National Curriculum Resource Tool

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National Curriculum Resource Tool

Materials to support teachers and schools in embedding the National Curriculum

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National Curriculum Tool
KS3 - Number

KS3 - Number

New Curriculum

Pupils should be taught to:

understand and use place value for decimals, measures and integers of any size
order positive and negative integers, decimals and fractions; use the number line as a model for ordering of the real numbers; use the symbols =, ≠, <, >, ≤, ≥
use the concepts and vocabulary of prime numbers, factors (or divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation property
use the four operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative
use conventional notation for the priority of operations, including brackets, powers, roots and reciprocals
recognise and use relationships between operations including inverse operations
use integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5 and distinguish between exact representations of roots and their decimal approximations
interpret and compare numbers in standard form A × 10n 1≤A>10, where n is a positive or negative integer or zero
work interchangeably with terminating decimals and their corresponding fractions
(such as 3.5 and 72 or 0.375 and 38)
define percentage as ‘number of parts per hundred’, interpret percentages and percentage changes as a fraction or a decimal, interpret these multiplicatively, express one quantity as a percentage of another, compare two quantities using percentages, and work with percentages greater than 100%
interpret fractions and percentages as operators
use standard units of mass, length, time, money and other measures, including with decimal quantities
round numbers and measures to an appropriate degree of accuracy [for example, to a number of decimal places or significant figures]
use approximation through rounding to estimate answers and calculate possible resulting errors expressed using inequality notation a<x≤b
use a calculator and other technologies to calculate results accurately and then interpret them appropriately
appreciate the infinite nature of the sets of integers, real and rational numbers

Non-Statutory Guidance

Links and Resources

Making connections to other topics within Key Stage 3

Algebra

There are strong connections with substituting into expressions and formulae and in manipulation of algebraic expressions.

Ratio, proportion and rates of change

Pupils will be required to use and apply number skills in fractions, percentages and decimals to solve problems involving ratio, proportion and rates of change.

Geometry and measures

Pupils will be required to use and apply number skills in fractions, percentages and decimals to solve problems involving calculating, angles, lengths, areas, perimeters and volumes.

Probability and Statistics

There is a strong connection with how to use and apply number skills in fractions, percentages and decimals less than 1 to solve problems involving probability.

Making connections to this topic in Year 6

In year 6 pupils will have used and applied number in a range of contexts:

Notes and guidance on number - addition, subtraction, multiplication and division (non-statutory) suggest:

Pupils practise addition, subtraction, multiplication and division for larger numbers, using the formal written methods of columnar addition and subtraction, short and long multiplication, and short and long division
They undertake mental calculations with increasingly large numbers and more complex calculations
Pupils continue to use all the multiplication tables to calculate mathematical statements in order to maintain their fluency
Pupils round answers to a specified degree of accuracy, for example, to the nearest 10, 20, 50 etc., but not to a specified number of significant figures
Pupils explore the order of operations using brackets; for example, 2+1 × 3 = 5 and (2+1) × 3 = 9
Common factors can be related to finding equivalent fractions

Notes and guidance on number - fractions (decimals and percentages) (non-statutory) suggest:

Pupils should practise, use and understand the addition and subtraction of fractions with different denominators by identifying equivalent fractions with the same denominator. They should start with fractions where the denominator of one fraction is a multiple of the other (for example, 12 + 18 = 58) and progress to varied and increasingly complex problems.
Pupils should use a variety of images to support their understanding of multiplication with fractions. This follows earlier work about fractions as operators (fractions of), as numbers, and as equal parts of objects, for example, as parts of a rectangle.
Pupils use their understanding of the relationship between unit fractions and division to work backwards by multiplying a quantity that represents a unit fraction to find the whole quantity (for example, if 14 of a length is 36cm, then the whole length is 36 × 4 = 144cm).
They practise calculations with simple fractions and decimal fraction equivalents to aid fluency, including listing equivalent fractions to identify fractions with common denominators.
Pupils can explore and make conjectures about converting a simple fraction to a decimal fraction (for example, 3 ÷ 8 = 0.375). For simple fractions with recurring decimal equivalents, pupils learn about rounding the decimal to three decimal places, or other appropriate approximations depending on the context. Pupils multiply and divide numbers with up to two decimal places by one-digit and two-digit whole numbers. Pupils multiply decimals by whole numbers, starting with the simplest cases, such as 0.4 × 2 = 0.8, and in practical contexts, such as measures and money.
Pupils are introduced to the division of decimal numbers by one-digit whole number, initially, in practical contexts involving measures and money. They recognise division calculations as the inverse of multiplication.
Pupils also develop their skills of rounding and estimating as a means of predicting and checking the order of magnitude of their answers to decimal calculations. This includes rounding answers to a specified degree of accuracy and checking the reasonableness of their answers.

