Best buys and fractions

These best buy questions were very in vogue when I started teaching every exam paper without fail had a best value question. I can still picture the washing powder boxes with their greyed out images those of you have been around awhile will still see them too. As I may have mentioned before one of my roles was to achieve 100% pass rate in a school that I taught in. As the best buy questions tended to be on the calculator paper we often felt these were easy marks that our lower sets could achieve.

It was my job to come up with a department wide approach for tackling best buy questions for those students lower set or higher set who just didn't get it. I cannot tell you how many times in my experience of working in schools marking mocks papers I have seen the division the wrong way round! 

Picture the morning of the exam. The free snacks or breakfast flowing and the whole of year 11 are looking at me for last minute tips. I was the lead presenter as the rest of the department floated around helping the panickers. I had been looking into all sorts of ways to solve the department wide problem of answering best buy questions so this wasn’t entirely out of the blue but I hadn’t firmed up my plans at this point. By opening my mouth I suddenly did firm up the department wide approach much to the shock of the head of department who was watching on! 

Price

Quantity

Yes there are best value questions where multiplying quantities up would give a more sensible answer but this became the approach for calculator best value questions. This may lead to fractional pence answers but it does give two prices that can be compared and a decision made on which is best value. I tell my learners that if they have their own way then they should use that but this may inspire some to tackle questions that they previously avoided. It also provides a nice opportunity to revisit rounding and significant figures. as I've said before rounding and significant figures is something that needs constant reminding and reinforcing and is always in my retrieval practice grids. 

In the department I was in we began to look at other opportunities for marks for our lower sets. We looked at two way tables. We found the learners enjoyed completing the tables but struggled to pull out the final fraction. 

I began by asking my students how they would work it out. The main question I was asked is “how do you know which number you want?” This was evident in their working out. Nearly all of the students had 100 as their denominator with a variety of 35 or 60 as the numerator. Keeping things as simple as possible I used the language of the students. I trialled what you get as a fraction of what you want. So in this example we want females and we get gym females. 

This not only works for two way tables but any time a fraction from probability is required. 

What is the fraction of red marbles in the bag that has 4 green and 6 red? 6/10. You want marbles, you get red. It’s quite simple and really helped us crack the using the wrong denominator in the two way tables. 

prime factor decomposition and fractions

I inherit every possible method for teaching fractions under the sun. My learners come from every school across the city and all have their own way of doing things. Kiss and flick, kris kross, X marks the spot and so on are all ways for multiplying and dividing fractions. Like I always say, if you have a way that works consistently for you then use that way. I never unteach a method unless it is incorrect. Often methods have become incorrect because the learner has become confused with the method shown to them. In my experience students are more likely to become confused if they lack knowledge of the basics. One of the ‘wins’ I enjoy teaching is product of primes, prime numbers, simplifying fractions and divisibility checks. 

I like to start with prime factor decomposition. I like to call it this in my intentions on the board. I like to discuss the anxieties the learners have when they see the title and then experience the joy on their faces when they crack it in the end! There’s no fancy trick in what I am teaching, this one I am afraid requires singing and dancing as it is all about the presentation. I am not above making a fool of myself to help a concept stick in the learners minds. It all starts with some audience participation. I write up 2, 3, 5, 7, 11 on the board. We spend a few minutes shouting it as loud as we can. We play finish my sentence and I will call out 2, 3 and pick a learner to finish it off. Then we move onto questions of what is the third prime number? What is the fifth? We spend quality time recalling the first 5 prime numbers.

I don’t think I am doing anything out of the ordinary at this point for the learners. Everyone I have ever seen teach the topic as well has started by spending quality time on the first 5 primes. I tend to just stick to 2 3 5 7 11 for these early stages. An old favourite from my cover supervisor days on my PGCE was the 10 x 10 grid and playing in pairs to find all the primes up to 100. I will, if we have time, move onto games like this but in the first instance I stick to the first 5 primes. 

Here comes the singing and dancing…

“If you like it then you should have put on a ring on it”

Here is an example of the prime factor decomposition of 420:

We begin with writing our first 5 primes. We then use these as divisibility checks. I am aware that learners can find the highest prime factor and solve this problem quicker. I am not aiming for speed here. I am aiming for a consistently correct method. We always begin by seeing if 2 goes into the number. We keep the primes on the left as well.

