REVISED GCSE Scheme of Work Mathematics Foundation Unit 5. For First Teaching September 2010 For First Examination Summer 2011 This Unit Summer 2012

Transcription

1 REVISED GCSE Scheme of Work Mathematics Foundation Unit 5 For First Teaching September 2010 For First Examination Summer 2011 This Unit Summer 2012

2 Version 1: 28 April 10

3 Version 1: 28 April 10 Unit T5

4 Unit T5 This is a working document for teachers to adapt to their own specific needs. Unit T5 Topic No. Topic Subject Content 1 Number Read, write and order numbers 2 Number The four operations 3 Number BODMAS 4 Algebra Unknowns 5 Algebra Formulae 6 Geometry and Measures Area/Surface Area 7 Algebra Indices 8 Number Number Calculations 9 Algebra Graphs 10 Geometry and Measures Imperial Measures 11 Geometry and Measures Compound Measures 12 Number Ratios 13 Algebra Inequalities 14 Algebra Quadratic Functions 15 Number Estimation 16 Number Degree of Accuracy 17 Geometry and Measures 2D shapes 18 Geometry and Measures Constructions 19 Geometry and Measures Symmetry 20 Geometry and Measures Transformations Version 1: 28 April 10 2

5 21 Number Language of Number 22 Number Multiplicative Inverses 23 Number Multiplying and Dividing by multiples of ten 24 Handling Data Probability experiments 25 Handling Data Calculating probabilities 26 Handling Data Estimating probabilities 27 Handling Data Probability outcomes Version 1: 28 April 10 3

6 TOPIC 1: NUMBER Read, write and order numbers read, write and order numbers. How to put numbers in order. Place value of digits. How to round off to the nearest ten, hundred, thousand. Readings from rulers, jugs and any other measuring instrument. Specimen P1 Q1(a) Specimen P2 Q1 Version 1: 28 April 10 4

7 TOPIC 2: NUMBER The four Operations understand the four operations and the relationship between them. Know addition and the effect it has. Know subtraction and the effect it has. Recognise that subtraction is the inverse of addition. Know multiplication and the effect it has. Know division and the effect it has. Recognise that division is the inverse of multiplication. Version 1: 28 April 10 5

8 TOPIC 3: NUMBER OPERATIONS AND APPLICATIONS BODMAS understand the effects of operations on numbers of any size; Class discussion: what effect addition, subtraction, multiplication and division have on numbers (less than and greater than one). apply order of precedence. Teacher demonstration: Introduce the term BODMAS. Explain what the letters BODMAS represent and complete a number of worked examples. First without brackets and then with brackets. Worked example: Distinguish correctly between and (3 + 2) 5. Solve problems requiring application of order of precedence. Example questions: (a) (b) (c) (10 + 2) ( ) Specimen P1, Q5 Distinguish correctly between: and Version 1: 28 April 10 6

9 TOPIC 4: TOPIC: ALGEBRA Unknowns understand key concepts and terms; appreciate the use of letters to represent unknowns. Terms such as solve, simplify, substitute. Distinguish between the different roles played by letter symbols in algebra. Knowing that letter symbols represent: definite unknown numbers in equations (e.g. 5x + 1 = 16), defined quantities or variables in formulae (e.g. V = IR), general unspecified numbers in identities (e.g. 3x + 2x = 5x for all values of x) and in functions they define new expressions or quantities by referring to known quantities (e.g. y = 2x). Specimen P1, Q6 Version 1: 28 April 10 7

10 TOPIC 5: TOPIC: ALGEBRA Formulae understand key concepts and terms; understand, construct and evaluate formulae related to Mathematics or other subjects or real-life situations; change the subject of linear formulae. Construct simple algebraic formulae. For example: Mr Smith bought a apples and some oranges. He had 4 times as many oranges as apples. How many oranges did he have? He used half the apples in a pie and his son ate one. How many apples were left? To cook a chicken, allow 20 minutes per lb plus 20 minutes. A chicken weighs x lb. Write an expression to show the number of minutes m to cook a chicken. Find an unknown where it is not the subject of the formula and where an equation must be solved. For example, the formula for the change C from 50 for d compact discs at 7 each is C = 50 7d. If C = 15, what is d? Specimen P2 Q3 Change the subject of a formula using inverse operations. For example: Make R the subject of V = IR Make u the subject of v = u + at Version 1: 28 April 10 8

11 TOPIC 6: GEOMETRY AND MEASURES Area/Surface Area recall and use the formula for the area of squares and triangles; Use to calculate the areas of compound shapes, parallelograms, rhombuses, kites and trapezia. Specimen P2, Q8 (c) Specimen P2, Q10 calculate the surface areas of cubes and cuboids. Calculate the surface area of a box. Illustrate the net of a cube as 6 joined squares. Calculate the area of one square and hence the surface area of the cube. Illustrate the net of a cuboid as 6 rectangles/squares, calculate area of each and hence the surface area. Version 1: 28 April 10 9

