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PRACTISE SOLVING QUADRATIC EQUATIONS BY FACTORISING WHILE WORKING ON A DEEPER PROBLEM TODAY YOU WILL... 46 www.teachsecondary.com LESSON PLAN CONNECTED QUADRATICS STARTER ACTIVITY Q: Do you remember how to solve a quadratic equation like: x 2 – 7x – 30 = 0 ? The easiest way is to factorise the left-hand side: (x + 3)(x – 10) = 0 (The 3 and the –10 come from the fact that their sum is –7, the coefficient of x, and their product is –30, the constant term.) Now if two numbers multiply to make zero, either the first one is zero or the second one is zero. So in this case, either x + 3 = 0 or x – 10 = 0, giving the two answers x = –3 or 10. Q: What if I change the starting equation by sticking a ‘2’ on the front: 2x 2 – 7x – 30 = 0 ? You can still solve it by factorising, but the factorising is harder now because the left- hand side is non-monic (i.e., the coefficient of x 2 is not 1). Sometimes learners do this by trial and error, but a more systematic method is to place x’s in two brackets and write the coefficient of x (which is 2, here) three times: (2x )(2x ) 2 Then put in two numbers that have a sum of –7 (the coefficient of x, as before), but this time they need to have a product of 2 × (–30) = –60, the product of the coefficient of x 2 and the constant term. The two numbers required here are 5 and –12, so you get: (2x +5 )(2x -12 ) 2 which simplifies to (2x + 5)(x – 6). There are other perfectly good ways of factorising non-monic quadratics, but this is a method learners (or even teachers) may not have seen [Note 1, below]. So (2x + 5)(x – 6) = 0, meaning that either 2x + 5 = 0 or x – 6 = 0, giving x = –2.5 or 6. Learners could try solving x 2 + 17x + 30 = 0 and 2x 2 + 17x + 30 = 0 to see if they can do them themselves. (The answers are x = –15 or –2 and x = –2.5 or 3, respectively.) How can learners become confident and fluent with techniques such as factorising quadratics without simply ploughing through pages of exercises? One way is to offer a scenario in which they need to construct their own quadratic equations in order to solve a bigger problem. Lots of practice happens along the way, while something a bit more interesting is going on. Q: Along each side of the triangle, make the two quadratic expressions equal and solve the equation. CONNECTED QUADRATICS 1 When learners are bored in maths lessons they sometimes ask, “When will we ever actually use this?” But even topics that are unlikely to feature in everyday life can still be fun for learners to work on if they experience them as a puzzle to solve, says Colin Foster5x 2 + 8x + 2 3x 2 + 5x + 1 2x 2 + 9x + 6 MAIN ACTIVITIES 1 This method comes from Lyszkowski, R. M. (1999) A simple method for factorizing expressions of the form ax 2 + bx+ c, Mathematics in School, 28, 34.

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Page 1: LESSON PLAN CONNECTED QUADRATICS - … Teach Secondary, Connected Quadratics.pdf · LESSON PLAN CONNECTED QUADRATICS ... experience them as a puzzle to solve, ... improvement practice

PRACTISE SOLVINGQUADRATIC EQUATIONSBY FACTORISING WHILE

WORKING ON A DEEPER PROBLEM

TODAY YOU

WILL...

46 www.teachsecondary.com

LESSONPLAN

CONNECTEDQUADRATICS

STARTER ACTIVITY

Q: Do you remember how to solve aquadratic equation like:x2 – 7x – 30 = 0 ?

The easiest way is to factorise theleft-hand side:

(x + 3)(x – 10) = 0

(The 3 and the –10 come fromthe fact that their sum is –7, thecoefficient of x, and their productis –30, the constant term.)

Now if two numbers multiply tomake zero, either the first one iszero or the second one is zero.So in this case, either x + 3 = 0 orx – 10 = 0, giving the twoanswers x = –3 or 10.

Q: What if I change the startingequation by sticking a ‘2’ on the front:2x2 – 7x – 30 = 0 ?

You can still solve it byfactorising, but the factorising isharder now because the left-hand side is non-monic (i.e., thecoefficient of x2 is not 1).Sometimes learners do this bytrial and error, but a moresystematic method is to place x’sin two brackets and write thecoefficient of x (which is 2, here)three times:

(2x )(2x )2

Then put in two numbers thathave a sum of –7 (the coefficientof x, as before), but this time theyneed to have a product of 2 × (–30) = –60, the product ofthe coefficient of x 2 and theconstant term. The two numbersrequired here are 5 and –12, soyou get:

(2x +5 )(2x -12 )2

which simplifies to (2x + 5)(x – 6).

There are other perfectly goodways of factorising non-monicquadratics, but this is a methodlearners (or even teachers) maynot have seen [Note 1, below].

So (2x + 5)(x – 6) = 0, meaningthat either 2x + 5 = 0 or x – 6 = 0,giving x = –2.5 or 6.

Learners could try solving x2 +17x + 30 = 0 and 2x2 + 17x + 30 =0 to see if they can do themthemselves. (The answers are x = –15 or –2 and x = –2.5 or 3,respectively.)

