A look at my new teaching and learning toy

In the spirit of encouraging personal positive thinking, I've decided to start a new Sunday blog series on the best lesson I've taught the previous week.

This week my pick actually covers two lessons of work, and was delivered to my Year 9s on Thursday and Friday. To provide a bit of context, we're trialling the Mastery Pathway with KS3 this year, and pupils sat the Elementary 4 test on Tuesday, covering negative numbers, substitution and lots of fractions and decimals work. They passed the previous three tests pretty well, but this one caused some difficulties, with marks ranging from 30% to 70%. After doing a bit of analysis on the results, I decided that my first topic for re-teaching would be working with fractions and decimals, particularly ordering and converting between the two representations.

Got a whole fortnight's worth of stuff to fit into this post as Easter holidays mean I've had a little break from frequent Twitter checking! Here are my picks from the last two weeks:

This is part of a series of blogs on my favourite way to teach adding and subtracting fractions in a way that sticks and really develops understanding of the process. If you haven't already done so, you can read about Lesson 1 here.

Lesson 2 - The importance of equal-sized bars

At the start of the next lesson, present pupils with two more sets of data, this time comparing two groups of unequal size. Using group sizes of 20 and 30 are particularly effective.

I usually get the class to first represent both sets of data on two more bar diagrams using the template from the previous lesson and get them to record the fractions of Year 8s and 9s that preferred each fruit, cancelling down to the simplest form.

The lesson(s) detailed below have been absolutely groundbreaking for me in terms of teaching adding and subtracting fractions in a way that makes the topic stick and that pupils really understand. The ideas behind it were introduced to me as part of the NCETM's Multiplicative Reasoning course (previously mentioned in my blog about bar modelling). Unfortunately, while I have the lesson materials, I'm still unsure about their status in terms of sharing - they were presented to us as trial materials, with the suggestion that they would be available to schools nationwide once the project had finished, but I can't find them anywhere on the NCETM's website (yet). So although I can't post a link to the PowerPoint and lesson materials I'm using, I thought I'd pop up a quick blog about the ideas behind the materials.

Just a quickie tonight to share the love for a few maths puzzle booklets I've found kicking around on the Internet. I'm not sure where they've come from originally, but here are the links:

Maths Practice Puzzles : Fractions and Decimals
Includes:

• Equivalent fractions 
• Comparing and ordering fractions 
• LCD and simplifying fractions
• Addition and subtraction (like denominators)
• Addition and subtraction (unlike denominators) 
• Multiplication and division of fractions
• Decimal addition and subtraction 
• Decimal multiplication and division

I've been using bar modelling quite extensively in my teaching since being involved with the NCETM's multiplicative reasoning project last year. Part of the project was to emphasise the importance of diagrammatic representations of problems in teaching maths for understanding; we were given materials to deliver to Key Stage 3 classes, some of which included use of Singapore bar modelling for topics such as fractions, percentages and ratio. I found the work we did really altered my teaching; I think that I managed to teach addition of fractions successfully for the first time since I started teaching, and I was amazed at just how well my students retained efficiency and accuracy with "traditional" written methods.

It seemed like a bit of a no-brainer to pick the bar modelling workshop at the Celebration of Maths, so I and my colleagues trotted along to the session, sat down with our mini-whiteboards and got ready to draw some bars. One thing I was really keen to get out of this session was to iron out some issues I still had with using the bar to solve problems with negative amounts, and I was still struggling to see how to apply bar modelling to exam technique (see my attempt with the Edexcel SAMs here).

First of all we looked at some simple problems, like fractions of amounts. I'm already pretty happy with this - there are plenty of examples of stuff like this in my Year 7's books at the moment. I was so proud of myself that I thought I'd add a really constructive "what went well" to my work too.

Having taught this to nearly all of my KS3 groups at some point in the last few months, I think I'm becoming something of an expert. This year, we've started the Mastery Pathway at KS3, and nearly all of our students in Years 7, 8 and 9 (unsurprisingly) had gaps in their understanding of fractions. So I've now taught this about five or six times; it's true what they say about practice making perfect (or pretty close).