Cross-curricular and real life connections

Opportunities for practising number skills occur in almost all other subjects. Science and the Social Scientists are the most obvious ones in the processing of data (including time, money, weights, areas, numbers of people); but there is much in Foreign Languages (converting currencies, arising in stories), PE (looking at distances run, averaging scores, times for different activities).

Night train from Teachers TV illustrates maths in real life. The twelve maths sequences were filmed on a sleeper train and include how fast and how far does the train travel, how much food is carried, how many sheets are there, how do the carriages get cleaned, and is that bed really big enough? Each sequence poses a question intended to stimulate mathematical discussion rather than to elicit a correct answer to a computation.
In the hotel from Teachers TV illustrates maths in real life. Mathematic sequences from behind the scenes at a London hotel. Each sequence poses a question intended to stimulate mathematical discussion rather than to elicit a correct answer to a computation. The mathematical content rewards repeated viewing and provides a rich source of mathematical imagery.
Cricket match from Teachers TV illustrates maths in real life. Ten maths sequences filmed during a cricket match which encourage pupils to think mathematically. Cricket is a game based on numbers and maths, offering questions about how fast a ball travels, how high, what the score is, and why cricket pitches are often oval.

Connecting Measurement and Decimals A. Watson, K. Jones, D. Pratt, 2013.

“Measures are numbers that are connected to a quantity, the connection being a ratio comparison (and not simply a count of a number of ‘things’). Even though measurement is intimately tied to diverse mathematical concepts from number through to geometry, algebra and statistics, it can be neglected in teaching. This can store up problems for students as they move on to more advanced mathematical topics, including graphs of functions, loci and vectors.”

Connecting Measurement and Decimals includes this Cre8ate maths activity, More Miles for Your Money from the STEM e-library (free registration)

Based on information and guides provided by the Department of Transport, introduces the cost benefits of improving the aerodynamics of the cab section of a truck. Mathematical connections involve using calculators and spreadsheets to work on conversions and percentages in the context of a complex multi-step problem. Considerable mathematical thinking is involved in breaking down the problem into simpler steps, keeping track of the constituent parts and analysing the final results.

Happy Numbers

In this problem pupils start with an integer (less than 1000), add the squares of its digits, and then go on repeating this until a pattern emerges.

I’m Eight

Pupils generate examples that use all the different mathematical ideas they know about.

Chocolate

Pupils work out fractions to maximise the amount of chocolate they would receive.

Consecutive Negative Numbers

This activity requires pupils to look at addition and subtraction of four negative numbers and seek generalisations.

Ordering Fractions and Decimals

(Improving Learning in Mathematics, N1) from the STEM e-library (free registration).

Pupils learn to interpret decimals and fractions using scales and areas, find equivalent fractions, order and relate fractions and decimals and to reflect on and discuss these processes.

Evaluating Statements about Number Operations

(Improving Learning in Mathematics, N2) from the STEM e-library (free registration).

Pupils learn to understand the properties of number operations, substitute integers, fractions, decimals and negative numbers into statements and equations in order to test their validity and address common misconceptions about the effect of addition, subtraction, multiplication, division, squaring and finding square roots.

Estimating Length Using Standard Form

(Improving Learning in Mathematics, N4) from the STEM e-library (free registration).

Pupils learn to interpret decimals using metric units, estimate lengths of real life objects and interpret standard form.

Understanding the Laws of Arithmetic

(Improving Learning in Mathematics, N5) from the STEM e-library (free registration).

Pupils learn to interpret numerical expressions using words and area representations, recognise the order of operations and equivalent expressions and understand the distributive laws of multiplication and division over addition (the expansion of brackets).

Using Percentages to Increase Quantities

(Improving Learning in Mathematics, N7) from the STEM e-library (free registration).

Pupils make links between percentages, decimals and fractions, represent percentage increase and decrease as multiplication and recognise the inverse relationship between increases and decreases using proportional reasoning.

Examples of what children should be able to do, to show a broad grasp of the content of this topic

Example A

Here is a list of seven numbers

3 6 8 9 12 14 20 23

From the list, write down
(i)a square number,
(ii)a number that is a multiple of 7,
(iii)two numbers that are factors of 40,
(iv)two numbers that are prime.
Express the following numbers as products of their prime factors.
(i)56
(ii)84
(iii)Find the Highest Common Factor of 56 and 84.
Tyrone says that “6 is a cube number because 23 = 6”.
Tyrone is wrong. Explain why.
Paul writes down two sums.
1 + 2 = 3
7 + 8 = 15
Paul says “The sum of two whole consecutive numbers is never a square number”.
Give an example to show that Paul is wrong.

