year 9 set 3

Calculations, checking and rounding

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

●         Add, subtract, multiply and divide decimals and whole numbers;

●         Multiply or divide by any number between 0 and 1;

●         Put digits in the correct place in a decimal calculation and use one calculation to find the answer to another;

●         Use the product rule for counting (i.e. if there are m ways of doing one task and for each of these, there are n ways of doing another task, then the total number of ways the two tasks can be done is m × n ways);

●         Round numbers to the nearest 10, 100, 1000;

●         Round to the nearest integer, to a given number of decimal places and to a given number of significant figures;

●         Estimate answers to one- or two-step calculations, including use of rounding numbers and formal estimation to 1 significant figure: mainly whole numbers and then decimals.

POSSIBLE SUCCESS CRITERIA

Given 5 digits, what is the largest even number, largest odd number, or largest or smallest answers when subtracting a two-digit number from a three-digit number?

Given 2.6 × 15.8 = 41.08 what is 26 × 0.158? What is 4108 ÷ 26?

COMMON MISCONCEPTIONS

Significant figure and decimal place rounding are often confused.

Some pupils may think 35 934 = 36 to two significant figures.

Graphs: the basics and real-life graphs  -

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

●         Identify and plot points in all four quadrants;

●         Draw and interpret straight-line graphs for real-life situations, including ready reckoner graphs, conversion graphs, fuel bills, fixed charge and cost per item;

●         Draw distance–time and velocity–time graphs;

●         Use graphs to calculate various measures (of individual sections), including: unit price (gradient), average speed, distance, time, acceleration; including using enclosed areas by counting squares or using areas of trapezia, rectangles and triangles;

●         Find the coordinates of the midpoint of a line segment with a diagram given and coordinates;

●         Find the coordinates of the midpoint of a line segment from coordinates;

POSSIBLE SUCCESS CRITERIA

Interpret a description of a journey into a distance–time or speed–time graph.

Calculate various measures given a graph.

Calculate an end point of a line segment given one coordinate and its midpoint.

COMMON MISCONCEPTIONS

Where line segments cross the y-axis, finding midpoints and lengths of segments is particularly challenging as students have to deal with negative numbers.

Polygons, angles and parallel lines

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;

POSSIBLE SUCCESS CRITERIA

Name all quadrilaterals that have a specific property.

Given the size of its exterior angle, how many sides does the polygon have?

COMMON MISCONCEPTIONS

Some students will think that all trapezia are isosceles, or a square is only square if ‘horizontal’, or a ‘non-horizontal’ square is called a diamond.

Pupils may believe, incorrectly, that:

●         perpendicular lines have to be horizontal/vertical;

●         all triangles have rotational symmetry of order 3;

●         all polygons are regular.

Incorrectly identifying the ‘base angles’ (i.e. the equal angles) of an isosceles triangle when not drawn horizontally.

Algebra: the basics

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

●         Use algebraic notation and symbols correctly;

●         Write an expression;

●         Know the difference between a term, expression, equation, formula and an identity;

●         Manipulate an expression by collecting like terms;

●         Substitute positive and negative numbers into expressions such as 3x + 4 and 2x3 and then into expressions involving brackets and powers;

●         Substitute numbers into formulae from mathematics and other subject using simple linear formulae, e.g. l × w, v = u + at;

●         Simplify expressions by cancelling, e.g.  = 2x

●         Use instances of index laws for positive integer powers;

●         Use index notation (positive powers) when multiplying or dividing algebraic terms;

●         Use instances of index laws, including use of zero, fractional and negative powers;

●         Multiply a single term over a bracket;

●         Recognise factors of algebraic terms involving single brackets and simplify expressions by factorising, including subsequently collecting like terms;

●         Expand the product of two linear expressions, i.e. double brackets working up to negatives in both brackets and also similar to (2x + 3y)(3x – y);

●         Know that squaring a linear expression is the same as expanding double brackets;

POSSIBLE SUCCESS CRITERIA

Simplify 4p – 2q2 + 1 – 3p + 5q2.  Evaluate 4x2 – 2x when x = –5. 