12 TOPIC 7: ALGEBRA Indices use the rules of indices for integral values. Use index notation for positive integer powers. Know that expressions involving repeated multiplication of the same number, such as n n; n n n; n n n n are written as n 2, n 3 and n 4 and are referred to as n squared, n cubed and n to the power of 4, etc. Specimen P2, Q11 Understand the different meanings of expressions such as 2n and n 2, 3n and n 3. Simplify expressions such as 2x 2 + 3x 2, n 2 n 3, p 3 p 2 Know and use the general forms of the index laws for multiplication and division of integer powers: p a p b = p a+b p a p b = p a-b (p a ) b = p ab Version 1: 28 April 10 10

13 TOPIC 8: NUMBER Money Calculations calculate with money. Solve problems in the context of finance (for e.g. currency exchange rates, loans, deposit accounts, credit cards, hire purchase, general bank accounts including overdrafts, interest rates and mortgages, simple interest, mail order sales, insurance, taxation, wages, salaries, unemployment benefit). Specimen P1 Q3 Specimen P2, Q8 (a), (b) Ability to read bank statements and bills for information. Work out total wages and salaries received given scenarios. Use given exchange rates to work out holiday money etc. Version 1: 28 April 10 11

14 TOPIC 9: ALGEBRA Graphs interpret and display information on graphs that describe real-life situations. Conversion graphs Discuss practical examples such as: The number of euros you can exchange for pounds sterling The number of km equal to miles The cost of books at 3 each Draw a graph of such information. Observe that the points lie in a straight line. Discuss, plot and interpret graphs (which may be non-linear) modelling real situations. Specimen P2 Q5 Distance-time graphs including intersecting travel graphs Plot a simple distance-time graph. Interpret a simple distance-time graph, giving plausible reasons for each section. Version 1: 28 April 10 12

15 TOPIC 10: GEOMETRY AND MEASURES Imperial Measures know imperial measures still in common use including foot, yard, mile, pound and pint and their approximate metric equivalents. Recall that 1 kg is about 2.2 lb, 8 km is approximately 5 miles, 1 litre is about 1.75 pints. Specimen P2, Q6 Version 1: 28 April 10 13

16 TOPIC 11: TOPIC: GEOMETRY AND MEASURES Compound Measures understand and use compound measures including speed and density. Work out average speed (distance/time) density (mass/volume) 30 mph for 3 hours = 90 miles 120 miles in 3 hours = 40 mph 400 miles at 50 mph = 8 hours Version 1: 28 April 10 14

17 TOPIC 12: NUMBER OPERATIONS AND APPLICATIONS Ratios use unitary ratios and calculate with ratios in a variety of situations; Unitary ratios: for example: 1:n or n:1. Know that if the ratio of cement to sand is 1:3 then the ratio of sand to cement is 3:1. The order of the objects is very important. Remember that the ratios are numbered in the same order as they are written. Example questions: In a recipe, the ratio of flour to currants is 5:1. If you need 10 ounces of flour, how many ounces of currants will you need? Simplify ratios. Use the approximation of 5 miles to 8 km to find the equivalent of 12 miles. State the lengths 8cm and 12cm in a drawing are in the ratio 2:3 Adapt a recipe for six people to one for eight people. divide a quantity in a given ratio. Understand the relationship between fractions and ratios. Divide 10 between two people in the ratio 2:3 Specimen P1, Q9 Version 1: 28 April 10 15

18 TOPIC 13: ALGEBRA Inequalities understand key concepts and terms; solve inequalities on a number line. Solve linear inequalities in one variable. Represent the solution set on a number line. For example: List the values of the integer n such 10 < 2n 20 Solve the inequality 2n 3 7 illustrating the solution on a number line. Solve x 3x 5 where x is a real number. Specimen P2 Q12 Version 1: 28 April 10 16

19 TOPIC 14: ALGEBRA Quadratic Functions explore the properties of quadratic functions. Make tables of such functions. Sketch and interpret their graphs using graphical calculators and computers to understand their behaviour. To include drawing graphs of y = ax 2 + bx + c Version 1: 28 April 10 17

20 TOPIC 15: NUMBER Estimation approximate numbers to the nearest unit, 10 or 100; Estimate that is about 1100 ( ) Estimate that is about 7 (280 40) Understand that is approximately Specimen P1 Q1(b) estimate within calculations (initially with numbers within 100 and extending to all whole numbers). Recognise that is about 3 or Version 1: 28 April 10 18

21 TOPIC 16: NUMBER Degree of Accuracy give solutions in the context of a problem to an appropriate degree of accuracy, recognising limitations on the accuracy of data and measurements. Comment on the upper and lower bounds of data. e.g. 10.5cm correct to the nearest tenth means a range from 10.45cm to 10.55cm. Version 1: 28 April 10 19

22 TOPIC 17: GEOMETRY AND MEASURES 2D Shapes explore shape through drawing and practical work using a wide range of materials. Recognise and describe 2-D shapes. Recognise right-angled corners in 2-D shapes. Specimen P1, Q8 Version 1: 28 April 10 20

23 TOPIC 18: GEOMETRY AND MEASURES Constructions use ruler and compasses for standard constructions; construct loci. To include triangles, quadrilaterals, the mid-point and perpendicular bisector of a line segment, the perpendicular from a point to a line, the perpendicular from a point on a line and the bisector of an angle. Region bounded by a circle and an intersecting line. Version 1: 28 April 10 21

