill in themselves. A great task to consolidate their work on sequences. Here are some pictures of their work (there is a small error on one card but it was discussed with pupils)

**Find of the week**

While sorting out some boxes after moving house I found some old school work. It was interesting to look at some of the resources I had made years ago. Here is one that I found:

I remember making this for a class who had struggled with fractions of a quantity and did not have great number skills. I am now thinking about how I can adapt this to use with future classes.

**Worksheet of the week**

Covering an S1 class for my colleague this week gave me the opportunity to use the angle worksheets I created last year. I spent a lot of time structuring thequestions to allow for variation of questions. It reminded me that it is worth spending time on making a resource that you like as it can be used again.

]]>It was a particularly busy week with a meeting after school on Tuesday with primary colleagues to discuss moderation, a parents evening on Wednesday and study club on Thursday after school. I also started taking an IDL group of S1 pupils. I will be working with them for 2 periods a week for the next 6 weeks.

Here’s what my classes were doing this week:

- S1 IDL – talking about transferable skills and practicing presenting in front of the class by reciting Twinkle Twinkle Little Star.
- S2 – arithmetic sequences including finding the next term, generating a sequence from a formula and finding the formula for the nth term
- S3 – surface area and volume of cubes and cuboids
- N5 – sketching trigonometric graphs involving vertical translation and phase angle
- Advanced Higher – curve sketching using asymptotes and sketching the modulus of a function

My lessons this week were mostly direct instruction followed by independent practise but there are a few nice resources I used that I would like to share.

Worksheet of the week

This one comes from www.mathspad.co.uk and was given to my S2 class to work out missing terms in arithmetic sequences.

I found it very informative as the sequences ranged from whole numbers, negative numbers, decimals and fractions. A nice way to review these topics in the context of sequences. There were useful conversations about writing fractions with common denominators in order to complete the sequence.

Question of the week

“What is the least amount and greatest amount of surface area possible on a cuboid with a volume of 64 cubic centimetres? (Use whole numbers only)”

This question is from www.openmiddle.com. It provided a great opportunity for my S3 class to explore how changing the dimensions of a cuboid alters the surface area. The class worked away on this task – competitively comparing the least and greatest amount they had found. Two interesting things came from this task – some pupils really struggled to find 3 numbers that multiplied to make 64 and some pupils did not like the idea that a cube is a cuboid and ruled out 4x4x4 as an option.

Don of the week

Of course there were a few questions taken from Don Steward’s website again this week. Rather than give lots of repetitive questions on finding volume and surface area of cuboids, I put together a worksheet that contained a mixture of problem solving questions. Here are two that I particularly liked:

The top problem let me see who understood the concept of surface area rather than simply following a set method. Some needed visual aids to see which faces not to include in their calculations.

The bottom problem allowed us to explore how algebra can be used to solve problems. What interested me the most was how some pupils knew it was ok to only find the area of three faces where as others found all 6 as they had done in previous questions. Some in the class used trial and error which I appreciated as it showed perserverance.

A side note – I finally booked my ticket to see Don Steward as part of the ATM meetings in Glasgow in February and I am so excited.

Thought of the week

I came across this tweet from Siobhan McKenna @ShivMcKenna55

This really hit home to me. I like to think that I plan my lessons so that pupils have to think mathematically but I’m not sure this is always the case. An example from this week would be teaching finding the formula of the nth term with my S2 class. I demonstrated how to do it, they did their own example so I could check understanding then they worked through a set of problems. These problems did not contain variation – simply find the formula of the nth term. The class did really well with them and I remember thinking “yes they’ve got this”. However, was there an opportunity to think mathematically?

Maybe I’m being a bit hard on myself but it can be very easy to throw together lessons where pupils follow procedures and have no real mathematical thinking to do. This can come from lack of time but for me I think it comes from being a little scared to give some pupils work I know they will react badly to. In my mind, a calm and peaceful lesson with all pupils working is better than a chaotic one where I am not sure if anyone is really getting anything other than confused. This is something I am definitely going to give a lot more thought to over the coming term.

]]>

The week started with a trip to court to see if I was required for jury duty. I wasn’t, but I missed most of Monday at school. The rest of the week was relatively uneventful with most pupils managing to keep working despite Christmas being around the corner.

One thing I’ve noticed is that I feel more tired than I remember from previous years. It could be that I’m older or that I’ve just moved house but I have been struggling to stay awake this week. Hopefully the holidays will allow me to relax, unwind and get some energy.

I wanted to end the term saying a massive thank you. This job is tough and I know it would be so much tougher without everyone who helps and supports me.

]]>My classes this week were looking at:

- S2 ratio calculations and Block 5 assessment
- S3 expanding triple brackets and area of 2D shapes
- N5 similarity (length, area and volume) and Block 4 assessment
- AH summation notation and proof by induction

**Don of the week**

I’ve noticed over the years that many pupils struggle to recognize the different 2D shapes despite regular reference to them. This task from Don Steward allowed my pupils to familiarize themselves with the properties of 2D shapes whilst tackling the problem of designing shapes that have an area of 8cm².

**Task of the week**

One of my favourite tasks to set when working on area is to sketch 2D shapes which all have a specific area. This time the area was 36cm². The pupils were encouraged to use different dimensions which allowed for exploration of factors. I was pleased to see some of the class use decimal lengths as well.

**Mistake of the week**

Being a little preoccupied this week with my house move I made a bit of a blunder in my Advanced Higher class. I put up a set of statements that were to be proved using proof by induction. I put together questions from a set of notes I found online. The first question set was this:

Prove by induction that n² + 2n is divisible by 3 for all n ∈ N.

Unfortunately this question was an example of a false conjecture. My poor pupils were very confused about why they couldn’t get the proof to work. Thankfully I teach lovely pupils who were very forgiving of my mistake (I think they are used to all my mistakes by now!!) I need to take my own advice when I tell pupils to read the full question.

**Future thinking of the week**

This week my N5 classes sat an assessment that was predominantly about quadratic graphs and equations. While they were quite successful with the quadratic equations most were still struggling with quadratic graphs. The main problem is distinguishing between quadratics of the form y = (ax – b)(cx – d) and y = (x + a)² + b. They try to use the same method for finding the turning point for both forms. For example, some identified the turning point from this form y = (x + a)² + b then found half way between the coordinates. I had spent a lot of time working on quadratic graphs so don’t know exactly what went wrong. I need to take some time to think how I can help my pupils learn the different ways to deal with the two forms. Any ideas would be welcome.

]]>