Solving Equations with Unknowns on Both Sides

As ever, a set of minimally different questions with unknowns on both sides! These were tricky to write – I’ve absolutely not covered all possibilities here but they should provide a useful starting point to introduce this concept with. Feedback would always be gratefully received – I’ve got a big long list of topics to create questions for over Easter, but if you have anything specific in mind, do let me know and I’ll see what I can manage!

Solving Equations with Unknowns on Both Sides – Minimally Different

Minimally Different – Solving Equations with Brackets

Latest minimally different questions are here – this time on solving equations with brackets! These become quite tricky at the end – students will need to be confident calculating with negative numbers/fractions etc., but I think the links between the questions are nice and enable pupils to make connections more easily than where the questions are sequenced randomly.

As ever, thoughts and feedback would be received gratefully!

Solving Equations with Brackets – Minimally Difference Practice

Minimally Different – Dividing Negative Numbers

Following on from the set of questions on multiplying negative numbers, I’ve attached a couple of exercises on dividing negative numbers. I have a very specific class in mind when these were created with a huge range of abilities, hence the different range of progression between the two. As ever – I hope you find them useful, and feedback would be great!


Minimally Different Practice – Dividing with Negatives

More minimally different practice – dividing with negatives

Minimally Different – Solving Equations (part 1!)

A set of equations to solve – I have intentionally kept the unknown as ‘x’ in order to draw my students attention to how each equation changes. I’ve deliberately included fractional/negative solutions fairly early on, so that students are aware that solutions aren’t always positive integers, so it may be worth assessing your students’ proficiency with those before beginning this. I hope this is helpful – as ever, feedback would be hugely welcome. Answers included.

Solving Equations – Minimally Different Practice

Ratio – One Part Given and Difference Given

As before, a selection of ‘minimally different’ questions on sharing in a ratio (including answers!).

The first sheet is on ratio questions where 1 part is given – the second is on ratio questions where the different is question.

N.B. Success with these questions does NOT make a student confident/competent at all ‘sharing in a ratio type’ questions! It is fundamental that they are then given the opportunity to practise selecting the question method based on the question type (some problems from would probably be ideal at this stage). This is only for introducing a concept to a class.

As ever, I hope you find these useful – feedback would be wonderful.

Minimally Different Practice –  Ratio – 1 part given

Minimally Different Practice – Ratio – difference given

Minimally Different Practice – Multiplying Negative Numbers

Inspired very much by reading Mr Barton’s How I Wish I’d Taught Maths and number of different conversations at the recent #mathsconf14, I’ve decided to start putting together a number of ‘minimally different’ exercises by applying the principles of variation theory. The rationale is that by ensuring that each question is related to the previous, pupils will be able to make connections which seem obvious to us (as experts), but which frequently elude the students I teach. It is worth pointing out that such exercises will be used when introducing a class to a new concept. This is very much going to be a work in progress – and I am by no means claiming that these exercises will be perfect, or even the only way of varying a question – but it seemed worth sharing in case they are useful to anyone else!

Feedback will always be very welcome.

Multiplying Negative Numbers – minimally different

Terrible Advice for Teachers #1

Recently, when talking to my NQT mentor about the behaviour in a challenging Year 10 class I teach, I expressed the following thought:

‘I completely understand the need to use praise and positivity, and develop successful relationships with the pupils I teach, but I’m finding it hard to be positive about anything with certain students. One of them always enters the room by going straight to the back to chat, and never to his seat. As such, I’m always having to remind him of the expectations and things are immediately negative.’

The response?
‘Maybe you could turn that into a positive.’

Okay. How?

‘Well, if you start by saying something like ‘Marcus, I much prefer it when you sit in your seat’ with a smile, he’s much less likely to react negatively.’
This advice seemed somewhat dubious, but in light of the fact that it was being given to me by someone with far more experience, I thought I’d give it a go. I will now conclude that this is one of the worst pieces of teaching advice I have ever been given.

First of all, the underlying problem remains, which is that I am wasting time trying to get a difficult pupil to cooperate, while simultaneously trying to make a start with the lesson for the students who actually want to be there. They receive a minimal amount of praise for sitting in their seats correctly – and so the behaviour that they see being rewarded with attention is negative behaviour, which is entirely unfair.

More importantly, though, is this: students sitting in their correct seat is the most minimal of expectations that a teacher can have. Regardless of uniform, equipment, rudeness, chattyness and punctuality, if students will not seat where they have been directed to, a successful lesson is entirely impossible. As such, students should not need praise or coercion to get them to sit where they should be – it should be automatic and unquestioned.

Any advice along the lines of ‘well, maybe you could let them choose their seats on a Friday if they work hard during the rest of the week’ is not simply wrong, it is dangerous for the effect that it has on school culture. A student who will not follow that instruction, immediately, without the need for reminders and rewards, should be sanctioned – and a school wide policy should be in place to deal with this.

