Transcript Numeracy Across the Curriculum (Toolkit)

Numeracy Across the
Curriculum
Toolkit
Addition
Subtraction
Addition and subtraction of decimals
time number lines
Worded Problems
Reading Numbers
Number
Relationships Multiplication
Division
Multiplication and division
patterns
Bidmas
Equivalent fractions, decimals and percentages
Percentage Checking Methods
Fractions
Proportion
Ratio
Negative numbers
Reading Scales
Metric Units Length
Metric Units Capacity and Mass
Perimeter
Area
Volume
Measure angles
Averages
What makes a good graph
Bar Chart Pictogram Grouped Frequency Diagram Frequency Polygon Pie Chart
Scatter Diagram metric and imperial units
Llanishen High School - Numeracy toolkit
Addition.
+70
+80
-1
+9
84
84
154
163
+70
150 154
+4
80
80 + 70 = 150
4 + 9 = 13
85 + 79 = 163
84 + 79
84 + 80 – 1 =163
80
163 164
163
+ 9
150 154 163
84
+ 79
163
1 1
Adding Words
•Sum
•Total
•Plus
•Altogether
•More than
+4
+24
- 50
-2
56 60
-4
84
Adding on to subtract
24 + 4 = 28
28 30
80
84
Subtraction
84 - 56
84 -50 – 6 =28
80
-50
-6
28
56 +4 +20 + 4 = 84
Answer = 28
150 154
34
163
7 1
84
Answer = 28
84
- 56
28
•
•
•
•
•
Subtracting Words
Difference
Take away
Take off
Minus
Less than
Worded questions.
The hardest thing about worded questions is deciding
which operation to use. Here are some ‘operation’ words
to help you .
Subtracting
Adding Words
Words
Division Words
• Sum
• Difference
• Share
• Total
• Take away
• Each
• Plus
• Take off
• Divide
• Altogether
• Minus
• Quotient
• More than
• Less
• Goes into
Multiplication words
• Times
• Product
• Multiply
• Of
• Lots of
Reading numbers.
It really helps to leave
gaps between the
hundred’s, thousand’s
and millions.
Numbers up to 999
345 – three hundred and forty two
902 – nine hundred and two
When reading decimals
remember that;
3.45 is three point four five
NOT three point forty five.
714 – seven hundred and fourteen
Numbers up to 999 999
6 703 – six thousand seven hundred and three.
15 243 – fifteen thousand two hundred and forty three
209 312 – two hundred and nine thousand, three hundred and twelve
745 697 – seven hundred and forty five thousand, six hundred and ninety seven
Numbers up to 999 999 999
6 703 769 – six million seven hundred and three thousand, seven hundred and sixty nine
89 503 102 – eighty nine million, five hundred and three thousand, one hundred and two.
607 132 617 – six hundred and seven
million, one hundred and thirty two, six hundred and seventeen.
Number Relationships and Divisibility tests.
The Multiples of a number are
all the numbers in that times
table.
E.G. The multiples of 6 are:
6, 12, 18,24 …
The Factors of a number are
all the numbers that goes into
it exactly
E.G. The factors of 12 are:
1, 2, 3, 4, 6, 12
The factor pairs of a number are
pairs of numbers that times to it.
E.G. The factor pairs of 20 are
1 x 20
2 x 10
4x5
Divisibility tests.
A number is divisible:
By 2 if it is even, i.e., it ends in 0, 2, 4, 6 or 8
By 3 if the sum of the digits is a multiple of 3
By 4 if half of the number is even
By 5 if it ends in 0 or 5
By 6 if half the number is divisible by 3
By 8 if half of half the number is even
By 9 if the sum of the digits is a multiple of 9
By 10 if it ends in 0.
A prime number is a number
with exactly two factors, one
and itself
E.G. 17 is prime because only 1
and 17 goes into it exactly.
The primes are 2, 3,5,7,11,13…
Remember that 1 is NOT a
prime number because it only
has one factor
324  6
3
6 324
5
3 2
6 324
54
6 3 322 4
6 into 3
doesn’t go
so carry
the 3
22117
1 x 17
2 x 17
4 x 17
8 x 17
= 17
= 34
= 68
= 136
13 x 17 = 221
6 into 32
goes 5
times
remainder
2
6 into 24
goes 4
times no
remainder
Check: 54 x 6 = 324
So,
221 17  13
Double to
divide!
