There is something very simple about a task which presents two numbers and simply asks “which is bigger?”. This should be done using mathematical notation, i.e. using the < > symbols. I have seen these being introduced successfully in Year 1 without any mention of crocodiles, or such similar unhelpful “stories”. But my Year 7 class still insist on calling them crocodiles and drawing teeth on them. But hey, I have bigger battles to fight…
As well as comparing 2 fractions we can put multiple fractions into order from smallest to largest. There is a significant range of difficulty in this apparently simple task.
- Comparing fractions of the same denominator
- Unitary fractions with different denominators
- Same numerator, different denominators
- Different denominators where one is a multiple of another.
- Different denominators where a common denominator needs to be found for both fractions.
Alongside all of these there may also be strategies where learners are using known facts or doing calculations to convert to decimals or percentages, e.g. 1/2=0.5, 2/5=0.4, therefore 1/2 > 2/5. That is not the intention of this task (it is of a different task here) but in the end we want learners to be able to play with all these ideas and I can’t really control, nor would I want to control, the order in which they coalesce in students’ mind.
Here is a simple set of cards that I used recently. I got the students to do the last bit of cutting to turn each strip into the 3 separate cards. I also told them that there is deliberately some space alongside the fraction to enable them to write equivalent fractions if they needed to.
I gave them out a strip at a time, the idea being that they were to “slot in” the subsequent fractions to maintain the order. The fractions are carefully chosen, so that each time they get a new strip they are having to apply the next level of reasoning. The first set are simple but this can end up quite challenging especially if they chose their own more “exotic” fractions.
It can be a bit of a hassle preparing and managing card sort exercises in the classroom. Whenever I see a resource that is created as a card sorts, I always think, could students get the same benefit by just writing in their books. But for some tasks such as this, I think it is worth it as it enables a richer discussion and the possibility for learners to easily changing their mind as they are building understanding.
These problems are ones that are made much clearer by drawing a rectangle to represent the “whole” and then deciding how to divide it into equal parts. The numbers are not too tricky but interpreting the question might be:
These are not intended to be fraction of an amount questions. An approach could be to decide upon an amount, but the intention is to direct students to drawing a representation of each question.
These questions aim to step through the various concepts needed to understand what is going on when we convert decimals to fractions with some generalizing questions at the end to get students exploring when decimals are equivalent to fractions that cancel and when they are not.
Available as a word document here.
Imagery is so important to help with conceptual understanding of fractions and I have seen some really powerful uses of imagery in lessons recently. So I thought I would create some kind of repository of fractions images that can easily be used when designing lessons involving fractions. The Windows snipping tool, Smart Notebook Screen Capture toolbar or Shift-command-4 on a Mac will all come in handy to quickly pull these images into your lesson.
A really quick way to create fraction images like these is on Excel (or Google Sheets). It’s much easier and more accurate than trying to create boxes in Powerpoint or on Smart Notebook. There are a random collection of these in this spreadsheet, all very easy to adjust by changing shading and/or borders of the cells as required.
You might be looking for something a bit more pictorial:
There is a large and also rather random collection of these in this PowerPoint. Many thanks to Declan Byrne from the London SE Maths Hub for agreeing to share these which have been compiled from his lessons.
Finally, there are some handy websites that enable you to create images which again can be quickly cropped into your lesson.
- National Strategies Virtual Manipulatives – there are a bunch of these that were created as part of the National Strategies and now hosted on eMaths. You can quickly and easily create bar fractions like these and add or remove the fraction, decimal, percentage and ratio alongside.
- NRICH Cuisenaire Environment – A simple to use tool for displaying Cuisenaire rods on screen with a grid.
3. Clip Art Kid – some images on this site which might be useful
A range of more complex fraction shapes including Tangrams:
If you have any more suggestions for places to find or ways to create fraction images, please let me know by leaving a comment below and I will update this blog post.
UPDATED POST. I used this task at my workshop at #mixedattainmentmaths on Saturday. I asked all teachers to have a go at this task but to do it in what they thought was the most obvious / simplest way. An interesting experiment: what is obvious to some is not to others. Of the solutions that I managed to take in, these were the choices:
This looks like a very useful open-ended task which provides an opportunity for creative solutions and rich discussion.
In my view, the value in this activity is in representing each area as a fraction calculation.
According the Australian blog where I first read about this task, this is one of the most common first solutions
I’d be looking for some rationalising as to why the red area is a quarter. For example:
There are 100 solutions posted here!
And on a Prezi here enabling you to zoom into each one individually.
This is potentially very high ceiling. If students are struggling to come up with suitably challenging solutions of their own, you could always ask:
Show why this is a quarter:
Have a go first yourself. I think this is a pretty mammoth task! This one caught my eye, but you might want to have a look at the 100 solutions to find something a bit easier!
This is a simple card sort activity where students fill in the blanks, practising converting between fractions, decimals and percentages and then placing them in order from smallest to largest.
What I particularly like about it is that you can hand out the cards in 3 sets of 5. This could provide differentiation, but more importantly (in my view) it gives the teacher the chance to assess the progress of the students as they go. It’s easy to glance at Set A to see if they have worked it out.
As you circulate the class, helping students you can give out Set B which interleave with Set A to produce this:
And then finally with Set C, you get this:
Here is the pdf as a set of 15 cards, but if you are printing a class set, then I strongly recommend using the Excel version here. This is set up so you print 5 at a time. They come out stacked so that as you cut you have a set of 5 already in a pile without needing to sort them.
Also, if you want to change the actual values and which values you show, you can do that on the spreadsheet too.
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.