Responding to a Twitter chat we were discussing the splitting the middle term method (which some students have difficulty with); if one must have a recipe to follow – try the box method. I always encourage students to check coefficients first, if the coefficients of x2 and the constant are prime they clearly do not need elaborate methods.
Working on Quadratic Grids from Underground Mathematics will help students develop and understand the method.
For instructions on the method:
Quadratic Factorisation Box method (pdf file)
An introductory post on Underground Mathematics is here and you can view all posts in the category Underground Mathematics.
Unless I’m missing something the box method doesn’t seem to work for 6x^2+33x-63?
A great example Vince – I have added your example to the document – I should have said this originally – take out any common factors first.
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I teach it by inspection, (or by the method above). Some of my 6th formers have been taught the splitting method – which they prefer.
Factorise 12y² – 20y + 3
= 12y² – 18y – 2y + 3 [here the 20y has been split up into two numbers whose multiple is 36. 36 was chosen because this is the product of 12 and 3, the other two numbers].
The first two terms, 12y² and -18y both divide by 6y, so ‘take out’ this factor of 6y.
6y(2y – 3) – 2y + 3 [we can do this because 6y(2y – 3) is the same as 12y² – 18y]
Now, make the last two expressions look like the expression in the bracket:
6y(2y – 3) -1(2y – 3)
The answer is (2y – 3)(6y – 1)
this eg from http://www.mathsrevision.net/gcse-maths-revision/algebra/factorising#BgyCbkEThspvYklO.99
Inspection is my favourite method and I’d be a little worried by students who wanted a recipe for simple cases. If one must have a method then I think the box method is more memorable than the splitting the middle term method which for many students is simply a string of rules to get muddled up with!
Well, I do it this way. To factor 12y² – 20y + 3, first multiply the number of the second degree term by the constant (12 x 3 = 36), and write.
y² – 20y + 36
Then factor it.
(y -18)(y – 2)
Now, you need to produce “12y²” and “3”. To be able to do that, divide the 2 in (y – 2) by 2 and multiply it back to the left hand y, and divide 18 in (y – 18) by 6 and multiply it back to the right hand y.
(2y – 3)(6y – 1)
But if you do it that way ,the value of the expression changes half way through.
This is a bad mathematical habit to get into.
I don’t really think that this method is “bad”. As long as the method is efficient and can be proven mathematically, it is considered a good solution. I specialize in number theory and combinatorics, so my opinion may be quite biased. As you may know, in those subjects, there can be literally more than a dozen solutions for solving a single problem and you would get the same answer if you do all your calculations correctly.
My point is that it’s alright to use any valid approach for solving a math problem provided that you know what you are doing… and I’m sure that I know what I’m doing.
Personally I have no objection to that method. I just think that changing the expression from 12y^2 to y^2 can confuse weaker students – the type who simplify x/3x to give 1/2x….
I understand your point. That’s why I teach several methods to my students and I would tell them to choose the method that’s easiest for them. So far, it was quite effective.
I think teaching more than one method (for any topic) and letting students choose one that works for them is a generally good idea, though I would aim for understanding and efficiency! I try to show that methods are actually equivalent too.
All students learn math differently. Some prefer very detailed methods while others prefer shortcuts. And of course, some conceptual learners prefer to solve problems using geometric and visual models (like the box method for factoring).
I personally prefer shortcuts but I can’t force them to use those shortcuts if they find the other methods more intuitive. So, I don’t have any problem no matter which method they use as long as they understand what they are doing.
And of course, showing them multiple ways of tackling a problem can help them realize that math is not a subject that hinders creativity.
So important for teachers to recognise that students learn differently and acknowledge what works for them, whilst striving for understanding of course. I think understanding can sometimes come much later actually – one of my favourite quotes:
“Young man, in mathematics you don’t understand things. You just get used to them.”
― John von Neumann
I’ve been teaching this for a couple of years now. It works well.
Good to have that confirmed Steven. I still like the just work it out approach where possible but this seems good where there are just too many possibilities!
The good thing about the box method is that it automatically double-checks that you have the right answer.
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