This unit supports students to form and use quadratic equations to solve ratio problems such as: The Square Xs Problem A square has an area, in m², that is a number nine less than six times the length of one side (in m). Find the dimensions of the square.
Many algebraic challenges are reducible to a quadratic problem. The techniques of solving for the two roots of a quadratic take many years to master. At the entry level, students should be able to find the two real roots of a factorised quadratic. The aim of this unit is to develop students' abilities to form and solve quadratic equations from word and context based problems.
The number and algebra skills embedded in this process of solution include knowledge of square numbers, factors, multiples, sums and differences of commonly used numbers. The ability to solve simple linear equations is also requires. If the students' prior knowledge needs some reinforcement, then many such exercises are available in textbooks and worksheets. Interspersing the challenges outlined in this unit with such reinforcement activities would be beneficial.
It is hoped that the students can be encouraged to appreciate the elegance of factorising to solve. To do this, it is worthwhile regularly checking back with groups and/or individuals in the class; "why do we write the equation in the form (x-a)(x-b) = 0" or "why do we write the equation in the form something times something = 0"? The answer should be along the lines of "Because one of the factors must be zero, so we can use this idea to write equations, x-a = 0 or x-b = 0". Focussed conversations around the algebra task a student is working on are vital for establishing deep understanding.
Because most of the calculations in this unit are fairly straightforward we think it is in the students’ best interest to not use a calculator.
Session 1
The main aim of this session is to remind students that there are two solutions to a quadratic. Underlying this is the need for students to develop the habit of factorising quadratics and recognising the difference between two squares. They should have prior knowledge and experience of expanding and factorising quadratic expressions.
Teachers’ Notes
What is the square root of 25? (Is there only one answer to this question?)
What happens if you square -5? (What is -5 x -5?)
There are two square roots for any number. Only a few special numbers have two real roots. In this unit we will focus on showing this algebraically, rather than just using guess and check. We need to treat the problem as a quadratic, put everything on one side of the = sign and factorise to solve for x.
The problem is what is x when x² = 25
x² = 25
x² – 25 = 0
(x-5)(x+5) = 0
x-5 = 0 or x+5 = 0
x = 5 or x = -5
Many students will invariably suggest that it was much easier just to guess the answers. You might suggest that this way you know there are two (and only two) solutions.
Teaching Sequence
Session 2
In this session we look at quadratics that can be factorised in the form (x-a)(x-b)=0 and use these to solve problems. The main aim of this session is to remind students that if a product is zero, one of the factors must also be zero. Then, to use this fact, to form and solve factorised quadratic equations.
Teachers’ Notes
The aim here is to be able to tackle simple quadratics, that arise in word problem situations and that can be factorised in the form (x-a)(x-b)=0. The students may be able to form a quadratic equation, given a word problem, but should be able to put every term on one side of the equation, factorise and solve.
Problem 1: The product of two consecutive whole numbers is 42, what are those numbers?
There are lots of ways of doing this problem.
Method 1: You might draw up a table of factors of 42 and find which have a difference of 1.
Method 2: You might multiply pairs of consecutive whole numbers until you have success.
Method 3: You might write the problem as an algebraic equation and then solve this. You will need to show the steps required to solve this problem.
x(x+1) = 42
x²+x = 42
x²+x-42 = 0
(x+7)(x-6) = 0
x+7 = 0 or x-6 = 0
x = -7 (not a whole number, so not a solution) or x = 6 (making the other number 5 or 7)
Of these possible solutions sets for the pair of whole numbers, (6,5), (6,7), which gives a product of 42?
So the possible solutions for the two numbers are 6 and 7
Discuss the pros and cons of each method. Can you be sure that your method gives all the possible solutions?
What if the numbers are bigger or fractions, would your method still be the best method to use?
Praise any correct method that works but keep leading them back to the algebraic method.
If the students have not yet mastered factorising and expanding quadratic expressions, they will need to spend time on these techniques at this stage. There are many available reinforcement exercises in textbooks and worksheets, or ten expansion and ten factorising tasks could be quickly put up on a whiteboard. Once the students are familiar with expanding and factorising, problems such as the one outlined above should be relatively straightforward to solve.
Problem 2: Two integers have a difference of 10 and a product of -16. What are those two integers?
There are lots of ways of doing this problem.
