Directed Numbers
Directed
Numbers
Curriculum Ready
ACMNA: 124, 134, 183, 186, 280
www.mathletics.com
Numbers come in all sizes and forms. They can be positive or negative, whole numbers,
fractions or decimals and rational or irrational.
Before you start, investigate these terms and write a brief description about them in the boxes.
Positive and negative numbers:
Whole numbers, fractions and decimals:
Rational and irrational numbers:
is
Give th
Q
a go!
Calculate the value of: (-2) 3 + {- 2 + [2 # (2 + 2)]}
Work through the book for a great way to solve this
Directed Numbers
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How does it work?
Your Turn
Directed Numbers
Types of numbers
Natural numbers are all the positive counting numbers "1, 2, 3, ... , .
Integers are all the positive (+) and negative (-) whole numbers (including zero).
Circle all the numbers below that belong to the family known as integers.
(psst!: some of them need to be simplified by calculating their value first)
1
2
- 10
5
41
2
0
4
5
- 2
- 9
1.19284750219...
4
5
- 5.0
13
-1
4
7
0.42
-16
9.86
- 8
3
36
8
-1
3
-0.3o
10
3
30
Place each of these numbers into their correct group below to classify them.
Irrational numbers can’t be written as a fraction
Rational numbers can be written as a fraction
Integers: the positive and negative whole rational numbers including 0.
Put the special
types of rational
numbers into their
correct group
-1
Natural numbers: the counting rational numbers (not including 0).
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Directed Numbers
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How does it work?
Directed Numbers
Measuring instruments and integers
There are many applications for directed numbers.
Thermometers are a great example because the temperature can go below zero.
The temperature at midday was 20 degrees Celsius. During the afternoon, the temperature rose 4
degrees before falling 9 degrees in the late afternoon.
(i) Shade in the thermometer showing the late afternoon temperature.
Starting at 20cC , temperature changed + 4cC then - 9cC
25o
24o
23o
22o
21o
20o
19o
18o
17o
16o
15
14o
13o
12o
11o
10o
9o
8o
7o
6o
5o
4o
3o
2o
1o
0o
Temperature at midday 20cC
Temperature by late afternoon 15cC
rose =+ and falling =-
+ 4cC
- 9cC
Centigrade
o
C
(ii) What directed number gives the overall change in temperature between the two readings?
Overall change =+4cC - 9cC
=-5cC
Digital scales need to use directed numbers to calculate changes in mass.
Use this information to calculate the mass of olives purchased by Maria at her local Deli:
Maria asked for 500 g of olives.
After first weighing 575 g of olives, the Deli owner took out 98 g and then put back 46 g.
The directed number sentence for this is: + 575 g - 98 g + 46 g
Increase = + and decrease = -
` Maria purchased + 575 g - 98 g + 46 g = 523 g of olives
If the first term is positive, we usually don't write in the plus sign.
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How does it work?
Your Turn
Directed Numbers
Measuring instruments and integers
1
Shade in each thermometer below to accurately display these temperatures:
(i) 11cC
(ii) -5cC
(iv) 0cC
(v) 8cC above zero (vi) 2cC below zero
20o
20o
20o
20o
20o
20o
15o
15o
15o
15o
15o
15o
10o
10o
10o
10o
10o
10o
5o
5o
5o
5o
5o
5o
0o
0o
0o
0o
0o
0o
-5o
-5o
-5o
-5o
-5o
-5o
-10o
-10o
-10o
-10o
-10o
-10o
Centigrade
o
C
2
(iii) 6cC
Centigrade
o
C
Centigrade
o
C
Centigrade
o
C
Centigrade
o
C
This thermometer shows a reading of 12cC at the start.