How can learners become confidentand fluent with techniques such asfactorising quadratics withoutsimply ploughing through pages ofexercises? One way is to offer ascenario in which they need to

construct their own quadraticequations in order to solve a biggerproblem. Lots of practice happensalong the way, while something a bitmore interesting is going on.

Q: Along each side of the triangle,make the two quadratic expressionsequal and solve the equation.

CONNECTEDQUADRATICS

1

When learners are bored in mathslessons they sometimes ask, “When

will we ever actually use this?” Buteven topics that are unlikely to

feature in everyday life can still befun for learners to work on if they

experience them as a puzzle to solve,says Colin Foster…

5x2 + 8x + 2

3x2 + 5x + 1 2x2 + 9x + 6

MAIN ACTIVITIES

1 This method comes from Lyszkowski, R. M. (1999) A simple method for factorizing expressions of the form ax2+ bx+ c, Mathematics in School, 28, 34.

Lesson plan maths Qx_Layout 1 21/12/2012 10:57 Page 1

Page 2: LESSON PLAN CONNECTED QUADRATICS - … Teach Secondary, Connected Quadratics.pdf · LESSON PLAN CONNECTED QUADRATICS ... experience them as a puzzle to solve, ... improvement practice

SUBSCRIBE AT TEACHSECONDARY.COM 47

s

MATHS | KS4

5x2 + 8x + 2

4/3 or –1–1 or –0.5

–1 or 5

3x2 + 5x + 1 2x2 + 9x + 6

2 MAKE UPYOUR OWN

Q: See if you can make up yourown triangle of connectedquadratics. Remember, all threeequations must be solvable by factorising!This is quite hard and will engagelearners in a lot of useful trial andimprovement practice at solvingquadratics (or trying to!) They willrealise the important fact that notall quadratics have real solutions,and even those that do cannot

This triangle of expressionsleads to three equations, all ofwhich can be solved byfactorising. The equationformed along the bottom side ofthe triangle is monic, so learnerscould start with that one, as it isthe easiest; the other two arenon-monic. Learners could writetheir solutions along the sidesof the triangle:

necessarily be solved by factorising.Additional (harder) challenges could be:Q: Can you make all thesolutions integers?Q: Can you make a square ofconnected quadratics, formingfour equations, each of whichcan be solved by factorising?Q: Can you make a square ofconnected quadratics, includingthe diagonals, so that there aresix equations, each of which canbe solved by factorising?

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48 SUBSCRIBE AT TEACHSECONDARY.COM

SUMMARY

LESSONPLAN

HOME LEARNINGLearners could try to construct atriangle in which one of theexpressions is quadratic and theother two are linear (i.e., so-manylots of x, plus or minus a number;no x2 term). This should lead totwo quadratic equations and onelinear equation, all of which shouldgive integer answers.

+ STRETCH THEM FURTHER

+ THE LESSON BEGAN WITH THEFACTORISABLE EXPRESSION x2 – 7x – 30, WHICH WAS ALSOFACTORISABLE WHEN A ‘2’ WASADDED ON THE FRONT TO GIVE2x2 – 7x – 30. LEARNERS MIGHTLIKE TO TRY TO FIND OTHERMONIC QUADRATICS WHICH AREFACTORISABLE BOTH BEFOREAND AFTER STICKING A ‘2’ (ORPERHAPS SOME OTHER INTEGER)ON THE FRONT.

YOU CAN FIND OUT MORE ABOUTTHIS AT FOSTER, C.(2012) QUADRATICDOUBLETS, MATHEMATICALGAZETTE, 96(30), 264–266.

INFO BAR

Colin Foster is aSenior ResearchFellow in the School ofEducation at theUniversity ofNottingham, andwrites books formathematics teachers(www.foster77.co.uk)

+ ABOUT THE EXPERT

Learners might like to designa clear and attractive posterto illustrate how to solvequadratic equations byfactorising. They could tackle monic or non-moniccases, or both.

3 POSTER EXPLANATIONS

+

You could conclude the lesson with a plenary in which learners share their creations and show thedifferent expressions that they have constructed. How did they go about it? What problems did theyencounter? Could they explain to someone else how to do it?

MATHS | KS4

+ ADDITIONALRESOURCES

+ A REALLY NICE RESOURCE,PARTICULARLY FOR USE ON ANELECTRONIC WHITEBOARD, ISTINYURL.COM/CBYW4YX THERE ARE SOME SUGGESTIONSABOUT WAYS OF USING IT ATTINYURL.COM/CZUUGLU, BUTTHE BASIC IDEA IS FORLEARNERS TO CONJECTUREABOUT THE CONTENTS OF THECELLS IN THE GRID, BASED ONTHEIR AWARENESS OF THESTRUCTURE, AS OTHER CELLSARE REVEALED. EACH CELLCONTAINS A QUADRATICEXPRESSION IN BOTHFACTORISED AND EXPANDED FORM.

(With additional thanks toDougald Tidswell)

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