Natural numbers in Year 8

One of the reasons I really like the MEI OCR A-Level Mathematics is the section 6 in the Core 1 unit entitled "the Language of Mathematics". I am a massive believer in mathematics being a tool we use to communicate abstract ideas and to translate real-life situations into mathematical models that can be logically analysed. I love working on proofs with pupils and using proper maths notation and symbology, including notation for different subsets of the complex numbers - including N for the natural numbers.

It was a few years into my teaching career that it suddenly dawned upon me the links between the set notation I had studied in my degree, and the algebra I was using with my pupils - x as a member of R, n s a member of N, or z as a member of C; and in particular how I could apply this to my teaching. I started off by making it clear to my upper set GCSE and A-Level pupils why certain letters were used in certain contexts (i.e. n is used in sequences because it refers to only whole numbers, whilst the assumption is that we use x when values could be fractions or irrational numbers) but more recently I have started using the idea with lower attaining pupils, particularly with sequences. I am writing about it today because I introduced it with Year 8 set 4 (of 5) with fabulous results. Starting with the natural numbers being n, we looked at defining the different times tables (2n, 3n etc) as well as sequences like the square numbers, all starting from the natural numbers. By the end of the lesson I had several people being able to find the first 5 terms of n³ + n² + n (remember, set 4 of 5 in Year 8!) and even some being able to explain why 154 was not in the sequence 3n - 1. The idea of the natural numbers, and building other sequences from them for me is much more powerful than some of the other approaches to position to term rule (ghost number is one that seemed to grow in popularity). Just be clear the idea is relatively straight forward and looks like this:

Question

"Write the first 5 terms of the sequence 4n - 3"

Answer

"4n is the 4 times table, so it goes 4, 8, 12, 16, 20, ...

So the sequence 4n - 3 is each number in the 4 times table subtract 3, i.e. 4-3, 8-3, 12-3, 16-3, 20-3, ...

So 4n - 3 = 1, 5, 9, 13, 17, ...."

So if you are teaching about ghost numbers, or another approach, consider this as an alternative that should hopefully lead to increased conceptual understanding.

Probability and Proportion

My year 9 bottom set have been doing probability recently (in fact they finished the topic today with some healthy topic assessment results - quite a few Level 6s in the old NC parlance). In order to help them prepare for the new GCSE, we have looked at some problems that link probability with proportion, beyond the idea of just fractional representation. I thought I would share a few of my favourites here.

1) "In a bag there are only red, white and yellow counters. I am going to take a counter out of the bag at random. The probability that it will be red is 1/4. It is twice as likely to be white as red. Give an example of how many counters of each colour there could be."

This was a great starter problem to link these ideas together. Many of the pupils did see the relatively obvious 1, 2, 1 solution; linking to 1 out of 4. But this problem allows us to explore other solutions, with the proportions linking the idea of solutions in the ratio 1:2:1. We can then talk about different probabilities for red (i like 1/6 with white being 3 times as likely so you get the ratio 1:3:2).

2) "A bag contains red, white and blue counters. The probability of taking a red counter is 1/6.
      (a) John says that there are 40 counters in the bag. Explain why John cannot be right.
      (b) There are 12 red counters in the bag. Work out the total number of counters in the bag."

This one is a great problem for exploring the links between probability and fraction calculations, in particular fraction of an amount and equivalent fractions. We can talk about mixed numbers, and that they are not appropriate for this context, What is nice then is to discuss changing the numbers in the question, and which numbers make it easier, harder, or the same? This allows the exploration of the idea that multiples of 6 in part (a) means that John can be right, that reducing the number of red counters to 1 makes it easier to answer the question, but that it doesn't necessarily need to be a multiple of 6, and that 12 is a 'coincidence' in that case - however change the fraction to a non-unit fraction and suddenly multiples of the numerator do become important.

3) "A die has three red faces, two blue faces, and one green face. Sam rolls the die 300 times and gets these results (in a table) Red = 153, Blue = 98, Green = 49. Is the die biased? Explain your answer."

I love this problem. Once you get past the idea of biased because it is more likely to land on red, you can talk about the relative proportions of the number of sides compared to the relative proportions of the results, and these being the same. Again it can be ratio, fractions of 6 compared to fractions of 300, or simply something like each one is nearly 50 times bigger (and in reality of course it would be nice if it were all of these).

There are others and again sharing is welcome through the comments or otherwise, but given the fact that proportional reasoning is such a big part of our new GCSE, in my opinion these sorts of problems are the sort that need exploring with pupils of all levels.