I am aware that it may be obvious for some learners to use 5 as the prime when breaking down 105. I have no problem with this, actually in making these images that’s what I did and had to start again! Once we have broken the number down. I will then sing a bit of Beyonce or show an image of her on my board. (Beyonce also sits on my wall as well as Terry Pratchett!) The blue writing is purely for the example here, we verablise this in class but don’t write it.

Once we have really solidly grasped these divisibility checks, we can use them in a variety of ways. We can then look at simplifying fractions.

Yes this is a long way round but it ensures a consistent approach and reinforces the use of 2, 3, 5, 7, 11. It's a nice introduction in a stepping-stone to unpicking that initial knowledge that's needed to tackle a wide range of questions. Like I always say if you have your own with working things out please use your own way, but for those who struggle 2, 3, 5, 7 11 may just help with developing those core divisibility skills.

Multiplication

I'm sure by now you have heard me mention that I teach a lot of resitters where they may be sitting the GCSE for the second, third, fourth or however many time. I've also talked about those quick wins in those early weeks to earn their trust help them gaining confidence and to deepen their mathematical understanding. One of the easiest ways I've found to do this is through teaching students how to multiply correctly. As part of my PGCE training I was asked to teach a year 7 group a series of lessons covering every possible type of multiplication that I could find available. It led me to Egyptian, Chinese and many other types of multiplication. I became familiar with a wide variety of multiplication techniques that I had to fully understand before I could explain them.

My own knowledge needed to be sound and secure before I explained them to the attentive year 7’s hanging off my every word. There are some elements of teaching year 7 that I miss greatly! Chinese multiplication became my new favourite thing. I know only multiply this way, much to my builder and husband’s annoyance when planning our new extension! I'm sure you've seen it before the lattice method the Geloise method or simply Chinese multiplication. 

What I want to discuss here is the importance of being able to multiply correctly including decimals. Like I always say if you have a way that works for you than that's the way you do it what I'm offering you is a way if you can't do it consistently correctly. Many of my learner's come to me unable to multiply correctly they have been taught the grid method in primary school and they don't know what to do when they're lining them up for the final addition.

Take a look at this work this is typical of my students work. The grid is correct but the place value when adding is appalling. Yes, I am aware I will need to re teach place value but will these mistakes arise again after we have done that unit? What would happen if we looked at Chinese multiplication instead? 

Look at the lattice method above. Here we have the correct answer in a much simpler way. Place value is correct. This makes me happy. I like the Chinese/lattice method because it promotes place value. I like the fact that it only uses the times tables up to 9 x 9 and although we teach up to 15 x 15 often up to 9x9 is secure and above that can be a bit sketchy in my experience. 

But the real reason I like the lattice method is because of the power it has to unlock learning for many resitters. I say to all my trainee teachers and staff that I mentor. Teach someone to multiply correctly and they will be set for life. It is true, multiplication is one of those skills people have had negative experiences with and switch off. To be able to show someone how to multiply correctly can sometimes be transforming. A colleague (not a student or a teacher) was trying to multiply the number of tickets and the cost of tickets to budget for the cost of a conference for her team. She was struggling. I showed her lattice multiplication ( I drew the grid) she loved it. She was so overwhelmed she began to tear up. Maths, multiplying decimals in particular, had always been a struggle for her but she felt that it wasn’t anymore. That is the power of being able to multiply correctly. Lattice may not be the best way for everyone but I think it is worth spending the time to teach learners how to multiply correctly, including with decimals, to provide the skills that are needed for life, not just GCSE.

SLED exam technique

I have mentioned before the challenge of answering A03 questions and some strategies I employ. A03 style questions are when our students are required to make connections interpret results and evaluate solutions. The easiest way I remember it and explain to student teachers is it's those questions that are open ended and they could go many different routes to find the answer generally with a full blank page underneath. I am forever looking for current relevant research that can be easily applied to my maths lessons. Last year while studying a course on a MOOC for the Friday Institute about learning differences I came across something that may help. Yes I can't quite see how course on learning differences lead me to this, but it did, and I am grateful! It just goes to show you should read as wide as possible because you never know what it will turn up. One of the things that came out of one of the pieces of reading for this course was SLED. 

sketch 

label 

explain 

discuss

It got me thinking about these complex a03 style questions where we could easily sketch a picture of what the answer might look like, the painting of the floor plan or the number of marbles in a bag whatever the question maybe about. 

