Stretford Grammar School
Key Stage 3
Numeracy Handbook
“Numeracy is a life skill. Being numerate goes beyond simply 'doing sums'; it means having the confidence and competence to use numbers and think mathematically in everyday life…”
http://www.nationalnumeracy.org.uk/what-is-numeracy/index.html
Checklist of equipment for Maths lessons
Students are expected to have the following items with them for every lesson:
Pen
Pencil (HB is usually best)
Ruler (15cm is adequate)
Eraser
Sharpener
Scientific Calculator
Not used as often, so the teacher might give a reminder when they are needed:
Protractor
Compasses
(Scissors and a glue-stick may also be useful sometimes)
Number
Reading and writing numbers
Students must be encouraged to write numbers simply and clearly. It is now common practice to use spaces rather than commas between each group of three figures e.g. 34 000 not 34,000 though the latter will still be found in many text books and cannot be considered incorrect.
E.g. 3 027 251 is three million, twenty seven thousand, two hundred and fifty one.
Written Calculations
Students often use the ‘ = ‘ sign incorrectly. When doing a series of operations they
sometimes write mathematical sentences which are untrue.
E.g. Calculate 5 × 4 + 3 – 8
5 x 4 = 20 + 3 = 23 – 8 = 15
While this does give a correct answer, students are encouraged to write such calculations correctly. Work from left to right: unless there are brackets, multiplication and division take precedence over addition and subtraction. Each step should be started on a new line, beginning with an equals sign.
E.g. Calculate 5 × 4 + 3 – 8
= 20 + 3 – 8
= 23 – 8
= 15
The ‘≈’ (approximately equal to) sign should be used when estimating answers.
E.g. Estimate 2 378 – 412 2 400 – 400
= 2 000 (‘=’ sign because this step is equal to the previous step)
Before completing any calculation, students should be encouraged to estimate a rough value for what they expect the answer to be. This should be done by rounding the numbers and mentally calculating the approximate answer.
After completing the calculation they should consider whether or not their answer is reasonable in the context of the question.
Methods for Written Calculations
The National Curriculum for 2014 onwards places a greater emphasis on formal written methods of calculation. Students may have already learned alternative methods.
E.g. Addition and Subtraction
Multiplication
“Short multiplication” “Long multiplication”
Students may already know the grid method:
x / 100 / 20 / 4 / Total20 / 2 000 / 400 / 80 / 2 480
6 / 600 / 120 / 24 / 744
3 224
Division
“Short division” “Long division”
Students may already know how to use “chunking” methods:
15 / 432- / 300 / 20
15 / 132
- / 120 / 8
12
Multiplication and Division by 10, 100, 1000…
When a number is multiplied by 10 its value has increased tenfold and each digit will move one place to the left so multiplying its value by 10. When multiplying by 100 each digit moves two places to the left, and so on…
Any empty columns will be filled with zeros so that place value is maintained when the numbers are written without column headings.
Note: we think about moving the digits not the decimal point
Percentages
Students should be aware that fractions, decimals and percentages are different ways of representing part of a whole and know the simple equivalents.
E.g. 10% = 1/10 12% = 0.12
Calculating percentages of a quantity
Methods for calculating percentages of a quantity vary depending upon the percentage required.
Where percentages have simple fraction equivalents, fractions of the amount can be calculated.
E.g. i) To find 50% of an amount, halve the amount.
ii) To find 75% of an amount, find a quarter by dividing by four and then multiply it by three.
Most other percentages can be found by finding 10%, by dividing by 10, and then finding multiples or fractions of that amount
E.g. To find 30% of an amount first find 10% by dividing the amount by 10 and then multiply this by three.
30% = 3x10%
Similarly: 5% = half of 10% and 15% = 10% + 5%
Most other percentages can be calculated in this way.
When using the calculator it is usual to think of the percentage as a decimal multiplying. Students should be encouraged to convert the question to a calculation containing mathematical symbols.