Two major changes this year have been use of the bar model (read about my discovery here) and linking cubes. I'm trying to do a lot more in concrete situations before moving to rules, and using manipulatives and models seems to get the ideas to stick.

1.  Start with a chocolate bar
Think Cadbury's Dairy Milk (single bar), Kinder Bueno, Hershey Bars - anything which comes in a single row of squares or pieces of chocolate. The first thing I did was show the students some pictures of these chocolate bars, and get them to tell me each piece size as a fraction - e.g. the Bueno has four pieces, so if I eat one piece, I've eaten 1/4, two pieces is 2/4 and so on. We spent some time discussing what the numerator and denominator tell us about the different chocolate bars, and emphasising that the denominator told us about the size of one whole bar.

2.  Move to a physical model
I then gave students different amounts of link cubes, explaining that they were pretending these were squares of chocolate. I asked them to make me as many whole chocolate bars as they could - we used the Bueno 4 square model to start with. There were a few comments about how I'd not given them enough cubes in some cases (they were expecting multiples of four); I kept quiet and just told them to build what they could. Once they'd built for a bit, a couple of the students realised that this was the point of the task.

We then collected ideas on the board and looked at different students' examples. I deliberately started using a bar model at this point to represent the cube pictures they were explaining. This is also a good point to get students to draw their models on the board themselves and explain what they've done.

3.  Introduce improper fraction notation
I took one example and modeled the use of fraction notation, drawing on our original chocolate examples. So 13/4 means 13 pieces of chocolate, and one whole bar has 4 pieces. We then discussed how this related to mixed number form:

• How many whole chocolate bars? This is the big number.
  
• How many pieces that don't make a whole bar? This is the numerator of the fraction part.
• How many pieces in a whole bar? This is the denominator.

4.  Practise problems using cubes and diagrams
I then gave students some problems to try for themselves. They started with more examples using four-square chocolate bars, then progressed to other sizes. I kept denominators fairly small to start with (2, 3, 4 and 5), just because modelling and drawing gets a little impractical with larger denominators. I encouraged students to use the cubes where necessary, and insisted they drew diagrams for the first few they attempted.

Some students quickly spotted the "rule" and then began working without diagrams. Towards the end of the activity, we discussed what they had discovered as a shortcut method and why it worked. However, I haven't been pushing rules too much this year - I'd rather students thought about what they were doing, and if necessary got the right answer through drawing a diagram, rather than learn a rule which is quickly forgotten.

5.  Work the opposite way

Some of the classes I've tried this with were ready to move on to converting back the other way, either with diagrams or by applying understanding gained from working from improper to mixed within the same lesson. 

Some of them needed a little more consolidation work one way before we moved on. One lesson I'm learning this year is how important it is not to push too quickly; a new idea takes time to cement thoroughly, and I think it's worth working on one thing properly at a time, rather than charging ahead and progressively losing students along the way.

If you're looking for questions quickly, Math Aids has a great worksheet generator, differentiable by easy, medium and hard denominators.

Last year, I attended the best CPD of my teaching career. It was a year-long course run by the NCETM; a project on multiplicative reasoning. Now don't get me wrong, training days can be useful, but I can honestly say that this course changed how I teach fractions and proportion on a fundamental level. We had six days of training, spread out over the year, and had to deliver certain materials to our KS3 classes, then evaluate their performance at the end of the year to assess the effects of the project materials. I'm going to blog more about this in later posts, but in this post I wanted to focus on one of the first questions they asked us on Day 1, because it really altered my thinking about teaching fractions. It's a simple question:

Just to give a bit of context to this, this was one of four diagnostic questions we had to give our classes prior to delivering the project materials. The students had to answer the questions in as much detail as they could, and explain their reasoning. 

We also had to try the questions. I suggest you give it a go now, too! Go on, before you peek at mine...