Numeracy across the curriculum teacher handbook · PDF fileMultiplying and dividing by 10, 100...
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1 NUMERACY ACROSS THE CURRICULUM
Transcript of Numeracy across the curriculum teacher handbook · PDF fileMultiplying and dividing by 10, 100...
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NUMERACY ACROSS
THE CURRICULUM
2
NUMERACY SKILLS
and
TEACHING METHODS
A Reference Handbook
for Teachers
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Contents Number Page Writing and Reading Numbers 4 Language of Operations 4 Order of operation (BODMAS) 5 Writing calculations 5 Addition whole numbers 7 Subtraction whole numbers 7 Addition decimal numbers 8 Subtraction decimal numbers 8 Multiplying and dividing by 10, 100 and 1000 8 Multiplication 9 Division 10 Multiplication and division of decimals 10 Using a calculator 10 Negative numbers 11 Fractions 12 Ratio and proportion 13 Decimal notation 14 Percentages 15 Rounding numbers 16 Standard form numbers 16 Algebra Using formulae 17 Plotting and drawing graphs 18 Geometry and measures 2D shapes 20 3D shapes 21 Length, mass and capacity unit conversions 21 Time 22 Reading scales 22 Area 23 Volume 23 Handling data Bar graphs and frequency diagrams 24 Line graphs 25 Pie charts 26 Using data – mean, median, mode and range 27 Correlation and scatter graphs 28 Types of graphs 29 When to use the different type of graphs 33 Numeracy glossary 34
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Number
Writing and Reading Numbers
Students need to be encouraged to write figures simply and clearly. Zeros with a line through (∅), and French sevens (7) should be discouraged. Encourage the correct shape for number and 0 which can sometimes look like six. Also decimal points shown as a “o” 4
e.g. 1o7 = 1.7
Most students are able to write numbers up to a thousand in words, but there are often problems with bigger numbers. It common practice to use spaces between each group of three figures in large numbers rather than commas, e.g. 34 000 not 34,000. In reading large numbers students should apply their knowledge of place value working from the right in groups of three digits. So the first group contains hundreds, tens and units, next group as thousands and the next group as millions. The number is then read from left to right.
So 2 084 142 is two million, eighty four thousand, one hundred and forty two.
Language of Operations
All students should be able to understand and use different terms for the four basic operations but some students with learning difficulties and EAL students may have difficulty in associating terms with symbols.
+ - x (.) ÷ (:) add
increase more plus sum total
decrease difference
less minus
reduce subtract takeaway
multiply of
product times lots of
multiplication
divide division quotient
share divisor
5 Order of Operations
When carrying out a series of arithmetic operations students can be confused about the order in which the operations should be done, e.g. does 6 + 5 x 7 mean 11 x 7 or 6 + 35? Some students will be familiar with the mnemonic BODMAS or BIDMAS
The important facts are that brackets are done first, then powers, multiplication and division and finally addition and subtraction, e.g.
(i) 6 + 5 x 7 = 6 + 35 = 41 (ii) (6 + 5) x 7 = 11 x 7 = 77 (iii) 2 + 43 ÷ 8 - 3 = 2 + 64 ÷ 8 - 3 = 2 + 8 - 3 = 7
Some non-scientific calculators will give incorrect answers because they do operations in the order in which they are put into the calculator rather than following the BODMAS order.
Writing calculations
Some pupils are over-dependent on the use of calculators for simple calculations. Wherever possible pupils should be encouraged to use mental or pencil and paper methods. It is, however, necessary to give consideration to the ability of the pupil and the objectives of the task in hand. In order to complete a task successfully it may be necessary for pupils to use a calculator for what you perceive to be a relatively simple calculation. This should be allowed if progress within the subject area is to be made. Before completing the calculation pupils should be encouraged to make an estimate of the answer. Having completed the calculation on the calculator they should consider whether the answer is reasonable in the context of the question.
Brackets, power Of, Division, Multiplication, Addition, Subtraction
Indices
6 Writing Calculations
Students have a tendency to use the ‘=’ sign incorrectly and write mathematical expressions that do not make sense, e.g.
3 x 2 = 6 + 4 = 10 - 7 = 3 X It is important that all teachers encourage students to write such calculations correctly, e.g.
3 x 2 = 6 6 + 4 = 10 10 - 7 = 3 P
The = sign should only be used when both sides of the equation have the same value. There is no problem with calculations such as
34 + 28 = 30 + 20 + 4 + 8 = 50 + 12 = 62
because each part of the working out has the same value.
The ‘≈’ (approximately equal to) sign should be used when students are estimating answers, e.g. 1 576 - 312 ≈ 1 600 - 300 = 1 300
Mental Calculations
All students should be able to carry out the following processes mentally, although some will need time to arrive at an answer:
• recall addition and subtraction facts up to 20 • recall multiplication and division facts for tables up to 10 x 10
Students should also be encouraged to carry out other calculations mentally using a variety of strategies but there will be significant differences in their ability to do so.