Method 1: You might draw up a table of factors of -16 and find which have a difference of 10.
Method 2: You might multiply pairs of integers, with a difference of 15, until you have success.
Method 3: You might write the problem as an algebraic equation and then solve this. You will need to show the steps required to solve this problem.
x(x+10) = -16
x²+10x = -16
x²+10x+16 = 0
(x+8)(x+2) = 0
x+8 = 0 or x+2 = 0
x = -8 (making the other number -18 or 2) or x = -2 (making the other number -12 or 8)
Of these possible solutions sets for the pair of integers, (-8, -18), (-8, 2), (-2, -10), (-2, 8), which give a product of -16?
So the possible solutions for the two numbers are (-8, 2) and (-2, 8).
Discuss the pros and cons of each method. Because the product is negative, what does that tell you about the factors? Can you be sure that your method gives all the possible solutions? What if the numbers are bigger or fractions, would your method still be the best method to use?
Praise any correct method that works but keep leading them back to the algebraic method.
Depending on skill level and understanding, further word problems or reinforcement exercises of forming, factorising and solving quadratic equations, found in textbooks or on worksheets, should be given to the students to work on individually or in pairs.
Teaching Sequence
Session 3
Here we look at simple problems like the Square Xs, where there is a unique solution. In these cases, the quadratic is a perfect square.
Teachers' Notes
The basis for the next two sessions is to familiarise students with factorised quadratics so that they can distinguish between quadratics with two real roots, the difference between two squares and perfect squares.
Quadratics with two real roots factorised: (x-a)(x-b) = 0 and expanded x²-(a+b)x+ab = 0
The difference between two squares factorised: (x-a)(x+a) = 0 and expanded x²-a² = 0
Perfect squares factorised (x-a)² = 0 and expanded x²-2ax+a² = 0
An example of a perfect square problem is:
Perfect Square Problem Two integers have a difference of 8 and a product of -16. What are these numbers?
The ability to recognise such quadratics is a very useful skill for solving advanced algebraic problems that the students will encounter in later years. The steps needed to solve such problems are:
x(x-8) = -16
x²-8x = -16
x² – 8x +16 = 0
(x-4)(x-4) = 0 or (x-4)² = 0
x-4 = 0
x = 4, so the other number is 4-8 = -4
The steps above are applicable to all factorisable quadratics. It should be the students' first approach to solving a problem, as soon as they have identified that it is indeed a quadratic. Once the students have grasped the concepts of these individually, it would be beneficial to provide a mixed group of problems in reinforcement activities, such as those found in textbooks, on worksheets or that can be written on the whiteboard.
Note: Since the numbers here are relatively straightforward we do not encourage the use of calculators in this work.
Teaching Sequence
Session 4
Here we look at students using their knowledge of quadratics to solve problems independently.
Teachers’ Notes
This session is based on problems reducible to a simple quadratic equations. Students should be encouraged to use their own method to find the solution, then the range of techniques can be discussed in the context of the initial problem.
Investigations involving quadratics can be found in the Rich Learning Activities.
An activity involving the translation of a written task into an algebraic equation is Three in a row.
A problem that can be modelled practically (to increase the number of ways a student could approach the problem, is the Parabolic Investigation.
An extensive algebraic investigation is Overlapping Powers.
For any of these activities, the students should be encouraged to solve the problem using whatever approach is most comfortable for them. This may be algebraically, with arithmetic methods, or by guess and check.
Teaching Sequence
Session 5 (Extension)
Here we look at 'completing the square' to solve quadratic. This is aimed at extending students' appreciation of the elegance of algebra and is the basis for deriving the quadratic formula that students will be introduced to in Level 7. At this level (5), the examples need to be carefully scaffolded to be as simple as possible. Completing the square allows students to find the solutions for a quadratic that is not easily factorised – but it for such cases, the students need to have an appreciation of square roots and expressing such irrational numbers in surd form.
Teachers’ Notes
This is aimed at extending students' appreciation of the elegance of algebra and is the basis for deriving the quadratic formula that students will be introduced to in Level 7. At this level (5), the examples need to be carefully scaffolded to be as simple as possible, eliminating any need for calculators.
Sample Problem 1 A teacher has given a student a problem that says x² - 8x = -7
Sample Problem 2 – not easily factorised A teacher has given a student a problem that says x² + 6x = 5
Teaching Sequence
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