Check: 13 x 17 = 221
612 18
6
18 6 1 2
18 into 6
doesn’t go
so carry
the 6
3
6
7
6
7
18 6 1 2
3 4
18 6 1 2
18 into 72
goes 4
times no
remainder
Check: 18 x 34 = 612
18 into 61
goes 3 times
remainder 7
Use a ready
reckoner.
1x18 = 18
2x18 = 36
3x18 = 54
4x18 = 72
5x18 = 90
6x18 = 108
7x18 = 126
8x18 = 144
9x18 = 162
add 20
takeaway 2!
Multiplication and division patterns
7 ´ 5 = 35
16 ´ 2 = 32
67 ´1 = 67
7 ´ 0.5 = 3.5
16 ´ 0.2 = 3.2
67 ´ 0.1 = 6.7
¸10 ¸10
¸10 ¸10
¸10 ¸10
You are multiplying by a number that is 10 times
smaller so the answer is ten times smaller.
65 ¸ 5 = 13
16 ¸ 2 = 8
18 ¸1 = 18
65 ¸ 0.5 = 130
16 ¸ 0.2 = 80
18 ¸ 0.1 = 180
¸10 ´10
¸10 ´10
¸10 ´10
You are dividing by a number that is 10 times smaller
so the answer is ten times bigger.
Bidmas is an
acronym
explaining the
order in which
operations
should be
carried out.
BIDMAS
Consider:
8 –3x2
8 – 6 =2
(8 – 3) x 2
5 x 2 = 10
5 +2 x 3
B
I
D
M
A
S
brackets
ndices
division
If you did this in the order
it was written the answer
is 21.
In fact the answer is 11.
This is because we must
multiply before we add
(8 – 3) x 2
5 x 2 = 10
multiply
(8 + 4) ÷ (10 - 7)
12 ÷
3 =4
addition
subtraction.
(8 – 3) x ( 4 +3)
5 x
7 = 35
Equivalent Fractions/Decimals/Percentages
1
= 0.5 = 50%
2
1
= 0.25 = 25%
4
1
= 0.1 = 10%
10
1
= 0.01 = 1%
100
1
 0.2  20%
5
1
= 0.3 = 33.3%
3
So 3 = 0.75 = 75%
4
9
 0.9  90%
So
10
So 17 = 0.17 = 17%
100
So
3
 0.6  60%
5
2
= 0.6 = 66.6%
So
3
25%
Half it and
half again
10%
Divide by 10
30%
10% x 3
Finding a percentage with a calculator.
50%
Half it
Percentage
Non
Calculator
5%
10% ÷ 2
Turn the percentage
into decimal and
multiply!
42% of 540
1%
Divide by
100
0.42 x 540 = 235.2
6% of 310
75%
50% + 25%
0.06 x 310 = 18.6
Find one number as a percentage of another with a calculator
I got 42 out of 60
in my test
What percentage
is that?
42
 100  70%
60
I do
42÷60 x 100
on the
calculator.
Fractions
Lowest Terms
Equivalent fractions
30
55
´3
5
15
=
7 ´3 21
Is there a number that goes
into 30 and 55 exactly?
¸5
Yes 5. So
¸5
25
5
=
40 ¸5 8
This is its lowest terms because
only 1 goes into 6 and 11 exactly
There are 6 equal
parts.
3 pieces are shaded.
There are 8 equal
parts.
3 pieces are shaded.
So,
3
8
is shaded
30
6
=
55 ¸5 11
So,
Or
3
6
is shaded
3
1
=
6
2
In its lowest terms
1
´ 42 = 6
7
42 ¸ 7
3
´ 60 = 36
5
60 ¸ 5´ 3
Proportion.
To make 12
pancakes
100g of plain flour
To make 6 pancakes I
would HALF all the
quantities
2 eggs
200ml of milk
60g of butter
To make 9 pancakes I would
divide by 4 to find the quantities
for 3 pancakes and then multiply
by 3 to find to quantities for 9
To make 24 pancakes I would
DOUBLE all of the quantities
To make 30 pancakes I would
add together the quantities for
24 and 6 pancakes.
To make 5 pancakes I would divide by
12 to find the quantities for 1 pancake
and then multiply by 5 to find to
quantities for 5
Negative
Numbers.
+
Direct number line sums.
-7 + 3 = -4
Start here
go up
three
end up here
4 - 6 = -2
−
Start here
go up
three
end up here
Adding and subtracting negative numbers.
