Year 11 set 3

Scheme of work

Pythagoras’ Theorem and trigonometry

OBJECTIVES

By the end of the sub-unit, students should be able to:

· Understand, recall and use Pythagoras’ Theorem in 2D;

· Given three sides of a triangle, justify if it is right-angled or not;

· Calculate the length of the hypotenuse in a right-angled triangle (including decimal lengths and a range of units);

· Find the length of a shorter side in a right-angled triangle;

· Calculate the length of a line segment AB given pairs of points;

Give an answer to the use of Pythagoras’ Theorem in surd form;

· Understand, use and recall the trigonometric ratios sine, cosine and tan, and apply them to find angles and lengths in general triangles in 2D figures;

· Use the trigonometric ratios to solve 2D problems;

· Find angles of elevation and depression;

· Know the exact values of sin θ and cos θ for θ = 0°, 30°, 45°, 60° and 90°; know the exact value of tan θ for θ = 0°, 30°, 45° and 60°.

Practice for mocks exams

Exam Preparations ( Mock set 3 only)

Constructions, loci and bearings

· Draw 3D shapes using isometric grids;

· Understand and draw front and side elevations and plans of shapes made from simple solids;

· Given the front and side elevations and the plan of a solid, draw a sketch of the 3D solid;

· Use and interpret maps and scale drawings, using a variety of scales and units;

· Read and construct scale drawings, drawing lines and shapes to scale;

· Estimate lengths using a scale diagram;

· Understand, draw and measure bearings;

· Calculate bearings and solve bearings problems, including on scaled maps, and find/mark and measure bearings

· Use the standard ruler and compass constructions:

· bisect a given angle;

· construct a perpendicular to a given line from/at a given point;

· construct angles of 90°, 45°;

· perpendicular bisector of a line segment;

Mocks

Exam preparation

Percentages

OBJECTIVES

By the end of the sub-unit, students should be able to:

· Convert between fractions, decimals and percentages;

· Express a given number as a percentage of another number;

· Express one quantity as a percentage of another where the percentage is greater than 100%

· Find a percentage of a quantity;

· Find the new amount after a percentage increase or decrease;

· Work out a percentage increase or decrease, including: simple interest, income tax calculations, value of profit or loss, percentage profit or loss;

· Compare two quantities using percentages, including a range of calculations and contexts such as those involving time or money;

· Find a percentage of a quantity using a multiplier;

· Use a multiplier to increase or decrease by a percentage in any scenario where percentages are used;

· Find the original amount given the final amount after a percentage increase or decrease (reverse percentages), including VAT;

· Use calculators for reverse percentage calculations by doing an appropriate division;

· Use percentages in real-life situations, including percentages greater than 100%;

· Describe percentage increase/decrease with fractions, e.g. 150% increase means times as big;

· Understand that fractions are more accurate in calculations than rounded percentage or decimal equivalents, and choose fractions, decimals or percentages appropriately for calculations.

Ratio and proportion

OBJECTIVES

By the end of the sub-unit, students should be able to:

· Express the division of a quantity into a number parts as a ratio;

· Write ratios in form 1 : m or m : 1 and to describe a situation;

· Write ratios in their simplest form, including three-part ratios;

· Divide a given quantity into two or more parts in a given part : part or part : whole ratio;

· Use a ratio to find one quantity when the other is known;

· Write a ratio as a fraction;

· Write a ratio as a linear function;

· Identify direct proportion from a table of values, by comparing ratios of values;

· Use a ratio to compare a scale model to real-life object;

· Use a ratio to convert between measures and currencies, e.g. £1.00 = €1.36;

· Scale up recipes;

· Convert between currencies.

Polygons, angles and parallel lines

OBJECTIVES

By the end of the sub-unit, students should be able to:

· Classify quadrilaterals by their geometric properties and distinguish between scalene, isosceles and equilateral triangles;

· Understand ‘regular’ and ‘irregular’ as applied to polygons;

· Understand the proof that the angle sum of a triangle is 180°, and derive and use the sum of angles in a triangle;

· Use symmetry property of an isosceles triangle to show that base angles are equal;

· Find missing angles in a triangle using the angle sum in a triangle AND the properties of an isosceles triangle;

· Understand a proof of, and use the fact that, the exterior angle of a triangle is equal to the sum of the interior angles at the other two vertices;

· Explain why the angle sum of a quadrilateral is 360°;

· Understand and use the angle properties of quadrilaterals and the fact that the angle sum of a quadrilateral is 360°;

· Understand and use the angle properties of parallel lines and find missing angles using the properties of corresponding and alternate angles, giving reasons;

Probability

· OBJECTIVES

By the end of the unit, students should be able to:

· Write probabilities using fractions, percentages or decimals;

· Understand and use experimental and theoretical measures of probability, including relative frequency to include outcomes using dice, spinners, coins, etc;

· Estimate the number of times an event will occur, given the probability and the number of trials;

· Find the probability of successive events, such as several throws of a single dice;

· List all outcomes for single events, and combined events, systematically;