34 + 28 = 34 + 30 - 2 163 - 47 = 163 - 50 + 3 23 x 3 = (20 x 3) + (3 x 3) 68 ÷ 4 = (68 ÷ 2) ÷ 2
It is helpful if teachers discuss with students who they might do mental calculations such as these. Any valid method that produces a correct answer is acceptable.
7 Pencil and Paper Calculations
All students should be able to use pencil and paper methods for calculations involving simple addition, subtraction, multiplication and division. The following methods are used by the majority of students, and are generally taught to those students who do not have a method that they can carry out correctly. It is important to encourage students to set out their work in columns when adding and subtracting.
Addition and Subtraction
Click on the ? to get extra help with column subtraction.
For weaker students the use of a number line may be useful particularly for subtraction.
3456 + 975 Estimate
3500 + 1000= 4500 3 4 5 6
+ 9 7 5 4 4 3 1
1 1 1
6286 -‐ 4857 Estimate
6300 -‐ 4900= 1400 56 127 8 16
-‐ 4 8 5 7 1 4 2 9
396 -‐ 178 3 9 16
-‐ 1 7 8 2 1 8 1
512 -‐ 386
8 Addition and subtraction of decimals are done using the same methods, but students may need to be reminded that it is necessary to ensure that the decimal points must be underneath each other.
Decimals
Multiplying and Dividing by 10, 100, 1000 . . .
The rule for multiplying by 10 is that each of the digits moves one place to the left. When multiplying by 100 each digit moves two places to the left and so on. In division the digits move to the right. This rule works for whole numbers and decimals. Decimal points do not move, e.g.
23 x 100 = 2300
Th H T U 2 3
2 3 0 0
Zeros are needed to fill the empty spaces in the tens and units columns, otherwise when the number is written down without the column headings it will appear as a different number i.e. 23 instead of 2 300
3.96 + 1.78
3. 9 6 + 1. 7 8
5 . 7 4 1 1
6. 2 – 4.57
6. 12 10 -‐ 4. 5 7 1. 6 3 1 1
6. 2 – 4.57
56. 12 10 (write the zero as a place filler) -‐ 4. 5 7 1. 6 3
9
3.45 x 10 = 34.5
Th H T U . 110
1100
3 . 4 5 3 4 . 5
Zeros are not needed in empty columns after the decimal point.
260 ÷ 10 = 26
H T U . 110
1100
2 6 0 . 2 6 .
439 ÷ 100 = 4.39
H T U . 110
1100
4 3 9 . 4 . 3 9
Multiplication and Division.
Multiplication Traditional Grid method
357 × 8 = 2400 + 400 + 56 = 2856
Click on the ? to get extra help with grid multiplication.
× 300 50 7 8 2400 400 56
357 × 8 Estimate
350 × 2 × 4 = 700 × 4 = 2 800
357 × 8 2856 4 5
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Division.
When multiplying decimal numbers
(i) Complete the calculation as if there are no decimal points (ii) Insert a decimal point in the answer so that there are the same
number of digits on the right of the decimal point in the answer as there are in the question.
(i) 3.2 x 2.4 becomes 32 x 24 = 768
(ii) 3.2 x 2.4 = 7.68
When dividing a decimal number by a whole number, the calculation is the same as a division without decimal points, with the decimal point in the answer being inserted directly above the decimal point in the question.
Using a Calculator
Many students believe that if they use a calculator, the answer they obtain must be correct. It is important that students are encouraged to make an estimate of the answer they expect to get and, having completed the calculation, complete the answer with their estimate. They also need to consider whether the answer they have got makes sense in the context of the question.
Students are taught how to use the various functions on a calculator including:
810 ÷ 6 Estimate 800÷ 5 = 160
1 3 5 6 8 21 3 0
Two decimal places in question
Two decimal places in answer
5832 ÷ 24 Estimate
5800 ÷ 20 = 2900 2 4 3
24 5 8 103 72
√ x2 ^ ⬚⬚
xy x3
11 Students are likely to have various types of scientific calculators that operate in different ways. In some cases an operation is entered before the number, in others the operation follows the number, e.g.
Finding √169
In older calculators this is done by
With Casio calculators the order is
Students should be encouraged to buy the Casio Fx 83 calculator and familiarise themselves with their own calculator and to learn to use if efficiently.
Negative Numbers
Some students have problems understanding the size of negative numbers and believe that -10 is bigger than – 5. A number line can be of help when ordering numbers. Moving to the right, numbers are getting bigger. Moving to the left, numbers are getting smaller.
Smaller Bigger
Students in lower ability groups may have difficulty with calculations involving negative numbers. For addition and subtraction a number line is useful.
Starting point Direction Distance to move
• Start at -3 • Move 8 places to right • So -3 + 8 = 5
- +
1 6 9 √ =
√ 6 1 9 =
-3 8 +
12 If there are two signs between the numbers, when the signs are the same the move is to the right and when the signs are different, the move is to the left.
So + + or - - is the same as + and means move to the right
+ - or - + is the same as - and means move to the left, e.g. (i) -5 + (-6) = -5 -6 = -11 (ii) 6 – (-3) = 6 + 3 = 9
When multiplying and dividing positive and negative numbers, if both numbers are positive or both numbers are negative, the answer will be positive. If one of the numbers is positive and the other is negative, the answer will be negative, e.g.
