Category Archives: working systematically

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:

download

…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.

 

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.

Feb-13-2017 09-26-28.gif

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.

screen-shot-2017-02-13-at-10-09-23So 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.

Bus Stop Division

Here’s a big number:

screen-shot-2017-01-31-at-06-48-48

Try different single-digit divisors.  No remainders.

This is an example of purposeful practice – exposing the wonder of mathematics whilst providing a reason to practise lots of  of bus stop division.

You might want to start by asking pupils to come up with their own dividend “in the tens of millions” and try different divisors. (Here for a quick primer on the mathematical language.) Inevitably they will end up with remainders, which they may or may not carry into decimal places. Then let show them this “magic” number.

Questions to ask:

  • What divisors does this work for and why? (Purposeful practice)
  • What other dividends could I make like this? (Purposeful practice + reasoning)
  • What smaller dividends could I make like this? (reasoning)
  • What is the smallest dividend I could make that all numbers 1-9 will divide into without remainders? (reasoning)

Whilst I would want everyone in the class to understand the reasoning through a whole-class discussion, you may have some learners who need the practice on bus stop long division and spend most of their time doing this. Those that are confident with this technique can spend their time exploring deeper into the structure of the number.

Whilst we are on the subject of “Bus Stop”, maybe this technique actually has nothing to do with standing in line waiting for a bus:

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:

checker_700

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.