Formative assessment strategies

It isn't often that I post here on subjects that aren't specific to Maths teaching; my subject is generally what gets me out of bed in the morning. However I was having a conversation with my trainee today about formative assessment, different strategies to use and different ways to manage the outcome and so I thought I would share some of the more interesting ones I have come across (either used, seen used, or a variant on one of those) over the years - I am striving for something a bit more interesting than just using mini-whiteboards (although I do use these all the time).

1) Lolly sticks - Now I know a lot of teachers use lolly sticks and most who use them have now got as far as having a different pot or section of the pot to put used ones back in, but how about having three separate pots to put them back in? One red, one amber, and one green depending on whether you think, based on the pupils input, whether they will need immediate attention in the activity that follows, or checking on after a few minutes, or are safe to proceed. Then when independent or group work starts, you just pick up your red pot lolly sticks and visit those pupils, then the amber and so on. Having the lolly sticks with you to hand also helps jog the memory of what they said or did that set off your 'in trouble' sensors in the first place!

2) Multichoice wall - Have a section of the wall where a multi-choice question is posed (give at least three options, and make one option a classic mistake or one that isn't obviously wrong would be my advice). Give pupils a post-it note, get them to write their name on it and stick it where they think on the wall. This again can let you know as a class how well pupils have understood, and will give you the names of the people who are very wrong, making common errors etc. This can also be an Always, Sometimes Never wall if the question is a good question for those sorts of responses. What can be nice then is to re-seat based on their response (either grouping people with the same response together for intervention, or mixing up one or two from each response for peer support) if the class can handle it quickly; if not why not try...

3) Move to the answer - Similar to above, but this time instead of a single wall, you have three or four possible answers in areas removed from each other around the room, and then pupils move to whichever answer they think in correct. This can be easier to do re-seating as pupils are already out of their seats collectively, so pairing them up (again either same response together for teacher intervention, or different responses for peer support) should be more straightforwad.

4) Red, Amber, Green sheets - A bit more work admittedly, but having my questions cut up into three sheets at three confidence levels (Red - I am struggling, Amber - I am OK but will need reassurance I am doing it right, Green - I am confident so at some point just come and checkwhat I am doing is right) and allow pupils to choose. I have seen this done with differet colours but actually my preference is for different sizes, and to make them smaller as they get harder. This battles the natural propensity for some pupils to take what they see is the easiest work (i.e. the red sheet) as it is clear that there is more to do on the red (the logic being that there is more to do but it is easier to do it, with green there is less but doing it involves more thinking and understanding). Of course then my interevention can be based around those pupils who have chosen the largest sheet, with some early checks on the ambers and then some later checks on the green (unless there is someone whom I don't think should be on green!).

5) Post-it head - This one is like the game we used to play when we were younger, where people would write the name of a famous person on a card and stick it on someone else's head, with the other person having to guess the word stuck to their own head. As well as being a great activity for key words in a topic when played in the traditional sense, it can also be used for formative assessment by pupils writing (discreetly) a RAG rating on their own confidence onto a post-it note and then sticking them to their own heads. I find the fun element to it can overcome the barriers some pupils have in admitting their own vulnerabilities, provided they don't write too large. If you don't fancy their heads (particularly if anyone in the class happens to be allergic to the glue used in post-it notes) then sticking it on their book or the end of their table can be nice; some pupils don't like having their planner blazing red for the whole class to see (or don't want to seem immodest by having a big green patch) and this can be a slightly discreeter alternative.

6) Heads down, Thumbs up (or middle, or down) - Another take on a classic game; everybody must have played heads down thumbs up at school, and gotten that secret thrill when the person you had a crush on turned out to be the one that had pinched your thumbs (or was that just me?) No such thrill this time I am afraid, but again I find that heads down first creates a bit more fun about the using thumbs to give feedback, and also reassures those people who are not confident in admitting their vulnerability a bit of anonymity. I normally do this by pinching thumbs (gently, no matter what the temptation is to do otherwise with certain pupils) to let them know I have noticed their feedback; but occasionally I will just scan the room and then tell them thumbs down, heads up.

So there we have some of my favourite alternative formative assessment strategies; feel free to take, adapt or if you have the time (and would be really great) share some of your own through the comments.

P.S don't forget the mini-whiteboards, they really are great too!

Forming and solving linear equations; a change of focus.

An interesting observation today; introduce a new variable and pupils tend to forget their basic skills. As a case in point take the triangle below.