Simplify z4 × z3, y3 ÷ y2, (a7)2, Expand and simplify 3(t – 1) + 57.   Factorise 15x2y – 35x2y2.

Expand and simplify (3x + 2)(4x – 1).  Factorise 6x2 – 7x + 1.

COMMON MISCONCEPTIONS

When expanding two linear expressions, poor number skills involving negatives and times tables will become evident.

Elevation, construction:

●         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;

●         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;

●          

POSSIBLE SUCCESS CRITERIA

Able to read and construct scale drawings.

When given the bearing of a point A from point B, can work out the bearing of B from A.

Know that scale diagrams, including bearings and maps, are ‘similar’ to the real-life examples.

Able to sketch the locus of point on a vertex of a rotating shape as it moves along a line, of a point on the circumference and at the centre of a wheel.

COMMON MISCONCEPTIONS

Correct use of a protractor may be an issue.

Fractions:

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

●         Express a given number as a fraction of another;

●         Find equivalent fractions and compare the size of fractions;

●         Write a fraction in its simplest form, including using it to simplify a calculation,
e.g. 50 ÷ 20 =  =  = 2.5;

●         Find a fraction of a quantity or measurement, including within a context;

●         Convert a fraction to a decimal to make a calculation easier;

●         Convert between mixed numbers and improper fractions;

●         Add, subtract, multiply and divide fractions;

●         Multiply and divide fractions, including mixed numbers and whole numbers and vice versa;

●         Add and subtract fractions, including mixed numbers;

●         Understand and use unit fractions as multiplicative inverses;

POSSIBLE SUCCESS CRITERIA

Express a given number as a fraction of another, including where the fraction is, for example, greater than 1, e.g.  =  = .

Answer the following: James delivers 56 newspapers.  of the newspapers have a magazine. How many of the newspapers have a magazine?

Prove whether a fraction is terminating or recurring.

Convert a fraction to a decimal including where the fraction is greater than 1.

COMMON MISCONCEPTIONS

The larger the denominator, the larger the fraction.

Collecting data, Representing and interpreting data:

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

●         Specify the problem and plan:

●         decide what data to collect and what analysis is needed;

●         understand primary and secondary data sources;

●         Understand what is meant by a sample and a population;

●         Know which charts to use for different types of data sets;

●         Produce and interpret composite bar charts;

●         Produce and interpret comparative and dual bar charts;

●         Produce and interpret pie charts:

●         find the mode and the frequency represented by each sector;

●         compare data from pie charts that represent different-sized samples;

●         Produce and interpret frequency polygons for grouped data:

●         from frequency polygons, read off frequency values, compare distributions, calculate total population, mean, estimate greatest and least possible values (and range);

●         Produce frequency diagrams for grouped discrete data:

●         read off frequency values, calculate total population, find greatest and least values;

POSSIBLE SUCCESS CRITERIA

Use a time–series data graph to make a prediction about a future value.

Explain why same-size sectors on pie charts with different data sets do not represent the same number of items, but do represent the same proportion.

Make comparisons between two data sets.

Factors, multiples and primes:

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.

POSSIBLE SUCCESS CRITERIA

Know how to test if a number up to 120 is prime.

Understand that every number can be written as a unique product of its prime factors.

Recall prime numbers up to 100.

Write a number as a product of its prime factors.

Use a Venn diagram to sort information.

COMMON MISCONCEPTIONS

1 is a prime number.

Particular emphasis should be made on the definition of “product” as multiplication, as many students get confused and think it relates to addition.

Sequences

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;

POSSIBLE SUCCESS CRITERIA

Given a sequence, ‘which is the 1st term greater than 50?’