Ultimately, ‘Marcus, I much prefer it when you sit in your seat’ undermines the authority of the teacher, as it undermines the expectation that students will follow instructions immediately. It is a responsive, placating strategy, which is representative of just how low standards of behaviour in British schools are becoming.

P.s. In the lesson where I put this advice into practice, Marcus later called me ‘fucking pathetic’. Who’d have thought that behaviour management would require more than trying to build up positive relationships with students?

The value of silent classrooms.

During my PGCE year, I used to teach far more ‘exciting’ lessons. I would have relay races and treasure hunts and group work and discussion and board games and team challenges and so, as you would expect, my classes were typically quite loud. As it happens, this was never an aspect of my teaching practice that was particularly criticised or commented upon. In fact, in one self-evaluation after a lesson observation, I noted that the class had been louder than I would have hoped for, to which the response was: ‘yes, but they were talking about Maths, which is excellent!’

Although my lessons, for the most part, are far less ‘exciting’ than when I first began, I’ve still rarely insisted upon silence in my class. I had frequently heard comments such as ‘oh, I don’t mind a bit of chatter as long as they’re getting on with the work’, and so this became part of my practice. Insisting on absolute silence during work happened rarely in my classes; I would witness that they were perfectly capable of being silent during exams and assemblies, but this did not transfer to my lessons.

Undoubtedly, one reason for this is that I was apprehensive of implementing such an unambiguous rule. After all, the only way in which rules have impact is if rewards and sanctions are in place for meeting them. When you instruct students to work quietly, you have some leeway and chances to offer reminders about what you expect. With silence, there is nowhere to hide. In fact, one of my tutors suggested it was always best to ask a new class for ‘quiet’ rather than ‘silence,’ so that you can build a positive relationship rather than immediately having to reprimand students for speaking.

During the past few months, though, I’ve increasingly insisted upon silence in my classrooms. Sometimes this is only for a few minutes; sometimes this is for the full lesson. I cannot stress the impact that this is having on students’ work output, which for many, has improved dramatically. I had lulled myself into a false sense of security regarding the noise in my classrooms, and I had told myself that silence was a feature of classrooms of the past – I want my students to enjoy their learning and be engaged, and noise was a necessary part of that.

This weekend I attended a Teach First conference. During one session I was in, numerous other attendees were continuing whispered conversations during the delivery by the speaker. This was overwhelmingly frustrating, as I was finding it impossible to focus on what was being discussed. When we were given the opportunity to complete some paired work, I really struggled, due to the increase noise level in the room. I could not work with my partner effectively, as I was unable to think about the directed task.

All the time I have been prepared to facilitate students talking while completing independent work, I have been doing the deepest disservice to a significant proportion of the students that I teach. My tolerant approach may work for a few students, but the majority require a calm, quiet environment which I have not consistently been successful in delivering. I am not suggesting there is no room for group work and discussion, or even peer support. However, I am going to be more decisive about when silence is required. Too frequently, silence is seen as a nice optional extra to a productive learning environment, rather than a fundamental component.


Why I Teach

I’ve recently been reflecting on my initial reasons to begin teacher training and my reasons for continuing to teach now.

I do not teach because of the love of my subject.

I do not teach because ‘every day is different’.

I do not teach because I love young people.

And I certainly do not teach because of the reasonable workload, the excellent pay and the long holidays, or because it was the only career available to me.

Now, the first three reasons might surprise anyone reading this. After all, surely these are prerequisites for being at least somewhat successful as a teacher. Allow me to reassure you that I do love my subject, I do love the variety of the job and I do love working with young people. I could write for pages about these things, but that is not my focus right now. Instead, I want to talk about the underlying reason as to why I am a teacher.

I teach because I believe that education is the best tool available for improving the life chances of vulnerable young people. I am so angry that 33% of pupils on Free School Meals achieve 5 A*-Cs at GCSE compared to 60.5% of all other pupils, and I am angry that pupils who attend fee-paying schools are quite significantly more likely to attend and graduate from the best universities. There are hundreds of statistics I could give to demonstrate this point; ultimately, I believe that one of the reasons that education in Britain today is so unsuccessful is because it ‘keeps poor children poor’ (Katharine Birbalsingh).

This is why I was so disappointed to read this in an article from The Times by Caitlin Moran.

‘My plan is very straightforward, and rests on two facts: (1) the 21st-century job market requires basically nothing of what is taught in 21st-century schools, and (2) everyone has a smartphone.

First, as anyone with a teenage/young adult child will know, the notion of them going into a full-time, long-term job with a pension at the end of it looks like something we left behind in the 20th century. The old pathway – learn a skill, use it for 40 years, then retire – is over. The jobs of the future require flexibility and self-motivation. Indeed, the jobs of the future increasingly require you to invent your own job. The majority of jobs our children will have – in just a few years’ time – have almost certainly not been invented yet.