I gave a similar triangle to Year 7 top set as part of their equation solving topic, but with only the bottom two angles only with an instruction to find x. Not all of them understood straight away, but as a class with the support of their peers they all figured out how that problem was solved.

Later in the topic (as a follow up target question after their topic assessment as it happens) I gave this triangle with the instruction to find y to those pupils who had performed strongly on the assessment and didn't need to target particular skills that were tested within the assessment. What very much surprised me is that event though these were the best equation solvers in the top set of Year 7, they still struggled to see their way through this problem.

I surmised from my conversations with them that it was the change of variable that threw them; that having the mix of x and y made it harder for them to see how the two related. I am glad I found this out now, as it allows me 5 years to explore more problems of this type with them (this particular problem is being written into my schemes of work as an example problem over all of the year groups, to spark the imagination of my other teachers). I suppose this just serves as a warning, particularly with those groups that will be taking the new GCSE exam in England; don't assume your pupils that can form and solve equations are confident in applying those skills to a problem with a small change of focus.

Properties of Diagonals of quadrilaterals.

Is it just me, or here in the UK did we stop teaching about the properties that the diagonals of different quadrilaterals have? My top set Year 7 can spot a square, rhombus, rectangle, kite, arrowhead, parallelogram, or trapezium. They know that a trapezium only has one pair of parallel sides, and to avoid pitfalls thinking parallelograms have lines of symmetry. But of one area they remain very ignorant, the properties of the diagonals of different quadrilaterals.

The activity linked here is designed to address that. It starts by getting pupils to focus on the different properties that diagonals can have, and which quadrilaterals have those particular properties. This can be done with pictures of the quadrilaterals for pupils to reference, or without if pupils can deal with it. This is followed by a richer activity with pupils having to try and write sentences that only identify a selection of quadrilaterals, which can be extension or differentiated starting point. I haven't tried this with my Year 7 yet, but I do think there is merit in this sort of exploration of diagonals.

Compound Area and the order of operations.

A colleague of mine reminded me today about an old idea that I must confess I had forgotten about. I believe it was 'originally' in the Standards Unit (I say originally as I am sure they got the idea from somewhere!), the idea of compound area and the order of operations. This led me to create the diagram below:

On its own it isn't much, but I very much like the two questions with it:

1. How many of these calculations give the area of the shape?
    - 3 x 5 + 2 x 4 [this one does]
    - 3(5 + 2) + 2 [this one does]
    - 5 x 2 x 4 x 3
    - 4 x 7 - 5 [this one does]
    - 5 + 2 x 3 + 2
    - 5 x 5 - 2 [this one does]
    - 5 x 7 - 3 x 4 [this one does]
    - 3 x (5 + 4) - 4 [this one does]

2. Can you explain how each one of the correct calculations is found from the shape (you may need to rearrange the two rectangles to see some of them). [I am leaving this one for you to answer for yourselves].

I really like this activity. In terms of applying linked areas or maths and being able to communicate their understanding (ears prick up for those working on the new GCSE and the KS3 curriculum) it links diagrams with explanations and it also gives pupils a discussion focus around the different calculations. These sort of activities are very easy to design and can be extended nicely into the formation and equivalence of algebraic expressions when pupils have developed their mathematics a little bit. You can adapt the questions, remove the links, or change the shapes slightly to provide differentiation if you need to.

Best approach for median from frequency table?

Discussion with my intern today about median from frequency table. Teaching it to Year 8 set 4 and she has had real trouble, particularly because of she has used the deleting extremes approach for finding median from raw data rather than direct identification of the middle position (note for others, consider the deleting extremes method carefully if you are going to want pupils to find median from frequency tables using identification of middle position). Got me wondering as to what approach do other teachers use when teaching median from frequency tables; I think I can see 3 approaches

1) Direct identification of median position, leading to the median.

2) Generating a list of raw data from the table, and finding the median from a raw list.

3) Deleting extremes in a table.

This third method I wanted to expand on a little, as it may not be clear to all.

Consider this table:

If we ask pupils to picture the list, we should be able to see that the list starts with seven ones, and ends with eight sixes. So if we were to delete extremes from the list, the first seven ones would be crossed off with seven sixes. This would leave one six at the far end of the list.

This six would then be crossed off with the first two in the list, leaving eight twos at the beginning.

The list now has eight twos and six fives at either end, so the six fives will be deleted with six of the twos, leaving two twos at the beginning.

These two twos will then be deleted with two of the fours, leaving four threes and four fours.

