## Ratio Tables

I have been a fan of ratio tables for a number of years and we have recently decided to use them consistently across the department. It has been nice to see that they have been (deservedly) celebrated a lot recently, in particular at the latest Le Salle Maths Conference, so I thought I would offer my experiences.

What is a ratio table?

A ratio table is a way of representing two or more things which have a multiplicative relationship. I first came across them around seven years ago when I was part of a multiplicative reasoning pilot scheme run by the NCETM. There, we were shown a horizontal table linking the costs of different lengths of ribbon.

Horizontal or Vertical?

It was relatively recently that we discussed using ratio tables as a consistent method throughout the faculty. We listed the different topics that could be represented using ratio tables (more on that later) and its effectiveness. One of the biggest discussions was about whether using a horizontal table would be best overall, as students are used to showing their calculations vertically when dealing with percentages and equations etc. Although, a horizontal table is more useful when calculating with recipes, as they tend to be set out that way. In addition, a horizontal table shows clearer links between fractions and percentages. The picture below shows an example of this

We decided that a vertical table would be best overall, given that students are more familiar with working down the page. Here are some examples of the vertical ratio table in action:

I have made an adapted version of the ribbons resource to reflect this.

Using a ratio table

When using a ratio table I insist on two things. The first is to ensure that the table has clear headings, so that there is an understanding of what each value represents. Included within the heading is the units, so that there isn’t a need to repeat the units on each line of the table.

The second expectation is the inclusion of arrows to show the intended calculation, to mitigate against making mistakes and ensure that working out is evident.

Exploring with Ratio Tables

I like to spend lots of time just practising using a ratio table to perform different calculations. In the Using Ribbons task (see below), you may have noticed that I prompt the students to find as many different ways as possible to solve the problem.

This shows that there is more than one way to skin a cat – students are invited to find their own way to a solution, rather than follow a prescribed model. All of the ideas are shared so that there are opportunities for everyone to explore the different connections. Of course, we would like students to work efficiently as well as effectively, so there is lots of discussion about which may be the best way.

In doing this, students also get to see that when using a ratio table, not only can you multiply and divide, but you can also add and subtract rows of the table in order to find the values of other amounts.

Consistency is Valuable

Ratio tables can be used in many areas of Maths, including:
– Percentage (amounts, increase/decrease, reverse, express as a percentage, percentage change)
– Compound measures (speed, density, pressure)
– Proportion (direct, inverse, currency, best buy, similarity)
– Circumference (see examples above)

If students can use a ratio table in all of these topics (and even more!), there are fewer methods to have to remember and they will quickly become even more fluent in using the table to perform calculations. Furthermore, by using this method consistently, it can help students make connections between topics as they see the same processes being used throughout. It can also help students make even deeper connections within a topic, and a perfect example of this is in calculations involving speed, distance and time. Not only does it eliminate the need for awful formula triangles (don’t get me started), but it also can help students understand more that speed represents the relationship between distance travelled and time taken.

Limitations

Although I am an advocate of ratio tables, I know that nothing is perfect. One of the biggest issues I have always had is that students have previously been taught to use a different method, so it has been counter-productive to undo prior learning. However, now that we have decided as a faculty to use a consistent method, this should (hopefully) be less of an issue. The next step I would like to take is to work with other faculties, in particular the Science department, to make it a whole school approach (and bin off the formula triangles for good).

## How Do You Know – Taking understanding to the next level

Last week I presented a workshop at the 5th #mixedattainmentmaths conference in London. In my session I shared a collection of tasks that can be used in Key Stages 2, 3 & 4 with with the aim of promoting fluency through practice, whilst developing a deeper understanding and promoting reasoning.

The resources I used can be found here:

How Do you Know – Taking Understanding to the Next Level

Expression Cards

Factors – Always, Sometimes, Never

Equivalent Fractions Always Sometimes Never

As part of my professional development I would really appreciate any feedback from the workshop. I would be eternally grateful if you could spend a couple of minutes completing the question below about the impact of my session:

## Improving Memory Using Spaced Repetition Through Weekly Quizzes

This year I have started to make weekly quizzes for Key Stage 3 classes across the faculty to help students to retain knowledge throughout the year. At the bottom of this post I have added links to example quizzes that I have used (feel free to have a go!).

Our Key Stage 3 students all have their own Chromebooks and at the start of the year I would have been the first to admit that I wasn’t sure how to use them effectively in lessons. However, once I had learned how to use Google Forms I quickly realised that it would provide a good platform for using weekly quizzes. They are easy to create – you can use EquatIO to include Maths notation, you can choose a variety of question types and you can set it up to self-mark (saving us a job!). As if that wasn’t enough to convince you, Google Forms provides you with analysis of answers so you can see how individual students, or groups of students, performed on each question. Of course not every school has Chromebooks, iPads or other technology for students, so these quizzes could just as easily be paper-based (I do this with my Year 11 class). Alternatively they could be set for homework instead.

