year 9 set 4 and 5

Integers and place value

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

●         Use and order positive and negative numbers (integers);

●         Order integers, decimals, use the symbols <, > and understand the ≠ symbol;

●         Add and subtract positive and negative numbers (integers);

●         Recall all multiplication facts to 10 × 10, and use them to derive quickly the corresponding division facts;

●         Multiply or divide any number by powers of 10;

●         Multiply and divide positive and negative numbers (integers);

●         Use brackets and the hierarchy of operations (not including powers);

POSSIBLE SUCCESS CRITERIA

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

Use inverse operations to justify answers, e.g. 9 x 23 = 207 so 207 ÷ 9 = 23.

Check answers by rounding to nearest 10, 100, or 1000 as appropriate, e.g. 29 × 31 ≈ 30 × 30

COMMON MISCONCEPTIONS

Stress the importance of knowing the multiplication tables to aid fluency.

Students may write statements such as 150 – 210 = 60.

Tables

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

●         Use suitable data collection techniques (data to be integer and decimal values);

●         Design and use data-collection sheets for grouped, discrete and continuous data, use inequalities for grouped data, and introduce ≤ and ≥ signs;

●         Interpret and discuss the data;

●         Sort, classify and tabulate data, both discrete and continuous quantitative data, and qualitative data;

●         Construct tables for time–series data;

●         Extract data from lists and tables;

●         Use correct notation for time, 12- and 24-hour clock;

●         Work out time taken for a journey from a timetable;

●         Calculate the total frequency from a frequency table;

●         Read off frequency values from a table;

●         Read off frequency values from a frequency table;

●         Find greatest and least values from a frequency table;

●         Identify the mode from a frequency table;

●         Identify the modal class from a grouped frequency table.

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

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

●          

POSSIBLE SUCCESS CRITERIA

Construct a frequency table for a continuous data set, deciding on appropriate intervals using inequalities

Plan a journey using timetables.

COMMON MISCONCEPTIONS

Students struggle to make the link between what the data in a frequency table represents, so for example may state the ‘frequency’ rather than the interval when asked for the modal group.

Statistics and sampling

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

●         Specify the problem and:

●         plan an investigation;

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

●         consider fairness;

●         Recognise types of data: primary secondary, quantitative and qualitative;

●         Identify which primary data they need to collect and in what format, including grouped data;

●         Collect data from a variety of suitable primary and secondary sources;

●         Understand how sources of data may be biased;

●         Explain why a sample may not be representative of a whole population;

●         Understand sample and population.

POSSIBLE SUCCESS CRITERIA

Explain why a sample may not be representative of a whole population.

Carry out a statistical investigation of their own and justify how sources of bias have been eliminated.

Show me an example of a situation in which biased data would result.

COMMON MISCONCEPTIONS

The concept of an unbiased sample is difficult for some students to understand.

Congruent shapes and construction

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

•         Understand congruence, as two shapes that are the same size and shape;

•         Visually identify shapes which are congruent;

•         Use straight edge and a pair of compasses to do standard constructions:

•         understand, from the experience of constructing them, that triangles satisfying SSS, SAS, ASA and RHS are unique, but SSA triangles are not;

POSSIBLE SUCCESS CRITERIA

Sketch the locus of point on a vertex of a rotating shape as it moves along a line, i.e. a point on the circumference or at the centre of a wheel.

COMMON MISCONCEPTIONS

Correct use of a protractor may be an issue.

Algebra

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

●         Use notation and symbols correctly;

●         Write an expression;

●         Select an expression/equation/formula/identity from a list;

●         Manipulate and simplify algebraic expressions by collecting ‘like’ terms;

●         Multiply together two simple algebraic expressions, e.g. 2a × 3b;

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

●         Use index notation when multiplying or dividing algebraic terms;

●         Use index laws in algebra;

●         Use index notation in algebra.

●         Understand the ≠ symbol and introduce the identity ≡ sign;

POSSIBLE SUCCESS CRITERIA

Simplify 4p – 2q + 3p + 5q.

COMMON MISCONCEPTIONS

Any poor number skills involving negatives and times tables will become evident.

