Contents. Number. Algebra. Exam board specification map Introduction Topic checker Topic checker answers. iv vi x xv


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1 Contents Exam board specification map Introduction Topic checker Topic checker answers iv vi x xv Number The decimal number system Fraction calculations Fractions, decimals and percentages Powers and roots Surds 0 Standard index form Ratio and proportion Percentage calculations Algebra Algebraic expressions Formulae and substitution 0 Rearranging formulae Using brackets in algebra Factorising quadratic expressions Algebraic fractions Solving equations 0 Equations of proportionality Trial and improvement Quadratic equations Functions Inequalities and regions 0 Number patterns and sequences Sequences and formulae Lines and equations Curved graphs Transforming graphs 0 Linear simultaneous equations Mixed simultaneous equations ii
2 Shape, space and measures Rounding and accuracy Dimensions Speed and motion graphs 0 Properties of shapes Circle geometry Pythagoras Rule and basic trigonometry The sine rule The cosine rule 0 Trigonometric graphs: angles of any size Problems in three dimensions Calculating areas Circle calculations Volume calculations 0 Congruence and similarity Constructions Loci Vectors Transformations 0 Handling data Pie charts Histograms Finding averages Cumulative frequency Comparing sets of data 00 Probability 0 Scatter diagrams and correlation 0 Exam questions and model answers 0 Complete the facts * Answers to complete the facts * Answers to practice questions Glossary Web links * Lastminute learner * Only available in the CDROM version of the book iii
3 The decimal number system Each place in a decimal number represents a power of ten:,,, 0, 00, etc A Place value The digit means something different in each of these numbers: Number Position of Value of tens 0 0 millions tenths 000 thousandths B Decimal calculations To multiply a number by 0, 00 or 000, simply move its digits, or places to the left: >> To divide by 0, 00 or 000, move digits, or places to the right: Multiplying by 0 is the same as dividing by 0 You can use the result of a whole number multiplication to find the answer to many decimal multiplications:, so and , etc If one of the numbers is made ten times smaller, the answer will be ten times smaller A similar rule works with division, but if the number you are dividing by gets smaller, the answer gets bigger:, so and , etc C Negative numbers: addition and subtraction Adding a negative number gives the same result as subtracting a positive number Examples: () brackets are used to make it clearer () you could use a number line to help
4 Another way to do the last example is to notice that when you swap the numbers in a subtraction, you change the sign of the answer: and so This can be useful when larger numbers are involved: (), so Subtracting a negative number gives the same result as adding a positive number Examples: () = = () () = () = use a rough number line to help >> D Negative numbers: multiplication and division The rules for multiplication and division are very simple: negative positive negative: negative negative positive negative positive negative: negative negative positive Examples: () ( ) () (0) 0 0 () ( ) () () You can remember this with the word SPON: Same (signs) Positive, Opposite (signs) Negative Otherwise, use this table: The square of a negative number is positive () () () / >> The cube of a negative number is negative () () () () () >> practice questions Do a whole number calculation first, then use the result to answer the question (a) (b) 0 (c) (d) 00 Work these out without a calculator (a) 0 (b) () (c) () (d) () (e) () () (f) () () (g) () (h) () () (i) (0) (j) () () (k) (0) (l) () Use your calculator to find these (a) (b) () (c) (0) (d) () () (e) () 0 (f) (0) (g) () (00) (h) (00)
5 Fraction calculations The numerator is the top number in a fraction and the denominator is the bottom number Create equivalent fractions by multiplying or dividing both numbers by the same thing Change mixed numbers to improper fractions before calculating A Equivalent fractions and mixed numbers Fractions that contain different numbers, but represent the same amount, are equivalent Fractions that contain the smallest possible whole numbers are in lowest terms lowest terms = = 0 Fractions in which the numerator is bigger than the denominator are called improper or topheavy Improper fractions can be written as mixed numbers: Changing mixed numbers to improper fractions is similar: = + = You can write fractions in order of size by writing them as equivalent fractions with a common denominator Order these:,,, 0, Look at the denominators:,,, 0 and will all divide into 0 Rewritten, the fractions are 0, 0, 0, 0, 0 0 So the order is,, 0,, B Adding and subtracting fractions To add or subtract fractions, write them using a common denominator If you can find the lowest common denominator (LCD), this keeps the numbers small and you are less likely to make a mistake Sometimes, the LCD will be the denominator of one of the fractions in the question 0 you can use as the LCD 0 The LCD is 0:
6 C Multiplying and dividing fractions To multiply fractions, just multiply the numerators and multiply the denominators Change mixed numbers to improper form first You can keep the numbers small by crosscancelling before you multiply : cancels with and cancels with To divide fractions, just invert the second fraction (replace it by its reciprocal) and multiply x = 0 D Fractions of an amount To find a fraction of an amount, divide by the denominator and multiply by the numerator What is of? 0 to find 0 0 to find To express an amount as a fraction of another, write the amounts as a fraction and cancel to lowest terms What fraction of km is km? top and bottom by 0: now it uses integers cancel >> practice questions Write in order of size, smallest to biggest: Calculate the following: 0 0 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l),,,, Find these amounts: (a) of km (b) of 0 ml (c) of 0 (d) times 0 grams What fraction of the second amount is the first? (a) cm, cm (b) 0, 0 (c) 0 cl, litre (d) 0 kg, tonnes
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