(‘of’ means ‘multiply’)
E.g. Find 27% of £350 becomes 0.27 X £350
and this is how it can be entered into the calculator.
Calculating the amount as a percentage
In every case the amount should be expressed as a fraction of the original amount and then converted to a percentage in one of the following ways:
E.g. What is 15 as a percentage of 60?
(using simple fractions)
15 = 1 = 25%
60 4
E.g. What is 27 out of 50 as a percentage?
(using equivalent fractions)
27 x 2 = 54 = 54%
50 x 2 100
E.g. What is 39 as a percentage of 57?
(Using a calculator)
39 = 39 ÷ 57 = 0.684 (to 3 d.p.) = 68.4%
57
Notes on Using a Calculator
We expect students to bring a scientific calculator to every Maths lesson, even if the work does not call for one.
Many other subjects will also expect them to have their calculators with them.
We recommend that students use a Casio fx83 (or compatible) scientific calculator.
Teaching will often focus on the functions as they are found on the Casio calculators. Students are encouraged to keep the instructions for their calculator (although they can be found online) and to understand how their calculator works. Sometimes, students might mistakenly put their calculator in to a mode that is unfamiliar to them.
If all else fails, Casio calculators can be re-set by pressing SHIFT, followed by 9 (CLR) and following the instructions on the display.
Interpreting the display
Modern calculators tend to use “natural display”, which means that calculations and answers appear like they might look when written down. This means that many answers will appear as a fraction, when a decimal might be required. The “SÛD” key can be used to switch between “symbolic” and “decimal” notation.
Money
When using the calculator for money calculations 4.7 on the calculator needs to be interpreted
as £4.70
Standard Form
Students meet standard form notation in Year 8, but before that, they might encounter answers on some calculators that use it.
E.g. on some calculators, 5 ÷ 200 = 2.5-2
This means 2.5 x 10-2 and is equivalent to 0.025.
Time
Students should never record 3hrs and 30 minutes as 3.30hrs but as 3.5hrs.
(When working with time it is possible to use the degrees/mins/secs key on many calculators)
Algebra
While some students may have encountered algebra in primary school, it is often new to many. They should be encouraged to understand that algebra is a generalisation of arithmetic and that many aspects follow the same rules as arithmetic.
Notation
Students must understand the difference between lower case and capital letters, as these may both appear in an algebraic expression, but standing for different quantities.
When a letter is multiplied by a number (coefficient), the number is placed in front of the letter, not after it.
E.g. For x + x + x, we simplify this to 3x (not x3)
When two letters are multiplied together, we would usually write them in alphabetical order (without a multiplication sign).
E.g. xy 2pq
For repeated multiplication, we would use indices as a shorthand
E.g. p × p × p × p = p4
Constant terms are numbers (not letters).
Expressions, Equations, Identities and Formulae
An expression is a combination of algebraic terms and constants, with operations. The letters are variables as they can take any value.
E.g. 2x + 1
In an equation, an expression is equal to a constant (another expression), for specific values of the letter terms. The letters represent unknowns that need to be found by solving the equation.
E.g. 2x + 1 = 3
In an identity, an expression is equivalent to another for all values for of the letter terms (variables).
E.g. 2x + 1 = x + x + 2 – 1
In a formula, a variable (the subject) is defined in terms variables in the form of an expression.
E.g. y = 2x + 1
Solving Equations
The terms "cross-multiply" and “swap sides – swap signs” can lead to misunderstandings, as part of any explanation of how to solve equations and so should be avoided. Once students understand and are
proficient with solving equations they may use these terms however.
To solve linear equations, the most common method is the 'balancing method':
E.g. To solve 3x – 7 = 5
(add 7 to both sides)
3x – 7 + 7 = 5 + 7
3x = 12
(divide both sides by 3)
3x = 12
3 3
x = 4
Graphs
Students should be encouraged to:
· use a sharp pencil.
