These are just some simple questions that I ended up writing because I couldn’t find any that contained a suitable mixture. I wanted worded problems that combined % increase, % decrease and % of an amount, but didn’t delve into reverse percentage. Enjoy, use, adapt…

# Simultaneous Equations, refining the procedure

Going over Simultaneous Equations today with Year 11, we all agreed that the thing that is most confusing about solving equations like these are the negative numbers:

4 - 2x = 9 + 6y 6 - 2x = 7 - 2y

We also agreed that we much prefer these types of questions when we heave a sigh of relief realizing that we can add instead of subtract.

4 + 2x = 9 + 6y 6 - 2x = 7 - 2y

Adding is easier. We have one less choice to make and we don’t need to keep track of which is the minuend and which is the subtrahend.

This is subtle. I would always *start* teaching simultaneous equations using some real-life examples (the ideas in this post).

I think real-life examples work in this case, because it is apparent what “extra” you are getting for the extra money. These problems create the need for the algebra – it becomes a way to represent what is going on by writing less. With these problems, everything is positive so it makes sense to think of the *difference* between the two situations or the two equations.

From there we can introduce the need to “multiply up” to get the same co-efficient for one of the terms. In my experience, most students don’t struggle too much with this concept.

What gets tricky is subtracting one equation from the other when negative signs are involved. We might identify the problem as being one of a lack of mastery of a fundamental concept – in this case negative numbers. So before teaching these types of simultaneous equations we ** could** do lots of practice and drills on negative numbers. But I’m not sure that is always a helpful approach. Some things are just more confusing (present a higher degree of cognitive load if you like). There is a lot going on, and lots of it needs to be done mentally. So, maybe there is a case here for explicitly teaching an easier technique which is less prone to error.

A simple way to avoid subtraction is by always ending up with equations with opposite coefficients of one of the variables.

So, for example when solving the following:

4x - 3y + 1 = 0 (A) 3x - 7y + 15 = 0 (B)

multiply (A) by 3 and (B) by **-4**. A valid shortcut / rule to remember here is that multiplying by a negative simply “flips the sign” of any term it is being applied to. So we end up with:

12x - 9y + 1 = 0 -12x + 28y - 60 = 0

It’s easy to check that the sign of each term has been flipped before then proceeding to add the equations.

As any Maths teacher would, I believe strongly in teaching for deep understanding of concepts, not blind memorisation of rules (Nix the Tricks is a frequent reference point). But there is also a place for remembering certain procedures to reduce cognitive load (times tables being the most obvious example). Dani Quinn has written a great post on this in relation to “moving the decimal point” here. Manipulating Simultaneous Equations like these is another example of when there is a case for a bit of explicit teaching of a method.

# Praise where it’s due

For regular readers of this blog, I’m afraid this post has nothing to do with maths. Writing helps clarify thoughts and that is my main reason for doing it. If others read it and find these clarified thoughts have some resonance, that makes me happy. If you feel moved to comment on these thoughts and add your own experiences, then that is really powerful as it hopefully moves us all forward.

This post is about school-wide systems for praise and sanctions. I am not SLT, and this year I don’t have a Pastoral leadership role. The reason I am involved in this is because I am a member of a Teaching and Learning Focus group. For the last couple of years, my school has run these groups in the directed time after school that traditionally might have been used for one-size-fits-all twilight INSETs. At the beginning of the year, teachers self-select which group to join, each group being led by a member of SLT. There are about 6 meetings throughout the year with the idea being that a particular issue is discussed, researched and some form of action taken. It’s a form of Action Research. From my perspective this has worked well this year. I see this as the leadership of the school saying, “OK, we do lots of good things here, but it’s not perfect. What can we improve? And how? Work as a team, do some research, consult with other staff and come back with a proposal.”

As a team, we are now at the proposal stage. I think it still needs some work and more consultation with staff. We haven’t done the “hearts and minds” bit yet. If we don’t do a good job of convincing the entire staff body that this is a good idea, the whole thing will have been a waste of time. My school is not the sort of organisation that issues diktats from senior management, I’m glad to say. Organisations that do rarely achieve the change they require. People may pay lip service the new initiative for a while, but if you cannot convince professional adults (i.e. teachers!) that this is something worth doing, it simply won’t happen.