When I made the first quiz I came up with 10 questions and informed students to take as long as they needed on the questions as it was important that they did not feel rushed. I had hoped the quiz would last around 10 minutes but instead it took some students around half an hour to complete, so I reduced the number of questions to 5 for the week after. Since then, it has fit into lessons much better and has been much more focused. The majority of the questions I use are diagnostic multiple choice, so I can quickly pinpoint misconceptions.

In order to keep track of the topics the quizzes have covered, I created a spreadsheet that I could edit weekly. An example can be downloaded here Weekly Quiz Ticklist Year 7

As you can see from in the linked document, I write down the week number under each topic when it is used in a question, so I can keep track of how I am spacing them.

The most important part of the quiz is the next step. This is important in 3 areas:

• For me: I use the analysis from the quizzes to create questions for the week after. For example, if a question is poorly answered I will include a similar question the week after
• For the teacher: the analysis should be used by the teacher to inform their lessons. If their class did not perform particularly well on a question they can either address it straight away, include it in starters or if needed, re-teach the skill.
• For the student: we subscribe to a website called Hegarty Maths which is a learning platform containing quizzes and linked videos for Maths topics. Each question has a linked Hegarty Clip Number and students pick out a question to work on each week, completing the Hegarty quiz for homework.

Below are some examples of quizzes I have made so far this year, including Google Forms quizzes for Years 7, 8 and 9, and paper-based quizzes for Year 11.

Year 7 Weekly Google Forms Quiz – https://goo.gl/forms/ebx2uYd2h8kjQFwp2

Year 8 Weekly Google Forms Quiz – https://goo.gl/forms/DWUFewfqWqdv76372

Year 9 Weekly Google Forms Quiz – https://goo.gl/forms/zSsLJJnXRnQXnZxz2

Year 11 Weekly Quiz – Year 11 Weekly Quiz 1

If you decide to try absolutely anything from this blog post I would really appreciate some feedback on how it went. Please could you fill in the document below and email it to MrE_Maths@hotmail.com

Weekly Quizzes Feedback

## Practising Mathematics: Substitution

Over the summer I invested in ATM’s Practising Mathematics by Tom Francome and Dave Hewitt, which is packed with interesting activities aiming to ‘develop the mathematician as well as the mathematics’. Having used several of these tasks during the first half term of the year it has quickly become one of my favourite resources for giving students practice, building on their knowledge by spotting relationships and probing them to develop a deeper understanding through powerful questions. These tasks have been a perfect fit for my mixed attainment classes, and if you haven’t got a copy of the book, I would highly recommend that you invest.

Just last week, whilst most of the country were on their half term break, I taught a lesson on substituting into expressions to a Year 9 class using the Expressions Cards task from Practising Mathematics. I was blown away with just how powerful it was so I wanted to share my experience, but I also wanted to reflect on what I could do even better next time.

The Lesson

The task revolves around a set of cards with expressions written on them. The suggested expressions in the book are:

I decided to add 4 other cards containing:

Working in pairs, students were prompted to choose a value for x and substitute it into the expressions. Once they worked out the value for each card they needed to put them in order from smallest to biggest. They then repeated the process and commented on which cards changed position and which stayed the same. Most students started with positive integers less than 10 and were already commenting on what they noticed as I questioned them.

At this point I suggested that students may want to try different types of numbers: large numbers, small numbers, decimals, fractions, negatives, to see what differences there may have been. As they did this I also prompted them to conjecture examples of cards where one would always be smaller than another, no matter what value of x. This prompted even more discussion as I circulated the room. I took several examples from students and we discussed them as a class. This allowed me to probe students even further as well as address any misconceptions. In particular, I was keen to highlight the difference between 2x2 and (2x)2.

The final task I set built on from the last one as I directed students to choose two cards: card A and card B. They had to find values for x so that:

This was particularly powerful as they had to think of the nature of the expressions on the cards. It also led them to find expressions where one was always bigger than the other.

Reflection

I was really pleased with the connections that the task allowed the students to make. However, as the lesson went on I found some opportunities to make it even better next time.

• I may change the cards from expressions to formulae by adding ‘y =’ to the start of each of them. Here I could instruct the students to calculate the size of y in each formula. I could even include a couple of formulae where y is not the subject, so some form or rearranging will need to be done. This resource could then link to linear graphs, simultaneous equations and many more.
• During the ordering I did not make it clear to everyone that they should write down the order each time so they can compare, although most students were savvy enough to do it anyway.