24 TOPIC 19: GEOMETRY AND MEASURES Symmetry recognise symmetry properties in a variety of shapes in two dimensions; recognise line symmetry; draw the axes of symmetry; reflect shapes in a mirror line; recognise rotational symmetry, its order and centre; know and use symmetry properties of triangles, quadrilaterals and other polygons; recognise planes of symmetry in practical situations. Find the centres and axes of symmetry in a variety of shapes. Rotate shapes using tracing paper. Reflection to include rectilinear figures only. Study shapes and identify some lines and planes of symmetry. Specimen P1, Q4 Version 1: 28 April 10 22

25 TOPIC 20: GEOMETRY AND MEASURES Transformations understand transformation of shapes; reflect shapes in a line, [for example, x = 1]; rotate shapes about a given centre; translate shapes; enlarge a shape through a given centre of enlargement recognise that enlargements preserve angle but not length; distinguish properties that are preserved under particular transformations. Transformations will include single or combined enlargements, reflections, rotations and translations. Rotations will be limited to 90 o and 180 o about a point. Reflections will be limited to reflections in lines parallel to the co-ordinate axes. Vector notation for translation. Enlarge a shape by a whole number scale factor only. Use transformations to create and analyse spatial patterns. Specimen P1, Q2 Specimen P1, Q12 Version 1: 28 April 10 23

26 TOPIC 21: NUMBER Language of Number understand and use the language of number. Reciprocal: understand the meaning of reciprocal and its place in number patterns. If the product of two numbers is 1 then each number is called the reciprocal of the other. Example: You know that 2 ½ = 1, so you say 2 is the reciprocal of ½ and ½ is the reciprocal of 2. Questions: What is the reciprocal of (a) 7 (b) 10 (c) ⅓ (d) ¼ Specimen P1 Q11(a) Version 1: 28 April 10 24

27 TOPIC 22: NUMBER Multiplicative inverses use unit fractions as multiplicative inverses; understand reciprocal as multiplicative inverse. Know that multiplication by 5 1 is equivalent to division by 5 Know that any non-zero number multiplied by its reciprocal is one and that zero has no reciprocal, because division by zero is not defined. Version 1: 28 April 10 25

28 TOPIC 23: NUMBER Multiplying and Dividing by multiples of ten multiply and divide mentally single digit multiples of any power of ten. Realise that, when multiplying by a number less than one, multiplication has a decreasing effect, and division an increasing effect. Multiply by numbers less than 1. Divide by numbers less than 1 (using and not using a calculator). Work out mentally and Version 1: 28 April 10 26

29 TOPIC 24: HANDLING DATA Probability experiments understand possible outcomes of random trials or experiments; Know what the term probability means. Know that you do not always get 5 heads in 10 tosses of a fair coin and very occasionally there will be none. Discussions as to whether or not all outcomes are equally likely. understand that there is a degree of uncertainty about the occurrence of some events, and others are certain or impossible; Place events in order of likelihood and use appropriate words to identify chance. Mark the likelihood of events on a probability scale. Specimen P2, Q9(a) know that when repeating the same experiment, different outcomes may result and that the possible outcomes may not be equally likely; Specimen P1 Q10(c) understand and use 0 and 1 as the limits of the probability scale. Specimen P2, Q7 Version 1: 28 April 10 27

30 TOPIC 25: HANDLING DATA Calculating Probability know that for equally likely outcomes, the probability of an event is the number of desirable outcomes divided by the number of possible outcomes. Know that if there are six identical beads numbered 1, 1, 2, 2, 3 and 4, the probability of selecting a bead labelled 1 is 2/6 = 1/3 Calculate probabilities. Write probabilities as fractions and simplify answers. Use probability to calculate expected frequency. Specimen P1 Q7(a),(b) Specimen P1 Q10(b) Specimen P2, Q9(b) Version 1: 28 April 10 28

31 TOPIC 26: HANDLING DATA Estimating Probability recognise situations where probabilities can be based on equally likely outcomes and others where estimates must be based on sufficient experimental evidence and make these estimates; understand and use relative frequency as an estimate of probability. Experimental probability. Experiment to find the contents of a bag to introduce relative frequency. Find relative frequency. Version 1: 28 April 10 29

32 TOPIC 27: HANDLING DATA Probability outcomes identify all the outcomes when dealing with a combination of two independent events using diagrams or tables, and use these to find probabilities; Complete sample space diagrams for various events. List all the outcomes when tossing two coins; HH, TT, TH, HT. Make a table of all the outcomes when throwing two dice and show the total sums arising. Find probabilities using sample spaces. know that if there are several possible outcomes (exhaustive and mutually exclusive), the total of these probabilities is 1. understand that the probability of something happening is 1 minus the probability of it not happening. Calculate the probability that an event will not happen. Recognise that if the probability of a machine failing is 0.05 then the probability of it not failing is 0.95 Specimen P1 Q10(a) Specimen P1 Q7(c) Version 1: 28 April 10 30