So first, let’s look at our current “Praise and Concern” system. I’m going to start with “Concern” because that is the bit we are not proposing to change, but it’s important to understand the context. As all schools do, we have a behaviour policy. I won’t go through the detail of it here, but it basically involves warnings, detentions and removal from the classroom to a “referral room”. Each of these “concerns” is to be recorded on SIMS by the teacher who issues the sanction. There is a workload implication here, but it is generally viewed as something worth doing. Recording the data centrally is important because it enables other staff (i.e. form tutor, head of year, etc.) to get an overview of issues pertaining to an individual student occurring across different subjects. It’s particularly useful for highlighting frequent low-level transgressions which might not result in detentions but which could be impacting learning.

At the moment, the same system is used for Praise. These are also recorded on SIMS. There are a bunch of categories and, as with Concerns, the teacher is expected to write a sentence or two of comments. Every few weeks, a Praise and Concern report is sent to tutors. This report lists, by student, each Praise and each Concern. How form tutors use this information varies widely, with some displaying it all in front of the form, some displaying just the praise bit, some having individual conversations and some not sharing it directly with students but just taking note of it for themselves. The raw numbers are also shared in end of term reports that go home to parents.

We feel that the main problem with this system is the idea that Praise and Concern are seen as two sides of the same coin. They are not. Often students look at their net score (i.e. Praises minus Concerns) and I know of students who have asked for Praises to offset a certain number of Concerns. An evident issue is that it is often the students with the poorest behaviour who end up with the most praises “Well done, you’ve got your pen out, have a praise”. That sort of thing. And the quiet ones who do what is expected of them day in, day out get ignored. That is borne out in the data and from focus groups with students – they know what’s going on.

It is also apparent that there is a lack of consistency about how often and how many teachers issue Praise. Some ignore it completely and use their own systems. Others use it frequently including issuing whole-class praise, which is nearly as bad as whole-class detentions in my view.

In parallel with Praise and Concern, we also have a system of housepoints. This culminates at the end of the year with the inter-house Sports Day (still my favourite day of the year!). Lots of points are awarded on Sports Day but these add to points collected throughout the year by individual students. Housepoints are more personal to students. They write them in their planners when they are awarded and each term the form tutor “collects” them and enters them into a central system. Our hypothesis is that because students write the reason for the Housepoint in their planner and then keep a record of them, these are more meaningful and motivating. From focus groups with students we believe that this is true for lower years (Year 7-8) but house points are less valued by older students (Year 10 up).

In looking for something better, we have reviewed educational research on the role of praise in teaching. We were looking at the role of extrinsic rewards impacting on intrinsic motivation (here and here), and on how to make praise “purposeful” (here). We defined purposeful praise as praise which would motivate a love of learning and challenge our students.

This is closely linked to ideas of growth mindset. How, when and what praise is given for *can* impact on a students’ mindset but it’s a highly complex picture and difficult to draw general conclusions to apply in the classroom.

However, among our group there was a fairly close consensus on the following:

- We should praise the process and the effort observed in the moment, not the individual.
- Extrinsic rewards (e.g. stickers, housepoints, postcards home, etc.) have a place but need to be valued by students and need to be issued for something specific, for going “above and beyond”
- Narrative, personal feedback given to students is more likely to motivate and challenge them than extrinsic reward
- We need to be more fair and notice and acknowledge those who are quietly engaged in the struggle of learning.
- The current housepoint system should be relaunched but there should be no attempt to track students’ individual scores. You are collecting them for your house!
- Centrally collecting data on praise issued is not valuable (although collecting it on Concerns is).

I would be really interested to hear from any others on their perspectives. How do the praise and sanctions systems sit together in your school? What do you do that works particularly effectively? Either comment below, or get in touch by e-mail (mark.horley@gmail.com)

# Mathematical debates, and Bounds

It’s possible that what I am about to explain is already completely obvious to many a maths teacher. But after 5 years of teaching, it’s only just dawned on me (a recurrent theme of this blog by the way!) so I thought I’d share.

