Numeracy Policy
Effective Date: September 2015
Ratified by LGB: 21st October 2015
Review Date:
July 2018
CONTENTS
1
2
3
4
5
6
7
8
9
PAGE
Introduction
Definition
Aims
Use of Calculators
Cross Curricular Agreements on Language and Routines
Notation and Terminology
Standard Mathematical Techniques
Mental Calculations
Presentation – graphs
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2
2
3
3
5
7
11
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1
1.1
Introduction
Literacy and numeracy are often the keystones to educational success in subject
areas beyond English and Mathematics lessons. It is important as teachers that we
aware of the skills that students may or may not bring to our lessons and that in
numeracy matters, we have a common approach to methods and terminology. If and
where differences in approach or terminology must appear in subject areas, we
should be able to highlight the differences and explain why they are used.
The overriding aim is to improve standards of numeracy across all subjects. This will
increase student attainment across the curriculum. It will also give the students more
confidence and opportunities on leaving school.
The whole school numeracy policy is on-going and developmental.
1.2
1.3
2
Definition
Numeracy is a proficiency, which involves confidence and competence with numbers
and measures. It requires an understanding of the number system, a repertoire of
computational skills and an inclination and ability to solve number problems in a
variety of contexts. Numeracy also demands practical understanding of the ways in
which information is gathered by counting and measuring, and is presented in
graphs, diagrams, charts and tables.
3
Aims
The strategy aims to ensure Goole Academy students:
have a sense of the size of a number and where it fits into the number system
know by heart number facts such as number bonds, multiplication tables, doubles
and halves, factors and multiples
use what they know by heart to figure out answers mentally,
calculate accurately and efficiently, both mentally and with pencil and paper, drawing
upon a range of calculation strategies,
recognise when it is appropriate to use a calculator and be able to do so efficiently
and effectively,
make sense of number problems, including non-routine problems, and recognise the
operations needed to solve them,
explain clearly their methods and reasoning using correct mathematical language,
judge whether their answers are reasonable and have strategies for checking them
when necessary,
suggest suitable units to use in measuring and make sensible estimates of
measurements,
collect, order and group data in suitable ways,
display data using suitable tables, diagrams and graphs,
interpret information from the numbers displayed in graphs, charts and tables,
make predictions using numbers from - number patterns, charts, graphs and
diagrams.
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
m)
4
a)
b)
c)
d)
e)
Use of Calculators
Students should have the opportunities to use a calculator appropriately but staff
should:
discuss why the use of a calculator may be appropriate,
take opportunities to show how a problem could be solved without a calculator (which
may justify its use or not),
encourage checking strategies to ensure that the answer is reasonable,
take opportunities to discuss efficient use of a calculator (e.g. use of memory
function),
encourage students to at least write down the calculation being done and not just the
answer,
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f)
encourage students to buy their own calculator and be familiar with its particular
functions and idiosyncrasies,
g) inform students of when a calculator may be used in public examinations,
5
Cross Curricular Agreements on Language and Routines
Use of Units of Measure
Length
millimetres
centimetres
metres
kilometres
mm
cm
m
km
inches
feet
yards
miles
in or ”
ft or ’
yd
miles
Weight
milligrams
grams
kilograms
tonnes
mg
g
kg
tonnes
ounces
pounds
stones
tons
oz
lb
st
ton
Area
cm2, m2, km2 etc. say ‘square metres’ etc. NOT ‘metres
squared’ etc.
hectares
ha
Volume
cm3, m3, km3 etc. say ‘cubic metres’ etc. NOT ‘metres cubed’
etc.
millilitres
ml
centilitres
cl
litres
l
Temperature
Celsius (science)
Fahrenheit
Time
seconds (s), minutes (min), hours (hr)
Speed
metres per second
kilometres per hour
miles per hour mph
m/s
km/h
Money
pounds
pence
£
p
Centigrade (maths)
0
F
0
C
6
Notation and Terminology
Directed numbers
Numbers on a number line are often prefixed by a positive or negative sign
to indicate their position either side of zero.
+4 say ‘positive 4’ NOT ‘plus 4’ or ‘add 4’.
-4 say ‘negative 4’ NOT ‘minus 4’ or ‘take away 4’.