Write the new readings that would be displayed after each of
these changes in temperature:
25o
20o
15o
(ii) Becomes 8cC cooler
cC
(iii) Decreases by 13cC
cC
(iv) Warms up by 9cC
cC
(v) Rose by 22cC
cC
(vi) Dropped 28cC
cC
(vii) Decrease by 7cC then increase by 12cC
cC
(viii)Warms by 3cC and then a further 9cC
cC
(ix) Cools 15cC before warming by 19cC
cC
(x) Changes to 0cC before dropping 4cC
cC
5o
-5o
o
-10
Centigrade
o
C
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cC
cC
10o
0o
12
(i) Increases by 3cC
30o
4
Centigrade
o
C
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Your Turn
Directed Numbers
3
The mass of a glass container with some water in it is shown below:
INSTRUM
TS AND
EN
Measuring instruments and integers
G
MEASURIN
How does it work?
..../...../20...
TEGERS *
IN
0402
grams
Write down the new reading on the scales after any one of the following changes occurred.
(i) 320 g of water poured into the glass
grams
(ii) 212 g of water was spilled out of the glass
grams
(iii) 88 g of flavour syrup was stirred in and then 250 g was drunk
grams
(iv) A similar glass with the same amount of water was placed on
the scales next to this one.
grams
(v) A cup filled with 216 g of water was emptied into the glass and
during the day, 7 g evaporated away because the room was hot.
grams
Try this tricky one!
4
A glass container holding cold water below was placed on the digital scales after being moved from
a 24cC room into a 32cC room.
The glass had collected 3 grams of water (through a process called condensation) on the outside for
every 2 degree rise in room temperature.
What was the original mass of the water filled glass if 24 grams of water was spilt during the move?
60o
55o
40o
30o
32cC
20o
10o
0o
Centigrade
o
C
0583
grams
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How does it work?
Directed Numbers
Magnitude and direction of integers
Integers are positive (+) and negative (-) whole numbers, including zeros.
They tell us the size (magnitude) and direction of values associated with change.
-1
It is important to first decide what will be the positive direction.
+3
• If up ( ) is positive, then the opposite direction down ( ) is negative
• If right ( ) is positive, then the opposite direction left ( ) is negative
• If hotter is positive (+), then the opposite colder is negative (-)
,
ER
ST
,
NEGATIVE
,
CE
DU
E, COOLER, S
EAS
OU
CR
TH
DE
,
BE
LO
RE
INCREASE, W
AR
ER,
ME
OV
R,
R,
INK, REVERSE
SHR
,
R, LESS, SLOW
ER
OSE
,
CL
FURTHER,
M
E,
OR
OV
E,
AB
FA
POSITIVE
,
ST
WE
W,
D,
G
EAST, ROW, F
OR
H,
T
WA
R
R
NO
CEND, LOWE
R
,
DES
UN
DE
R
RA, ASCEND
,
EXT
HI
GH
E
These are some other words usually associated with positive and negative directed numbers
Only negative numbers must have the sign written in front.
Write down the directed number for each of these statements:
(i) 5 degrees below the zero:
=-5c
The magnitude of the temperature is 5 degrees
(ii) 30 degrees above the zero:
=+30c
30c
=
The magnitude of the temperature is 30 degrees
The positive sign (+) can be hidden
(iii) $12 extra money:
The magnitude of the money is $12
=+$12
= $12
The directed number of an amount left after some changes represents the overall change.
One day Matt started with no money and was given $40. He spent $14 of it downloading music.
What is his overall change of money on this day?
The change in money for Matt during this day was: $0 + $40 - $14 = $26 left
` the overall change in money for Matt on this day is +$26 The final directed value = overall change
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Your Turn
Directed Numbers
Magnitude and direction of integers
For each of these statements, write down:
(i) the directed number that matches it and, (ii) the magnitude of the directed number.
a
..../...../20...
EGERS *
NT
1
The tree in Jamie’s yard grew 3 metres in one year.
(i) Directed number:
AND DIRE
ION OF I
CT
E
NITUD
MAG
How does it work?