Fractions

●         Use diagrams to find equivalent fractions or compare fractions;

●         Write fractions to describe shaded parts of diagrams;

●         Express a given number as a fraction of another, using very simple numbers, some cancelling, and where the fraction is both < 1 and > 1;

●         Write a fraction in its simplest form and find equivalent fractions;

●         Order fractions, by using a common denominator;

●         Compare fractions, use inequality signs, compare unit fractions;

●         Convert between mixed numbers and improper fractions;

●         Add and subtract fractions;

●         Add fractions and write the answer as a mixed number;

●         Multiply and divide an integer by a fraction;

●         Multiply and divide a fraction by an integer, including finding fractions of quantities or measurements, and apply this by finding the size of each category from a pie chart using fractions;

●         Understand and use unit fractions as multiplicative inverses;

●         Multiply fractions: simplify calculations by cancelling first;

●         Divide a fraction by a whole number;

●         Divide fractions by fractions.

●          

POSSIBLE SUCCESS CRITERIA

Express a given number as a fraction of another, including where the fraction > 1.

Simplify .

 × 15, 20 × .

 of 36 m,  of £20.

Find the size of each category from a pie chart using fractions.

Calculate:  × ,  ÷ 3.

COMMON MISCONCEPTIONS

The larger the denominator the larger the fraction.

Charts

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

●         Plotting coordinates in first quadrant and read graph scales in multiples;

●         Produce:

●         pictograms;

●         composite bar charts;

●         dual/comparative bar charts for categorical and ungrouped discrete data;

●         bar-line charts;

●         vertical line charts;

●         line graphs;

●         line graphs for time–series data;

●         histograms with equal class intervals;

●         stem and leaf (including back-to-back);

●         Interpret data shown in

●         pictograms;

●         composite bar charts;

●         dual/comparative bar charts;

●         line graphs;

●         line graphs for time–series data;

●         histograms with equal class intervals;

●         stem and leaf;

●         Calculate total population from a bar chart or table;

●         Find greatest and least values from a bar chart or table;

●         Find the mode from a stem and leaf diagram;

●         Identify the mode from a bar chart;

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

POSSIBLE SUCCESS CRITERIA

Decide the most appropriate chart or table given a data set.

State the mode, smallest value or largest value from a stem and leaf diagram.

NOTES

Ensure that you include a variety of scales, including decimal numbers of millions and thousands, time scales in hours, minutes, seconds.

Misleading graphs are a useful life skill.

Factors, multiples and primes:

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

●         List all three-digit numbers that can be made from three given integers;

●         Recognise odd and even numbers;

●         Identify factors, multiples and prime numbers;

●         Recognise two-digit prime numbers;

●         List all factors of a number and list multiples systematically;

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

POSSIBLE SUCCESS CRITERIA

Given the digits 1, 2 and 3, find how many numbers can be made using all the digits.

Convince me that 8 is not prime.

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

Recall prime numbers up to 100.

Understand the meaning of prime factor.

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 sequences of odd and even numbers, and other sequences including Fibonacci sequences;

●         Use function machines to find terms of a sequence;

●         Write the term-to-term definition of a sequence in words;

●         Find a specific term in the sequence using position-to-term or term-to-term rules;

●         Generate arithmetic sequences of numbers, triangular number, square and cube integers and sequences derived from diagrams;

●         Recognise such sequences from diagrams and draw the next term in a pattern sequence;

●         Find the next term in a sequence, including negative values;

●         Find the nth term for a pattern sequence;

●         Find the nth term of a linear sequence;

●         Find the nth term of an arithmetic sequence;

●         Use the nth term of an arithmetic sequence to generate terms;

POSSIBLE SUCCESS CRITERIA

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

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

What are the next terms in the following sequences?

1, 3, 9, …              100, 50, 25, …                     2, 4, 8, 16, …

Write down an expression for the nth term of the arithmetic sequence 2, 5, 8, 11, …

Is 67 a term in the sequence 4, 7, 10, 13, …?

NOTES

Emphasise use of 3n meaning 3 × n.

Students need to be clear on the description of the pattern in words, the difference between the terms and the algebraic description of the nth term.

Decimals

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

●         Use decimal notation and place value;

●         Identify the value of digits in a decimal or whole number;

●         Compare and order decimal numbers using the symbols <, >;

●         Understand the ≠ symbol (not equal);

●         Write decimal numbers of millions, e.g. 2 300 000 = 2.3 million;

●         Add, subtract, multiply and divide decimals;

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

●         Round to the nearest integer;

●         Round to a given number of decimal places;

●         Round to any given number of significant figures;

●         Estimate answers to calculations by rounding numbers to 1 significant figure;

●         Use one calculation to find the answer to another.

