CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE

Description and purpose: It is appropriate for the work in this chapter that students know how to find the points of intersection of a line and a curve, as these will be related to conditions for the nature of the roots of quadratic equations. The methodologies used to solve simultaneous equations are standard. These methods are tried and tested and students usually understand them well and use them with proficiency. However, we all forget things from time to time, and so the Coursebook offers plenty of revision of the key concepts. PowerPoint 2 recap a animates the recap of solving a pair of linear simultaneous equations at the start of the chapter in the Coursebook. PowerPoint 2.1a starts with the example in the Coursebook at the start of section 2.1 and then works through worked example 1. These techniques are sufficient to be able to solve simultaneous equations where at least one equation is linear and so lead into Exercise 2.1 of the Coursebook.

Students need to be familiar with solving simple simultaneous equations where neither equation is linear. PowerPoint 2.1b has some examples of these and includes an example where students are asked to solve a quartic equation that is quadratic in x 2. This can be used as an introduction for Chapter 5 section 5.5 or simply as a forerunner to the ideas which are explored more fully in Chapter 5.

Differentiation:

Support:

• Some students try to use the elimination method, when the method of substitution is far simpler when solving simultaneous equations. If this is the case, try to encourage your students to stick to one method of solution. Make sure that they know that, at this level, substitution is very much more useful as a method as it is more universal.

• Try to ensure that students use their calculator as a checking tool. If they have made an error, work with them through their solution to help them find it and correct it successfully.

Challenge: Some more challenging material is also provided in the exercises, to allow some students to develop their skills; for example, Exercise 2.1 Q19 to Q26.

Assessment for Learning: As well as the discussion opportunities which will naturally arise through the use of the resource materials and when your students are working through questions, assessment for learning can be carried out using a lesson review by students. At the end of the lesson, ask them what they have learned. Write their responses down for them to refer to in future work. This could then be used as part of a revision session later in the course.

5 Nature of roots

Learning intention:

• Know the conditions for ax 2 + 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.

Resources:

• PowerPoint 2.3b: Roots and intersections

• PowerPoint 2.5a: Worked examples 8 to 12

• PowerPoint 2.5b: Connecting the nature of roots with intersections of graphs

• Coursebook Exercise 2.5

• Coursebook Exercise 2.6

Description and purpose: You may choose to link the final sections on the nature of roots and the intersections of lines and curves. The resources and the Coursebook enable you to choose to separate or combine them as you prefer. PowerPoint 2.5b looks at six specific curves, their graphs and hence their roots and then identifies the discriminant in each case. Students are asked all through to explain what is happening on the basis of what they can see in the formula. The results are summarised. In the final three slides, more consideration of the connection between roots and intersections is given. Worked examples 8 to 12 have been combined in one resource, PowerPoint 2.5a. Again, you may wish to use part of it and come back to it at another point. It is not suggested that you use it all in one session. Exercise 2.5 of the Coursebook may be worked through after worked example 9, and Exercise 2.6 follows once the PowerPoint is complete, or both exercises may be looked at after the full set of worked examples in the PowerPoint has been considered.