If you can put a circle around two signs use the following rules
+
--
+
++
+-
-+
“If the signs are the same we like it!”
ADD
“If the signs are different we don’t like it!”
SUBTRACT
-7 + - 3 = 10
+
9 - - 3 = 12
+
-14 - - 3 = -11
6 - + 9 = -3
Multiplying negative numbers
Neg No. x Neg No. = Pos. Answer
Neg No. x Pos. No. = Pos. Answer
Pos. No. x Pos. No. = Pos. Answer
Pos. No. x Neg. No. = Pos. Answer
The same rules apply when dividing.
Reading Scales.
There are 5 small divisions between 0
and 250 so each small division is 50
So 750 and 3 small
divisions equals 900
Metric Units: Length
10mm = 1cm
100cm = 1m
6 cm
Is the same as
60 mm
6.2 cm
Is the same as
62 mm
5m
Is the same as
500cm
0.2cm
Is the same as
2 mm
Multiply it by 10, so
every digit becomes
10 times BIGGER.
0.3m
Multiply it by 100, so
Is the same as every digit becomes
30cm
100 times BIGGER.
EVERY digit moves
one place to the
LEFT.
5.4m
Is the same as
540cm
EVERY digit moves
two places to the
LEFT.
2 mm
Is the same as
0.2 cm
53 mm
Is the same as
5.3 cm
200 cm
Is the same as
2m
50 mm
Is the same as
5 cm
Divide it by 10, so
every digit becomes
10 times SMALLER.
5 cm
Divide it by 100, so
Is the same as every digit becomes
0.05 m
100 times SMALLER.
EVERY digit moves
one place to the
RIGHT.
62 cm
Is the same as
0.62 m
EVERY digit moves
two places to the
RIGHT.
1000m = 1km
7 km
Is the same as
7000km
0.75km
Is the same as
750m
6.9km
Is the same as
6900km
Multiply it by 1000,
so every digit
becomes 1000
times BIGGER.
EVERY digit moves
three places to the
LEFT.
3000m
Is the same as
3km
8800m
Is the same as
8.8km
51 m
Is the same as
0.051km
Divide it by 1000, so
every digit becomes
1000times SMALLER.
EVERY digit moves
three places to the
RIGHT.
Metric Units: Capacity and Mass
1000ml = 1l
4l
Is the same as
4000ml
7.4l
Is the same as
7400ml
0.89l
Is the same as
890ml
Multiply it by 1000, so every
digit becomes 1000 times
BIGGER.
EVERY digit moves three
places to the LEFT.
6700ml
Is the same as
6.7l
35ml
Is the same as
0.035l
580ml
Is the same as
0.580l
Divide it by 1000, so every
digit becomes 1000
times SMALLER.
EVERY digit moves three
places to the RIGHT.
1000g = 1Kg
6 kg
Is the same as
6000g
0.54Kg
Is the same as
540g
6.9Kg
Is the same as
6900g
Multiply it by 1000, so
every digit becomes 1000
times BIGGER.
EVERY digit moves three
places to the LEFT.
3700g
Is the same as
3.7Kg
62g
Is the same as
0.062Kg
530g
Is the same as
0.530Kg
Divide it by 1000, so every
digit becomes 1000 times
SMALLER.
EVERY digit moves three
places to the RIGHT.
Perimeter is the
distance around
the outside
edge of a 2D
shape
Perimeter.
8cm
10cm
P = 2 x10 +2 x 6
= 32cm
P = 8 + 9 +7 + 5
5cm
= 29cm
9cm
The perimeter of
a circle is called
the
CIRCUMFERENCE
7cm
6cm
P=4x6
= 24cm
d
C = pd
or
C = 2p r
6cm
Area.
Area is the
number of unit
squares that fit
onto a 2D
shape
A =l´w
a
1
A = ´ (a + b) ´ h
2
1. Check all values have the same unit
before you start.
w
2. Always write down the correct formula
first.
3. Rewrite the formula with values
l
h
b
r
A = p r2
4. Use Bodmas when doing the
calculation
h
b´h
A=
2
b
5. Give correct units.
A = b´h
b
2
(cm )
h
The surface
area of a 3D
shape is the
total area of
all its faces
h
w
l
Vol = l ´ w ´ h
Volume.