From here will pupils spot that the middle is halfway between 3 and 4? Or would they write out the list of 8 numbers? Or would we take the method to its natural end to say that if we cross off three of the threes, and three of the fours, that this leaves a single three and a single four?

I will admit that this is a much longer approach than the identification of the middle position, but I wonder if it is more natural for pupils? Or if it is at least a viable alternative for pupils that understand deletion of extremes for raw data and struggle with identification of middle position. Any feedback from other teachers about their approaches with pupils on finding median from table would be greatly appreciated.

Fill in the box algebra - not for me thanks!

Recently I have been talking about introducing algebra to pupils with a number of my colleagues. Most of them use a variant on the 'fill in the box' approach i.e. [  ] x 4 = 28, what number goes in the box? Some use other symbols such as the playing card suits, but the approach is the same. I have started to wonder if this approach doesn't lead to a couple of problems, namely:

1) Does this approach lead to or reinforce the misconception that there is only one value for a variable (the dreaded "what is x?" question)?

2) Does this approach start part-way down the journey by focusing on the equation before ensuring a proper understanding of the more fundamental idea, that of the expression, has been explored.

Forgive me for expanding on this second point, but to me the fundamental algebraic idea is that of the expression. By equating expressions with values or other expressions we arrive at equations and formulae; by evaluating at integer intervals we generate sequences and from here we can explore graphical representations and other algebraic constructs. By introducing algebra using equations I wonder if we start ahead of ourselves and then have to try and move pupils backwards before we go forward again. I think it might be better to approach introducing algebra with the expression rather than the equation and I have found a way that I think works; the think of a number problem.

Don't get me wrong, I don't mean the classic "I think of a number, times it by 3 and add 5, and get an answer of 14." This is just an equation in a different form. No I mean the magic number problems like the 2, 4, 8 trick - Pick any whole number, multiply it by 4, add 8, divide by 2, then subtract your original number. Your answer is 4." These sort of number tricks do tend to capture the imagination of pupils as they like the idea of performing tricks on people; and for me they give the perfect vehicle for exploring what happens to expressions. It allows for the reinforcement that algebra is used to deal with unknowns and relationships between them and allows pupils to explore what an expression looks like through different manipulations (multiply by 4 etc).

I have a compendium of Mathemagic tricks here which can be used at varying levels with pupils to explore expressions or to go as far as using difference of two squares in formal proof; feel free to use to introduce the idea of an algebraic expression as the fundamental algebraic representation.

Dice of Ultimate Power

On my shelf at school is a big inflatable die, it takes pride of place and whenever I walk towards that area of my classroom my pupils eyes widen in anticipation; for this is a special die, the die of Ultimate Power!

The die is my favourite plenary, and the kids love it too. The crux of the idea is that the pupils throw the die to each other, and whichever number they catch it on they have to contribute in a certain way. Of course people can pick any 6 options at this point, but the ones I use are:

1) Give a key word used in the lesson.
2) Summarise a key point of the lesson.
3) Explain a key idea from the lesson (and I insist on full explanations).
4) Give me something you don't understand following this lesson.
5) Give me something you would like to find out following this lesson.
6) Free choice from 1 to 5.

If the pupil gives a suitable response, they get to line up by the door ready to go; if they don't or they don't pass it on sensibly (stops them trying to bat it around or throw it forcefully at someones head) then they have to sit down. The lined up people get to leave first, then anyone that is still stood up (I have them stood behind chairs so I can see who hasn't had the die yet) gets to leave, then the people who have sat down leave last. Pupils either really want the die, or have that 'I don't want it' but smile when they get it. You get some really quality feedback and the idea can be adapted to question difficulty selection or a whole host of other things (not to mention having a big die in your room just tends to come in handy; I used it for demonstrating mutual exclusivity the other day by asking a pupil to put the die on the table so both the 5 and the 2 were visible).

Must give a nod at this point to Jamie Butler at Oxford Spires Academy who originally gave me the idea.

Hats off to the Welsh...

With no disrespect intended at all; in recent times with all of the adverse media attention, the word "Wales" has not exactly been synonymous with the words "great education", at least in the minds of the public (please notice how careful I am trying to be not to judge a situation I have no knowledge of beyond the print media). However a colleague today forwarded a resource to me on the Welsh government website (hwb.wales.gov.uk). Created by Brian Sharp and Matthew Nixon (I believe through Aberystwyth), there are a series of questions and tasks that are beautifully geared towards the new GCSE (certainly English, I cannot comment on the new Welsh GCSE, as I am not particularly familiar with it, other than I believe it is going back to a 3-tier system). What is particularly impressive about these resources are the very clear notes for teachers that go along with them, which expertly highlight the sort of questions that teachers can ask around the resource and some great commentary around the resource and likely pupil response to it. I know my NQTs for next year will really value these resources as will my experienced teachers and I will definitely be adding most if not all of the resources to my GCSE Scheme for next year.