POSSIBLE SUCCESS CRITERIA

Use mental methods for × and ÷, e.g. 5 × 0.6, 1.8 ÷ 3.

Solve a problem involving division by a decimal (up to 2 decimal places).

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

Calculate, e.g. 5.2 million + 4.3 million.

COMMON MISCONCEPTIONS

Significant figures and decimal place rounding are often confused.

Some students may think 35 877 = 36 to two significant figures.

Straight Line graphs

●          

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

●         Use function machines to find coordinates (i.e. given the input x, find the output y);

●         Plot and draw graphs of y = a, x = a, y = x and y = –x;

●         Recognise straight-line graphs parallel to the axes;

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

●         Plot and draw graphs of straight lines of the form y = mx + c using a table of values;

●         Sketch a graph of a linear function, using the gradient and y-intercept;

●         Identify and interpret gradient from an equation y = mx + c;

●         Identify parallel lines from their equations;

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

●         Find the equation of a straight line from a graph;

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

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

●         Find the gradient of a straight line from real-life graphs too.

POSSIBLE SUCCESS CRITERIA

Plot and draw the graph for y = 2x – 4.

Which of these lines are parallel:  y = 2x + 3,   y = 5x + 3,   y = 2x – 9,   2y = 4x – 8

COMMON MISCONCEPTIONS

When not given a table of values, students rarely see the relationship between the coordinate axes.

Converting between fractions-decimals-percentages

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

●         Recall the fraction-to-decimal conversion;

●         Convert between fractions and decimals;

●         Convert a fraction to a decimal to make a calculation easier, e.g. 0.25 × 8 =  × 8, or
  × 10 = 0.375 × 10;

●         Recognise recurring decimals and convert fractions such as ,  and  into recurring decimals;

●         Compare and order fractions, decimals and integers, using inequality signs;

●         Understand that a percentage is a fraction in hundredths;

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

●         Convert between fractions, decimals and percentages;

●         Order fractions, decimals and percentages, including use of inequality signs.

POSSIBLE SUCCESS CRITERIA

Write terminating decimals (up to 3 d.p.) as fractions.

Convert between fractions, decimals and percentages, common ones such as , , ,
  and .

Order integers, decimals and fractions.

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%.

Percentages

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

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

●         Find a percentage of a quantity without a calculator: 50%, 25% and multiples of 10% and 5%;

●         Find a percentage of a quantity or measurement (use measurements they should know from Key Stage 3 only);

●         Calculate amount of increase/decrease;

●         Use percentages to solve problems, including comparisons of two quantities using percentages;

●         Percentages over 100%;

POSSIBLE SUCCESS CRITERIA

What is 10%, 15%, 17.5% of £30?

COMMON MISCONCEPTIONS

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

Averages and Range

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

●         Calculate the mean, mode, median and range for discrete data;

●         Can interpret and find a range of averages as follows:

●         median, mean and range from a (discrete) frequency table;

●         range, modal class, interval containing the median, and estimate of the mean from a grouped data frequency table;

●         mode and range from a bar chart;

●         median, mode and range from stem and leaf diagrams;

●         mean from a bar chart;

●         Understand that the expression 'estimate' will be used where appropriate, when finding the mean of grouped data using mid-interval values;

●         Compare the mean, median, mode and range (as appropriate) of two distributions using bar charts, dual bar charts, pictograms and back-to-back stem and leaf;

●         Recognise the advantages and disadvantages between measures of average.

POSSIBLE SUCCESS CRITERIA

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

Often the ∑(m × f) is divided by the number of classes rather than ∑f when estimating the mean.

Indices, powers and roots

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

●         Find squares and cubes:

●         recall integer squares up to 10 x 10 and the corresponding square roots;

●         understand the difference between positive and negative square roots;

●         recall the cubes of 1, 2, 3, 4, 5 and 10;

●         Use index notation for squares and cubes;

●         Recognise powers of 2, 3, 4, 5;

●         Evaluate expressions involving squares, cubes and roots:

POSSIBLE SUCCESS CRITERIA

What is the value of 23?

