Skills in Algebra and Arithmetic

References

Online revision texts for pre-university mathematics

[D]      sections 1.1 - 1.4, 1.8, 2.1 - 2.2, 1.5, 2.5;
[EG]   section 1.4;

and the text below.

Keywords

Arithmetic with numbers, fractions and square roots. How to manipulate and reduce algebraic expressions. How to solve linear equations. How to solve quadratic equations.

Introduction

We continue in this block the revision of some important pre-university topics which we started in Block 1 and we enhance your skills by solving harder exercises in these topics. The topics include arithmetic with integers and fractions, reduction of algebraic expressions and the solution of linear equations. We shall also revise basic work with functions. Finally it is very useful to be able to solve quadratic equations, so we shall give you a formula for obtaining such solutions. As mentioned in Block 1, if your mathematical skills are a bit old and rusty, you might need to spend more than the average of 10 hours for this block as you will need time to get the hang of things again. Solve lots of the recommended exercises from the coursebook [D]. You learn much more by solving exercises yourself than by reading examples and exercise solutions made by other people.

If at any point in the reading below, you find the coursebook incomprehensible or to hard, you should go back to your 'gymnasium'-books and read about the same topics or you can use the revision resources for pre-university mathematics which I introduced you to in Block 1.

1. Arithmetic with numbers and symbols

At this point in the reading it is time to open the other coursebook by Dunkels et al, it is an excellent text which trains the reader in the arithmetic and algebraic skills required for a university mathematics course at level A. You should study [D] according to the following guidelines. You should not use a calculator for any of the exercises in [D].

[D] section 1.1 trains you in doing calculation with integers. The section starts with a box listing the rules and then there is a whole array of exercises to which there are answers at the back of the book. I recommend that you do all of these exercises (1.1 - 1.5).

[D] section 1.2 trains you in doing calculation with fractions (Swedish: bråk) and laws of exponents. The section starts with two boxes showing how to do arithmetic with fractions and later a third box revising the laws of exponents. Again you should do all exercises (1.6 - 1.14) or at least so many that you are absolutely certain that you know how to do these kind of calculations.

[D] section 1.3 trains you in manipulating algebraic expressions. In order to solve mathematical exercises, it is often crucial to be able to rewrite an expression in a simpler manner to be able to use it in future calculations. It is a highly skilled process to be able to do so, and this section in [D] takes you through the basic rules, but it also tests that you know how to use the rules through a serious of very rewarding exercises. You should be able to do exercises 1.15 - 1.19 and exercises 1.21 - 1.28.

[D] section 1.4 trains you in manipulating algebraic expressions involving square roots. You should be able to do exercises 1.29 - 1.31.

2. Functions

You learnt the basics about functions in your 'gymnasium'-course(s). Revise this by reading section 1.8 in [D] and solving exercises 1.49, 1.51 and 1.52.

Section 2.1 in [D] revises polynomials and polynomial functions (functions in which the function rule is given by a polynomial). You should be able to solve exercises 2.1 - 2.2.

Section 2.2 in [D] revises rational functions, that is functions whose function rule is a quotient of two polynomials. You should be able to solve exercise 2.3.

3. Solving equations

You learnt how to solve linear equations (Swedish: förstagradsekvationer) in your 'gymnasium'-course(s). Revise this topic by reading section 1.5 and solving exercises 1.34 - 1.35 in [D].

It is also useful to be able to solve quadratic equations (Swedish: andragradsekvationer). Dunkels covers this topic in section 2.5, where he gives a formula for solving a general quadratic equation. You should be able to use this formula, but you need not remember it by rote as it is contained in the formula collection which you may use at the examination. The revision resources for pre-university mathematics which I introduced you to in Block 1 contains three very nice summary texts about quadratic polynomials and quadratic equations in the section about Basic Algebra. These summary texts are less compact than the text in [D] and thus easier to read even though they are in English. I recommend that you click here and then on 'view online' to read the summary texts named 'Factorising quadratics', 'Transposition of formulae' and 'Quadratic equations'.
Solve exercises 2.6 and 2.8 in [D].

Finally read section 1.4 in [EG] again. Hopefully it is now easily read and the exercises seem less complicated to you!

Week Exercise 2

1. Showing all your working, solve the following equations without using a calculator:

(a) x² - 9 = 0;
(b) x² + 9 = 0;
(c) x² - 9 = 3;
(d) 2x² + 3x + 1 = 0
where x is a real number.

2. Showing all your working, solve without using a calculator:

(a) exercise 1.32 in [D];
(b) exercise 1.51(b) in [D];
(c) exercise 1.58(a) in [D];
(d) exercise 1.61(c) in [D].