CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE

Possible functions to draw are

To type these into Geogebra or Desmos, for example, enter y

This activity could possibly lead into: the study of modulus functions whose initial form was quadratic or revision of solving modulus equations.

This activity could be adapted: If technology is not available, draw the graph or graphs on a flip chart or a display board. Also, the exercise could be extended to include simple quadratic functions such as y = x2 , which could be a challenge.

Main teaching ideas

This topic could be taught with or without a calculator.

Much of the simultaneous equation solving students meet in this course requires them to be able to solve a quadratic equation to be able to complete the task. This is why these syllabus areas have been grouped into one chapter. You may choose to start with this, as the Coursebook does, and build on skills your students should already have. Alternatively, you could start with a recap of solving quadratic equations and build on that. Either approach is well supported by the Coursebook and teacher resources.

Students may have a good understanding of the methods used to solve quadratic equations. It is a good idea to make sure of this before progressing through the rest of the material in the chapter. Students need a good foundation on which to build their skills. Some of these ideas will last for more than one lesson. All the suggestions made have assessment for learning activities embedded within them.

1 Quadratic equations and the parabola

Learning intention:

• Solve quadratic equations for real roots.

• Make a simple sketch of the graph of a quadratic function using any roots and the y-intercept.

• Find the maximum or minimum value of the quadratic function f : x ↦ ax 2 + bx + c by completing the square.

• Use the maximum or minimum value of f (x) = ax 2 + bx + c to sketch the graph or determine the range for a given domain.

Resources:

• PowerPoint 2 recap b: Factorising and quadratic formula

• PowerPoint 2.2a: Worked examples 2 and 4

• PowerPoint 2.2b: Completing the square recap

• PowerPoint 2.2c: The parabola and quadratic function forms

• Coursebook Exercise 2.2

Description and purpose: Solving quadratic equations, which is an essential skill for the solution of the simultaneous equations, has been split into two sections. In PowerPoint 2 recap b the methods of factorising and using the quadratic formula are revised. Factorising is demonstrated using a reverse grid approach. This method reduces the amount of purely mental processing and allows the visual to help with thinking. Students will be able to ‘see’ the factors of the first and last terms in place in the grid. These skills should be sufficient for the work on solving simultaneous equations, which could then be studied if you wish. Before moving on to applications of solving quadratic equations, it is sensible for students to investigate the possible shapes of the graphs. This, as well as the different ways of presenting a quadratic function (vertex form, standard form, factorised form) are looked at in PowerPoint 2.2c.

CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE

The method of completing the square is required as a skill in its own right, as well as being a useful method of solving equations. It also is useful for drawing graphs and for finding least and greatest values of functions, for example. A slightly different method to those described in the Coursebook is demonstrated in PowerPoint 2.2b which recaps the method of solving quadratic equations by completing the square using the first two terms of a 3-term quadratic expression only. The method used is called the square and compare method for completing the square. As with all the alternative methods given, it is offered as a useful alternative to support students who have not engaged with other approaches. Worked examples 2 and 4 have been put together in one resource in PowerPoint 2.2a. This is to allow you to dip into it, or not, as you wish. This leads neatly into Exercise 2.2 of the Coursebook.

Differentiation:

Support:

• Factorising using a reverse grid approach reduces the amount of purely mental processing and allows the visual to help with thinking.

• The square and compare method of completing the square also is visually supportive for students who have not engaged with other methods of doing this.

Challenge:

• Rearranging equations which include algebraic fractions and then solving.

• Deriving equations first and then solving − in a real-world context such as business. These could be given as investigation tasks for some students to use to self-study while other students master the more basic skills.

Assessment for Learning: There are many opportunities for discussion using the discussion points in the PowerPoints provided. There should also be opportunities for students to ask questions of each other and of the teacher, whilst working. Many of these skills will be knowledge that students already have, but try not to assume that they will all easily recall how to apply each technique. Allow students time to revise these skills and repair any skills that have not been recalled correctly.

2 Modulus functions

Learning intention:

• Understand the relationship between y = f(x) and y = |f(x)|, where f(x) is quadratic.

Resources:

• PowerPoint 2.3a: Modulus of a quadratic including Worked example 5

• Coursebook Exercise 2.3

Description and purpose: The section on modulus functions builds on the processes studied in Chapter 1. You may choose to look at it in Chapter 1, when the linear functions are considered, or even later in the course when other functions may then be included. The PowerPoint available for this section works through the examples in section 2.3. Students will need to be able to find the roots and y-intercept for a quadratic function and then apply their knowledge of the modulus to it when drawing graphs. Sometimes, students will need to be able to find the coordinates of the turning point to solve simple problems about points of intersection of the graph of the modulus function with another function. This may lead into a problemsolving exercise on solving equations of the type shown in question 7 of Exercise 2.3 in the Coursebook, where students could find the number of solutions using graphing software or by drawing accurately and then going on to solve.

Differentiation:

Support:

• It is vital that students understand the basic skill of solving a quadratic equation and its application to finding the roots here.

• It is also vital that students understand the shape of a parabola and how the modulus function acts on this.

• Try to use visual support, such as graphing software, whenever possible.

CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE

Challenge:

• Coursebook Exercise 2.3, Q7 and Q8 and similar questions should be a good challenge for students.

• The Purplemath website offers an example similar to those in Exercise 2.3 Q7 and also gives an interesting example of nested absolute value functions.

Assessment for Learning: Some opportunities for discussion arise through the use of the PowerPoint. Students may also be encouraged to peer mark and assess their answers to Exercise 2.3. Natural opportunities for Q&A sessions should arise when this exercise is being carried out. Try to ask questions that allow students to show you that they have understood the mathematics they are studying. You could review their knowledge using a piece of work that they had to mark and grade. Can they find errors? Can they discuss the impact of those errors?

3 Quadratic inequalities

Learning intention:

• Find the solution set for quadratic inequalities.

Resources:

• PowerPoint 2.4: Worked examples 6 and 7

• Chapter 2 Teacher notes: class discussion section 2.4

• Chapter 2 Lesson plan: Solving inequalities

• Coursebook Exercise 2.4

Description and purpose: A demonstration lesson plan has been given for a possible lesson covering quadratic inequalities. The lesson incorporates the recap and class discussion in section 2.4. The Chapter 2 teacher notes give some support for managing the discussion direction, if it is needed. Some ideas about students developing their own explanations are also given. There are links in the document to a video that may be useful to challenge students and a Wolfram inequality checker tool. The lesson leads into Exercise 2.4 of the Coursebook.

Differentiation:

Support: The focus should be on the algebraic process here but if students need visual support, access to graphing software may be supportive for some.

Challenge: The Khan Academy has a video on rearranging inequalities with algebraic fractions and then solving.

Assessment for Learning: Assessment for learning opportunities should arise naturally through observation, peer checking of answers, Q&A sessions and whole class discussion as well as the discussion points which arise in the PowerPoint of worked examples.

4 Simultaneous equations

Learning intention:

• Solve simple simultaneous equations in two unknowns, with at least one linear, by elimination or substitution.

Resources:

• PowerPoint 2 recap a: Solving two linear simultaneous equations

• PowerPoint 2.1a: Solving two simultaneous equations with one linear including worked example 1

• PowerPoint 2.1b: Solving two simultaneous equations both non-linear

• Coursebook Exercise 2.1