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.

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Decimal Place Value Charts

I use these a lot at the early stages of understanding of place value.  In my opinion, place value is the single most important mathematical concept that children need to master in upper KS2 to prepare them for secondary school maths. Many don’t, so it is incumbent on secondary maths teachers to ensure that any gaps are filled in Year 7.

As a pdf.  Or if you want to change things, as a Google Sheets or MS Excel file.

If you print them out on card, you can put them inside transparent A4 plastic wallets and then use mini-whiteboard pens to write on them.  There are different types of plastic wallet – you want the ones that have relatively thick plastic which is smooth, not textured.  Otherwise you will have a hard time rubbing off the pen!

Initial Activities

1. Show me 3 tens.  Now show me Thirty. Why do we need to put the zero in there?
2. Show me 5 tenths. Show me 5 hundreths.  Why do we need the zeros?
3. Show me 34.  What is 34 made up of?
4. Show me 6.75.  What is 6.75 made up of?

From here, of course, you might want to look at multiplying and dividing by powers of 10 and then eventually 4 operations involving decimals.  But don’t rush into that until you are confident they have secured a depth of understanding.  These questions are good for really testing that:

For me these are a crucial AfL tool to ensure the building blocks are in place before doing more complicated things with decimals.

 

 

 

The Wisdom of the Crowd

Look, stats isn’t boring! I love doing these Wisdom of the Crowd exercises when teaching averages. Another way of doing it is to display a random scattering of dots on the screen (about 40 or so). Get everyone individually to provide an estimate then calculate the mean.
A really interesting addition to this is to do it once getting everybody to call out their estimate. Then, before calculating the mean, give everyone the chance to change their estimate and record them all a second time.
This can lead to regression to the mean, and standard deviation if you like.
See – stats isn’t boring!

The World Is Maths

Sir Francis Galton was a statistician in the 19th century. Thanks to him we have concepts such as correlation and standard deviation.  Galton, it would seem, thought through the filter of statistics, a genius who produced hundreds of papers and books on fields as diverse as meteorology, historiometry and psychometrics and who pioneered the use of questionnaires to gather better information for his statistical analyses.

Last week, at my school’s Open Evening, we conducted a mathematical experiment based on one of Galton’s observations.  

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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!

The Power of Boxes and Circles

Here is a presentation that I gave at today’s #teachmeet held in Oakwood School, Surrey organised by Paul Collins, @mrprcollins.

I have been using this technique a lot recently so I collated some example questions from Year 1 to Year 12 (!) really just to show how versatile this technique can be. Here is a sample:

The full set are here as a pdf . For any Lyx.org fans out there, the original Lyx file.

 

 

 

 

 

Reflections on Pythagoras

I was with a Year 10 class doing Pythagoras’ Theorem recently.  This was a low attaining group who had encountered it before but were shaky.  They had recently been doing rounding to d.p. and s.f.  They had made progress with that but were getting a bit bored so the teacher decided to go over Pythagoras which I thought was a nice way of interleaving topics.  Calculator answers needed to be rounded (for those triangles which were not pythagorean triples).

It got me thinking about the knowledge and understanding required to apply Pythagoras’ Theorem and also about planning topics. In this lesson, they stuck to finding the hypotenuse and didn’t do any problems where they had to find the shorter side. This decision was made by the teacher because she knew the class and the context. By the end all students had practised it a few times and had built self-belief that they could do it. In other contexts, another teacher might decide to introduce both cases side-by-side.

Is there a “best” approach?  Has there been educational research looking into such a finer point of teaching this one topic? If there has, please point me to it because I wouldn’t have the time to look for it.  Nor the belief that if I did eventually find something, reading it would actually improve the outcome of my learners.  There is no single perfect way of teaching any topic. As teachers, we need to keep our eyes and minds open to approaches that we hadn’t thought of or used before. But we must not agonise over trying to find the holy grail, the golden nugget that will suddenly enlighten our students.

As I wasn’t actually teaching the lesson, just helping out, I had time to draw a quick mindmap in the lesson.  I was trying to put myself in one of the students shoes.

What are the things I need to know, understand and be able to do to be successful in this topic?

This, fundamentally is what they care about.

I’ve included a picture of my mindmap here, not because it is a stunningly useful breakdown of this topic, but just to show the complexity of what our students need to learn.  And this is without much page space taken up on the “And then…” topics of proofs, pythagorean triples, etc., some may say the “interesting bit”.

I am going to try to do this mindmapping exercise more often in my lesson planning. I found it quite easy to do when I was sitting in this lesson, but I find it harder when faced with a blank sheet and maybe some resources and ideas that I might have used before.  The point is, I think, not to agonise over the perfect mindmap showing the perfect sequencing of “learning nuggets”. It needs to be done whilst thinking about the class and the context. And if there is ever time for such luxuries, doing it with a colleague surely makes the process more satisfying and enjoyable.

 

The ones that didn’t get away

This week, most of Year 7 are away on a residential trip leaving 20 pupils who, for whatever reason, are in school.  I taught them on Monday and much to my relief they didn’t seem particularly aggrieved by this state of affairs.

