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.

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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.

]]>This week my classes have been working on:

- S2 drawing and interpreting graphs, ratio
- S3 expanding double brackets including quadratic factors
- N5 angles, perpendicular bisectors, 3d Pythagoras
- AH Maclaurin series

**Method of the week**

Expanding brackets with my S3 class this week reminded me how much I love using a grid rather than FOIL. The structure allows pupils to see clearly what to do and they can adapt the grid to expand different sizes of brackets.

We started off expanding brackets like (x + 2)(x + 3), (x – 4)(x – 5), (x + 7)(x – 8). Very quickly many were able to expand brackets like (2x + 5)(3x – 7). There were the usual problems when it came to simplifying the x term. Pupils struggling with negative numbers, thinking -3x + -4x = 7x as two negatives make a positive. This did lead to some positive conversations about using a number line. On an encouraging note, there were some pupils who expanded brackets like (x^{2 }+ 2x – 6)(x^{2 }– 4x – 7) without direct explanation as they knew just to expand the grid.

**Answer of the week**

While tackling Maclaurin expansions I was pleased to see the different way a pupil solved this problem.

I assumed that they would use the chain rule to differentiate f(x) then continue. However, one pupil chose to find the Maclaurin series for y = sinx then multiply the series by itself to get y = sin^{2}x. It’s always nice to see pupils thinking differently from yourself.

**Numeracy of the week**

I had a cover class in Home Economics. It was for N5 Hospitality. The class were writing time plans for cooking their three recipes for their prelim. This is essentially a problem solving task trying to fit in different stages of the recipes in appropriate times so that the dishes were all served on time. I was amazed/concerned about how difficult many of them found it. I will definitely be thinking about how I can build in skills to help as part of our maths courses.

**Questions of the week**

I introduced the concept of ratio to my S2 class this week and wanted to focus on simplifying first. I knew they wouldn’t need an excessive number of practice questions so I set this from Increasingly Difficult Questions by @Taylorda01

What I love about the question sequence is the increase in numerical difficulty, the need to convert between different units and the link to algebra. I’m not sure I have ever written a ratio question involving algebra.

]]>My classes have been learning about:

- S2 – drawing and interpreting graphs
- S3 – appreciation and depreciation
- N5 – quadratic equations
- AH – sequences and series

**Don of the week**

Yip, another Don Steward mention. Before looking at how to draw pie charts I showed this question to my S2 class.

It was a really great way to assess whether they understood how the frequency related to the size of the segment in the pie chart.

**Frustration of the week**

My frustration is the inability of pupils to use their calculators correctly. Whilst working on the quadratic formula, despite me pointing out exactly how to use the calculator properly, many pupils did this:

When working out the highlighted calculation, by not putting a bracket around the -2, the calculator worked out the answer as 4 instead of 12.

This happened with several questions which led to incorrect answers as well as many MATH ERROR displays due to trying to find the square root of a negative number. Why do my pupils ignore me?

**Joke of the week**

Teaching the quadratic formula allowed me to share this joke with my N5 classes: “what do you feed baby parabolas? Quadratic formula”. It was received with a few smiles and lots of groans.

**Resource of the week**

My S3 lesson on Thursday ended up being a review lesson as half the class were away on a trip. I decided to recap non-calculator percentages with the help of this Code breaker by Andy Lutwyche (@andylutwyche).

It worked great. The pupils loved trying to figure out the punchline to the joke and then tried to create their own when finished.

**Starter of the week**

I used two task from Open Middle this week. One about multiplication and one about the quadratic formula.

I was pleasantly surprised by my pupils when tackling the multiplication that they began to spot patterns and connections, essentially using the commutative property of multiplication.

**Personal challenge of the week**

This week I had to face one of my fears – presenting a workshop to teachers. I was asked to share my thoughts on lesson planning with the secondary NQTs in my region. Even though it was a small group of 16, I was still nervous about standing up and talking to them. But the strangest thing happened, I actually got over my nerves and didn’t feel a complete fool. My workshop was based around evaluating how we plan lessons and sharing some thoughts on planning for learning. Hopefully the NQTs took something away from the workshop. I took inspiration from the blog by Mark Enser (@EnserMark) and from the book “Lean Lesson Planning” by Peps Mccrea (@PepsMccrea).

]]>Here’s what my classes have been learning about this week:

- S2 – finishing averages and range, drawing graphs
- S3 – percentages
- N5 – quadratic graphs
- AH – complex numbers

**Best moment of the week**

Whenever I introduce complex numbers I love to show the class “John and Betty’s Journey to Complex Numbers”. It is a delightful cartoon showing the development of numbers with guides John and Betty. I particularly like the giant cookies!! You can read the cartoon here.