(ii) Magnitude of the tree’s growth:
The same tree was then trimmed which reduced the height of the tree by 4 metres.
(iii) Directed number:
b
(iv) Magnitude of the reduction in height:
Aki walked 250 m West from her starting point.
(i) Directed number:
(ii) Magnitude of the distance Aki walked:
Aki then turned around and walked another 600 m East
(iii) Directed number:
c
(iv) Magnitude of this distance walked by Aki:
Sean’s bank account balance earned interest and increased by $4.
(i) Directed number:
(ii) Magnitude of Sean's bank balance increase:
Sean’s bank account balance then changed due to being charged $3 in bank fees.
(iii) Directed number:
2
(iv) Magnitude of the change in Sean's bank balance:
Think carefully for each of these statements and write down the directed number that matches
the overall change.
a
Pip was given $24 and then spent $15 on a dress the same day.
Directed number for the overall change in money that day:
b
+$9
The temperature initially increased by 6 degrees and then increased by a further 4 degrees.
Directed number for the overall change in temperature:
c
Nigel went down 6 rungs on a ladder and then up 2 rungs.
Directed number for the overall change in ladder rungs:
d
Cameron's coolness rose by 4 points when he played guitar, and a further 20 points
when he did maths.
Directed number for the overall change in Cameron's coolness after doing both activities:
e
Shiromee hiked 12 km North, 2 km East and then 19 km South. North is the positive direction.
Directed number for the overall change in Shiromee's North-South movement:
f
Adele's hair grew from 20 cm to 30 cm long at the back. After a hair cut it was only 17 cm long.
Directed number for the overall change in Adele's hair length from the initial 20 cm:
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How does it work?
Directed Numbers
Ascending and descending order
When comparing directed numbers, it often helps to arrange them into numerical order.
De
sce
ng
ndi
e
Asc
Ascending order = lowest to highest
ndi
ng
Descending order = highest to lowest
Don’t be tricked by the magnitude of a number.
Which of these two values is higher?
4
2cC
- 10cC
2cC
- 10cC represents a very cold temperature (below/less than 0cC )
2cC is a cold temperature, however it is (above/greater than 0cC )
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
` 2cC is a higher value than - 10cC
-8
-10cC
-9
-10
-11
-12
-10 is a bigger number in magnitude than 2, however the negative sign makes it lower in value
Centigrade
o
C
Let’s look at an ascending and descending order example.
For the numbers -4 , 0 , 14.5 , -6 , 5.2 , 3 , -10 , 5.4 :
(i) Arrange them into ascending order
-10 , -6 , -4 , 0 , 3 , 5.2 , 5.4 , 14.5
Lowest negative
Highest positive
(ii) Arrange them into descending order
14.5 , 5.4 , 5.2 , 3 , 0 , -4 , -6 , -10
Highest positive
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Lowest negative
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Order is opposite to ascending
How does it work?
Your Turn
Directed Numbers
2
3
Circle the word that represents the order of the values in these statements:
a
Shortest to Tallest
Ascending
Descending
b
Longest to shortest
Ascending
Descending
c
Closest to farthest
Ascending
Descending
d
Warmest to Coolest
Ascending
Descending
e
Heaviest to lightest
Ascending
Descending
f
Thinnest to widest
Ascending
Descending
..../...../20.
..
G ORDER *
IN
1
ASCEN
Ascending and descending order
END
SC
NG AND D
DI
E
Arrange the following groups of numbers into ascending order (lowest to highest).
a
14 , 0 , 17 , 3
,
,
,
b
21 , 25 , 20 , 19 , 22
,
,
,
,
c
-10 , -8 , -12 , -4 , -16
,
,
,
,
d
2.2 , 2.4 , -2.6 , 0 , -2
,
,
,
,
e
-1 , 11 , 1 , -1.5 , 1 , 1
5
4
,
,
,
,
,
Arrange the following groups of numbers into descending order (highest to lowest).
a
2 , 16 , 5 , 8
,
,
,
b
40 , 31 , 32 , 38 , 29
,
,
,
,
c
-19 , -11 , -16 , -18 , -13
,
,
,
,
d
1.6 , 1.9 , -1.3 , -1.0 , 1.2
,
,
,
,
e
0 , - 1 , -2 1 , -2.1 , 1.2 , -30
4
2
,
,
,
,
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How does it work?