· label both axes and give a title or equation
· use independent variable on x-axis (horizontal), and dependent variable on the y-axis (vertical)
E.g. if graphing temperature of a cooling liquid, time should go on the x-axis and temperature on the y-axis. [The temperature of the liquid is dependent on the time of the reading.]
· label lines not spaces, unless a bar-chart with discrete data
· use equally spaced intervals
· use convenient scales
· mark points by a small cross not a dot
· draw graphs on squared or graph paper
· to draw graphs of a sensible size, maximising the use of the page.
· If axes do not start from zero, a break represented by a zig-zag line should be shown on the axis.
Shape, Space and Measures
Rough Conversions between Metric and Imperial Systems
Although most teaching will use the Metric system, many Imperial units are still in common use.
We use the following approximate conversions:
1 inch ≈ 2.5 cm 1 yard ≈ 1 m 1 kg ≈ 2 lbs
1¾ pints ≈ 1 litre 5 miles ≈ 8 km 1 oz ≈ 25 g
Students should be expected to record the units they are using when answering a question.
Please note that we use cm2 (not sq cm) for area and cm3 (not cc) for volume.
Data Handling
Many subjects will call upon students to collect, process and analyse data.
Most students will already be familiar with the Data Handling Cycle as below:
Specifying the problem and planning
In order to specify a problem, students need to suggest a hypothesis that could be investigated. A hypothesis is a statement about something they are going to investigate.
E.g. most students walk to school; tallest athletes jump best; the cost of a car has an effect on its top speed.
Collecting Data
It is important that data* are collected for a purpose.
(* “Data” is the plural; the singular is “datum”, but we might refer to a “piece of data”)
Data can be quantitative (numerical) or qualitative (non-numerical). Quantitative data may be discrete (taking individual values, e.g. shoe size) or continuous (taking any value in a range, e.g. foot length).
Data is found as either:
Primary data – data you collect yourself using a survey or experiment
Secondary data – data that is already collected for you (e.g. in books or on the internet)
A sample might be used, rather than conducting a census (collecting data for a whole population). For a sample to be reliable there needs to be a sample size of at least 30. The larger the sample, the more reliable it can be said to be.
Representing data and interpretation
Representing data in an orderly and easy-to-read/understand form is paramount to Handling Data. Charts and diagrams without headings, labels and an appropriate scale are useless.
Averages
An average is value (or item) that is representative of a set of data. There are three main averages used in different circumstances.
Mean The sum of all the values divided by the number of values. Many people use the term average when they are actually referring to the mean.
Median The value in the middle of the data after it has been arranged in size order. If we have an even number of data values, then we find the mean of the middle two values.
Mode The value in the data that occurs most frequently. There may be more than one mode.
Spread
Spread means how far apart (or inconsistent) the data is. For data with a large spread, the mean might not be a reliable average to use and the median might be better.
The simplest measure of spread is the range: the difference between the largest and the smallest values.
What you can do at home to help your child make progress beyond Level 6
At Level 6 and above the nature of maths becomes more algebraic and abstract. This involves making and using formulae and developing knowledge of sequences and graphs.
You could ask your child to explain their understanding of some of the maths problems they are working on and solving at school. This will help reinforce and consolidate what they know.
You could also encourage your child to:
· work out the best value for money when shopping
· watch documentaries and discuss the maths involved in climate change or other environmental concerns
· talk about their work with reference to a textbook or online resource such as BBC Bitesize or MyMaths
High achieving students could be encouraged to:
· consider the maths involved in modelling real-life situations, such as building a bridge or the arc a ball makes when thrown
· find out why gambling is likely to cost you money
· explore the interest earned on a range of savings accounts, the cost of obtaining money for a mortgage or the cost involved in using credit, for example, children can be encouraged to use an ICT spreadsheet to calculate and compare interest rates