Example B

Write the number seventeen thousand, two hundred and two in figures.
Write the number 5262 correct to the nearest hundred.
Write down the value of the 4 in the number 274 663
Write 37 591 correct to 1 significant figure.
Write 0.000 755 9 correct to 1 significant figure.

Example C

Write \(\frac{1}{5}\) as a percentage.
Write \(\frac{2}{5}\) as a decimal.
Write 0.3 as a fraction.
Write 11% as a fraction.
Write 0.25 as a percentage.
Write 8% as a decimal.

Example D

Write these numbers in order of size.

Start with the smallest number.

a.	-3,	0,	-7,	-1,	5
b.	0.72,	0.7,	0.07,	0.702,	0.072
c.	67%	\(\frac{3}{4}\)	0.72	\(\frac{2}{3}\)	\(\frac{3}{5}\)
d.	\(\frac{3}{4}\)	\(\frac{1}{2}\)	\(\frac{1}{6}\)	\(\frac{2}{3}\)	\(\frac{3}{8}\)

Example E

Some students each chose one PE activity.
\(\frac{1}{5}\) of the students chose hockey.
\(\frac{3}{8}\) of the students chose football.
All the rest of these students chose basketball.
What fraction of the students chose basketball?
A school has 1200 pupils.
625 of these pupils are boys.
\(\frac{2}{5}\) of the girls like sport.
\(\frac{3}{5}\) of the boys like sport.
Work out the total number of pupils in the school who like sport.

Example F

Ron went to Spain.
He changed pounds (£) into Euros (€). The exchange rate was £1 = €1.22
The value of the pound has decreased from €1.22 to €1.13
Calculate the percentage decrease in the value of the pound.
Hugh went on holiday to Italy.
While on holiday, he went shopping.
He bought a tee shirt and a belt.
The tee shirt cost 25 euros.
The belt cost 14 euros.
The exchange rate was £1 = 1.18 euros.
Work out the total cost of the tee shirt and the belt.
Give the total cost in pounds.

Example G

In a sale all the normal prices are reduced by 18%.
In the sale Manim pays £12.71 for a bag.
Calculate the normal price of the bag.
Bill bought a box of 40 oranges for £2.
\(\frac{3}{10}\) of the 40 oranges were damaged so he threw them away.
He sold the remaining oranges at x pence each.
He made a profit of 40%.
Calculate the value of x.
A year ago, Donna weighed 51.5 kg.
Donna now weighs 8.5% less.
Work out how much Donna now weighs.
Give your answer to an appropriate degree of accuracy.

Example H

Mr Holland uses 367 units of electricity in one month.
He pays 5.84p for each unit of electricity.
Mr Holland also pays a fixed charge of £6.14 for the month.
Work out the total amount he pays.

Example I

Here is part of a railway timetable.

Aytoun	11 30	11 55	–	–	12 30	12 55
Beeston	11 34	11 59	12 04	12 29	12 34	12 59
Carton	11 39	12 04	12 09	12 34	12 39	13 04
Dunton	11 53	–	12 23	–	12 53	–
Eastleigh	11 59	–	12 29	–	12 59	–
Farnton	12 17	12 30	12 48	13 05	13 17	13 31

A train leaves Aytoun at 11 55

At what time should this train arrive in Farnton?
Another train leaves Aytoun at 12 55
Work out how many minutes it should take this train to get to Farnton.
James is going to Farnton.
He will catch the train in Dunton.
Jerry needs to arrive in Farnton before 13 00
Write down the time of the latest train he can catch from Dunton.

Example J

Mr White stayed some time at the South Pole.

The highest temperature there was –28 °C.
The lowest temperature there was –55 °C.

Work out the difference between the highest temperature and the lowest temperature at the South Pole.

Mr White returned to his house in England.
The temperature outside his house was –1 °C.
The temperature inside his house was 15 °C higher.

Work out the temperature inside his house.

Example K

Work out the value of 4 + 5 × (2 + 3)
Add brackets ( ) to make each statement below correct.
You may use more than one pair of brackets in each statement.
(i) 2 + 3 × 4 + 5 = 29
(ii) 2 + 3 × 4 + 5 = 45
Use your calculator to work out (2.7 + 1.9)² × 1.07
Write down all the figures on your calculator display.
Put brackets in the expression below so that its value is 45.024
1.6 + 3.8 × 2.4 × 4.2

Example L

One gigabyte is 1 073 741 824 bytes.

Write the number 1 073 741 824 in standard form.
A hard disk can store 500 gigabytes of data.
How many bytes is this? Write your answer in standard form.

Prime numbers

Presenter Matthew Tosh looks at the complex subject of prime numbers, from the theories relating to them, to some of their uses, in an information-packed video for maths teachers.

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