**Resources of the week**

I introduced solving quadratic equations by discussing the zero product property. Then after a few example I set my N5 class this matching sheet.

The sheet allowed pupils to self check their work and meant my time could be spent supporting pupils rather than marking their answers. Of course, as always, I ended up having a mistake on the sheet – there was a + 12 where there was meant to be a – 12. The mistake has now been fixed and the sheet is available here.

Last week I blogged about the sheet I used for drawing quadratic graphs by completing the square. This week my N5 class were sketching quadratics by factorising. I made this sheet for them to use to provide some structure.

Some of the pupils chose not to use the sheets but those who did seemed to find the layout really helpful. Unfortunately I never got any pictures of pupil work to show. A copy of the sheet can be found here.

**Mistakes of the week**

I made two other mistakes this week (other than my worksheet) – I messed up completing a symmetrical picture on the marking scheme of an assessment and I forgot to go to an observation lesson I had arranged. Luckily I have a lovely department who’s only comment was “at least we know you are human too”.

]]>My classes this week have been working on:

- S2 – mean, mode, median and range
- S3 – circle properties and arcs and sectors
- N5 – graphs of quadratic functions
- AH – binomial theorem

**Mini Whiteboard Task of the week**

After an introduction to the vocabulary associated with graphs of quadratic functions I wanted my N5 classes to get familiar with the vocabulary by reading information from some graphs. I displayed these graphs one at a time and, in pairs, my pupils had to write down the turning point, y-intercept, roots and equation of the axis of symmetry.

Using the mini whiteboards gave me instant feedback about my pupils understanding of each term. It was finding the equation of the axis of symmetry which required the most support from me.

**Worksheet of the week**

This worksheet I put together myself (based on one I had seen online). Quadratic graphs are always a scary topic for some pupils and my aim was to provide some structure. After working through an example together at the board, I gave out the sheet for my pupils to complete.

Some chose to work across the way and complete one graph at a time whilst others chose to work out all the y-intercepts first then the turning points and so on. It took awhile for most of the pupils to get into the task – I think this was because there were six different parts to work out and they felt overwhelmed. I’m hoping that the difficulty built in will lead to better understanding. A copy of this sheet can be found here.

**New website of the week**

So I looked at the website Starting Points when it was first launched by @ChrisMcGrane84 but this week was the first time I used a resource from the site. My S3 class have been working on circle properties for a few weeks now and this week it was time to explore perpendicular bisectors. I found this sheet (written by @mpcopland):

This is great resource where Pythagoras in circles has been split into 3 different strategies. These types of questions in textbooks are usually in a context and have very cluttered diagrams. These clear diagrams allowed my S3 class to tackle these problems without having to navigate through all the clutter. I will definitely be checking out other resources on this site in the future.

**Oops of the week**

I had this question to set for my AH class to try as a starter.

Unfortunately, I made a typo. This is what I actually typed.

My pupils asked for help with part b and I said to try to substitute in a value for x. A few minutes later they still seemed puzzled. Then I actually looked at the question as it was written on the board. It became clear why they were struggling. What was slightly concerning is that none of the class said to me that the question made no sense at all. I think they were all too polite to say that 0.95 cannot possible be equal to 0.59049.

]]>So this week I had a 380 mile round trip to Glasgow with 4 pupils. After winning the Moray competition back in June, our team from KGS headed to Glasgow for the 2018 Enterprising Maths in Scotland final, where approximately 270 S3 & 4 pupils in 68 teams from all across Scotland were competing. The trip down involved a stop at McDonalds, lots of snacks, a night in a hotel, a continental breakfast and a race around a supermarket. The competition was held at the Glasgow Science Centre and had four different rounds including a practical round and a relay round. The whole event was amazing. From the superb organisation and the visit around the Science Centre, I was so impressed that 270 pupils had so much fun doing maths all day and their behaviour was exemplary. The trip home involved a Sat Nav disaster, a change of route and a dinner break in Dundee. We arrived back at 9pm feeling very tired indeed.

Well, I missed a day of school but have still been very busy with my classes. This week:

- S2 – percentage increase/decrease and averages
- S3 – circle properties including tangents and perpendicular bisectors
- N5 – simultaneous equations
- AH – binomial theorem

__Task of the week__

My N5 classes have been finishing up their work on simultaneous equations so I wanted to see how they coped with a variety of exam questions that all required slightly different ways of forming the equations. This is the sheet I put together:

The pupils worked in pairs through the set of problems, some choosing to use whiteboards and others writing solutions on paper. I made sure to emphasise communicating the final answer in terms of the context of the question.