Volume is the
number of unit
cubes that fit
onto a 3D
shape
1. Check all values have the same unit before
you start.
2. Always write down the correct formula first.
3. Rewrite the formula with values
4. Use Bodmas when doing the calculation
5. Give correct units.
V = A ´l
V=
b´h
´l
2
(cm3 )
A prism is a shape with a uniform cross section
.
Vol. of a prism = area of uniform cross section x length
V = CSA ´l
V = A ´l
V = p r2 ´ l
Mean
Median
Mode
The ‘average’, or arithmetic mean,
of a set of discrete data is the sum
of quantities divided by the
number of quantities.
The Median is the middle
number or value when all
values in a set of data are
arranged in ascending order.
The Mode is the most
commonly occurring
value.
Find the mean of the following data:
Find the Median of the following data:
Find the Mode of the following data:
2, 3, 4, 5, 4, 6
4, 7, 8, 2, 3, 5, 6, 10, 6, 5, 6
Add the
numbers and
then divide
your answer by
how many
numbers there
are?
2  3  4  5  4  6 24

4
6
6
You need to
order the data
and then find
the number in
the middle.
2, 3, 4, 5, 5,
6, 6, 6, 7, 8, 10
5, 7, 8, 2, 3, 6, 6, 10, 6, 5, 6
Find the
number that
appears the
most.
2, 3, 5, 5, 6, 6, 6, 6, 7, 8, 10
6 is the mode.
The Range is the difference between the greatest value and the least value in a
set of numerical data. Remember it is not an average
What makes a good graph?
A Graph Showing the Results of …………………………
title
label
with
units
Temperature (oc)
arrow
100
90
80
Each graph must have
• A title
• A label on both axes
• An arrow at the end of each axes
• A sensible scale for both axes
×
×
70
60
×
50
×
40
×
×
30
×
Plot each point as a neat cross ×
20
10
arrow
0
0
10
20
30
40
50
60
70
80
Time (s)
label with units
Bar Chart
Favourite Number topic
24-
Check list:
•Title
•A label on each axis
•A sensible scale
2220-
F
r
e
q
u
e
n
c
y
18161412-
A bar chart is for
discrete data; there
should be gaps between
the bars
108 6 4 2 0 -
%
long
division
Bodmas
Number topic
F’s, D’s
and P’s
All the bars must
be the same
width
and
All the gaps
between the bars
should be equal
Pictogram
Vehicle
Car
Frequency
When drawing a pictogram:
Van
All pictures must be the same
shape, size and colour
Bike
All gaps between the pictures
must be the same size.
There must be a key
Bus
Key
= 40 Vehicles
There are 90 vans
And
130 bikes
Grouped Frequency Diagram
Height of year 9 girls.
242220-
F
r
e
q
u
e
n
c
y
Check list:
•Title
•A label on each axis
•A sensible scale
181614-
A Grouped
Frequency
Diagram is for
Continuous Data;
there should be
no gaps between
the bars
12108 6 4 2 0 -
1.4m
1.5m
1.6m
Height (M)
1.7m
1.8m
Notice that the
each height
interval is equal
Frequency
Polygon
Height of yr9 girls.
Frequency
1.4 < m £1.6
9
1.6 < m £ 1.8
1.6 < m 1.8
1.6 < m £ 1.8
21
16
5
£
Height of year 9 girls.
Check list:
•Title
•A label on each axis
•A sensible scale
242220-
F
r
e
q
u
e
n
c
y
1816-
A Frequency
polygon is for
continuous data.
Points are plotted
on the mid-point
1412108 6 4 -
Notice that the
each height
interval is equal
2 0 -
1.4m
1.5m
1.6m
Height (M)
1.7m
1.8m
Pie Chart
Sixty people were asked their favourite number topic.
First step is to find out how many degrees will represent
each person
60 people = 360
o
360 o
1person =
= 6o
60
Favourite
Number topic.
Frequency
Angle
Percentage
14
14 x 6o = 84 o
Long Division
24
24 x 6o = 144 o
Bodmas
17
17 x 6o = 102 o
Factors,
multiples and
primes
5
5 x 6o = 30 o
Remember that pie charts
show proportions not the
actual numbers so can be
misleading.
If you look at this chart you
can see that Long division in
is the most popular topic, but
you can not tell how many
people said long division
Percentage
Bodmas
Long Division
Check the
angles add
to 360o
A Scatter Diagram helps us to determine whether there is a link between two sets of data .