· Draw sample space diagrams and use them for adding simple probabilities;

· Know that the sum of the probabilities of all outcomes is 1;

· Use 1 – p as the probability of an event not occurring where p is the probability of the event occurring;

· Work out probabilities from Venn diagrams to represent real-life situations and also ‘abstract’ sets of numbers/values;

· Use union and intersection notation;

· Find a missing probability from a list or two-way table, including algebraic terms;

· Understand conditional probabilities and decide if two events are independent;

· Draw a probability tree diagram based on given information, and use this to find probability and expected number of outcome;

· Understand selection with or without replacement;

· Calculate the probability of independent and dependent combined events;

· Use a two-way table to calculate conditional probability;

· Use a tree diagram to calculate conditional probability;

· Use a Venn diagram to calculate conditional probability;

· Compare experimental data and theoretical probabilities;

· Compare relative frequencies from samples of different sizes.

Sequences

OBJECTIVES

By the end of the sub-unit, students should be able to:

· Recognise simple sequences including at the most basic level odd, even, triangular, square and cube numbers and Fibonacci-type sequences;

· Generate sequences of numbers, squared integers and sequences derived from diagrams;

· Describe in words a term-to-term sequence and identify which terms cannot be in a sequence;

· Generate specific terms in a sequence using the position-to-term rule and term-to-term rule;

· Find and use (to generate terms) the nth term of an arithmetic sequence;

· Use the nth term of an arithmetic sequence to decide if a given number is a term in the sequence, or find the first term above or below a given number;

· Identify which terms cannot be in a sequence by finding the nth term;

· Continue a quadratic sequence and use the nth term to generate terms;

Cumulative frequency, box plots and histograms

OBJECTIVES

By the end of the sub-unit, students should be able to:

· Use statistics found in all graphs/charts in this unit to describe a population;

· Know the appropriate uses of cumulative frequency diagrams;

· Construct and interpret cumulative frequency tables;

· Construct and interpret cumulative frequency graphs/diagrams and from the graph:

· estimate frequency greater/less than a given value;

· find the median and quartile values and interquartile range;

· Compare the mean and range of two distributions, or median and interquartile range, as appropriate;

· Interpret box plots to find median, quartiles, range and interquartile range and draw conclusions;

· Produce box plots from raw data and when given quartiles, median and identify any outliers;

· Know the appropriate uses of histograms;

· Construct and interpret histograms from class intervals with unequal width;

· Use and understand frequency density;

· From histograms:

· complete a grouped frequency table;

· understand and define frequency density;

· Estimate the mean from a histogram;

Estimate the median from a histogram with unequal class widths or any other information from a histogram, such as the number of people in a given interval

Averages and range

OBJECTIVES

By the end of the sub-unit, students should be able to:

· Design and use two-way tables for discrete and grouped data;

· Use information provided to complete a two-way table;

· Sort, classify and tabulate data and discrete or continuous quantitative data;

· Calculate mean and range, find median and mode from small data set;

· Use a spreadsheet to calculate mean and range, and find median and mode;

· Recognise the advantages and disadvantages between measures of average;

· Construct and interpret stem and leaf diagrams (including back-to-back diagrams):

· find the mode, median, range, as well as the greatest and least values from stem and leaf diagrams, and compare two distributions from stem and leaf diagrams (mode, median, range);

· Calculate the mean, mode, median and range from a frequency table (discrete data);

· Construct and interpret grouped frequency tables for continuous data:

· for grouped data, find the interval which contains the median and the modal class;

· estimate the mean with grouped data;

· understand that the expression ‘estimate’ will be used where appropriate, when finding the mean of grouped data using mid-interval values.

Standard form and surds

· Understand surd notation, e.g. calculator gives answer to sq rt 8 as 4 rt 2;

· Simplify surd expressions involving squares (e.g. √12 = √(4 × 3) = √4 × √3 = 2√3).

Transformations

OBJECTIVES

By the end of the sub-unit, students should be able to:

· Distinguish properties that are preserved under particular transformations;

· Recognise and describe rotations – know that that they are specified by a centre and an angle;

· Rotate 2D shapes using the origin or any other point (not necessarily on a coordinate grid);

· Identify the equation of a line of symmetry;

· Recognise and describe reflections on a coordinate grid – know to include the mirror line as a simple algebraic equation, x = a, y = a, y = x, y = –x and lines not parallel to the axes;

· Reflect 2D shapes using specified mirror lines including lines parallel to the axes and also
y = x and y = –x;

· Recognise and describe single translations using column vectors on a coordinate grid;

· Translate a given shape by a vector;

· Understand the effect of one translation followed by another, in terms of column vectors (to introduce vectors in a concrete way);

· Enlarge a shape on a grid without a centre specified;

· Describe and transform 2D shapes using enlargements by a positive integer, positive fractional, and negative scale factor;

· Know that an enlargement on a grid is specified by a centre and a scale factor;

· Identify the scale factor of an enlargement of a shape;

· Enlarge a given shape using a given centre as the centre of enlargement by counting distances from centre, and find the centre of enlargement by drawing;

· Find areas after enlargement and compare with before enlargement, to deduce multiplicative relationship (area scale factor); given the areas of two shapes, one an enlargement of the other, find the scale factor of the enlargement (whole number values only);

· 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;

· Describe and transform 2D shapes using combined rotations, reflections, translations, or enlargements;

· Describe the changes and invariance achieved by combinations of rotations, reflections and translations.

