Four operations of fractions by folding paper

Another idea from Mike Ollerton’s workshops.  This file gives a comprehensive explanation of the activity, which starts like this:

I have used a similar task before but I realised that I had missed a key step which is to label each fraction after folding:

The file then goes on to describe how to use this for demonstrating all four operations: add, subtract, multiply, divide. It’s a lovely way of reinforcing the concept of equivalent fractions at every stage.

My only reservation with this task is that doing the folding in the first place might be a barrier for some learners.  Especially folding something into thirds – it’s not straightforward.

I have added some Powerpoint printables that provide guidelines along which to fold. Note these are set up as A4, so print them 2 to a page and then cut.

In a sense, I can see that this might detract from the notion of “folding in half” because it becomes “fold along that line”.  I haven’t had enough experience of which is the “better” way to do this – I’d be very happy if anyone wanted to share their thoughts!

 

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Comparing Fractions

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.

1. Comparing fractions of the same denominator
2. Unitary fractions with different denominators
3. Same numerator, different denominators
4. Different denominators where one is a multiple of another.
5. 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.

 

Fraction problems

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.

 

Converting Decimals to Fractions

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.

Fraction images

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.

1. 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.
2. 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

4. fractiontalks.com

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.

Quarter the Cross

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.  I have produce some printouts here.

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!

Since this post was originally written back in January, I have used this task a couple of times at conferences and had some really good discussions on this example.  If you want a rather big hint, scroll down to see an animation.  Or you can find the Geogebra file here.

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FDP ordering activity

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.

 

Sofia’s Ribbons

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?
• etc.

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:

Now ask:

• 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.

 

 

Multiplying fractions – showing why

It was a long time ago, so I can’t be certain, but when I first learned to multiply fractions, it was a procedure that involved turning mixed numbers into improper fractions, multiplying numerators and denominators with maybe some cancelling down along the way. It was a procedure with no understanding.

You could just apply that procedure to these questions. But there is scope for a greater depth of understanding not to mention some creativity in showing why these work.  Bar models are one way to demonstrate and calculate. Here are two examples:

A worthwhile exercise is to go through each of these questions attempting a drawing to show why (squared paper is a must).

Depending on your class, you will probably need to show some examples first. Or maybe you would prefer to give the completed statements so the focus is on drawing the representation rather doing the calculations.

Here is the lyx file for these questions and the pdf.

Fractions made of sequences

This is an update of the original post.  There were a couple of mistakes in the first one, which are now corrected.  And I’ve learned some new maths in the process. I’ve credited those below who helped me.  The power of Twitter – Thank you!

What happens when you take sequences of odd or even numbers and make fractions out of them?

These investigations provide some low threshold, high ceiling (in some cases too high for me!) rich investigations. I have been playing with these and encourage you to do so too!

These tasks have lots of benefits in the classroom if used well:

• Purposeful practise. In continuing the sequence and generating subsequent terms, students will repeatedly practice key skills. But rather than just working down a boring list of questions in a text book, they are practicing with a greater sense of purpose, i.e. to try to spot something else. There is a big range here from addition of consecutive numbers and cancelling down fractions to finding the nth term of a quadratic sequence (and harder).
• At key points stop everyone and get a whole classroom discussion going.  Ask students to explain their “noticings”.  By verbalising their reasoning pupils can grow their individual mathematical confidence. And it builds a classroom culture where other pupils’ noticings are highly valued. Children realise that we can learn by collaborating and listening to each others’ ideas.
• Always look for patterns and aim to generalise.  Ask why this happens, does it happen every time and can we build a proof?  This is hard. Sometimes what happens is that the students that aren’t ready to make this leap yet.  Often they just continue generating more examples.  This isn’t a problem as hopefully they are continuing to benefit from purposeful practice of an underlying mathematical skill.

I really encourage you to do some Mathematics and play with them before using them, but if time is short, here are some notes on each one.  I have put them in order of difficulty.

Sum of Odd Numbers

This is probably the most accessible of the four in terms of getting to a generalisation, although actually proving that algebraically is no mean feat!

Before we even get to the fractions, there is some good discussion to be had on mental methods for adding series of odd numbers and spotting that this generates square numbers:

1+3=4
1+3+5=9
1+3+5+7=8+8=16
1+3+5+7+9=10+10+5=25

To generalise this, we need to know that the nth odd number is (2n-1). Working from the last term backwards, we can write out the sequence as:

1, 3, 5, ..., 2n-5, 2n-3, 2n-1

By adopting the standard approach to find the sum of an arithmetic series, i.e. adding the first to last, second to second last, etc. we see that we get a whole bunch of “2n”s.  How many “2n”s? Well there are n numbers so there must be n/2 pairs. So:

2n × n/2 = n²

Now you can start examining the fractions themselves.  There is some good practice here of cancelling down fractions and students will realise quite quickly that they all cancel down to 1/3

At this point you might ask some students to generalise whilst some might prefer to continue generating examples.

The generalisation for the denominator builds on the generalisation for numerator. This time we with start with the odd number after the nth odd number and then add a series of odd numbers. Again think about what the last term would be and work backwards.

2n+1, 2n+3, 2n+5, ..., 2n+(2n-5), 2n+(2n-3), 2n+(2n-1)

By combining first and last, second and second last, etc. we can see we now have pairs of “6n”. How many “6n”s? Again, n/2.  So the denominator becomes:

6n × n/2 = 3n²

and:

3n²/n² = 3

Sum of Even numbers

With this one, cancelling down the fractions doesn’t help.

You end up with a pair of quadratic sequences (now corrected – thanks @solvemymaths)

Which neatly cancels down to:

Product of Even Numbers

This one provides lots of practice in “cancelling down” of fractions. Each time you end up with a unitary fraction (i.e. a numerator of one). But does this always happen? And why?

I made a mistake first time on this so I couldn’t find any pattern in the numbers that formed the denominators. Thanks to @mathforge and @wjhornby for pointing out by error.

So the sequence of denominators is 2, 6, 20, 70,…  That’s beyond my knowledge of Integer sequences (I did Engineering, you know, not pure Maths!).  But @mathsforge sent me this link to oeis.org. That’s another web-site I’ve learned about through this process!

If I had played around a bit longer with this and thought Factorials! then I might have eventually got here (thanks to @MrMattock for sending me this)

Product of Odd Numbers

This one again provides some cancelling down practice although you are going to be reaching for the calculator pretty quickly.

I’m struggling to spot any pattern in here (no, the next term doesn’t have a 9 for its numerator…), but there must be something, right?

And this is no bad place to take a discussion with your class.

There must be something here to be found.  We haven’t found it today. Your maths teacher is finding this very hard.  Maybe nobody has ever found it. But if we start off with something so simple there must be a way of generalising it.  Surely…?

Postscript: Again, I got a helpful response from @MrMattock.  You can see it here, but don’t spoil it, have a go for yourself.  The clue is to look for factorials again and don’t express all fractions in their simplest form. Good luck, but I warn you – it’s not pretty!