Large numbers
Spaces rather than commas are used to separate each group of three
digits e.g. 23 456 312 NOT 23,456,312
Billions
We seem to have accepted now the American and French concept of a
billion as one thousand million not a million million as it used to be
traditionally in England.
Therefore one billion = 1 000 000 000
This system makes reading large numbers much easier.
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Decimals
4.23 say ‘four point two three’ not ‘four point twenty three.’
Rounded Answers
These should always include (2dp), (3sf) etc. when rounding to a number
of decimal places or significant figures. Other methods include stating –
nearest metre etc.
Time
am and pm should not be used with the 24 hour clock
e.g 10:30am is written 1030 or 10:30hr and 3:30pm is written 1530 or
15:30hr NOT 15:30pm.
Standard Form
3.6 x 105 say ‘three point six times ten to the power five.’ Note some
calculators display 3.605 and students must be encouraged to write 3.6 x
105.
Average
We must be clear when we talk about averages - better to be precise and
use either MEAN, MODE or MEDIAN.
Mean – sum of items divided by number of items
Mode – the most frequently occurring item
Median - the middle item, once the items are placed in rank order.
Graphs and charts
Examples of the suggested terminology and the production of graphs and
charts are given in a separate section.
7
Standard Mathematical Techniques
When encountering the four rules of number, most students in secondary school will
fall back on a standard mathematical technique or algorithm to solve problems, if a
mental method cannot be applied. Early work in primary school should have enabled
students to have a good knowledge of:
Number Bonds i.e. 6 + 3 = 9 etc. and the related subtractions 9 – 3 = 6 and 9 – 6 = 3
Multiplication tables e.g. 6 x 3 = 18 and the related divisions 18 ÷ 6 = 3 etc.
7.1
Addition
Most students can handle addition but the most common problems are: students who
forget ‘place value’ when setting out an addition e.g.
427
30 +
students who add from the left hand column first, and students who carry the wrong
digit to the next column.
7.2
Subtraction
Where possible students should have mental strategies for subtraction but the
algorithm taught is called decomposition and looks like this for 263 – 127
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Students being unable to take 7 from 3 in the units column move a ten from the ten’s
column. The number 263 has been decomposed into two hundred, fifty and thirteen which
still is 263. The subtraction can now be done as above.
Problems occur with children who not being able to subtract 7 from 3 take 3 from 7.
4000 – 23 should be done mentally but using the algorithm looks like this
Weaker students may successfully use informal techniques using number lines
e.g.
Calculate 1032 – 786
Often weaker students do this
14
786
So 1032 – 786 =
200
800
32
1000
14 + 200 + 32 =
1032
246
7.3
Multiplication
Tables should take care of simple multiplications and strategies for multiplying by 4 (double
and double again), 5, 25, and multiples of 10 are encouraged.
Long multiplication has the standard algorithm as below, but several other methods are
widely used by students successfully. If students have a successful method, which is
reasonably efficient, then this should not be changed.
For 124 x 26
Standard algorithm
Other methods
Remember an estimate of 100 x 30 = 3 000 is advised.
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7.4
Division
Division is a stumbling block for many students and this is where they will immediately reach
for a calculator, particularly if their multiplication tables are poor. Using the table square in
reverse is a useful skill we encourage. In early years students are taught division by
repeated subtraction e. g.
57 ÷ 9
57 – 9 = 48
48 – 9 = 39
39 – 9 = 30
30 – 9 = 21
21 – 9 = 12
12 – 9 = 3
therefore 57 ÷ 9 = 6 r 3
The standard algorithm for 12 563 ÷ 3 is
Advanced students may give the answer as 4 187 2 or as a rounded decimal.
3
For long division most of us reach for a calculator but either of the following are acceptable.
Note many students complain that they do not understand the left hand example.
In both cases it is useful for students to list the multiples of the divisor so they can see what
they are doing.
Another standard algorithm for division is division by “chunking”.
Chunking method for division
e.g.
387 24
387
240
147
96
51
48
3
Answer:
10  24
4  24
2  24
16 remainder 3
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8
Mental Calculations
Students use a wide range of mental calculation strategies. They are encouraged to use and
explain the techniques they understand.