Your Turn
Directed Numbers
Ascending and descending order
Combo Time!
4
Elevator Riding
A department store elevator operator started work on the 3rd floor of the 15 storey building.
For the first 15 minutes of work, the operator travelled to the following floors in the order written:
12 th floor
8 th floor
13 14 15
Ground floor
10 11 12
2nd floor
7
8
9
4
5
6
15 th floor
1
2
3
4 th floor
Ground floor
10 th floor
G >|< >|<
5 th floor
a
If up is positive and down is negative, write numbers that represent the movement of the
elevator operator during the first 15 minutes of work.
,
b
,
,
,
,
,
,
,
,
,
Arrange the directed movements into descending order.
,
c
,
,
,
,
,
During the first 15 minutes, did the operator mostly ascend or descend in the elevator?
Explain you answer.
20...
../
..../...
ME *
TI
C
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* COMB
O
BO TIME
OM
COMBO TI
ME
*
How does it work?
Directed Numbers
The number line
Directed numbers can be plotted on a number line to instantly see their order of value.
Values decrease (descending)
Values increase (ascending)
-15-14-13-12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 1112 1314 15
Lower numbers are further left and higher numbers are further right.
Display the numbers -4, -1, 7 on a number line.
-9
-8
-7
-6
-5
-4
-3
-2
-1012345 67 8 9
Use the plotted points to compare the values of:
(i) -4, -1:
-4 is further left than -1, ` -4 1 -1
(ii) -1, 7:
-1 is further left than 7,
Remember:
1 means 'less than' and
2 means 'greater than'
` -1 1 7
This example requests numbers to be plotted using a given rule.
Display all the even positive integers between -5 and 13 on a number line.
Even and positive
-6
-5
-4
-3
-2
-1012345678910
11
12
13
Write these numbers in descending order.
Read the numbers from right to left (
)
` The numbers in descending order are: 12 , 10 , 8 , 6 , 4 , 2
Half values are plotted by placing the dot half-way between the integers on either side.
Display the numbers 1, -2, 4 1 , -5.5, 9, 0 on a number line.
2
-9
-8
-7
-6
-5
-4
-3
-2
-1012345 67 8 9
4 1 is half way between 4 and 5
-5.5 is half way between -5 and -6
2
Write these numbers in ascending order.
Read the numbers from left to right (
)
` The numbers in ascending order are: -5.5 , -2 , 0 , 1 , 4 1 , 9
2
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Directed Numbers
2
Insert the correct symbol 1 (less than) or 2 (greater than) for each of these.
a
6
d
-11
g
-1
2
j
- 1.8
..../...../20...
LINE
*
ER
1
THE NUMB
The number line
THE NUMB
LINE
*
ER
b
3
-12
c
-5
-2
e
8
-8
f
-7 1
2
-7
0
h
91
2
i
-5.1
-5.8
k
-12.5
l
11 1
2
11 1
3
4
1.2
9.9
-21.5
4
List the numbers displayed below in ascending order
a
-15-14-13-12-11-10-9-8 -7 -6 -5-4 -3 -2 -1012345678910
11
121314
15
,
,
,
,
b
-9 -8
-7
-6
-5
-4
-3
-2
,
3
-10123456789
,
,
,
List the numbers displayed below in descending order
a
-18-17-16-15-14-13-12-11-10 -9-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 1112
,
,
,
,
b
-9 -8
-7
-6
-5
-4
-3
,
12
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-2
-10123456789
,
,
Directed Numbers
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,
How does it work?