Evaluate (23 × 25) ÷ 24.

COMMON MISCONCEPTIONS

The order of operations is often not applied correctly when squaring negative numbers, and many calculators will reinforce this misconception.

103, for example, is interpreted as 10 × 3.

Pythagoras

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

●         Understand, recall and use Pythagoras’ Theorem in 2D, including leaving answers in surd form;

●         Given 3 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;

●         Apply Pythagoras’ Theorem with a triangle drawn on a coordinate grid;

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

POSSIBLE SUCCESS CRITERIA

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

Justify when to use Pythagoras’ Theorem

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

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

●         Indicate given values on a scale, including decimal value;

●         Know that measurements using real numbers depend upon the choice of unit;

●         Convert between units of measure within one system, including time;

●         Convert metric units to metric units;

●         Make sensible estimates of a range of measures in everyday settings;

●         Measure shapes to find perimeters and areas using a range of scales;

●         Find the perimeter of rectangles and triangles;

●         Find the perimeter of parallelograms and trapezia;

●         Find the perimeter of compound shapes;

●         Recall and use the formulae for the area of a triangle and rectangle;

●         Find the area of a rectangle and triangle;

POSSIBLE SUCCESS CRITERIA

Find the area/perimeter of a given shape, stating the correct units.

COMMON MISCONCEPTIONS

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

●         Students often confuse perimeter and area.

Ratio and proportion

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

●         Understand and express the division of a quantity into a of number parts as a ratio;

●         Write ratios in their simplest form;

●         Write/interpret a ratio to describe a situation;

●         Share a quantity in a given ratio including three-part ratios;

●         Solve a ratio problem in context:

●         use a ratio to find one quantity when the other is known;

●         use a ratio to compare a scale model to a real-life object;

●         use a ratio to convert between measures and currencies;

●         problems involving mixing, e.g. paint colours, cement and drawn conclusions;

●         Compare ratios;

●         Write ratios in form 1 : m or m : 1;

●         Write a ratio as a fraction;

POSSIBLE SUCCESS CRITERIA

Write a ratio to describe a situation such as 1 blue for every 2 red, or 3 adults for every 10 children.

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

Express the statement ‘There are twice as many girls as boys’ as the ratio 2 : 1 or the linear function y = 2x, where x is the number of boys and y is the number of girls.

COMMON MISCONCEPTIONS

Students find three-part ratios difficult.

Using a ratio to find one quantity when the other is known often results in students ‘sharing’ the known amount.

Transformation

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

●         Identify congruent shapes by eye;

●         Understand clockwise and anticlockwise;

●         Understand that rotations are specified by a centre, an angle and a direction of rotation;

●         Find the centre of rotation, angle and direction of rotation and describe rotations;

●         Describe a rotation fully using the angle, direction of turn, and centre;

●         Rotate a shape about the origin or any other point on a coordinate grid;

●         Draw the position of a shape after rotation about a centre (not on a coordinate grid);

●         Identify correct rotations from a choice of diagrams;

●         Understand that translations are specified by a distance and direction using a vector;

●         Translate a given shape by a vector;

●         Describe and transform 2D shapes using single translations on a coordinate grid;

●         Use column vectors to describe translations;

●         Understand that distances and angles are preserved under rotations and translations, so that any figure is congruent under either of these transformations.

POSSIBLE SUCCESS CRITERIA

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

Describe and transform a given shape by either a rotation or a translation.

COMMON MISCONCEPTIONS

The directions on a column vector often get mixed up.

Student need to understand that the ‘units of movement’ are those on the axes, and care needs to be taken to check the scale.

Correct language must be used: students often use ‘turn’ rather than

‘rotate’.

Probability

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

●         Distinguish between events which are impossible, unlikely, even chance, likely, and certain to occur;

●         Mark events and/or probabilities on a probability scale of 0 to 1;

●         Write probabilities in words or fractions, decimals and percentages;

●         Find the probability of an event happening using theoretical probability;

●         Use theoretical models to include outcomes using dice, spinners, coins;

●         List all outcomes for single events systematically;

●         Work out probabilities from frequency tables;

●         Work out probabilities from two-way tables;

●         Record outcomes of probability experiments in tables;

●         Add simple probabilities;

●         Identify different mutually exclusive outcomes and know that the sum of the probabilities of all outcomes is 1;

●         Using 1 – p as the probability of an event not occurring where p is the probability of the event occurring;

●         Find a missing probability from a list or table including algebraic terms.