Here are some examples of mental calculations that typical year 7 students should be able
to do.
a) Calculate
42 + 35
b) Calculate
2005 – 1996
c) Find 2.5% of £3000
d) How might you estimate a quarter of 57.9
e) How many CDs at £3.99 can you buy with £25
f) Roughly how long will it take you to travel 50 miles at 30 mph
9
Presentation – graphs
All graphs and charts should have a title.
If points are plotted these should be marked by a small x or a dot.
Axes on graphs should:
(i)
have scales marked at regular intervals
(ii)
labelled to describe what their numbers represent (including units).
Remember: S.A.L.T. Scales, Axes Labelled & Title.
Examples follow of graphs and charts commonly used by most students across the
curriculum and covered in mathematics lessons in KS3.
Graph A:
Tally Chart
Tally Charts are used to collect and organise data
Tally Chart Showing the Favourite Football Teams of 8IGT
Favourite Team
Tally
Frequency
Manchester Utd
6
West Ham Utd
8
Leeds
3
Liverpool
2
Hull City
4
25
Graph B:
Pictogram
The pictogram shows what a small café sold on a particular day.
Each symbol in the pictogram represents 10.
Sandwiches
Ice creams
Hot drinks
Cold drinks
Hot
Hot
Hot
Cold
Cold
Cold
Cold
Cold
Monday
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Cold
Graph C:
Pie Chart
The pie chart shows how the 24 members of 8IGT travel to school. The numbers in each
section may be calculated once the angle of each person is found.
This is calculated by sharing the 3600 in the pie chart between the 24 students
Angle of one student = 3600  24
= 150
For instance, the number that went to school by bus is 60  15 = 4
In some pie charts, the frequency within each group may be labelled rather than the angles.
Pie Chart Showing how 8IGT travel to School
Car 90*
Walk 180*
Bus 60*
Bike
30*
Note that when constructing pie charts, the calculations are much easier if the total number
of items is a number such as 20, 24, 30 or 36 divides into 360.
Graph D:
Bar Chart (Discrete Data)
Bar Chart Showing the Number of Students in each Year
300
Frequency
200
100
7
8
Year Group
9
10
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11
As the data is discrete (see notes on discrete and continuous data) the bars should be
separated by gaps. The gaps are of equal width, as are the bars. The numbers label the
centre of each bar.
Note that a bar chart with lines rather than bars is called a line graph.
Graph E:
Bar Chart (Continuous Data)
Bar Chart Showing the height of students in 8IGT
15
10
Frequency
5
130
140
150
height in cm
160
170
Note that the bars are of equal width and as the data is continuous the numbers label the
boundaries between the bars.
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Graph F:
Frequency Polygon
Frequency Polygon showing the height of students in 8IGT
15
10
Frequency
5
130
140
height in cm
150
160
170
This is essentially a continuous bar chart with the midpoints at the top of each bar joined with
a straight line. They can also be constructed without the bars. Frequency polygons help to
identify trends of how the data is changing.
Graph G:
Scatter Diagram
Scatter diagram showing the heights and masses of some horses
700
600
Mass
(kg)
500
400
300
140
150
160
Height (cm)
170
A scatter diagram is used to establish if a relationship, called a correlation, exists between
two variables (in this case height and mass). This example shows a positive correlation
(generally the larger the height, the larger the mass).
A line of best fit may be drawn and in mathematics this is always a straight line. However,
scientists may plot similar graphs and their line of best fit may be a curve.
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9.1
Discrete and Continuous Data
This often causes difficulty in data presentation.
Discrete data is where data can only take certain values e.g. shoe size (you can have a size
6 but not a size 6.3).
Continuous data can take any value within a range e.g. a person’s height could be anything
between 130 and 190cm such as 143.2cm or 167.342cm.
9.2
Tables
Numbers are often placed in tables to display information. A tally chart has been shown in
Example A.
Another very popular table used to display information is called a TWO WAY TABLE. It has
different categories both horizontally and vertically. Here is an example.
How students in 7RR have lunch
Boys
Girls
Totals
Canteen
8
6
14
Pack up
4
2
6
Go Home
3
1
4
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Totals
15
9
24