I don’t know what they are doing in other lessons, but in Maths, well, they’re going to do some mathematics!

I actually love the freedom of not having to follow a scheme of work for a bit (although I’m sure I would eventually feel lost without it!)  I wanted to do something that was proper maths but felt a bit different from a normal maths lesson.

As luck would have it, I attended an fantastic session run by John Mason and Anne Watson on Friday.  I’m not going to attempt to write about that session. All I will say is that if you ever get the chance to meet these two, you should.  A great opportunity to reflect deeper on the nature of mathematics and mathematics teaching.

So, like all things you pick up at CPD, I decided I would use something soon before I forgot it.  And here is what occupied us for nearly an hour last thing on Monday afternoon.

This is a classic low ceiling, high threshold task.  Everyone can see some patterns in this, some will notice that the first term of the nth row is n², some will determine that the last term in every row is…

On Friday, however, I picked up something more valuable than this piece of mathematics.  I got to play around with techniques of how to present it to students.  To start with a blank board and write very deliberately and quite slowly stopping and asking questions like, “What am I going to write next?”, “What should we do now?”, “What are you thinking now?”

I really felt that we began to see the power of getting students to explain their reasoning, especially in a mixed group like this.

So, moving on.  What else can I do in the remaining lessons I have with this group?

My first instinct was to look at my own blog under investigations.  This is, after all the main reason I do this blog – as a reference for me!

Some good stuff, and it triggered memories of lessons I would otherwise have forgotten about.  And then I thought, what about those CPD events I went on in the years before I started this blog?  I looked back through my notes and found this from the 2012 ATM conference.

 

Factor Game

Write out numbers 1-12.

1st person choses a number.  2nd person gets all the remaining factors of that number.

1st person choses another number (but it has to be one that still has factors remaining).  2nd person gets all the remaining factors of that number.  Keep going until there are no numbers left to chose that still have factors – the second person gets all the remaining number.  Sum the totals of each – highest score wins.

Nice way to practice factors of a number as well as introduce Gauss’s trick to sum an arithmetic progression of n terms: n(n+1)/2 as you only need to calculate one of the totals, you can then take it away from the sum to get the other total.  So, in this case the total is always 78 because the sum of the first 12 numbers is 78 and that’s all we are doing.

e.g.

Person 1	Person 2
11	1
4	2
9	3
10	5
12	6
	7,8
Total = 46	Total = 32

Can you always win? What is your strategy?  What’s the perfect game, i.e. highest possible score.  Try it for other totals.

I tried:

• 8.  Nice, easy one to start with.
• 16 a bit more tricky, but the player choosing the numbers should still win
• 20 was interesting as it was a draw – 95 each!
• 19 was easy to win

 

Mathematical Behaviour

An interesting Twitter chat tonight on behaviour in Maths lessons. And my first time hosting!

Lots of questions and even a few answers in the Storify of the chat here.

 

http://tinyurl.com/zztmfqr

Rounding Errors

To investigate rounding more deeply, here is a structure for some questions that hopefully prompt the question, why does this happen?

Here are three examples as a pdf and as a Google doc.  It doesn’t happen in the first example, but it does in the second two.

Next, see if students can create their own to investigate why this happens.  I have created a blank sheet (3 on each page).  Again, as a pdf document and as a Google doc.

 

 

Formal methods – the devil’s work!

I’ve been thinking a lot recently about “formal methods”, e.g. Column Addition or Long Multiplication.

As secondary maths teachers we don’t pay much attention to these.  I assess their ability to carry out the procedure and have sometimes attempted to teach long multiplication explicitly. But more often than not it is assumed that these methods have been learnt at primary school have been pretty well-practised and are therefore secure.

We have been doing a lot of number work with Year 7 so far this term including exploring in some detail the Laws of Arithmetic.

Here is an example of the type of question we have been looking at:

127 + 54 + 73

Most of my students’ first instinct with a question like this is to draw up a nice column addition and solve it.  Would you agree that most students do this?

I want them to look at the structure of the numbers first.  To realise that addition is commutative and that it’s much easier if we add the 127 to the 73 first because that’s a number bond to 200.

I am having 2 issues with this approach:

1. My students are not convinced that this is a valid exercise.  Basically they think it is contrived (which, of course all exercises are). They think I have chosen numbers to make this work, it will only work for those certain numbers and basically they feel like I have tricked them.
2. Some students perceive the message that the formal methods they spent hours practising at primary school are now not the way we do things at secondary school. I have said that this is not the case, they are not the devil’s work, and that those methods still have a place. But we need to have smarter ways to work. The fact that we have spent many lessons not solving calculations using formal methods leads me to believe that they are confused about what I want them to do.

Fundamentally I want my students to rely less on the authority of the teacher and rely more on their own understanding.  I want them to see the structure in everything they do, to be certain that what they have done is valid because the mathematics tells them it is so, not because I tell them that they are right.

How should we bridge the gap between informal methods which require (and develop) depth of understanding and formal methods which are efficient and accurate?

I have an idea using manipulatives.  I am not sure if it will work, but if I get chance to try it with the class, I will write about it again later.

In the meantime, does this resonate with anyone else? How do you bridge the gap? I’d love to read your comments.