I like teaching the concept of Bounds. It highlights a fundamental difference between concepts that exist in Maths and the real world. We measure things using numbers as continuous variables. When we measure the length of a table, say, as being 120cm, we are rounding to the nearest centimetre. We could be more accurate and say it is 120.4cm or even 120.4187234cm but this is still an approximation. It reminds me of my engineering days, looking at milling machines that were capable of cutting to the nearest 0.1mm. I used to wonder about how those milling machines were made and the tolerances required in the dimensions of their components. What machines were used to make the components? And what about the machines that were used to make those machines? Where does it stop?!

But one of the stumbling blocks with bounds is always with the Upper bound.

“But you told us that 5 rounds up!” is the usual complaint. And then we get into a heated arguement about whether:

Which of course it is, for the same reason:

Earlier this week, I was marking this GCSE question, which most of my students, including those sitting Foundation were answering no problem.

I’m pretty sure that I have previously suggested that students state the “Lower Bound” and the “Upper Bound” almost as though they are two separate answers. But expressing the **range** of possible values in this way makes more sense. It is also entirely consistent with the well-understood rule that “5 rounds up”. In the question above, if the actual length was 13.5, it **would** be rounded up to 14, but a value below 13.5 would not. Whereas a measured value of 12.5 would be recorded as 13.

All makes perfect sense now!

# Comparing Fractions

There is something very simple about a task which presents two numbers and simply asks “which is bigger?”. This should be done using mathematical notation, i.e. using the < > symbols. I have seen these being introduced successfully in Year 1 without any mention of crocodiles, or such similar unhelpful “stories”. But my Year 7 class still insist on calling them crocodiles and drawing teeth on them. But hey, I have bigger battles to fight…

As well as comparing 2 fractions we can put multiple fractions into order from smallest to largest. There is a significant range of difficulty in this apparently simple task.

- Comparing fractions of the same denominator
- Unitary fractions with different denominators
- Same numerator, different denominators
- Different denominators where one is a multiple of another.
- Different denominators where a common denominator needs to be found for both fractions.

Alongside all of these there may also be strategies where learners are using known facts or doing calculations to convert to decimals or percentages, e.g. 1/2=0.5, 2/5=0.4, therefore 1/2 > 2/5. That is not the intention of *this* task (it is of a different task here) but in the end we want learners to be able to play with all these ideas and I can’t really control, nor would I want to control, the order in which they coalesce in students’ mind.

Here is a simple set of cards that I used recently. I got the students to do the last bit of cutting to turn each strip into the 3 separate cards. I also told them that there is deliberately some space alongside the fraction to enable them to write equivalent fractions if they needed to.

I gave them out a strip at a time, the idea being that they were to “slot in” the subsequent fractions to maintain the order. The fractions are carefully chosen, so that each time they get a new strip they are having to apply the next level of reasoning. The first set are simple but this can end up quite challenging especially if they chose their own more “exotic” fractions.

It can be a bit of a hassle preparing and managing card sort exercises in the classroom. Whenever I see a resource that is created as a card sorts, I always think, could students get the same benefit by just writing in their books. But for some tasks such as this, I think it is worth it as it enables a richer discussion and the possibility for learners to easily changing their mind as they are building understanding.

# Zoomable Number Line

This little gadget on Mathisfun.com is so illuminating for teaching decimals. I always get a positive response from students when I show it.

I think it can be used it in a number of ways:

- Predict what happens when I zoom in. As you zoom in, first the tick marks appear and then at the next level of zoom, numbers start appearing. It’s a great way of “bridging” from the familiar to the new. A number line is likely to be a familiar concept. However, what is “in between” 0 and 1? An intentionally ambiguous question. Students are likely to say “half” or 0.5. How else can we show half? What other numbers are in between 0 and 1?
- Why are there 10 ticks between 0 and 1? We have divided 1 into 10 equal parts, what is each part. How else can we represent 1/10?

From there we “zoom” into the next level of hundredths

This table might be useful as some practice to relate fractions to decimals. I would love to hear some comments on this.

# Problem Solving

A nice collection of Problem solving resources arranged by topics on this site by @LeanneShawAHS

Source: Home