Sequences in square numbers

List the first 6 square numbers.

What do you notice about the difference between them?

Express the difference between the square numbers as a sequence in terms of n?

Express algebraically the difference between the nth square number and the (n+1)th square number?

Use you algebra skills to show that they are the same thing.

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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!

Simple sequences starters

Here is a simple task which would work well as a starter which practises negative numbers at the same time as (hopefully) leading into a nice discussion about sequences and term-to-term rules.

I created this on a spreadsheet here, so you can change the questions or the order.  If you are feeling ambitious you could do this with decimals or fractions, or also include multiplication, which would generate quadratic sequences.

And here is another one which basically just seeks to reinforce the idea of what “n” means by substituting it into nth term expressions.

 

 

Generalising with Areas of Circles

So, at #MathsMeetGlyn last weekend Don Steward briefly put this picture up on the screen but then decided to talk about something else as time was pressing.

That’s a bit like a “Wet Paint” sign.  You have to touch it just to see if it really is wet…

So I had a little play with these. I couldn’t find anything about it on his blog but of course there is lots of other lovely stuff there.

A few different questions you could pose here:

• What is the ratio of red area to blue area?
• What is the red area as a fraction of the blue area?
• What is the red area as a fraction of the whole circle?
• What is the blue area as a fraction of the whole circle?

And then of course, can we generalise?  What about when there are n small circles.  What happens as n gets bigger? Why? Can we find a general expression for the area?

Have a go.  It actually turns out to be quite simple but depending on which question you start with, you can get into a lot of practice with ratio and dividing fractions.

Finding the nth term

I’ve used this simple booklet with a range of classes and it just works! Many thanks to my former colleague Nick McIvor for creating it and sharing it.  A well structured set of examples that enables students to see the patterns. I would still teach the nth term more formally after using this, but it’s a great introduction and it builds confidence.

For quadratic sequences, here’s a similar approach from @mathsjem

Triangle numbers

In common with most Maths teachers, I love Don Steward’s site Median for the fantastic range of quality resources and ideas.  All the ideas in post come from here. I have taken some of them and put them into a PowerPoint with some solutions.  As teachers, we should do the Maths that we are asking our students to do in lessons. Which is, I guess, why most of the resources on Median don’t have solutions with them.  However, in the midst of a busy term, lesson planning has to be time-efficient and so I have added a slide in here with some possible solutions (there may be more elegant options).  The idea being that lesson planning is then a case choosing how to present this and involves deleting pieces of this solution which is much quicker that creating them from scratch.

Adding colour to Sequences

After browsing the ever excellent Don Steward’s Median site for sequences, I found this on Linear and Quadratic growths.

It got me thinking about using shapes to represent sequences and in particular using different colours to represent different sequences laid on top of each other.  For example, the following patterns produce a linear sequence.

However, there are a couple of ways you could look at this which combine sequences which are arguably simpler.

Or:

We can then extend this on to quadratic sequences.  This is nice: one way we can see the sequence of square numbers, the other way we can see how multiplying one dimension by the other leading to some brackets which can be multiplied out.

There are a bunch of these in this file.

There are also over a hundred sequences at visualpatterns.org a website entirely dedicated to, well, visual patterns.

Get your students to create their own once they have got the idea.  Could make for some great wall displays!

On quadratic sequences specifically, a nice worksheet here from @solvemymaths.

Finally, back to Don Steward with this which is actually just a series of terms stacked on top of each other.