(i) (-3) x (-2) = 9 (ii) (-8) ÷ 4 = -2
Common Fractions
Most students should be able to calculate simple fractions of quantities. They should know that to find half of an amount they need to divide by 2. To find a third of something they divide by 3 etc. To find a quarter of an amount it is sometimes easier to divide by two twice rather than to divide by 4.
They should know that the denominator (bottom number of a fraction) indicates the number of parts the amount is to be divided into and the numerator (top number) indicates the number of parts required, e.g.
To find !! of 24
• First find !! of 24 = 24 ÷ 4 = 6
• So !! = 3 × 6 = 18
Generally, students should be able to do this type of calculation mentally, but some may need prompting in order to do so. The rule divide by the bottom number and multiply by the top number does work, but if this rule is used without understanding, it can cause problems because students tend to forget which numbers they use to divide or multiply.
13 Most students should be able to make equivalent fractions by multiplying or dividing the numerator or denominator of a fraction by the same number.
×2 ÷5
e.g. !! = !
! !"
!" = !
!
×2 ÷5
Ratio and Proportion
Ratios compare numbers or measurements, e.g.
When making pastry the ration of fat to flour is 1 : 2. This means that for every gram of fat used, 2 grams of flour are required, or for every ounce of fat there must be 2 ounces of flour.
Ratios can be simplified in the same way as fractions by dividing all of the parts of the ratio by the same number, e.g.
The ration 3 : 12 : 6 can be simplified by dividing by 3 giving 1 : 4 : 2
If an amount is to be split into a given ratio, it is first necessary to begin by finding out how many parts are needed, e.g.
Three people hold shares in a company in the ratio 60 : 100 : 20 and a dividend of £3 600 is divided between them in the same ratio as their shares. How much does each person receive?
The three people hold a total of 60 + 100 + 20 = 180 shares. £3 600 ÷ 180 = £20 so the dividend for each share is £20 and the payments are:
60 x £20 = £1 200 100 x £20 = £2 000 20 x £20 = £400
Ratios are often used to show the scale of a map or plan,
e.g. If the scale of a map is 1 : 50 000 this means that 1 centimetre on the map represents a distance of 50 000 centimetres on the ground.
50 000 centimetres = 500 metres = 0.5 kilometre
so a scale of 1 : 50 000 means that 1 centimetre on the map represents 0.5 kilometre on the ground, or 2 cm represents 1 kilometre.
14 Students may experience difficulties in adapting recipes to produce larger or smaller quantities. They may need practice in modifying recipes and should be encouraged to adopt a method that is suitable for the particular problem.
If a recipe for four people is to be modified to cater for five people it may be easiest to find the quantities required for one person and multiply these by five. In other cases it will be sensible to use other methods to adjust quantities, e.g.
The following recipe will produce 16 flapjacks: 40 grams of margarine 50 grams of sugar 200 grams of rolled oats 100 grams of golden syrup If 24 flapjacks are required, it is probably easiest to find the recipe for 8 flapjacks (by halving the given quantities) and add these amounts to the original recipe.
Decimal Notation
All students should be familiar with decimal notation for money although they may use incorrect notation.
P X £2.35 £2.35p £3.33 £3.33333.. £4.60 £4.6 £0.25 0.25p 32p 0.32p
Students should also be familiar with the use of decimal notation for metric measures, but sometimes misinterpret the decimal part of the number. They may need to be reminded for example that 1.5 metres is 150 centimetres not 105 centimetres.
Students often read decimal numbers incorrectly, e.g. 8.72 is read as eight point seventy two instead of eight point seven two.
They may also have problems with comparing the size of decimal numbers and may believe that 2.36 is bigger than 2.8 because 36 is bigger than 8. If they need to compare numbers it may help to write all of the numbers to the same decimal places, e.g. 2.36 and 2.80
15 Percentages
Calculating Percentages of a Quantity
Methods for calculating percentages of amounts vary depending on the percentage required. Students should know that fractions, decimals and percentages are different ways of representing part of a whole and know simple equivalents.
e.g. find 10% divide by 10
find 1% divide by 100
So to find 12% find 1% then times by 12.
Where percentages have simple fraction equivalents, fractions of amounts 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 of the amount and multiply it by three.
10% of an amount can be calculated by dividing by 10. Amounts that are multimples of 10 can then be calculated from this, e.g.
30% is 3 x 10% 5% is half of 10% etc.
When using a calculator it is usually easiest to think of a percentage as a decimal number and ‘of’ as x, e.g.
17.5% of £36 = 0.175 x £36 = £6.30
The most able students in Years 9, 10 and 11 should be able to increase or decrease amounts using one operation. This is particularly useful in spreadsheets, e.g.
(i) Increase 52 by 14% 114% of 52 = 1.14 x 52 = 59.28
(ii) Reduce 175 by 30% 70% of 175 = 0.7 x 175 = 122.5
16 Calculating an Amount as a Percentage
The most straightforward way to do this is to think of the problem as a common fraction and then to convert this to a percentage if the fraction is simple. For more complex questions the fraction can be converted to a decimal number and then a percentage.
e.g. What is 20 as a percentage of 80? !"!"= !
