I’m not sure who Sofia is or where this originated, but it was presented by Liz Henning at the recent MTN hosted by La Salle Education in London and it struck me as a great way to introduce bar modelling at all levels, and could really help with ratio and fractions.
Start with 2 equal strips of paper, “ribbons”. Ask questions like:
- If each ribbon cost 10p how much do they cost altogether?
- If both ribbons together weigh 6g, how much does one weigh?
- If both ribbons represent one hour, how much time is one ribbon?
Representation is a key concept here. The ribbons can represent something else but that representation can be useful to work things out. It also uses the idea of part-part-whole.
Next, take one of the ribbons and fold it in half. Tear along the fold, so you have this:
Now you can ask questions like:
- If the orange ribbon is 10p, how much is the white ribbon?
- If the ribbons weigh 15g overall, how much does the orange one weigh?
- What fraction of this is the total?
- If white represents 12 hours how much does orange represent?
- What is the ratio of orange to white?
Next, take the orange ribbon and fold it in half again, so you have this:
- If orange is now worth 10p, how much is white?
- If both represent 2½ hours, how much does white represent?
and from here you can get into drawing bar model to represent what is going, e.g.
I used the bar modelling tool Thinking Blocks to create these images. Once you get used to the interface, it is a quick way of creating bar models for use in the classroom and contains a number of problems that you can use with learners.
I walked into my classroom this morning and noticed my clock was broken. Not just stopped but really broken, can you see why?
It reminded me of a nice problem solving task which is sort of to do with angles but actually much more to do with ratio and proportion.
I wrote the following on the board:
For a normal clock, what is the angle between the hour hand and the minute hand at the following times:
There is a significant range of challenge in these questions. 15:00 – straightforward, right. As soon as you start moving the minute hand away from 12, you need to consider the fraction of (360/12) degrees that the hour hand moves. 12.30 might be the best option for a question 2 if you really want to scaffold it. You also might want to squeeze a few more in between Qu 4 and 5. e.g. 14.40, 15.20.
Next time I do it, I won’t write them up all in one go, but will keep adding to them as I can see learners making progress. Or ask students to challenge themselves by creating their times which might work better in a mixed attainment classroom.
These can all be done without a calculator. It demonstrates how useful it is to have 360 degrees in a circle and 60 minutes in an hour because they have so many factors.
A nice build on this is this question from an OCR Booklet of problem-solving questions.
This sheet practices getting ratios into 1:n for the Unitary Method but also interleaves sequences. It’s low threshold, high ceiling in that students can just manipulate ratios (simplify in the first set) but you can ask some fairly challenging questions including:
|Write a rule explaining the sequence
|What would happen if you kept this sequence going?
Here is a simple pdf document (with them printed 2 to a page):
Here is the original spreadsheet where you can hide / reveal the answers or tinker with the questions. Pretty tricky to work out the generalised 1:x form for some of these!
This week saw 4 lessons with Year 9 on “Proportional Reasoning”. It’s a skill that pervades lots of topics, obviously ratio but also underlies algebra, fractions, shape, measures, statistics, everything really!
Our school has been trialling some resources developed by the ICCAMS project. I liked the look of some of them for example, this one:
However, I was a bit nervous as it seemed like there were lots of examples that would involve whole class, teacher-led discussion and not enough for students to do. So I had these from Don Steward’s Median, ready on a slide.
It also seemed to me that all the various examples and contexts used fundamentally the same skills and that students (as I had a fairly high achieving class) would “get it” and then quickly get bored.
I was wrong. As we looked at the different examples it became clear that the change of context was not straightforward. In the example above many students initially added 10cm to the 6cm and 7cm to get 16cm and 17cm. Once we had examined it further and introduced the concept of a Double Number Line, they fully appreciated why this was wrong. So then we looked at this one:
Again, many fell into the trap of adding 2 to the 30m to get 32m rather than 35m by taking a multiplicative reasoning approach.
The power of presenting different contexts for the same basic skills both provides interesting ways to practise that skill as well as giving the student (and teacher) an assessment of whether they have mastered it or not yet. Sometimes misconceptions can be strongly engrained, maybe even more so in top set kids who are used to being right most of the time! It takes time to develop the right instincts when approaching these problems and gain that depth of understanding. With topics as fundamental as Proportional Reasoning, that is time well spent.