Unit 3: Number, Algebra, Geometry 2 Number Use standard form, expressed in standard notation and on a calculator display Calculate with standard form Convert between ordinary and standard form representations Convert to standard form to make sensible estimates for calculations involving multiplication and/or division Recognise that each terminating decimal is a fraction Recognise that recurring decimals are exact fractions, and that some exact fractions are recurring decimals Convert any recurring decimal to a fraction (proof) Use percentages in real-life situations Solve percentage problems, including increase and decrease, and reverse percentage Represent repeated proportional change using a multiplier raised to a power Use repeated proportional change Use calculators for reverse percentages calculations by doing an appropriate division Use calculators to explore exponential growth and decay, using a multiplier and power key Calculate a given fraction of a quantity, expressing the answer as a fraction Multiply and divide a given fraction by an integer, by a unit fraction or by a general fraction Understand reciprocal as multiplicative inverse By using calculators, or written methods, calculate the upper and lower bounds, particularly when working with measurements Recognise that measurements given to the nearest whole unit may be inaccurate by up to one half in either direction Find the upper and lower bounds of calculations involving perimeter, ares and volumes of 2-D and 3-D shapes Use a calculator to solve real-life problems involving fractions Use standard form display and know how to enter numbers in standard form [when using a calculator]
Algebra Simplify terms, products and sums Multiply a single term over a bracket Take out common factors Expand the product of two linear expressions Factorise quadratic expressions Manipulate algebraic fractions Solve equations by using inverse operations or by transforming both sides in the same way Solve linear equations with integer or fractional coefficients, in which the unknown appears on either side or on both sides of the equation Find the exact solution of two (linear) simultaneous equations in two unknowns by eliminating a variable and interpreting the equation as lines and their common solution as the point of intersection Solve linear equations that require prior simplification of brackets, including those that have negative signs occurring anywhere in the equation, and those with a negative solution Factorise quadratic expressions Solve simple quadratic equations by factorising, completing the square and using the quadratic formula Substitute numbers into formulae Use formulae from mathematics and other subjects that require prior simplification of brackets, including those that have negative signs occurring anywhere in the equation, and those with a negative solution Change the subject of a formula including where the subject occurs once or more than once Generate a formula Solve several linear inequalities in two variables and finding the solution set Solve simple linear inequalities in one variable, and represent the solution set on a number line Use systematic trial and improvement to find approximate solutions of equations where there is no simple analytical method of solving them Calculate the length of a line segment between two coordinates. Recognise and plot equations that correspond to straight-line graphs in the coordinate plane, including finding gradients. Find the gradient of lines given equations of the form y = mx + c values are given for m and c)
Analyse problems and use gradients to see how one variable changes in relation to another Find the intersection points of the graphs of linear and quadratic functions, knowing that these are the approximate solutions of the corresponding simultaneous equation representing the linear and quadratic functions Find graphically the intersection points of a given straight line and a circle and knowing that this corresponds to solving the simultaneous equations representing the line and the circle Solve exactly, by elimination of an unknown, two simultaneous equations in two unknowns, one of which is linear in each unknown, and the other is linear in one unknown and quadratic in the other, of where the second is of the form x 2 + y 2 = r 2 Plot graphs of simple cubic functions, the reciprocal function y = x 1 with x 0, the exponential function y = k x for integer values of x and simple positive values of k, the circular functions y = sin x and y = cos x Recognise the characteristic shapes of all these functions Construct the graph of x 2 + y 2 = r 2 for a circle of radius r centred at the origin of coordinates Generate points and plot graphs of simple quadratic functions Generate points and plot graphs of quadratic functions Find approximate solutions of a quadratic equation from the graph of the corresponding quadratic function Set up and use equations to solve word and other problems involving direct proportion or inverse proportion and relating algebraic solutions to graphical representations of the equations Calculate an unknown quantity from quantities that vary in direct or inverse proportion Apply to the graph of y = f(x) the transformations y = f(x) + a, y = f(ax), y = f(x + a), y = af(x) for linear, quadratic, sine and cosine functions f(x) Draw, sketch and describe the graphs of trigonometric functions for angles of any size, including transformations involving scalings in either or both the x and y directions
Geometry and Measures Understand similarity of triangles and of other plane figures and use this to make geometric inferences Recognise that enlargements preserve angle but not length Understand, recall and use Pythagoras theorem in 2-D Understand, recall and use trigonometrical relationships in right-angled triangles, and use these to solve problems, including those involving bearings Find angles of elevation and angles of depression Understand and recall and use trigonometric relationships in right angled triangles and use these to solve problems in 3-D (include the use of Pythagoras theorem) Use the trigonometric ratios and the sine and cosine rules to solve 3-D problems Find the angles between a line and a plane Use the sine and cosine rules to solve 2-D problems Use the sine and cosine rules to solve 3-D problems Explain why the perpendicular from the centre to a chord bisect the chord Prove and use the fact that the angle subtended by an arc at the centre of a circle is twice the angle subtended at any point on the circumference Prove and use the fact that the angle subtended at the circumference by a semicircle is a right angle Prove and use the fact that angles in the same segments are equal Prove and use the fact that opposite angles of a cyclic quadrilateral sum to 180 Prove and use the alternate segment theorem Transform triangles and other 2-D shapes by translation, rotation and reflection and combinations of these transformations Understand that translations are specified by a distance and direction (or a column vector), and enlarging by a centre and a scale factor Rotate a shape about the origin, or any other point Measure the angle of rotation using right angles, simple fractions of a turn or degrees Understand that rotations are specified by a centre and an (anticlockwise) angle Understand that reflections are specified by a mirror line, at first using a line parallel to any axis, then a mirror line such as y = x or y = x Recognise, visualise and construct enlargements of objects using positive and negative scale factors greater and less than one Use congruence to show that translations, rotations and reflections preserve length and angle, so that any figure is congruent to its image under any of these transformations
Understand and use SSS, SAS, ASA and RHS conditions to prove the congruence of triangles by using formal arguments, and also to verify standard ruler and compass constructions Identify the scale factor of an enlargement as a ratio of the lengths of any two corresponding line segments Understand the implications of enlargement for perimeter Understand and use the effect of enlargement on areas and volumes of shapes and solids Convert measurements from one unit to another (area and volume) Understand and use compound measures, including density Draw triangles and other 2-D shapes, using a ruler and protractor, given information about side lengths and angles Understand, from their experience of constructing them, that triangles satisfying SSS, SAS, ASA and RHS are unique, but SSA are not Use straight edge and compasses to do standard constructions Construct loci Calculate the area of a triangle using 1 absinc 2 Find circumferences of circles and areas enclosed by circles, recalling relevant formulae Solve problems involving surface areas and volumes of prisms and cylinders Use surds and π in exact calculations, without a calculator Solve problems involving more complex shapes, including segments of circles, length of a chord and frustums of cones Solve problems involving volumes of prisms and cylinders Solve problems involving surface areas and volumes of cones and pyramids Find the surface areas and volumes of compound solids constructed from; spheres, hemispheres and cylinders (include examples of solids in everyday use) Understand and use vector notation (revise column vectors) Calculate, and represent graphically the sum of two vectors, the difference of two vectors and a scalar multiple of a vector Calculate the resultant of two vectors Understand and use the commutative and associative properties of vector addition Solve simple geometrical problems in 2-D using vector methods