!= 25%
×2
What is 43 out 50 as a percentage? !"!"= !"
!""= 86%
×2
What is 123 out 375 as a percentage? !"#!"#
= 123 ÷ 375 ×100 = 32.8%
Rounding Numbers
When rounding numbers, digits below 5 round down and digits of 5 or above round up, e.g.
344 ≈ 340 345 ≈ 350 346 ≈ 350
Students are likely to find it easier to understand rounding in terms of rounding to the nearest hundred, whole number, tenth, etc than rounding to a specified number of significant figures or decimal places.
Standard Form
Only the students in Set 1 in Years 8 and 9 and Sets 1 and 2 in Years 10 and 11 are likely to be familiar with standard form notation. They should be able to use calculators for calculation in standard form using the EXP or the xy button. Students often have difficulty in interpreting the result of a standard form calculation on some calculators and need to be encouraged to write for example 3.42 x 108 and not 3.428.
17
ALGEBRA
Using Formulae
Most students should be able to cope with using simple formulae. Those who find it difficult might find it helpful if the formula is written with boxes to insert the relevant numbers. S T e.g. D = S T If S = 40 and T =2 find D. D = × = 80
Rules such as the triangle rule, work for specific types of questions and may be useful for weaker students in some subjects as Science, PE
So with Distance, Speed and Time:
Speed = Distance , Distance = Speed x Time , Time = Distance Time Speed
Because speed is calculated by dividing a distance by a time the units of speed involve a distance and a time e.g. metres per second or m/s or m s-1.
40 2
18
Also with Mass, Density and Volume:
Density = Mass , Mass = Density x Volume , Volume = Mass Volume Density
Because density is calculated by dividing a mass by a volume the units of density involve a mass and a volume e.g. grams per cubic centimetre or g/cm3 or g cm-3
Plotting Points and Drawing Graphs
When drawing a diagram on which points are to be plotted, some students will need to be reminded that numbers on the axes are written on the lines and not in the spaces, e.g.
P 0 1 2 3 4 5
Not X
0 1 2 3 4
And equally spaced
X
0 1 2 3 4 5 6
Not
X
0 1 2 3 4
19 When drawing graphs for experimental data it is customary to use the horizontal axes for the variable which has a regular interval, e.g.
(i) In an experiment in which the temperature is taken every 5 minutes the horizontal axis would be used for time and the vertical axis for temperature/
(ii) If the depth of a river is measured every metre, the horizontal axis would be used for the distance from the bank and the vertical axis for the depth.
Having plotted points correctly students can be confused about whether they should join the points or draw a line of best fit. Generally when the students have calculated the points from an equation the points should be joined. If the points have been obtained through experiment, a line of best fit is probably needed. Further details appear in the following section on Data Handling.
20
GEOMETRY & MEASURES
Shapes
It is important to use correct names of shapes. 2D and 3D shapes and their properties are below.
2D Shapes
A polygon is a 2D shape consisting of 3 or more straight sides. A regular polygon has all sides and angles the same size. Specific names of polygons are shown below the table.
Number of sides Name of polygon 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon
Some triangles have special names:
Equilateral triangle Isosceles triangle Scalene triangle
All sides and angles Two sides and two All sides and angles are
are equal. angles are equal. different.
Right-angled triangle
One angle is a right angle (90º)
21 3D Shapes
Students should know all the names of the shapes below
Length, Mass and Capacity
All students should be familiar with metric units of length, mass and capacity but some students will have difficulty with notation and converting from one unit to another.
They should know the following facts:
Many students will be unfamiliar with imperial units and the metric equivalents.
Length Mass Capacity
10mm = 1 cm 1 000 g = 1 kg 1 000 ml = 1 litre 100 cm = 1 m 1 000 kg = 1 tonne 1 000 m = 1 km
Length Mass Capacity
1 inch = 2.5 cm 1 ounce = 30 g 1 pint = 570 ml 1 foot = 30 cm 1 pound = 450 g 1 gallon = 4.5 litres 1 yard = 90 cm 1 stone = 6.4 kg 1 mile = 1.6 km 1 m = 40 inches 1 kg = 2.2 pounds 1 km = 58 mile
22 Students are encouraged to develop an awareness of the sizes of units and an ability to make estimates in everyday contexts. They are also expected to consider the appropriateness of their solution to a problem in the particular context of the question.
Time
All Students should be able to tell the time but a small number of students with learning difficulties may experience problems in using analogue clocks and equating analogue and digital times. Some students will need help with problems involving time, e.g.