As expected, the forming of equations proved challenging. The most interesting question was this one:

Most of the class glanced at the question, saw the graph and started to find a graphical solution. This gave me a good opportunity to remind them about reading the full question and about the meaning of “solving algebraically”.

__Question of the week__

I set this question for my S2 class.

It is taken from a worksheet I found on Craig Barton’s site (now available on TES resources). It was given prior to discussing averages. The pupils were mixed in their response. Some went with consistency and reliability where others wanted to gamble. This allowed us to talk about data and why we want to have information that will help us to make decisions.

**Worksheet of the week**

I was teaching my S2 class about finding the mean of a set of data. After discussing the definition of the mean and completing a few example problem pairs I set the class this worksheet, which I took from Craig Barton’s website Variation Theory.

I did not use the sheet in quite the same way as suggested by Craig. I have had several comments made to me from science teachers that pupils have no real concept of what the average is and that they don’t know what answer would be realistic. This occurs when an error is made in calculation giving an answer that couldn’t possibly be correct. To try to address this I spent time with my S2 class discussing how to estimate the mean from the data before calculating it.

This is what I used the “expect” column for. Hopefully taking the time to consider the answer before working it out will allow my pupils to be more aware about reasonable expectations and when an answer is clearly wrong.

__Don of the Week__

There isn’t a week that goes by without me using a task from Don Steward’s wonderful website. This week was no exception. As a follow on task from the worksheet mentioned above, I set this for the pupils to work on.

A lovely little task which was accessible to all. Some tried to estimate the answer before calculation to narrow down the choices whilst others just used the process of elimination. Nice for them to see the impact a number has on the mean.

__Treat of the week__

The art teacher at my school offered to teach staff how to create a stained glass picture. Last week I designed mine and this week I started to cut the glass for it. I have opted for a geometric design and I have really enjoyed the time learning a new skill and hopefully will have a beautiful creation to put up in my new house.

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My classes have been working on:

- S2 Percentages
- S3 Angles and Circles
- National 5 Simultaneous Equations
- Advanced Higher Second Order Non-Homogeneous Differential Equations

**Task of the week**

As is often the case, the tasks I find are most successful with my pupils are ones that I find on Don Steward’s website. This week was no exception. Here is the task that I set my S2 class to work on percentages without a calculator.

This set of questions allowed my pupils to see the connections between different percentages and how you can use one percentage calculation to find another. After we had explored the task above I set my pupils the challenge of creating their own percentage spiders when only given the number in the centre. I chose to do this for two reasons:

- my class is mixed ability so this allowed all pupils to work on the same task but working out different percentages
- without anymore information given, other than the central number, I wanted to see how far my pupils would push themselves

Here’s a few of pictures of their work:

**Starter of the week**

My S3 class were going to be solving problems involving right angled triangles in semi-circles, some of which would involve the Theorem of Pythagoras. I found this starter on Transum

It was such a great little task. We recapped the definitions of square numbers and consecutive numbers and it gave the class the chance to notice and conjecture. Whilst no-one proved the result algebraically, we did discuss it as a whole class the difference between something being true for a few cases and proving it for all numbers.

**Worksheet of the week**

As mentioned above my S3 class have been working with angles in circles. Part of this involves isosceles triangles formed from two radii. I looked in a few of the textbooks we have but couldn’t find a set of questions that I wanted and I didn’t fancy making my own as circle diagrams are so time consuming to draw. Then I found this one from Maths4Everyone and @Maths4Everyone on Twitter.

It was exactly what I was looking for – a few straightforward questions followed by some that involved several different angle calculations. The questions were set out clearly and uncluttered and not too many, which I have learned can be off-putting to some in my class. Once the pupils had completed the sheet they were asked to create some questions of their own that involved more than one angle calculation. This proved tricky for some but perseverance led to some brilliant questions being written.

**Wobble of the week**

This week I experienced a bit of a wobble. I was in two minds whether to write about but I want to be completely open about my teaching life. My wobble happened on Tuesday morning. I woke up and the thought of going to work actually made me cry. This is something that hasn’t happened to me since my probationer year back in 2005. I took the day off and spent the time sleeping and then reflecting on what caused this. I came to the conclusion that there were four main reasons:

- I’m in the middle of selling a property and buying another which has added a lot of extra stress.
- At the moment I’m not feeling like the best teacher.
- At the moment I’m not feeling like the best principal teacher.
- My workload is huge and felt unmanageable.

Why do I feel like such a poor teacher at the moment? As much as Twitter inspires me with ideas, resources, strategies and connections, it also can make me feel pretty rubbish. There are so many teachers talking about maths teaching in ways that I could never think of, designing tasks in ways that I could never think of and planning curriculum in ways that I could never think of. This is not a comment about other teachers but a reflection about my self-esteem. I see the great work of others and beat myself up that I’m not able to come up with such inspiring ideas.