10 learners take a quiz in two subjects Maths and Science.
The results are recorded below.
Science
10
6
3
9
1
9
4
7
5
6
9
7
2
8
0
7
3
7
5
4
Science
Maths
10
9
8
7
6
5
4
3
2
1
0
Negative Correlation
Is there a
link
(correlation)
between the
maths and
science score
for each
child?
×
×
×
Plot the
points
You can draw a
scatter diagram
to decide
(6,7)
(9,8)
×
×
×
×
×
As one value increases the
other value decreases
×
× × ×
×
The data values move in
opposite directions
(10,9)
(3,2)
× ×
×
×
(1,0)
(9,7)
×
Positive Correlation
As one value increases the
other value increases
The data values move in the
same direction
×
×
×
×
×
×
×
(4,3)
×
(7,7)
(5,5)
×
×
(6,4)
0 1 2 3 4 5 6 7 8 9 10
Maths
There is a positive correlation – As the maths
mark increases so does the science mark..
×
×
×
×
×
×
××
×
×
No Correlation
The statistics are not linked
at all.
The one value has no effect
on the other value
Metric Units for
length:
Millimetres
Centimetres
Metre
Kilometres
Metric units for
capacity:
Mililitres
Litres
Metric Units for
mass:
Grams
Kilograms
Metric and Imperial Units
1 inch
»2.5 centimetres
1 foot = 12 inches
1 mile
1 pint
» 1.6 Kilometre
» 0.568 litre
Imperial Units
for length:
Inch
Foot
Yard
Mile
Imperial units
for capacity:
Pint
Gallon
Imperial Units
1 gallon = 8 pints
1 pound
» 0.45 kilograms
1 stone = 14 pounds
for mass:
Ounce
Pound
Stone
Addition and Subtraction of Decimals.
18.46 + 9.71
18.46
+ 9.71
28.17
1 1
14.6 - 9.17
Line up
the
decimal
points!
5 1
14.60
- 9.17
5.43
Fill in the
gap with a
zero!
Line up the
decimal
points!
I buy an ice cream for £1.74. How much change do I get
from £5 ?
4
9
1 1
5. 0 0
- 1 .7 4
3.26
£3.00
6p
74p 80p
20p
£2.00
£5.00
Adding to subtract: £3.00 + 20p +6p = £3.26
How do I find out how much time has passed between 1947 and 2015
3+50+15 = 68 years
4+30+900+15 = 949 years
2015
+ 900
2000
+ 30
1100
+4
1070
1066
How do I find out how much time has passed between 1066 and 2013
+ 13
+ 15
2015
2000
+3
+ 50
1950
1947
The easiest way is to use a number line.
Checking methods
Check by substitution
Check by estimating
Find a, when b= 6 and t=18
68 X 71= 4828
2b + 3a = t
2x6 + 3xa = 18
So a=2
Check: 70 x 70 = 4900
So 4928 is a sensible
answer.
Check by substitution;
2x6 + 3x2 = 18
Check using inverse operations
71 X 6 + 5 = 431
4.56 x 5.7 = 25.992
So using inverse operations
If you did 431 – 5 and then
divided that answer by 6 you
should get 71.
Three numbers after the
decimal point in the
question, three numbers
after the decimal point in
the answer.
We are measuring
in a clockwise
direction so use
the outside scale.
Measuring angles
30o
50o
We are measuring in an
anti-clockwise direction so
use the inside scale.
60o
We are measuring
in a clockwise
direction so use
the outside scale.
Ratio
Mixing blue and yellow paint in the ratio of 2:3 produces green paint.
2parts blue +
3 parts yellow
=
makes 5 parts green
If I need 300ml of green paint,
how much blue and yellow paint
would I need?
If I need 1.5 litres of green paint, how
much blue and yellow paint would I
need?
The 5 parts green = 300ml
So 1 part = 300 ÷ 5 = 60 ml
The 5 parts green = 1.5 litres
So, 1 part = 1.5 ÷ 5 = 0.3 litres
I need 2 parts blue = 120ml
And
I need 3 parts green = 180ml
I need 2 parts blue = 0.6 litres
And
I need 3 parts green = 0.9 litres
If I had 50ml of blue paint how much yellow paint would I need to make green paint?
The two parts blue = 50 ml
So 1 part = 50÷2 = 25 ml
I need 3 parts yellow 75 ml