The link to the resources http://hwb.wales.gov.uk/Resources/browse?sort=recommendation&language=en&query=GCSE%202015%20-%20Maths and I would very much recommend a look.

Probability problems...

Year 9 bottom set today. 25 pupils in the room. Question "How many boys in the room?" "13". "What is the probability of choosing a boy at random from the class?" "13/25". "What does that mean?" Blank stares...

Fortunately this was anticipated, and the point of the lesson - what does it actually mean to write down a probability? What information does that fraction give us? To help illustrate the idea I used the random name generator that I have for each of my classes and we did some basic probability experiments, cycling the random name generator 25 times and seeing how many times a boy came out (actually happened 14 times, which was lucky!). We then did some further fun data collection - pupils had to find out which of their classmates, could roll their tongue, had ever broken a bone, won a competition or similar. Having done this activity before I had learnt from previous mistakes - I only had 6 pieces of data collection where previously I might have done 10 (understanding that 6 bits of data x 24 pupils = 144 data points altogether for each pupils to collect), I structured their time a little more so that they had certain time around the room, then time at their own tables to share the data they have gathered, then separate time to write down the probabilities - whereas before I had previously just given time for the whole activity without the structure. We then did the same sort of experiments again, looking at how many times in the random name generator produced the names of people that did or did not meet the criteria, and compared to the probability that the data had given us (thankfully they compared favourably). By the end pupils did have an understanding of the idea of probability as a measure of chance when outcomes could be selected at random, that this also told us about how many times outcomes occurred when a number of events are possible and they also gained more comfort with the idea of probability telling us if something is biasing an experiment if the experimental results are significantly different to those that the probability suggests. Although there are still things to tweak, particularly with the data collection, I think that overall this lesson was a success in generating an extra level of understanding about probability in pupils.

Making mode and median meaningful

So today I started the averages topic with my Year 8 set 4 of 5, and so faced that age old conundrum known by all Maths teachers - how to engage pupils in something they have seen before but only half remembered?

I decided to test a hypothesis of mine, that these pupils will have an instrumental understanding (to borrow the language from Skemp) of finding mode and median, but no understanding of what the values mean or how to use them to form judgements. So the focus of my lesson today was not finding mode or median, it was what these averages actually tell us and making judgements about which average measure we are going to use.

We looked at the idea of average age (in years) to justify mode, and then average height of a group of 10 pupils to justify median. From there we looked at trying to convince a boss to give us a pay rise (or alternatively, being the boss and being able to explain why you are not giving a rise), and similar situations (convincing parents to increase an allowance, or to purchase an extra pet), as well as the classic situation of having to choose stock for a shoe shop based on average shoe size. In these situations the averages become what they are always supposed to be, a means to an end rather than and end in themselves. Pupils were much more engaged in the processes because they could see the point in them, and it would appear that the processes themselves have become more embedded through being applied. I think from now on I will stop teaching "how to find mode, median and mean" and only teach applying averages, and trust that the calculations will take care of themselves.

AQA Problem Solving and Guidance

Like most teachers, the Easter holiday is not so much a break as a chance to catchup -  both on quality time with the family (hence my absence from blogging and twitter for the last few days) and on jobs for school.  As Director of Maths and Numeracy I have spent much of the year preparing for the new GCSE next year; I have designed our two-year,  three-tier scheme of work and started the ball rolling on involving all departments in mathematics across the curriculum.  My big job this Easter holiday is linking the resource bank into the new GCSE scheme,  particularly those that will give pupils the opportunity to practice and consolidate skills at AO2 and AO3. In this vein I thought I would share the AQA GCSE Problem Solving Questions and Guidance. Published in 2010, I have found this a truly excellent source of in-class material and have been referencing it heavily in my new scheme.  For anyone out there that is finding preparing for the demands of the new GCSE I can heartily recommend getting hold of a copy and introducing these problems into your GCSE and KS3 lessons and sharing with your team. A link to a copy is here http://aqamaths.aqa.org.uk/attachments/2050.pdf and as long as you are an AQA centre you have licence to use in your school.