A recipe says that a cake should be in the oven for 35 minutes. If it is put into the oven at 10.50 a.m. when will it be ready? With students who have difficulty this is usually best approached by splitting up the time, e.g. 10.50 + 10 mins à 11.00 11.00 + 25 mins à 11.25 Most students should be familiar with 12 and 24 hour clock times but support may be necessary when interpreting timetables. When writing 24 hour clock times, four figures are used without any punctuation. e.g. 0650 or 1427 24 hour clock times often appear on digital clocks with a colon between the hours and minutes. e.g. or 12 hour clock times are usually written with a dot between the hours and the minutes, e.g. 6.50 am or 2.27 p.m. Reading Scales
Students in lower ability groups often have difficulty in reading scales on graphs. They are inclined to assume that one division on the scale represents one unit. It may help if they begin by counting the number of divisions between each number on the scale and then determine what each division of the scale represents.
There is 5 units between each end of the scale
that has 10 divisions. So each division is worth !!"
=
0.5
06:50 14:57
23 Area and Volume
Problems can arise when calculating areas and volumes if the lengths give are not in the same units. It is usually easiest to begin by converting all of the units of length to the units that are required for the answer, before doing any calculation, e.g.
16 mm Either
Area = 16 x 20 = 320 mm2 2 cm or Area = 1.6 x 2 = 3.2 cm2 A common error when converting square and cubic units is to use the wrong conversion factor. Some higher ability students will be aware that 1 litre = 1 000 cm3
Area 1cm 10 mm
1cm = 10mm 1cm × 1cm = 1 cm² 10mm × 10mm = 100 mm² Volume 1m 100cm 1m = 100cm 1m 100cm 1m × 1m × 1m = 1m³ 100 cm × 100cm × 100cm = 1 000 000cm³
Area Volume 1 cm2 = 100mm2 1 cm3 = 1 000 mm3 1 m2 = 10 00 0 cm2 1 m3 = 1 000 000 cm3 1 km2 = 1 000 000 m2
24
HANDLING DATA
Graphs and Diagrams
Bar graphs and line graphs should always be drawn on squared paper or graph paper.
Bar Graphs
All students should be able to draw bar graphs, but they will frequently draw them incorrectly. The way in which a graph is drawn depends on the type of data being represented. Graphs should be drawn with gaps between the bars if the data categories are not numeric (colours, types of vehicle, names of football teams etc). There should also be gaps between the bars if the categories are numeric but can only take particular values (shoe size, Key Stage 3 level etc). If the data categories are numeric and continuous they can take any value within the range (heights of children, time taken to complete a marathon etc). In this case the data will be grouped, the horizontal axis should be labelled as when plotting co-ordinates rather than under the bars and there should be no gaps between the bars.
It is also important to emphasise the need to label axes correctly. In a bar graph with separate columns, the labels should be underneath the columns on the horizontal axis. On the vertical axis the labels should be on the lines, e.g.
0 2 4 6 8 10
iphone HTC Samsung Sony
Freq
uenc
y
Phone
Favourite mobile phone
25 Where data are continuous e.g. lengths, the horizontal scale should be like the scale used for a graph on which points are plotted, e.g.
P 0 10 20 30 40 50
not x 0-10 11-20 21-30 31-40 41-50
Line Graphs
Line graphs are used to show trends. They should only be used with data in which the order in which the categories are written is significant.
P
0
5
10
15
20
1 2 3 4 5 6 7
Num
ber o
f pup
ils
Text messages sent in 1 hour
Graph to show number of text messages sent
26
Computer Drawn Graphs
Students throughout the school should be able to use Excel to draw graphs to represent data, although students in lower ability groups will need assistance. Because it is easy to produce a wide variety of graphs, there is a tendency to produce diagrams that have little relevance. In most cases diagrams other than bar charts, column graphs, line graphs, scatter graphs and pie charts should be avoided.
Pie Charts
The way in which students should be expected to calculate the angles for pie charts will depend on the complexity of the question. If the numbers involved are simple it will be possible to calculate simple fractions of 360o e.g.
Type of computer Frequency Fraction Angle Desk top 20 20
40=12 180º
Lap top 10 1040
=14 90º
Tablet 6 640
=320
54º Tower 4 4
40=
110
36º Total 40 360º
With more difficult numbers, students should first work out the share of 360o to be allotted to one item and then multiply this by the number in each category, e.g. 180 students were asked what they intended to do after leaving school.
360o ÷ 180 = 2o so each student has 2o of the pie chart.
0
2
4
6
8
10
Mars Flake Aero Twix Twirl
Num
ber o
f pup
ils
Favourite chocolate bar
Graph to show favourite chocolate bar bought
27 If data is in percentage form is to presented as a pie chart it is probable easiest to multiply 360o by the decimal equivalent, e.g.
32% = 0.32
0.32 x 360o = 115.20 ≈ 1150
Alternatively a circular protractor, scaled in percentage divisions(pie chart scale) rather than degrees, can be used.
Using Data
Range
The range of a set of data is the difference between the highest and lowest data values, e.g. If in an examination the highest mark is 80% and the lowest mark is 45%, the range is 35% because 80% - 45% = 35% The range is always a positive number, so it is not 45% - 80%. Averages
Three different averages are commonly used:
• The mean is calculated by adding up all of the values and dividing by the number of values
• The median is the middle value when the values have been arranged in order • The mode is the most common value. It is sometimes called the modal value,
e.g.
for the values
The mean is !!!!!!!!!!!!!!!!!