If you work better sleeping until noon, then working until 2am – as most teenagers do – congratulations! You no longer have to deny your own biology! And if, working at 2am, you have no teacher to help you, discovering how to research on your own is, frankly, going to be far more useful than the thing you’re supposed to be learning. Because, (2) my education policy would be to stop bothering kids with anything you can access on a smartphone.’

I’m not qualified enough to talk about her claim that the majority of young people will go on to do jobs that have not been invented yet, but I want to challenge the idea that teaching knowledge is pointless, because everything is ‘Googleable’.  We need to challenge this sort of thinking as it is so harmful to the poorest, most vulnerable students.

In my first year of teaching, I had been challenged to ensure my pupils were developing ‘independent learning’ skills. Some of the feedback I had received informed me that my lessons involved too much ‘teacher talk’, and that I needed to let the students discover Mathematics for themselves. The school I worked in was making full use of a ‘Bring Your Own Technology’ policy, and so I also needed to incorporate this. This sounded somewhat plausible, so I planned a series of lessons for my Year 9 class making use of this feedback.

The series of lessons were on Scatter Graphs – by the end, I wanted pupils to be able to:

  • identify the purpose of scatter graphs and when they are appropriate to be used
  • plot points on scatter graphs
  • identify negative, positive and no correlation
  • draw lines of best fit
  • estimate values using a line of best fit
  • discuss the dangers of using a line of best fit to estimate values outside of the data range

I gave them task cards, which explained the success criteria for this topic. They had full access to the Internet. Each group had 3 ‘Teacher Help cards’ which they were allowed to redeem if they decided they required my assistance. Each group had to present at the end of the week (using either a PowerPoint or a poster or whatever they wanted) to explain their findings on Scatter Graphs. I really cannot stress enough how unsuccessful these lessons were – by the end of it, although most pupils could identify positive and negative correlation, their learning had not extended beyond this.

Why, then, was it so unsuccessful? After all, pupils had access to the Internet which contains a wealth of information about Scatter Graphs. They were in groups, which should have allowed for excellent collaboration and developing their communication skills. Instead, pupils became overwhelmed by the task in front of them and simply could not make sense of it.

Their basic method was to type in Scatter Graphs to Google, and then spend all lesson copying and pasting from Wikipedia onto a PowerPoint, which they then mumbled to the class during their presentation time.

A scatter plot can be used either when one continuous variable that is under the control of the experimenter and the other depends on it or when both continuous variables are independent. If a parameter exists that is systematically incremented and/or decremented by the other, it is called the control parameter or independent variable and is customarily plotted along the horizontal axis. The measured or dependent variable is customarily plotted along the vertical axis. If no dependent variable exists, either type of variable can be plotted on either axis and a scatter plot will illustrate only the degree of correlation (not causation) between two variables.

Now, I can make sense of this chunk of text, but only because I have a wealth of knowledge already. I already understand clearly the definition of a ‘continuous variable’ and what ‘customarily’ means and the difference between correlation and causation. They didn’t. As such, I spent much of the lessons rushing around trying to explain to pupils what was relevant and what wasn’t, and it was chaotic and unproductive. At the end, I retaught this unit in a more ‘traditional’ fashion – I stood at the front, and explained what pupils would be learning. Pupils then practised these skills in a variety of ways. It was undeniably more successful in terms of their learning.

If you are a student from an affluent background, with parents who attended university and have high aspirations for your future, then this approach to learning may be somewhat successful. If such a student is given a task, they are likely to have a higher level of vocabulary which makes it easier to break down more complex pieces of writing. They are more likely to want to succeed, and so will be prepared to think harder and persevere for long in the hope of being successful. Such students are more likely to have been praised by class teachers for previous excellent work, and so this provides an additional level of motivation.

I do not think that the series of lessons on Scatter Graphs would ever be the most successful method of pupils learning about Scatter Graphs – a well-crafted quality exposition by a teacher who has carefully planned the sequence of learning would lead to greater retention and pupil learning.

The key point is that it is the poorest students who have the most to lose from this ‘just Google it’ approach to education, and we should be challenging this approach whenever we see it creeping in. This is because these students are most unlikely to attain highly at GCSE currently, and this has an overwhelming impact on their life chances. Our focus needs to be on improving attainment and outcomes, and we need to be drawing on the best educational research about ‘what works’ in learning to enable all students to excel. We do not need to be focusing on vague criteria about ’21st century skills’ and suggesting that Google makes teaching a redundant profession.

I now teach because I believe that education is the best tool for improving the life chances of vulnerable young people – I no longer facilitate ‘independent’ learning.