Be able to solve problems involving sequences from real-life situations, such as:

●         1 grain of rice on first square, 2 grains on second, 4 grains on third, etc (geometric progression), or person saves £10 one week, £20 the next, £30 the next, etc;

●         What is the amount of money after x months saving the same amount, or the height of tree that grows 6 m per year;

●         Compare two pocket money options, e.g. same number of £ per week as your age from 5 until 21, or starting with £5 a week aged 5 and increasing by 15% a year until 21.

COMMON MISCONCEPTIONS

Students struggle to relate the position of the term to “n”.

Setting up, rearranging and solving equations: -

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;

POSSIBLE SUCCESS CRITERIA

A room is 2 m longer than it is wide. If its area is 30 m2 what is its perimeter?

Use fractions when working in algebraic situations.

Substitute positive and negative numbers into formulae.

Be aware of common scientific formulae.

Know the meaning of the ‘subject’ of a formula.

Change the subject of a formula when one step is required.

Change the subject of a formula when two steps are required.

COMMON MISCONCEPTIONS

Hierarchy of operations applied in the wrong order when changing the subject of a formula.

a0 = 0.

3xy and 5yx are different “types of term” and cannot be “collected” when simplifying expressions.

The square and cube operations on a calculator may not be similar on all makes.

Not using brackets with negative numbers on a calculator.

Not writing down all the digits on the display.

Linear graphs and coordinate geometry

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

●         Plot and draw graphs of y = a, x = a, y = x and y = –x, drawing and recognising lines parallel to axes, plus y = x and y = –x;

●         Identify and interpret the gradient of a line segment;

●         Recognise that equations of the form y = mx + c correspond to straight-line graphs in the coordinate plane;

●         Identify and interpret the gradient and y-intercept of a linear graph given by equations of the form y = mx + c;

●         Find the equation of a straight line from a graph in the form y = mx + c;

●         Plot and draw graphs of straight lines of the form y = mx + c with and without a table of values;

●         Sketch a graph of a linear function, using the gradient and y-intercept (i.e. without a table of values);

●         Find the equation of the line through one point with a given gradient;

●         Identify and interpret gradient from an equation ax + by = c;

●         Find the equation of a straight line from a graph in the form ax + by = c;

●         Plot and draw graphs of straight lines in the form ax + by = c;

●         Interpret and analyse information presented in a range of linear graphs:

●         use gradients to interpret how one variable changes in relation to another;

●         find approximate solutions to a linear equation from a graph;

●         identify direct proportion from a graph;

●         find the equation of a line of best fit (scatter graphs) to model the relationship between quantities;

●         Explore the gradients of parallel lines and lines perpendicular to each other;

●         Interpret and analyse a straight-line graph and generate equations of lines parallel and perpendicular to the given line;

●         Select and use the fact that when y = mx + c is the equation of a straight line, then the gradient of a line parallel to it will have a gradient of m and a line perpendicular to this line will have a gradient of .

POSSIBLE SUCCESS CRITERIA

Find the equation of the line passing through two coordinates by calculating the gradient first.

Understand that the form y = mx + c or ax + by = c represents a straight line.

COMMON MISCONCEPTIONS

Students can find visualisation of a question difficult, especially when dealing with gradients resulting from negative coordinates.

Percentages:

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;

●         Simple and compound interest

●         Reverse percentage

POSSIBLE SUCCESS CRITERIA

Be able to work out the price of a deposit, given the price of a sofa is £480 and the deposit is 15% of the price, without a calculator.

Find fractional percentages of amounts, with and without using a calculator.

Convince me that 0.125 is .

COMMON MISCONCEPTIONS

Incorrect links between fractions and decimals, such as thinking that  = 0.15, 5% = 0.5,
4% = 0.4, etc.

It is not possible to have a percentage greater than 100%.

Substitution, rearranging formulae, Iteration

●         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.

POSSIBLE SUCCESS CRITERIA

A room is 2 m longer than it is wide. If its area is 30 m2 what is its perimeter?

Use fractions when working in algebraic situations.