. Quadratic, cubic and other graphs

OBJECTIVES

By the end of the sub-unit, students should be able to:

· Recognise a linear, quadratic, cubic, reciprocal and circle graph from its shape;

· Generate points and plot graphs of simple quadratic functions, then more general quadratic functions;

· Find approximate solutions of a quadratic equation from the graph of the corresponding quadratic function;

· Interpret graphs of quadratic functions from real-life problems;

· Draw graphs of simple cubic functions using tables of values;

· Interpret graphs of simple cubic functions, including finding solutions to cubic equations;

3D forms and volume, cylinders, cones and spheres

OBJECTIVES

By the end of the sub-unit, students should be able to:

· Find the surface area of prisms using the formulae for triangles and rectangles, and other (simple) shapes with and without a diagram;

· Draw sketches of 3D solids;

· Identify planes of symmetry of 3D solids, and sketch planes of symmetry;

· Recall and use the formula for the volume of a cuboid or prism made from composite 3D solids using a variety of metric measures;

· Convert between metric volume measures;

· Convert between metric measures of volume and capacity, e.g. 1 ml = 1 cm3;

· Use volume to solve problems;

· Estimating surface area, perimeter and volume by rounding measurements to 1 significant figure to check reasonableness of answers.

· Use π ≈ 3.142 or use the π button on a calculator;

Find the volume and surface area of a cylinder;

Standard form and surds

By the end of the sub-unit, students should be able to:

· Convert large and small numbers into standard form and vice versa;

· Add and subtract numbers in standard form;

· Multiply and divide numbers in standard form;

· Interpret a calculator display using standard form and know how to enter numbers in standard form;

· Understand surd notation, e.g. calculator gives answer to sq rt 8 as 4 rt 2;

· Simplify surd expressions involving squares (e.g. √12 = √(4 × 3) = √4 × √3 = 2√3).

Factors, multiples and primes

OBJECTIVES

By the end of the sub-unit, students should be able to:

· Identify factors, multiples and prime numbers;

· Find the prime factor decomposition of positive integers – write as a product using index notation;

· Find common factors and common multiples of two numbers;

· Find the LCM and HCF of two numbers, by listing, Venn diagrams and using prime factors – include finding LCM and HCF given the prime factorisation of two numbers;

· Solve problems using HCF and LCM, and prime numbers;

· Understand that the prime factor decomposition of a positive integer is unique, whichever factor pair you start with, and that every number can be written as a product of prime factors.

Mocks only for set 3

Setting up, rearranging and solving equations

OBJECTIVES

By the end of the sub-unit, students should be able to:

· Set up simple equations from word problems and derive simple formulae;

· Understand the ≠ symbol (not equal), e.g. 6x + 4 ≠ 3(x + 2), and introduce identity ≡ sign;

· Solve linear equations, with integer coefficients, in which the unknown appears on either side or on both sides of the equation;

· Solve linear equations which contain brackets, including those that have negative signs occurring anywhere in the equation, and those with a negative solution;

· Solve linear equations in one unknown, with integer or fractional coefficients;

· Set up and solve linear equations to solve to solve a problem;

· Derive a formula and set up simple equations from word problems, then solve these equations, interpreting the solution in the context of the problem;

· Substitute positive and negative numbers into a formula, solve the resulting equation including brackets, powers or standard form;

· Use and substitute formulae from mathematics and other subjects, including the kinematics formulae v = u + at, v2 – u2 = 2as, and s = ut + at2;

· Change the subject of a simple formula, i.e. linear one-step, such as x = 4y;

· Change the subject of a formula, including cases where the subject is on both sides of the original formula, or involving fractions and small powers of the subject;

· Simple proofs and use of ≡ in “show that” style questions; know the difference between an equation and an identity;

· Use iteration to find approximate solutions to equations, for simple equations in the first instance, then quadratic and cubic equations.

Solving quadratics and simultaneous equations

· Find the exact solutions of two simultaneous equations in two unknowns;

· Use elimination or substitution to solve simultaneous equations;

· Solve exactly, by elimination of an unknown, two simultaneous equations in two unknowns:

· linear / linear, including where both need multiplying;

Inequalities

OBJECTIVES

By the end of the sub-unit, students should be able to:

· Show inequalities on number lines;

· Write down whole number values that satisfy an inequality;

· Solve simple linear inequalities in one variable, and represent the solution set on a number line;

· Solve two linear inequalities in x, find the solution sets and compare them to see which value of x satisfies both solve linear inequalities in two variables algebraically;

· Use the correct notation to show inclusive and exclusive inequalities.