We can label underline key parts of the question. Now instead of explain I switch this to estimate. This was to relieve the pressure of the learners to immediately hit upon the right answer. They are allowed to roughly approximate first. Generally they are correct but with the pressure being off it really improved their confidence in giving these types of questions a go. So the learners could estimate how many tubs of paint how many marbles in a bag and then we could discuss our answers with our peers and then choose the correct strategy for solving the question. The power in relieving the pressure in making it an estimate was immediately evident with my groups. Learners openly admitted that previously they would have skipped this type of question but were now willing to give it a go. We now had a strategy where we could pose, pause, pounce and bounce! We could now have those meaningful discussion that deepen understanding that I keep seeing in other subjects lessons! 

This is a rough example of a simple A03 question using SLED:

Being brave I asked my learner's to SLED an A03 question before the lesson. We then discussed on tables and then as a class different approaches to the same problem. Again I believe powerful learning takes place where misconceptions or different approaches to the same problem can be explored. It was an instant hit with my learners. We then began thinking about how they could adopt this in an exam. We changed discuss to DECIDE for exam situations. This is when after you have estimated, you may have created 2 or 3 different options you must decide on the correct working out and clearly state that as your answer. I like to remind students at this point that multiple workings out will attract lowest marks available and remember each question is a new opportunity to leave an impression on an examiner. 

Here is the same ratio SLED problem with the decide element added, the original working has been left in for this examples purpose but would normally be crossed out:

I like this approach as it gives confidence to tackling complex A03 problems. It provides a crutch for learners to hold on to. It promotes discussion in maths lessons, it’s not always right or wrong! It promotes deeper understanding by looking at common errors and misconceptions. One mature learner, I won’t say her age, said that this strategy gave her the confidence at work to plan her response to requests and improved her stress levels, so it worked outside the maths classroom too!

Challenge for all2

As I said in the previous post the 'challenge for all' really changed my teaching from stretching the more able to challenging everyone and not by giving them more activities to do. Dylan Wiliam is just a treasure trove of hints and tips every time he writes or speaks. In one piece I read (the full article is here) he talks about activating students as instructional resources for one another. In its simplest terms he talks about before a piece of work is handed in a peer marks it and signs to say they have done so. If there are any remedial actions needed the peer must let the learner know, as if there’s anything missing when the teacher marks the work it is the peer that is responsible not the original learner. I liked this as I could easily see this in maths working well. You could give some learners a mark sheet to help them and for others you could ask them to do it without. Everyone could be involved. I was then browsing the brilliant trythisteaching and came across this Doctor’s Diagnosis It takes Wiliam’s idea and asks learners to prescribe remedial work to the originator. 

I took some prescriptions from the trythisteaching templates into my lesson. We were looking at sequences. 

Ben has completed the task. Stacey has marked his work and corrected his error. What I like about this is the level of differentiation available. Depending on my instruction Stacey could have just asked Ben to look again at part B, she could indicate the date we looked at the topic, or she could research a resource online that would help Ben and share that with him. I also like that they don’t need to be in the same class. Stacey is from group B and Ben group A, same set just opposite timetable. The possibilities of this are endless. You could do it verbally in class. However, when I tried this all I heard was “good try” “you did this bit great”. I found that when students had to put their comments in writing they became more focussed in their feedback. I find that many great teaching and feedback ideas are tricky to fit with maths as a subject but this one worked for me!

I am known when mentoring to say “if it isn’t in the plan it won’t happen” I am not pushing lesson plans, please don’t mistake me! What I mean is, whatever you use to plan, make sure all your activities and episodes are planned. For example, for progress checks against the learning intentions I encourage student teachers to have a slide in their presentation to ensure that they happen. (It’s not just student teachers that need reminding of this, I am more than aware!) I needed to ensure that I remembered to include a 'challenge for all' element in my lessons. As I have an open door policy for all staff, and especially mentees, I always need to be on point. I like doing one thing once and doing it well so that it is always the same level of planning from me. I found that whilst creating all these 'challenge for all' activities I found that some worked and some didn’t. So I hit upon a choice board. I created a Padlet of my 'challenge for all' activities. If you are unfamiliar with Padlet, it can be used as a virtual noticeboard with ideas pinned to it. If you are out of free usage of Padlet there are alternatives, try Linoit. My padlet looks like this:

I loved Fighting Talk on BBC Radio Five Live when I was younger. It was part of my pre match routine. The game ends with defend the indefensible. Two players were given opposing sides of a statement to defend. The statement was purposely controversial or opposing to the participant views. For example, one participant was asked to defend the statement that his wife’s cooking was terrible. He refused to engage! I have found that this format plays great in maths lessons. 