Substitute positive and negative numbers into formulae.

Be aware of common scientific formulae.

Know the meaning of the ‘subject’ of a formula.

Change the subject of a formula when one step is required.

Change the subject of a formula when two steps are required.

COMMON MISCONCEPTIONS

Hierarchy of operations applied in the wrong order when changing the subject of a formula.

a0 = 0.

3xy and 5yx are different “types of term” and cannot be “collected” when simplifying expressions.

The square and cube operations on a calculator may not be similar on all makes.

Not using brackets with negative numbers on a calculator.

Not writing down all the digits on the display.

Averages and range

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.

POSSIBLE SUCCESS CRITERIA

Be able to state the median, mode, mean and range from a small data set.

Extract the averages from a stem and leaf diagram.

Estimate the mean from a table.

COMMON MISCONCEPTIONS

Students often forget the difference between continuous and discrete data.

Pythagoras and trigonometry

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°.

POSSIBLE SUCCESS CRITERIA

Does 2, 3, 6 give a right-angled triangle?

Justify when to use Pythagoras’ Theorem and when to use trigonometry.

COMMON MISCONCEPTIONS

Answers may be displayed on a calculator in surd form.

Students forget to square root their final answer, or round their answer prematurely.

Perimeter, Area

●         Recall and use the formulae for the area of a triangle, rectangle, trapezium and parallelogram using a variety of metric measures;

●         Calculate the area of compound shapes made from triangles, rectangles, trapezia and parallelograms using a variety of metric measures;

●         Find the perimeter of a rectangle, trapezium and parallelogram using a variety of metric measures;

●         Calculate the perimeter of compound shapes made from triangles and rectangles;

●         Estimate area and perimeter by rounding measurements to 1 significant figure to check reasonableness of answers.

POSSIBLE SUCCESS CRITERIA

Calculate the area and/or perimeter of shapes with different units of measurement.

Students often get the concepts of area and perimeter confused.

Shapes involving missing lengths of sides often result in incorrect answers.

Circle

●         Recall the definition of a circle and name and draw parts of a circle;

●         Recall and use formulae for the circumference of a circle and the area enclosed by a circle (using circumference = 2πr = πd and area of a circle = πr2) using a variety of metric measures;

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

●         Calculate perimeters and areas of composite shapes made from circles and parts of circles (including semicircles, quarter-circles, combinations of these and also incorporating other polygons);

●         Calculate arc lengths, angles and areas of sectors of circles;

●         Find radius or diameter, given area or circumference of circles in a variety of metric measures;

●         Give answers in terms of π;

●         Form equations involving more complex shapes and solve these equations.

POSSIBLE SUCCESS CRITERIA

Understand that answers in terms of π are more accurate.

Calculate the perimeters and/or areas of circles, semicircles and quarter-circles given the radius or diameter and vice versa.

COMMON MISCONCEPTIONS

Diameter and radius are often confused, and recollection of area and circumference of circles involves incorrect radius or diameter.

 Transformation

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.

POSSIBLE SUCCESS CRITERIA

Recognise similar shapes because they have equal corresponding angles and/or sides scaled up in same ratio.

Understand that translations are specified by a distance and direction (using a vector).

Recognise that enlargements preserve angle but not length.

Understand that distances and angles are preserved under rotations, reflections and translations so that any shape is congruent to its image.

Understand that similar shapes are enlargements of each other and angles are preserved.

COMMON MISCONCEPTIONS

Students often use the term ‘transformation’ when describing transformations instead of the required information.

Lines parallel to the coordinate axes often get confused.

Probability

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;

3D shapes, volume, and surface area

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.

POSSIBLE SUCCESS CRITERIA

Given dimensions of a rectangle and a pictorial representation of it when folded, work out the dimensions of the new shape.

Work out the length given the area of the cross-section and volume of a cuboid.

COMMON MISCONCEPTIONS

Students often get the concepts of surface area and volume confused.