Here are some photos I took of students’ work after the lesson:

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## #mixedattainmentmaths Conference

I am proud to have run my first conference workshop yesterday at the 3rd #mixedattainmentmaths conference in London. My session was about my school’s transition from setting to mixed attainment classes in Years 7 and 8. I discussed the nuts and bolts of how we got off the mark, what barriers we faced and how we overcame them, as well as the impact of advice and ideas from an array of advocates of mixed attainment teaching from the likes of Helen Hindle, Zeb Friedman, Mike Ollerton, Bruno Reddy and Mark Horley to name a few.

As promised, I have provided the resources from my workshop for anyone who wants more information or couldn’t make the conference. If you have any further questions I would love to hear them, and if you want to delve further into the conference then resources, information and links can be found at the brilliant http://www.mixedattainmentmaths.com that is run by conference organiser Helen Hindle. And if you would like to come and see us in action, or you want arrange to speak to our SLT for advice please get in contact.

Here are my resources:

Transition to Mixed Attainment

Key Stage 3 Learning Journey – Directed Number

Year 7 Unit 3 Assessment – Directed Number

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NTM Examples

Notes To Myself – Directed Number

## How Many Ways…

Just like everyone else who has used them, I am a big fan of Don Steward’s resources, particularly because of the way they get students thinking. They allow students to explore relationships and patterns at their own pace, often increasing in difficulty as they go on.

I have used some of his ideas to create my own ‘How many ways…’ questions. Here are some of them:

## Would I Lie To You?

A few years ago I was obsessed with the idea of creating resources based on TV gameshows, including such things as Catchphrase, Pointless and A Question of Sport. Some of these I still use on a regular basis but one show I didn’t continue to use was Would I Lie To You? I made one resource for expanding single brackets where 3 students were selected to stand at the front and read out 3 answers – one of which was true – and the rest of the students needed to decide who was telling the truth.

Over Christmas I was doing some (last-minute!) Christmas shopping and I came across the ‘Would I Lie To You?’ board game and decided to buy it. I played it with the family on Christmas day and we all really enjoyed it. So it has got me thinking about how I can use the idea in the classroom to identify misconceptions.

The game is designed for 2 teams of up to 4 players and has 3 types of challenge:

• Quickfire Lie – the player is given a card with the first half of an interesting fact. Underneath there is a correct answer and a blank space for the player to make up their own false answer. The aim is to decide which is the true answer.
• Ring of Truth – the player is given a card, again with the first half of an interesting fact and a correct answer. Each of the players in the team has to come up with a false answer. The other team have to decide which is the truth.
• This is… – the player is given a card with the first half of an interesting fact. If it is a TRUE card there is the second half of the fact and a picture which links to it. If it is a FALSE card there is no picture or second half of the fact. The player has to describe the picture and state the fact (if FALSE, they have to make up the picture and the second half of the fact). The other team then interrogate them about the picture before deciding whether it is true or false.

I am really excited to see how any of these ideas could be used effectively in the classroom and I will share any resources I make. If anyone has already made resources like this I would love to see them and find out how they went.

## The Super 9

A resource I use religiously as a way of showing progressively difficult questions, the Super 9 has differentiated questions in a 3×3 grid so that, in theory, questions get more difficult as you move right and/or down. I got the idea from a colleague (@miss_jobacon on Twitter) early in my teaching career and I have made lots of my own in the last few years. I have taken a snapshot of a few examples and included the files of some others. Thanks must go to various sources such as Corbett Maths and Mathed Up for some of the questions! If you notice one of your resources and would like acknowledgement please let me know.

Super 9 – Volume and Surface Area

Super 9 – Angles

The Super 9 – Direct Proportion

The Super 9 – Perimeter and Area

The Super 9 – Polygons

The Super 9 – Ratio

## Developing the Use of Learning Journeys

It has been a long first half term of the academic year for my school and I am sure I am not the only one who was ready for a break after a typically busy start to the year. Whilst I am currently enjoying recharging my batteries it is also a nice opportunity to reflect on the changes we have made in the last couple of months – something we don’t often get the chance to do when in the thick of it. At the start of the academic year we started to use learning journeys with all of our Key Stage 3 classes in line with our transition to mixed attainment classes. In addition I have been using them with Key Stage 4 classes as these are something I have been trying for the last year or so since I came across them at the first mixed attainment maths conference organised by Helen Hindle and feedback from my students has been largely positive, with most seeing value in seeing how they have made progress over time (check out http://www.growthmindsetmaths.com for more information and resources).

We chose to introduce learning journeys for a number of reasons, but the main one is because we want students to have a clear picture of their starting point and what where they could potentially progress to. Helen Hindle likes to structure learning journeys with outcomes written in different columns depending on difficulty. The inclusion of an arrow to show the direction of the journey also represents continual progress which isn’t limited to the outcomes. In the example below the students aren’t necessarily limited to the outcomes shown, but they could progress onto more difficult skills not represented in the learning journey e.g. circle theorems.