Imagine we are looking at solving equations x^2 = 36

Ben is asked to defend that x=6

Stacey is asked to defend that x=18

What I like about this is that both need to calculate the correct answer. Stacey needs to identify where the misconception has come from in order to defend it. I am a big fan of learners knowing the common misconceptions in order to avoid them. All to often in maths I see the power of 2 mis-interpreted as multiplying by 2! 

I found that defend the indefensible worked in many lessons and I could generally think of them on the spot! Any time solving or rearranging with algebra was involved I would challenge the whole class with defend the indefensible. We would do it as 2 teams against each other or lots of small pairs debating to the class. I found it a nice way to promote respect and listening along with developing debate and argument forming skills.

Spot my error and tick or trash were brought to my attention via TES. Spot my error came from the fabulous alutwyche https://www.tes.com/teaching-resources/shop/alutwyche  (he is also on Twitter https://twitter.com/andylutwyche) and tick or trash I originally found on the old but brilliant Number Loving from Laura Rees Hughes A lot of the Number loving resources are now to pay for on TES. Tick or trash has a question in the middle and to the left and right are two answers, one is correct, to be ticked, and one is incorrect, to be trashed.

The spot my error tasks from alutwyche are quite comprehensive. I have enjoyed adapting this idea. I will present and incorrectly answered question on the board and ask the learners to spot my error. This also helps with the inevitable board work mistakes I make daily as it gives me a great cover story! The spot my error from alutwyche and my own ones all focus on displaying a common misconception. Again if learners know the common misconception and can realise where it comes from they can avoid it. Both STEM and NCETM have resources and evidence explaining the common misconceptions in maths. You can find them here and here. The NCTEM also has a great paper on inspection worthy mistakes made by students here. It makes a great read. Broadly it suggests that “Conducting appropriate class inspections of mistakes in student work can create pedagogically powerful moments in the classroom.” One of the most powerful types of mistake to investigate, the article says, are those that approach the problem from a different way that the broad solutions from the class, as if the problem had been interpreted differently. This is my aim when I create either defend the indefensible, spot my error or tick or trash activities. How could the problem be interpreted and how different will the answer be?

My final activity in challenge for all is to create cheat sheets. I was shown these by my mentor in my NQT year and they have been with me every since. Do you remember the old, make a poster lesson? I hated them in school as I would rather be at home watching MTV Select than in school making posters! As a teacher I also dislike them as really, what are the learners learning apart from how to use fineliners of contrasting colours to make 3D lettering? A cheat sheet is like a poster but different! Have you ever had this... You ask the learners after a lesson, how much of this are you going to remember? Queue awkward faces and panic! It happens multiple times a day to me! A cheat sheet gives the learners a way out of this. You ask learners to imagine that they have 2 minutes before the exam and their present self wants to tell their future self all about this topic and they need to make a cheat sheet to record it. I normally model one as an example. I follow a thought or spider diagram. I encourage the learners to make it personal to them. A specific error they made in class they may want to clarify or they may want to emphasise a specific stage of the process. 

This rough example (apologies!) is personal to the learner. They are clearly wanting to remember interior means inside the shape angles. They are also keen to remember it’s angles around a point rather than in a circle! With my adult learners, evening class learners specifically, we do this weekly as they worry they won’t be able to remember it. I don’t blame them, maths is hard, many are working and raising families as well as coming to maths. We end up with a bank of 20 or so personalised cheat sheets. The learners then have a revision guide ready to go when exam day comes.

I am now pleased with where I have got to with my 'challenge for all'. There is challenge for all available in every learning episode via my padlet or prepared activities (if I remember!) It has been powerful in helping me develop skills to enable understanding rather than offering more tasks for my learners to complete. My learners were initially sceptical but embrace the challenge now and enjoy unpicking where the incorrect answers have come from.