A decimal investigation

Inspired by playing the excellent Sumaze! 2 game from MEI, here is an investigation that aims to provide some purposeful practice on decimals. The aim is to provide an accessible entry point for all learners with opportunities for depth through generalisation. This slide presentation steps through it although exactly how you move from one part to the next will, of course, depend on the class.

I have included solutions in this spreadsheet although I would be hesitant to display them in this form, as I would prefer that the results are found and discussed as we go along rather than just revealing them at the end.

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When is a Quadratic “factorisable”?

There are 3 standard ways of solving quadratic equations once they are in the form:

ax² + bx + c = 0

They are:

1. Factorise
2. Complete the square
3. Use the formula

I think I generally teach them in that order probably without much thought as to why. I guess the formula needs to be derived by using completing the square and factorising seems to follow on from multiplying out double brackets, which comes before all of this.  The I question that I sometimes get from students is “what’s the point in learning factorising if the two other methods always work?”.  Well, it’s quicker and you can do it on a non-calc exam is probably a standard response.

But have you tried using the formula without a calculator to solve a quadratic that you know will factorise?  Have a go.  Plug this:

x² - 3x - 28 = 0

into this:

…and solve without a calculator.

It’s a surprisingly satisfying experience, one that I would not want to deny my students.

You’ll need to know your square numbers because b² – 4ac will always give you a square number for quadratics that factorise. But the arithmetic is perfectly reasonable and is likely to be so for most quadratics that can be factorised.

When I did this recently, I had a great question from one of my students, as I was aching my brain trying to make up a quadratic that I knew would factorise.   “If you just picked one randomly, what are the chances that you would be able to factorise it?”

I’ve since had chance to investigate this further.  It’s a great question and there is a lesson in here, or at least an extension question to explore once the fundamentals of using the formula are secure.

I started approaching it by using the formula and focussing on b² – 4ac and what values of a, b and c would yield square numbers.  To simplify the problem, I started with a=1, so I was looking for when b² – 4c = 1,4,9,16,25, etc.

I then looked at it from the other end, i.e. starting with e.g. (x+1)(x+n) what values of a, b, and c are yielded.  Then vary further to look at (x+m)(x+n).  I just started with a few values of n and m to see if I could spot patterns.  I won’t spoil the fun by revealing those patterns, but this is very open-ended and could provide some intrigue for the right learners.

 

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.

Algebra

What other integer values for a and b produce rational values for c?

Cubes Cubed

No 5 in a series of posts based on Thinking Mathematically (1985) by Mason, Burton, Stacey

I might use this task as an introduction before doing anything involving 3D visualisation. I’ve always felt that being able to conceptualise physical shapes is a skill that is very hard to teach in an instructive way. Tasks like this provide experiences that help to develop those skills. Here is the task:

I have 8 cubes all the same size. Two of them are painted red, two green, two blue and two yellow.
I wish to assemble them into one large cube with each colour appearing on each face.
In how many different ways can I assemble the cube?

The first challenge of this task is to interpret what it is asking and construct a mental image of that. The above image may help.

Then the question comes what do we mean by “different” large cubes, which is a conversation starter in itself. If we have the physical cubes in the form of multilink cubes we are likely to see that “different” means that one large cube cannot be simply rotated to form another one. The position of the small cubes actually have to be changed.

This 3D animation created on Geogebra (file here) may also help if real cubes are not available.

As John Mason points out in the book, whichever representation is used, it is important to record carefully the “different” cubes so they can be compared.

I came to the realisation that I need only look at the 4 small cubes on the front face because I can deduce where other 4 cubes on the rear face will be. Because no two small cubes of the same colour can share a face, the small cubes must be arranged so that they on diagonally opposite corners of the large cube.

So then it was a case of working out how many ways there are of arranging 4 things in a 2×2 arrangement where rotation is not allowed. I actually looked at it like this.

I convinced myself there are six ways of doing this.  But I’m still not convinced that this means 6 is the answer to the question.  I feel I would need to construct it to be sure I haven’t got any repeats within these 6.

Chessboard Squares

No 3 in a series of posts based on Thinking Mathematically (1985) by Mason, Burton, Stacey

This is a classic task for working systematically:

It was once claimed that there are 204 squares on an ordinary chessboard. 
Can you justify this claim?

 

I like this way of stating the problem, rather than just “how many squares on a chessboard?” because it gets straight to the nub of the issue – we are looking at different sized squares, some of which overlap.