! = !"!
= 4
The median is 3 because 3 is the middle number when the values are put in order.
2 , 2 , 3 , 3 , 3 , 4 , 5 , 6 , 8
The mode is 3 because 3 occurs most often.
3, 2, 5, 8, 4, 3, 6, 3, 2
28 Correlation
Correlation is a measure of how strongly two sets of data are connected. (It is important to remember that a high correlation might occur coincidentally and does not necessarily indicate a relationship between the two sets of data).
A high positive correlation (0.8 to 1.0) indicates that generally, as one data value increases the other increases. A high negative correlation (-0.8 to -1.0) indicates that generally as one value gets bigger the other value gets smaller. A correlation that is near to zero suggests that there is no connection between the two sets of data.
Scatter graphs are used to indicate the degree of correlation. If the points are near to the line of best fit there is a good correlation between the two sets of data.
If there is a good correlation the line of best fit can be used to estimate an unknown data value if one value is known.
Ice cream sales Line of best fit
Temperature
0
1
2
0 0.5 1 1.5 2
Good posi<ve correla<on
0
1
2
0 0.5 1 1.5 2 Good nega<ve correla<on
0
1
2
0 0.5 1 1.5 2 No correla<on
29 Types of graphs
Bar Graph
• Useful for comparing data in different categories
• The length of bars indicate the size of the data categories
• For qualitative data eg. colour, names and discrete quantitative data eg. number of children in family gaps are left between the bars.
Frequency polygon
• Useful for comparing data in different categories and showing trends.
• The mid-points of the frequencies of each group are joined.
Vertical line graph
• Use for comparing data in different categories.
• This type of graph is generally just seen in Maths.
• The lengths of the lines indicates the frequency of the data category.
• For quantitative and discrete quantitative data. • Really a bar chart with narrow columns.
0 2 4 6 8
10
iphone HTC Samsung Sony
Freq
uenc
y
Phone
Favourite mobile phone
30 Line graph
• Useful for showing upward And downward trends in data.
• Intermediate points might be unreliable but might be used for estimates.
Histograms
• Useful for showing data in grouped frequencies when the class intervals are not all the same.
• The area of the columns represent the frequencies.
Pie chart
• Used to show how something is divided up. • Useful when proportions are more important
than numbers.
29 30 31 32 33 34 35 36 37 38
Mon am Mon pm Tues am Tues pm Tem
pera
ture
(deg
rees
Cel
sius
)
Time
31 Cumulative frequency diagram(polygon)
• Useful for finding the number of values above or below a specified point and to find the median, quartiles and percentiles.
• A cumulative frequency table is created from a frequency table by adding each frequency to the sum of its previous value.
• In a cumulative frequency graph points are plotted at the top of the range for the class.
Scatter graph
• Used to examine the relationship between two variables.
• If the points generally lie near to a line of best fit(trend line) there is a good correlation between the variables.
• The line of best fit can be used to estimate one variable given the other variable.
• Estimates of variables found by extending the line of best fit beyond the range of the data are unreliable.
Pictogram
• Pictograms are charts in which icons represent numbers to make it more interesting and easier to understand.
• A key is often included to indicate what each icon represents. All icons must be of the same size, but a fraction of an icon can be used to show the respective fraction of that amount.
32
Function graph
• Used to show the relationship between two variables.
• The function is represented by an equation which can be used to produce a table of values for the graph
• Rienforce the correct plotting of coordinates, particularly those involving negatives.
Travel graph
• Used to show changes in distance or velocity over a period of time,
• The gradient of a distance time graph gives the velocity.
• The gradient of a velocity time graph gives the accelration.
• The area under a velocity time graph gives the distance travelled.
