CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE

Syllabus learning objectives / learning intentions Success criteria

Know the conditions for ax2 + bx + c = 0 to have:

(i) two real roots, (ii) two equal roots, (iii) no real roots and the related conditions for a given line to:

(i) intersect a given curve, (ii) be a tangent to a given curve, (iii) not intersect a given curve.

Background Knowledge

Students understand the relevance of the discriminant and are able to apply knowledge of the appropriate condition to solve simple algebraic problems. Students are able to combine all the necessary skills to solve simultaneous equations and connect the conditions for the nature of the roots of a quadratic equation to determine how a line intersects with a curve.

• The following table details what knowledge it is assumed that students already have from studying Cambridge IGCSE or O Level Mathematics. In Additional Mathematics, it is expected that students will be able to use these skills as part of a solution in a multi-step process, and the interpretation needed to do this should be of a greater challenge than that generally expected in the mathematics course.

What your students should be able to doExamples

Solve simultaneous equations using the elimination method.

Use the elimination method to solve these simultaneous equations.

a 4x + 3y = 1; 2x – 3y = 14 b 3x + 2y = 19; x + 2y = 13 a y = 3x – 10; x + y = –2 b x + 2y = 11; 4y – x = –2 a x2 + x – 6 = 0 b x2 – 10x + 16 = 0 c 6x2 + 11x – 10 = 0 a Write 2x2 + 7x + 3 in the form a ( x + b ) 2 + c b Use your answer to part a to solve the equation 2x2 + 7x + 3 = 0.

Solve simultaneous equations using the substitution method.

Use the substitution method to solve these simultaneous equations.

Solve quadratic equations using the factorisation method.

Factorise and solve these equations.

Solve quadratic equations by completing the square.

Solve quadratic equations using the quadratic formula.

Solve 2x2 – 9x + 8 = 0.

Give your answers correct to 2 decimal places.

• The work in this chapter is essential to the whole course. The skill of solving quadratic equations or of factorising a quadratic expression is required in several other syllabus areas. It is highly recommended that this chapter is covered as soon as possible in the course. The skill of solving a pair of simultaneous equations also appears in other syllabus areas. For example, the work on the straight line, in equations, inequalities and graphs and in sequences and series. Some of the questions in this chapter require the use of skills that are considered in Chapter 6, Straight-line graphs. It may be sensible, therefore, to have looked at this chapter first or to work on them in sections, together.

Background Knowledge

• This chapter starts with solving pairs of simultaneous linear equations by elimination and substitution. Solving quadratic equations is briefly recapped before solving pairs of equations in which only one of the equations is linear is studied. Quadratic expressions and functions are then considered more fully, including the shape of the graphs, maximum and minimum values, symmetry and modulus of the quadratic function. This is all essential to what comes after, that is, the solution of quadratic inequalities and using the discriminant to study the nature of the roots of quadratic equations and the points of intersections of graphs.

Language Support

The definitions of key words and phrases are given in the glossary.

When considering the nature of the roots of quadratic equations it is important to model the correct language for the possible cases. These cases are:

• two roots that are real and distinct (sometimes written as real and different)

• two roots that are equal (sometimes written as real and equal or repeated)

• two roots that are real (this includes those that are real and distinct and real and equal)

• no real roots.

Model this language and these ideas for students as much as possible so that the interpretation needed to be successful is instinctive for them.

Links to Digital Resources

In worked examples 3 and 4, the completing of the square is done using the algebraic structure which has been given and then forming and solving equations. The language used in the worked examples is such that this method is fine, as students are simply required to write down the correct form or find the values of the constants given. However, if students need to show that a quadratic expression has a particular completed square form, then they should not form and solve equations in this way. Students should understand that using what you are trying to show as part of your solution is invalid. They should derive the correct completed square form using an approach similar to that used in the Coursebook prior to worked example 3 or as demonstrated in PowerPoint 2.2b. This is a key and important difference in the language used.

• WolframAlpha has a systems of equations solver and some step-by-step example solutions

• Purplemath has examples for solving systems of non-linear equations by considering graphs

• There are many useful videos on quadratics to be found at The Khan Academy.

• Maths is Fun has some real-world examples of quadratic equations

REAL-LIFE CONTEXT

Simultaneous equations can be used to represent and solve a variety of everyday problems in the real world. For example, deciding whether one mobile phone deal is better value than another, or finding the maximum profit available from making and selling goods. They are used in many applications in the study of various sciences and are an essential tool for any student of science or engineering, for example.

Quadratic equations are everywhere. They are used in business and finance, physics, architecture and the natural world, not just

Common Misconceptions and Issues

algebra classes. To engage your students in this topic, it may be helpful to start this section by looking at the sort of real-world situations that can be modelled by quadratic equations. The properties of the parabola give us satellite dishes and car headlamps, for example. The Sydney Opera House is distinctly parabolic in appearance. The motion of a pebble thrown up in the air and falling to the ground is also parabolic. Many features in design and modelling require the skills that are introduced in this syllabus. Knowing this may help students understand the importance of what they are studying.

Students are expected to have developed proficient algebraic methods of solving equations and inequalities. Graphs support the learning and help understanding, but algebra is the main key to a successful, efficient and, most importantly, accurate solution.

Misconception/issue How to identify How to avoid or overcome

Students are too dependent on their calculator to solve equations and do not demonstrate that they have mastered the techniques in the syllabus.

For example, students often find factors by first using their calculator to find roots and then working back.

Students often need to make a sketch of a function. Students need to be clear that drawing a sketch is not the same as drawing an accurate graph.

Students commonly reach for their calculator to solve quadratic equations. Calculators are an excellent checking tool, but no substitute for showing proper method.

Very often, factors such as 2x − 1 are written as x − 0.5, which is incorrect.

Students who are not sure about sketching often plot points and join them together. This can result in some very poor graph shapes.

Some icons have been used in the PowerPoint presentations to try to indicate when it is a useful time to check your working with a calculator, to allow you to emphasise this with your students.

Students should be clear that when a sketch is needed, the command word in the question will be sketch.

To draw a good sketch, it should be approximately correctly positioned. Any key points, such as intercepts or turning points should be marked if possible. All key features of the curve should be present.

Students should also be clear that when an accurate drawing is expected, the command word is likely to be draw.

This is likely should the graph then be used to solve an equation or inequality, for example.