# Algebra: Indices, linear and quadratic functions

**AS Level**

- Understand and use the laws of indices for all rational exponents
- Use and manipulate surds, including rationalising the denominator
- Work with quadratic functions and their graphs; the discriminant of a quadratic function, including the conditions for real and repeated roots; completing the square; solution of quadratic equations including solving quadratic equations in a function of the unknown
- Solve simultaneous equations in two variables by elimination and by substitution, including one linear and one quadratic equation
- Solve linear and quadratic inequalities in a single variable and interpret such inequalities graphically, including inequalities with brackets and fractions
- Express solutions through correct use of ‘and’ and ‘or’, or through set notation]
- Represent linear and quadratic inequalities such as y >x+1 and y>ax
^{2}+bx+c graphically - Manipulate polynomials algebraically, including expanding brackets and collecting like terms, factorisation and simple algebraic division; use of the factor theorem

**A Level**

- Simplify rational expressions including by factorising and cancelling, and algebraic division (by linear expressions only)

### Indices

Five activities designed to enable students to explore indices both numerically and algebraically. The resource features a number of activities dealing with negative indices and fractional indices.

Starter activities: contains two starter activities. In the first activity students have to represent an integer value in as many ways as they can, each way to include an index value. The second activity, ‘Why does?’ students have to explain mathematical statements involving indices.

Matching pairs game: Students are required to match a number in index form with its integer value.

Ordering indices: Students compare the values written in index form and place the cards in numerical order.

Odd one out: Given three cards, students find the odd one out and make up their own card to match the odd one.

True or false: Aimed at addressing common misconceptions, in this activity students discuss whether the statements given are true or false.

### Further Indices

Four activities designed to explore the rules of indices as well as differentiating and integrating functions containing indices.

The rules of indices with algebraic expressions: This activity is a matching exercise with algebraic statements involving negative indices and fractional indices.

Dominoes: is a loop card activity involving negative indices and fractional indices.

Differentiation and integration involving indices: A large, multi-tiered activity matching equivalent functions, differentials and integrals.

Marking: In this activity, students are given questions and solutions to a number of differentiation and integration questions. Students are required to mark the work. The solutions contain many common errors that are made. Students should mark the work for accuracy, correct solutions where necessary and give advice to help the candidate therefore explain what the error was and how to correct it.

### Indices and surds

This resource contains ten problems which require students to think, explore and explain the mathematical topic of indices and surds. The problems require students to not only be able to manipulate expressions containing indices and surds but also be able to explain and justify how conclusions are arrived at. A number of the questions require students to show that they can create a number of expressions which all mean the same thing mathematically.

### Indices and Surds

This activity from can be used when either consolidating or revising ideas of curve-sketching and indices. The numbers phi, e and pi are used in this investigation where students are asked to estimate the size numbers generated when raising these numbers to different powers. It is suggested that a graphing package would prove useful when investigating the set problem.

### Quadratic Functions

This resource contains eleven problems requiring students to explore the graphs of quadratic equations, intersection points of quadratics and straight lines, explain what will be the same and what will be different about the graphs of two quadratic functions, transform quadratic graphs, find the odd one out, complete the square and deal with quadratic inequalities. In each problem students are encouraged to explain and justify their solutions.

### Quadratics

This resource contains four excel programs with interactive spreadsheets dealing with topics relating to quadratic expressions, their graphs and solving quadratic equations.

Quadratics: Rearranging and Graphing

The first sheet shows a set of quadratic functions and the student is asked for the general shape of each one. The second sheet requires the student to expand brackets and arrange the expressions into the form ax2+bx+c. The next sheet gives a set of quadratic functions and the student needs to state the x and y intercepts. The workbook contains a further four printable worksheets providing further questions for use in the classroom.

Quadratics: Solving

This interactive workbook begins with examples of solving quadratic equations using the formula. Roots are given in surd form or shown on a graph in decimal form. The discriminant and the nature of the roots is also dealt with. Sets of questions require the student to decide if the roots are real and if so rational or irrational. Two further sheets give quadratic equations which can be solved by factorising. There are also eight printable worksheets of questions.

Quadratics: Tables of Values

This interactive workbook deals with how graphs can be plotted. Suitable tables of values for quadratic functions can be revealed and the graphs are shown. Various forms of the function can be selected including -x2+bx and 2x2+bx+c. There are three further worksheets of questions suitable for use in the classroom.

Quadratics: Turning point(XP)

This interactive workbook deals with completing the square when the coefficient of x2 is 1. It begins with a series of (x+a}2 expressions to expand which can be swapped to become perfect squares to be factorised. Subsequent sheets show each step in the process of completing the square alongside graphs showing the turning points. There are three printable sheets of questions for use in the classroom.

### Quadratic Equations

This resource is designed to enable students explore what is meant by a quadratic equation, the meaning of the coefficients of a quadratic equation and to be able to solve quadratic equations. An introduction page gives examples of where quadratic equations can be found which is useful for class discussion. There follows an explanation of the first task in which students have to investigate how changing the coefficients a, b and c in the function f(x) =a(x+b)2+c affect the graph. The second activity requires students to change the values of a, b and c so that the green graph matches the blue graph on the screen. The third activity asks students to form equations to match the paths shown on the screen. The fourth activity explores the roots of a quadratic equation. This is followed by pages of notes explaining the nature of quadratic equations including the formula for solving quadratic equations, the determinant, factorising a quadratic and completing the square. The final exercise asks students to solve a series of quadratic equations.

### Simultaneous Equations (Non-Linear)

This file provides practice at solving pairs of simultaneous equations where at least one is not linear. The pairs of graphs and the coordinates of the points of intersection can be revealed.

The first sheet deals with the intersection of a straight line with y=x2 where all the solutions have integer values. The second sheet is similar but there may be half values in the answers. The next two sheets deal with the intersection of a straight line with a more general quadratic and the intersection of two parabolas. The final interactive sheet gives practice at finding the points of intersection of a straight line and a circle.

### Inequalities

This activity begins with inequalities associated with the properties of triangles. Students are challenged to develop proofs of the inequalities found in a variety of situations, using the arithmetic, geometric and harmonic means.

The resource is designed to explore aspects of the subject which may not normally be encountered, to encourage new ways to approach a problem mathematically and to broaden the range of tools that an A Level mathematician can call upon if necessary.

### Algebraic Long Division

In this activity students explore how cubic polynomials can be divided by linear factors. The first four activities generate cubic expressions and the steps in the algebraic long division method can be gradually revealed. The constant term in the cubic can be altered to show how it affects the remainder. Some of these activities always generate factors of the cubic leading to a zero remainder. The next activity requires students to use the Remainder Theorem. The final activity sets of examples of factorising cubic expressions where the Factor Theorem can be used.