The first lesson of each unit is spent looking at the learning journey and answering ‘Super 9’ questions relating to the outcomes in order to find their starting point. For Key Stage 4 classes I use GCSE questions. During this lesson my students will ask for help and I reply that I am not teaching them…yet, as I would end up teaching umpteen different skills throughout the lesson and the students would have a false idea of their starting points. At the end of the lesson I share the answers and students use what they have done to highlight any of the outcomes they are totally confident they can already achieve. At this point I stress that it is important to be honest and even if no outcomes are highlighted at this stage, that is fine, in fact those students could make the most progress.

Learning journeys give me some useful information about the students and it is important as the teacher that I read through them to inform my planning. Over the course of the subsequent lessons the students regularly reflect on their new learning and highlight outcomes as they go along. Whilst students are getting used to the process they may need some guidance but generally, as they become more effective in reflecting on their learning they develop consistency.

At the end of the unit students have the opportunity to revisit the ‘Super 9’ questions posed at the start, in the form of an assessment. At the time of writing we have completed one unit and the students have completed the assessment. We tailored a mark scheme to promote thorough working out so that we recognise the efforts of students at each stage of the question. We did not give time restrictions for completion as we want students to take their time to get questions right, without the pressure of a deadline. Students then mark their own answers and we record their score as a percentage. On reflection, I have decided that in future I will not ask students to find their score as it does not add value to the assessment. Instead, I will ask students to write a detailed reflection of their work in the form of a ‘Dear Sir’, relating to the outcomes of their learning journey (see previous blog post on this – I will be adding to this soon). The most integral part of this process will be their areas for improvement – I will train my students to reflect on their work and identify their next steps. I can then assign them a homework task using the wonderful Hegarty Maths. I feel it is important that they reflect on the outcomes they have and haven’t highlighted in their learning journey. I want this to be a working document which they can edit; I anticipate that students will make mistakes when highlighting outcomes and make errors in judgement.

Please find below some examples of learning journeys I have used this year, for both Key Stage 3 and 4 classes, including the related questions. There is also loads more information available at www.growthmindsetmaths.com

Key Stage 3 Learning Journey – Factors and Multiples

Key Stage 3 Learning Journey – Fractions

Key Stage 4 Learning Journey – Percentages

Key Stage 4 Learning Journey – Pythagoras’ Theorem and Trigonometry

GCSE Questions Pythagoras’ Theorem and Trigonometry

Key Stage 4 Learning Journey – Angles

GCSE Questions Angles

## Engaging the Disengaged

In my first year of teaching I had 2 challenging Year 9 classes which I struggled to engage. I made countless mistakes as I looked for the best strategies. 3 years on, I have come up with some of the things I have found most effective in engaging the disengaged:

1. Praise those who are on task – praise is a powerful tool when used effectively. By acknowledging the students who are doing what you expect of them, you are showing the disengaged students the level to aim for.
2. Use competition – every teacher has their preferences but some of the things I have found particularly useful are:
• Relays – a good way to facilitate group work. A selection of relays I have made can be found here
• Game show-style competitions – my favourite is Catchphrase. I use mini whiteboards to ask questions and then randomly select numbered cards which correspond to a square to remove. Download my versions here
• Kahoot quizzes – if you have access to tablets, this is a fantastic online multiple choice quiz. You can make your own; check out g.evans for all of my quizzes.
• Puzzles – students like a challenge. I have found sudokus and Don Steward problems to be popular.
• Point-per-question challenges – I like to have 1-point, 2-point, 3-point etc questions. Solving Equations Challenge Answers Solving Equations Challenge
3. Give deadlines – use a timer on the board if you can, and refer to it throughout a task
4. Get them up to the board – a bit of responsibility goes a long way. Plus, they get to be centre of attention for a bit.
5. Use manipulatives – multilink cubes, counters, playing cards, dice. There are an infinite number of manipulatives you could get your hands on
6. Variety – it is the spice of life, so don’t be afraid to shake things up a bit. Group work, discussions, worksheets, quizzes, open-ended tasks, bingo, match up tasks, videos, treasure hunts, code-breakers. Routines are important in lessons, but that does not mean you cannot have a variety.
7. Get to know your students – take the time to find out about each young person and find common interests. You can then use this to your advantage when they inevitably try to engage in conversation about last night’s football or the latest episode of Bake Off – “answer 3 more questions and I will talk to you about it”.
8. Have a laugh – teaching is the best job in the world but it is also up there with the most stressful, so it is important that, particularly with the more challenging classes, you enjoy what you are doing. As teachers we often feel under pressure and so do the students, so I like to take a couple of minutes in the lesson to go off the script and have a laugh – providing that you remain professional of course!