POSSIBLE SUCCESS CRITERIA

Mark events on a probability scale and use the language of probability.

If the probability of outcomes are x, 2x, 4x, 3x calculate x.

Calculate the probability of an event from a two-way table or frequency table.

Decide if a coin, spinner or game is fair.

Angles 1-2

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

●         Estimate sizes of angles;

●         Measure angles using a protractor;

●         Use geometric language appropriately;

●         Use letters to identify points, lines and angles;

●         Use two-letter notation for a line and three-letter notation for an angle;

●         Describe angles as turns and in degrees;

●         Understand clockwise and anticlockwise;

●         Know that there are 360° in a full turn, 180° in a half turn and 90° in a quarter turn;

●         Identify a line perpendicular to a given line;

●         Mark perpendicular lines on a diagram and use their properties;

●         Identify parallel lines;

●         Mark parallel lines on a diagram and use their properties;

●         Finding missing angles: on a straight line, around the point, triangle, quadrilateral

●         Recognise alternate, corresponding, co-interior and opposite angles

POSSIBLE SUCCESS CRITERIA

Name all quadrilaterals that have a specific property.

Use geometric reasoning to answer problems giving detailed reasons.

Find the size of missing angles at a point or at a point on a straight line.

COMMON MISCONCEPTIONS

Pupils may believe, incorrectly, that perpendicular lines have to be horizontal/vertical or all triangles have rotational symmetry of order 3.

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.

Some students may think that the equal angles in an isosceles triangle are the ‘base angles’.

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

Pie chart

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

●         Draw circles and arcs to a given radius;

●         Know there are 360 degrees in a full turn, 180 degrees in a half turn, and 90 degrees in a quarter turn;

●         Measure and draw angles, to the nearest degree;

●         Interpret tables; represent data in tables and charts;

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

●         Construct pie charts for categorical data and discrete/continuous numerical data;

●         Interpret simple pie charts using simple fractions and percentages; ,  and multiples of 10% sections;

●         From a pie chart:

•      find the mode;

•      find the total frequency;

●         Understand that the frequency represented by corresponding sectors in two pie charts is dependent upon the total populations represented by each of the pie charts.

POSSIBLE SUCCESS CRITERIA

From a simple pie chart identify the frequency represented by  and  sections.

From a simple pie chart identify the mode.

Find the angle for one item.

COMMON MISCONCEPTIONS

Same size sectors for different sized data sets represent the same number rather than the same proportion.

HCF, LCM

●         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;

●         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 two factors;

●         Solve simple problems using HCF, LCM and prime numbers.

POSSIBLE SUCCESS CRITERIA

Given the digits 1, 2 and 3, find how many numbers can be made using all the digits.

Convince me that 8 is not prime.

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

Recall prime numbers up to 100.

Understand the meaning of prime factor.

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.

Expanding brackets

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

●         Multiply a single number term over a bracket;

●         Write and simplify expressions using squares and cubes;

●         Simplify expressions involving brackets, i.e. expand the brackets, then add/subtract;

●         Argue mathematically to show algebraic expressions are equivalent;

●         Recognise factors of algebraic terms involving single brackets;

●         Factorise algebraic expressions by taking out common factors.

POSSIBLE SUCCESS CRITERIA

Expand and simplify 3(t – 1).

Understand 6x + 4 ≠ 3(x + 2).

Argue mathematically that 2(x + 5) = 2x + 10.

COMMON MISCONCEPTIONS

3(x + 4) = 3x + 4.

The convention of not writing a coefficient with a single value, i.e. x instead of 1x, may cause confusion.

Index Laws

●         Evaluate expressions involving squares, cubes and roots:

●         add, subtract, multiply and divide numbers in index form;

●         cancel to simplify a calculation;

●         Use index notation for powers of 10, including negative powers;

●         Use the laws of indices to multiply and divide numbers written in index notation;

●         Use the square, cube and power keys on a calculator;

●         Use brackets and the hierarchy of operations with powers inside the brackets, or raising brackets to powers;

●         Use calculators for all calculations: positive and negative numbers, brackets, powers and roots, four operations.