Why do I feel like such a poor PT at the moment? Just like my teaching I worry about being the best PT that I can be and get anxious when I don’t feel successful. We had our attainment meeting on Monday, and despite already knowing what we need to work on as a department, hearing it said again piled on the pressure. I already have a huge list of development tasks and teaching and learning strategies I want to implement but time is not my friend. The days tick over filled with teaching, marking, administration, meetings, etc and I can’t seem to find time to devote to the development tasks and teaching and learning strategies. I should stress here that none of the pressure is coming from management, it is all me putting pressure on myself.

All of this, along with having a cold and feeling really tired, led to my little wobble. When I returned to work on Wednesday I was still feeling a bit fragile so had a nice long chat with my line manager and started to feel better and more relaxed. I think just saying it all out loud released a lot of my worries and allowed me to see things more clearly. I know I am not a poor teacher or a poor PT but sometimes the doubts creep in and take over. My note to myself is that I need to find ways to not put so much pressure on myself to be perfect.

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So here’s what my classes have been up to:

- S2 – converting fractions to percentages and decimals and calculating percentages without a calculator
- S3 – area and circumference of a circle and circle properties
- N5 – solving equations and inequations
- AH – solving second order differential equations

As the week has been very busy I did not have much time to make many of my own resources but have a few different thoughts to share this week.

**Question of the week**

My N5 classes were looking at how to solve equations involving fractions and I found this question from the website Open Middle.

As the pupils worked on this problem, I walked around the class listening in to their conversations. Some were determined that they would find the greatest value and got spurred on when someone found a value higher than theirs. What I was pleased about was that most of the class pretty quickly figured out that the numerator of the fraction should be 1 so that no division is required. After the class agreed that this was the best solution:

I asked them to tell me what the smallest value could be. Almost instantly, one pupil said “-56, just switch the nine and the two”. I love these kind of questions that promote thinking as well as practice of solving equations.

**Game of the week**

I alternate from being someone who loves having classes take part in maths games to someone who thinks games detract from actual learning. This week I love maths games. My N5 classes were solving equations with fractions so I made this question sheet for them to work on in small groups.

The game part – this is an idea I’ve seen on Twitter in various forms and originally from Kim Hughey’s blog about Ghosts in a Graveyard. I adapted it slightly and simple put four shapes up on the board.

Each shape was allocated a number of points (10, 5, 0, -5) but this was kept secret from the class. Each group was given a number and when they correctly solved an equation they wrote their group number in a shape. Towards the end of the period I stopped the class and revealed the value of each shape. Groups then worked out their total points. The competition element of the game motivated the class and they solved many equations successfully.

**Worksheet of the week**

After spending a few periods reviewing circumference and area of a circle with my S3 class we were moving onto properties of circles involving angles. I decided to start this with some measuring tasks and exploring circles. This was the sheet I gave to the class.

I did not write this sheet myself, it was one I found in our department files. As the class worked through this several issues arose:

- not being able to read instructions carefully
- incorrect use of a protractor

This led to some problems when answering “What do you notice?”. However, I still found it a valuable task as we discussed how to use a protractor and how to make and check conjectures. Next week we will spend time formally using these properties to solve angle problems.

**Discussion of the week**

For our department meeting this week I had set three articles about memory to be read. Each member of the department read one article then fed back to the department about the article and what practical applications there could be for our department. The three articles were:

- Making Things Hard on Yourself, But in a Good Way: Creating Desirable Difficulties to Enhance Learning
**Author(s):**Elizabeth L. Bjork and Robert Bjork - Both Multiple-Choice and Short-Answer Quizzes Enhance Later Exam Performance in Middle and High School Classes
**Author(s):**Kathleen B. McDermott, Pooja K. Agarwal, Laura D’Antonio, Henry L. Roediger, III, and Mark A. McDaniel - What will improve a student’s memory?
**Author(s):**Daniel Willingham

All of these articles I found on Craig Barton’s Podcast notes.

The main ideas coming from these articles were:

- learning is improved by introducing “desirable difficulties”
- spacing and interleaving are necessary
- low stakes quizzes are invaluable
- the format (Multiple Choice or Short Answer) doesn’t have a big impact
- pupils gain valuable feedback simply from taking the quizzes
- the difference between learning and performance
- pupils remember what they think about
- effective revision should be challenging

The most exciting part of the meeting for me was the way in which the teachers had engaged in the process of reading and discussion. We spent most of the meeting talking about memory and teaching and practical strategies we could implement. This time was so useful and inspiring. I will definitely be having more reading and discussion time built into department meetings.

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