Standard Form

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;

POSSIBLE SUCCESS CRITERIA

Write 51080 in standard form.

Write 3.74 x 10–6 as an ordinary number.

COMMON MISCONCEPTIONS

Some students may think that any number multiplied by a power of ten qualifies as a number written in standard form.

When rounding to significant figures some students may think, for example, that 6729 rounded to one significant figure is 7.

Ratio and proportion

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;

POSSIBLE SUCCESS CRITERIA

Write/interpret a ratio to describe a situation such as 1 blue for every 2 red …, 3 adults for every 10 children …

Recognise that two paints mixed red to yellow 5 : 4 and 20 : 16 are the same colour.

When a quantity is split in the ratio 3:5, what fraction does each person get?

Find amounts for three people when amount for one given.

Construction, loci, bearing

●         Construct:

●         SSS, SAS, ASA triangles

●         Bisect a line and angle

●         a region bounded by a circle and an intersecting line;

●         a given distance from a point and a given distance from a line;

●         equal distances from two points or two line segments;

●         regions which may be defined by ‘nearer to’ or ‘greater than’;

●         Find and describe regions satisfying a combination of loci, including in 3D;

●         Use constructions to solve loci problems including with bearings;

●         Know that the perpendicular distance from a point to a line is the shortest distance to the line.

POSSIBLE SUCCESS CRITERIA

Able to read and construct scale drawings.

When given the bearing of a point A from point B, can work out the bearing of B from A.

Know that scale diagrams, including bearings and maps, are ‘similar’ to the real-life examples.

Able to sketch the locus of point on a vertex of a rotating shape as it moves along a line, of a point on the circumference and at the centre of a wheel.

COMMON MISCONCEPTIONS

Correct use of a protractor may be an issue.

Cumulative frequency, box plots

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;

POSSIBLE SUCCESS CRITERIA

Construct cumulative frequency graphs, box plots from frequency tables.

Compare two data sets and justify their comparisons based on measures extracted from their diagrams where appropriate in terms of the context of the data.

COMMON MISCONCEPTIONS

Labelling axes incorrectly in terms of the scales, and also using ‘Frequency’ instead of ‘Cumulative Frequency’.

Students often confuse the methods involved with cumulative frequency, estimating the mean when dealing with data tables.

Scatter graphs:

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

●         Draw and interpret scatter graphs;

●         Interpret scatter graphs in terms of the relationship between two variables;

●         Draw lines of best fit by eye, understanding what these represent;

●         Identify outliers and ignore them on scatter graphs;

●         Use a line of best fit, or otherwise, to predict values of a variable given values of the other variable;

●         Distinguish between positive, negative and zero correlation using lines of best fit, and interpret correlation in terms of the problem;

●         Understand that correlation does not imply causality, and appreciate that correlation is a measure of the strength of the association between two variables and that zero correlation does not necessarily imply ‘no relationship’ but merely ‘no linear correlation’;

●         Explain an isolated point on a scatter graph;

●         Use the line of best fit make predictions; interpolate and extrapolate apparent trends whilst knowing the dangers of so doing.

POSSIBLE SUCCESS CRITERIA

Be able to justify an estimate they have made using a line of best fit.

Identify outliers and explain why they may occur.

Given two sets of data in a table, model the relationship and make predictions.

COMMON MISCONCEPTIONS

Students often forget the difference between continuous and discrete data.

Lines of best fit are often forgotten, but correct answers still obtained by sight.

Histogram and graphs

●         Produce histograms with equal class intervals:

●         estimate the median from a histogram with equal class width or any other information, such as the number of people in a given interval;

●         Produce line graphs:

●         read off frequency values, calculate total population, find greatest and least values;

●         Construct and interpret time–series graphs, comment on trends;

●         Compare the mean and range of two distributions, or median or mode as appropriate;

●         Recognise simple patterns, characteristics relationships in bar charts, line graphs and frequency polygons.