 

Palindromes

No 2 in a series of posts based on Thinking Mathematically (1985) by Mason, Burton, Stacey

This is the sort of exercise I can envisage taking a number of different paths depending on what my students do with it which is exciting. The book walks through a generalisation by looking for the lowest 4-digit palindromic number, 1001 and then noting that subsequent palindromic numbers can be found by adding 110. Since 1001 and 110 are multiples of 11, then all numbers in this series are multiples of 11. However, this series misses out other palindromic numbers, e.g. 7557 so we need to refine it further.

I am intrigued to see if this is indeed a path my students would follow or if we would discover something else in these numbers. Depending on the class, I might start by asking “how many 4 digit palindromic numbers are there?”  Before getting into the general, I would see this as an opportunity for purposeful practice of long division if that was something that my students require.  Some students might need a fair amount of direction to reach a proof, but I would aim to make sure that all students left this lesson with an appreciation of that proof even if I had to lead them through it.

 

Why stop at 3 dimensions?

A friend of mine (who is not a maths teacher) recently sent me this article that Marcus Du Sautoy wrote in 2013 when he was president of the MA. He makes the case for a “Mathematical Literature” GCSE to sit alongside the utilitarian curriculum a bit like English Literature sits alongside English Language GCSE. It should aim to develop a student’s love of mathematics through doing maths for enjoyment to foster an appreciation of our rich and varied mathematical heritage.

At the end of the article Du Sautoy he says he is a mathematician not an educationalist. But I think the example he gives in here is a great one. There is definitely a lesson in it, and not just for higher attaining classes. Once the basics of plotting coordinates in a Cartesian system are secure, I think this would be accessible to any KS2/3 class upwards. But this comes with a health warning: I haven’t tried it yet,  so (as with anything you find on the Internet) proceed at your own risk!

The lesson plan

Prerequisites:

• Pupils need to know how to plot coordinates in 2D (1st quadrant only)
• They need to know what a square is and be able to reason why the basic square has coordinates (0,0), (0,1), (1,0), (1,1).  You could maybe start with something like this from Don Steward, although this is already maybe more than is required.
• They need a systematic way of finding arrangements of things.  e.g. how many ways are there of arranging the letters ABC? ABCD? etc.

And that is about it.

So next, we can introduce the 3D coordinate system.  Visualising things in 3d can be hard; often it is not closely correlated to mathematical attainment. But there are a wealth of options now to show and “spin” a 3D object on a screen. Here is one I created on Geogebra:

I would spend some time on this, looking at systematic ways to make sure we have the coordinates of all 8 vertices.  You might want to take the approach I wrote about here.  I would try to convince students that they could have done this systematically with having had to see the cube in 3D space.  Because what they are going to do next is find the coordinates of an object that they definitely can’t see – a 4D cube.

Although it’s impossible to see in our 3D world, we can use maths to work out the coordinates of each vertex of a 4D cube.  Again, if this is done systematically students will hopefully soon see that the number of vertices doubles each time we add another dimension. Here is 2D, 3D and 4D:

The next part of the article is where it really gets interesting. Because as well as this being an important piece of maths in its own right, it also has an application in computing.  It is used in error correction when sending digital signals. The rules are reasonably straightforward and the article provides an example of a piece of code that contains an error.  Can your students apply the rule to find the incorrect bit?

 

 

Do you really understand addition and subtraction?

This was the question Mark McCourt was getting us to ponder in the first session of Maths Teacher Network.  He started off by running through the classic 1089 “trick” which I have written about before here.

These types of activity are such a powerful way to get students to carry out repeated practice to build fluency. If you handed out a sheet with 20 column subtractions and 20 column additions, you would get groans and do no favours for the general popularity of maths as a subject.  If those repeated calculations have a purpose as they do here, the dynamic and energy in the classroom is completely different.

I’d used this before but had never given it the time that it deserves, nor had I extended it:

• Try it with 4 digits, 5 digits, etc.
• Go back to three digits and try it with a, b, c. In other words prove it algebraically.
• Now try in a different base, e.g. Base 7!

I actually got a bit obsessed with this when I got home.  I do love a good spreadsheet challenge, so I attempted to build something that would provide an algebraic proof of different numbers of digits and bases. Here is the result…

My reflection from this session was that even if I have “seen” something before, have I really investigated it deeply and could I use it more effectively with my classes? To which my answers were no and yes!

Another approach, back to 3 digits. The prime factorisation of 1089 is 9×11×11. Why is this? And what are the prime factorisations of the 4 digit magic numbers. Can these be explained?

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!