-9
2
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Numeracy Glossary Acute angle – An angle measuring less than 90° Add/addition – To join two or more quantities to get the sum or total Adjacent – Next to Algebra – An area of maths where unknown quantities are represented by letters Alternate angles – Equal angles within parallel lines that are identified by a Z shape Angle – The amount of turning between two lines meeting at the same point Anti-clockwise – The opposite direction to which hands move round a clock Approximate – To estimate a number, usually through rounding Arc – A section of the circumference of a circle Area – The size of the space a surface takes up, measured in units² Ascending – Going up Average – A summary of a set of data, either mode, median and mean Axis – Reference lines on a graph Bar graph – A graph using bars to show quantities for easy comparison Bisect – To divide into two equal sections Box plot – A diagram that uses a number line to show the distribution of data through the minimum, lower quartile, median, upper quartile and maximum Brackets – Symbols used to enclose an expression, ( ) Calculate – Work out, find the value of Calculator – A device that performs mathematical operations Capacity – The amount a container can hold Centimetre – A metric unit for measuring length (10 millimetres) Centre – The middle Certain – Inevitable, will definitely happen Chance – The likelihood that a particular outcome will occur Circle – A 2D shape whose edge is always the same distance from the centre Circumference – The perimeter of the circle Chord – A straight line joining two points at the edge of the circle, not through the centre Clockwise - The direction which hands move round a clock Common denominator – A denominator which is a multiple of the other denominators Compasses (pair of) – A mathematical instrument used to draw circles Cone – A 3D shape with a circular base which tapers to a single vertex at the top Congruent – Having the same shape and the same size Continuous data – Data which could have an infinite number of values with a particular range Coordinates – Pairs of numbers used to show a position of a graph with axes, eg (2,-4) Corresponding angles– Equal angles within parallel lines that are identified by a F shape Cross section – The face that results from slicing through a prism Cube – A 3D shape with 6 square faces Cuboid A 3D with 3 pairs of rectangular faces Cube number – A number found by multiply a number by itself 3 times, eg 43 = 4 x 4 x 4 = 64 Cylinder – A prism whose cross section is a circle Data – A collection of information Decagon – A 2D shape with 10 sides Decimal – A part of a number or a whole, 0.4 or 3.279 Decrease – To make smaller Degree – The unit with which angles are measured, eg 67°
35 Denominator – The bottom number of a fraction Density - The degree of compactness of a substance, found by mass ÷ volume Descending – Going down Diagonal – A straight line joining two non-adjacent vertices Diameter – A line going through a circle edge to edge that passes through the centre Dice – A cube marked with dots or numbers Digit – A symbol used to show a number, 1 2 3... Discrete data - Data which has only a finite number of values Divide/division – To share equally, ÷ Double – To multiply by 2 Edge – The part of a 3D shape where 2 faces meet Equal to/equals – To have the same value, = Equation - Two expressions that are equal to each other Equilateral triangle – A triangle with 3 equal sides and 3 equal angles Equivalent fractions – Two fractions representing the same proportion Estimate – To find a close answer by rounding Even number – A number in the 2x table Even chance – An outcome shares the same probability of occurring with another Expression (algebraic) – Made up of terms and operations (algebra) Exterior angle – The angle formed outside a polygon when a side is extended Face – The flat part of a 3D shape Factor – A number that divides exactly into another Formula – A mathematical rule to describe a relationship between quantities Fraction – A part of a number or a whole, !
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Frequency – The number of times a particular value appears in a set of data Gradient – The slope of a line Gram – A metric unit for measuring mass Graph – A drawing or diagram used to record information Half – To divide by 2 Hexagon – A 2D shape with 6 sides Heptagon – A 2D shape with 7 sides Highest common factor – The greatest of all the factors shared by a pair of numbers Horizontal – A straight line parallel to the horizon Hypotenuse – The longest side of a right-angled triangle Impossible – Will not happen Improper fraction – A fraction with a larger numerator than denominator Increase – To make bigger Index/indices – Numbers or letters raised to a power, 4² or a6
Inequality – Two amounts not equal to each other, < ≤ ≥ > Infinite/infinity – Unlimited, goes on forever Integer – A whole number Interior angle – An angle inside a polygon Intersect – The point where two lines cross Inverse operations – Opposite operations, + inverse to -, x inverse to ÷ Irregular (polygon) – A polygon with different sized sides and angles Isometric (paper) – equal dimensions between dots Isosceles triangle – A triangle with 2 equal sides and 2 equal angles Kilogram – A metric unit for measuring mass (1000 grams)
36 Kilometre – A metric unit for measuring length (1000 metres) Kite – A 2D shape with two pairs of equal sides and one pair of opposite angles that are equal Line of symmetry – Divides a shape into two congruent sides Linear – Has one dimension Litre – A metric unit for measuring capacity (1000 millilitres) Lowest common multiple - The smallest of all the multiples shared by a pair of numbers Maximum – The greatest possible value Mean – An average found by finding the sum of the data and dividing by the number of values Median – An average found by locating the middle value of an ordered set of data Metre – A metric unit for measuring length (100 centimetres, 1000 millimetres) Midpoint – The middles point between 2 values or 2 coordinates Millilitre – A metric unit for measuring capacity Millimetre – A metric unit for measuring length Minimum – The smallest possible value Minus - Negative Mixed number – A number comprised of an integer and a fraction Mode – An average found by identifying the value with the highest frequency Multiply/multiplication – A number is added to itself a number of times, x Multiple – A number in another number’s times table Negative – Below/less than zero/0, -4 Net – A 2D shape that can be folded into a 3D shape Nonagon – A 2D shape with 9 sides Number line – A line marked