Is this a graph of a quadratic function?

Here’s an idea that came to me in Anne Watson’s plenary today at the MEI conference in Keele.  I think it’s quite challenging but I haven’t used it with any classes yet, so have no idea really!

I expect students to think of translating the graphs along x and y so that the turning point ends up at the origin.  Then they can test whether y = x² .  What other methods might you use? What other prompts might you give  to learners that are stuck? What additional questions could you ask?

Here is the above image as a pdf.  I created it in Excel here as a scatter graph with a smoothed curve, so can tweak the values if you wish.

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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.

 

Algebra

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

The paper napkin trigonometry trick with a smattering of Pythagorean triples.

Take a piece of paper and do the following:

1. Make it into a square (interesting discussion on best way to do this).
2. Fold in half then unfold so you have created crease along a vertical line of symmetry
3. Then take any corner and fold to the midpoint of the opposite edge. Press down to make a crease along the fold line
4. Unfold and now investigate all the triangles you have created, i.e. can you work out their lengths?

Here are some pictures, which also give some hints, although not a complete solution.  The result is very satisfying although I would love to find a way to show this that doesn’t require reams of algebra.  Any takers??

Going off piste with the difference of two squares

Sometimes it’s worth taking a risk and changing course of a lesson halfway through. And sometimes it pays off.

Today’s lesson was supposed to be about algebraic proof and we started with these nice questions from Don Steward. I thought they looked like good practice for multiplying out double brackets at the same time as introducing algebraic proof.

By the way question 3 is particularly tricky.

Following on from the previous lesson, lots of them started by trying values for n.  Great to then have the discussion on what makes a proof vs. an example.

We got into a good discussion on Question 5 and I wanted to know if they were familiar with the difference of two squares.  This is where the lesson changed course completely.  None of them could tell me what it was called but I got the sense that they had seen it before. So next, I wrote these questions on the board:

Factorise:

1. n² – 9
2. 4n² – 25
3. 81 – n²
4. 100 – 81

They raced through these because they had all spotted the short cut.  So next, I put this up, again from Don Steward:

Which followed on nicely from the final of the 4 questions I had put on the board.  Many of them initially struggled to see the link immediately or see the pattern in the numbers.  But with a bit of time and just the right amount of help (i.e. not much actual help, just encouragement!) they started to find others.  I heard them forming statements like: “I need to number that multiply to give 2016, then I need another two numbers that add to give one of those number and subtract to give the other.”

About 5 minutes before it was time to pack away there was a great buzz in the room as 3 students found some other numbers and then the race was on. Lots of solutions started coming, but nobody got all 12 (including me!).

I went away and built a spreadsheet to investigate further, but I still didn’t find all 12.  Can you help???

Introduction to Factorising Quadratics

Useful as a starter / drills for a lesson teaching how to factorise quadratics.  Should provide some number confidence before then tackling actual quadratics, although beware it doesn’t just become another method to learn without the understanding.

Using a quadratic and a straight line to do simple multiplication

So, here’s something cool that Johnny Ball showed me that I hadn’t seen before.  I love it when this happens although I fear that as the years roll on and I become a wizened old maths teacher it may happen less and less…

Anyway, here’s a picture to explain, click on it to move the points:

Multiply the moduli of the x-coordinates of the two points where the line intersects the curve and you get the y coordinate where the line intersects the y-axis. (Or another way of looking at it, multiply the x co-ordinates and you get the negative of the y-axis intersect.)

Neat.

The proof for this is very satisfying and would make a nice extension exercise for Year 12 C1 class. Here are my scribbles. Enjoy!