with numbers Numerator – The top number of a fraction Obtuse angle - An angle measuring more than 90° but less than 180° Octagon – A 2D shape with 8 sides Odd number – A number not in the 2 x table Operations – Add, subtract, multiply, divide Opposite angles – A pair of equal angles directly opposite each other formed by the intersection of 2 straight lines Origin – Coordinate (0,0) Outcome – One of the possible results of a probability experiment Outlier – A value far away from the others in a set of data (also called anomaly) Parallel – Lines that are the same distance apart Parallelogram – A 2D shape with 2 pairs of parallel lines Pentagon – A 2D shape with 5 sides Percent/percentage – A part of a number or a whole. Per cent means out of 100, 46% Perimeter – The distance around the edge of a 2D shape Perpendicular – Two lines meeting at a right-angle Pi – Ratio of the circumference to a circle’s diameter, �, 3.141592... Pictogram – A graph using pictures to represent frequency Pie chart – A graph using a divided circle where each section represents a part of the total Place value – The value of a digit depending on its place in the number Plan – A diagram showing the view from directly above Plane – A flat surface Polygon – A 2D shape with straight sides Population – Whole set from which a sample is taken Positive – Above/greater than zero/0 Prime – a number with only two factors, 1 and itself
37 Prime factor – A number which is both a factor of something and a prime Prism – A 3D shape with a constant cross section throughout Probability – The chance that a particular outcome will occur Product – The result of multiplying Proportion – A part to whole comparison Protractor – An instrument used to measure the size of angles Pyramid - A 3D shape with a polygon base which tapers to a single vertex at the top Pythagoras – In any right-angled triangle where c is the hypotenuse, a² + b² = c² Quadrant – Any quarter of a plane divided by an x- and y-axis Quadrilateral – A 2D shape with 4 sides Qualitative data – Non-numerical data Quantitative data – Numerical data Quantity – A number of something Radius – The distance from the centre of a circle to its edge Random – A chance pick from a number of items Range – The smallest value subtracted from the greatest value Ratio – Comparative value of 2 or more amounts Reciprocal – One of two numbers whose product is 1, ½ and 2 Rectangle – A quadrilateral with two pairs of parallel sides with different lengths and all vertices are right-angles Recurring decimal – A decimal which has repeating digits or a repeating pattern of digits Reflection – A mirror view Reflex angle – An angle measuring more than 180° and less than 360° Regular polygon – A polygon with all sides and angles equal Remainder – The remaining amount after dividing a quantity by a number that is not a factor Rhombus – A parallelogram with all sides equal Right-angle – An angle measuring exactly 90° Right-angled triangle – A triangle with one right-angle Rotation – To turn an object Rotational symmetry – When a turning shape has the same outline as the original shape Round/rounding – Change the number to a more convenient value Sample – A part of the population to be used Scale factor – The ratio of two corresponding edges on a scaled drawing Scalene triangle – A triangle with all different sides and all different angles Scatter diagram – A diagram with coordinates plotted to show the relationship between two variables Sector – A section of a circle bounded by two radii and an arc Segment – A section of a circle bounded by a chord and an arc Semi-circle – Half a circle Sequence – An ordered set of numbers or objects arranged according to a rule Set (of data) – A collection of items Similar - Having the same shape but a different size Simplify (algebra) – To remove brackets, unnecessary terms and numbers Simplify (fractions) – To reduce the numerator and denominator in a fraction to the smallest numbers possible Solve/solution – To work out the answer Sphere – A 3D shape that is perfectly round, a ball Square – A 2D shape with all equal sides and all angles 90º Square number – A number that results by multiplying another number by itself
38 Square root – The opposite of squaring a number Subtract/subtraction – To take one quantity away from another, - Sum – The result of adding Surface area – The area of the surface of a 3D shape Symmetry – An object is symmetrical when one half is a mirror image of the other Tally – Use of sets of 5 marks to record a total, Term (nth) – One of the numbers in a sequence Tessellation – Patterns of shapes that fit together without any gaps Tetrahedron – A 3D shape with four triangular faces, a triangular-based pyramid Three-dimensional (3D) – Having three dimensions, length, width and height Transformation – A change in position or size Translation – To move an item in any direction without rotating it Trapezium – A 2D shape with four sides, two of them being parallel Tree diagram – A diagram used to display the probability of different outcomes with each branch representing one possible outcome Triangle – A 2D shape with three sides Triple/treble – To multiply by three Two-dimensional (2D) - Having two dimensions, length and width Unit - One Unit of measure – Standard amount or quantity Variable – Something that varies, represented by a letter in algebra Venn diagram – A diagram using circles to show relationships between sets Vertex/vertices – The point where two sides meet, or three or more faces Vertical – Perpendicular to the horizon Volume – The amount of space occupied by a 3D object X-axis – The horizontal axis on a graph Y-axis – The vertical axis on a graph Y-intercept – Where a line intersects the y-axis
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Index 2D shapes 20 3D shapes 21 Addition decimal numbers 8 Addition whole numbers 7 Area 23 Bar graphs and frequency diagrams 24 Correlation and scatter graphs 28 Decimal notation 14 Division 10 Fractions 12 Language of Operations 4 Length, mass and capacity 21 Conversions 21 Line graphs 25 Multiplication 9 Multiplication and division of decimals 10 Multiplying and dividing by 10, 100 and 1000 8 Negative numbers 11 Numeracy glossary 34 Order of operation (BODMAS) 5 Percentages 15 Pie charts 26 Plotting and drawing graphs 18 Ratio and proportion 13 Reading scales 22 Rounding numbers 16 Standard form numbers 16 Subtraction decimal numbers 8 Subtraction whole numbers 7 Time 22 Types of graphs 29 Using a calculator 10 Using data – mean, median, mode and range 27 Using formulae 17 Volume 23 When to use the different type of graphs 33 Writing and Reading Numbers 4 Writing calculations 5