Your Turn
Directed Numbers
The number line
4
Display the following sets of numbers on a number line.
a
-5, 8, -1, 3, 6.
-9 -8
b
-5
-4
-3
-2
-10123456789
-7
-6
-5
-4
-3
-2
-10123456789
-7
-6
-5
-4
-3
-2
-10123456789
-3
-2
-10123456789
- 7, 7.5, 3 1 , 0, 2 1 , 2.
2
2
-9 -8
5
-6
-4 1 , 4, 6 1 , - 1 , 5.
2
2
2
-9 -8
d
-7
-8.5, 1.5, -2, 7.5, 5.
-9 -8
c
1 = 0.5
2
-7
-6
-5
-4
Display the following sets of numbers on a number line.
a
The first six odd integers above -8.
-9 -8
b
-6
-5
-4
-3
-2
-10123456789
All the negative even integers less than 7 but greater than negative 8.
-9 -8
c
-7
-7
-6
-5
-4
-3
-2
-10123456789
All the multiples of 3 higher than 7 but lower than 18.
0 1 2 3 4 5 6 7 8 9 10 11 1213 14 15 16 17 18
d
All the integers that are 3 or 7 whole numbers away from -2.
-9 -8
-7
-6
-5
-4
-3
-2
-10123456789
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How does it work?
Directed Numbers
Addition and subtraction using a number line
Starting from zero each time, calculations involving directed numbers can be made using a number line.
Negative direction
Positive direction
Remember
Always start from 0 and move using each directed number one at a time.
Use a number line to calculate these:
(i)
8+4
+4
-15-14-13-12-11-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 101112 131415
Start
Starting at +8, 4 to the right finishes at 12 right (+12)
Finish
` 8 + 4 = 12
(ii) -6 - 7
-7
-18-17-16-15-14-13-12-11-10 -9-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 1112
Finish
Start
Starting at -6, then 7 to the left finishes at 13 left (-13)
` -6 - 7 =-13
(iii) 3 + (-2)
(iv) (-2) + 3
Negative numbers are often written inside parentheses
-2
-3
-2
-101 2 3
Finish
+3
-3
Start
Starting at +3 then 2 to the left finishes at 1 right (+1)
-2
-101 2 3
Start
Finish
Starting at -2, 3 to the right finishes at 1 right (+1)
` 3 + (-2) = 1
(v) 10 - 26+4
Same as (iii) but in different order
` (-2) + 3 = 1
-26
+4
-18-17-16-15-14-13-12-11-10 -9-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 1112
Finish
Starting at +10, then 26 to the left followed by 4 to the right finishes at 12 left (-12)
` 10 - 26 + 4 =-12
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Start
How does it work?
Your Turn
Directed Numbers
Addition and subtraction using a number line
1
Show the calculation for each of these on a number line and write down the answer.
a
3+5 =
-9 -8
b
-4
-3
-2
-10123456789
-7
-6
-5
-4
-3
-2
-10123456789
-7
-6
-5
-4
-3
-2
-10123456789
-7
-6
-5
-4
-3
-2
-10123456789
7 - 15 =
-9 -8
e
-5
(-3) - 1 =
-9 -8
d
-6
(-4) + 8 =
-9 -8
c
-7
14 - 6 + 3 =
012345678910
11
12
13
14
1516
1718
2
Write down the calculation to get the given answer shown on each of these number lines:
a
=-12
-15 -14 -13 -12 -11 -10 -9
b
-8
-7
-6
-5
-4
-3
-2
-1 01 2 3
-3
-2
-10123456789
-3
-2
-10123456789
=-3
-9 -8
-7
-6
c
-5
-4
=4
-9 -8
-7
-6
-5
-4
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How does it work?
Your Turn
Directed Numbers
Addition and subtraction using a number line
3
Show the calculation for each of these on a number line and state the answer.
a
1.5 + 2 1 =
2
-9 -8
b
-7
-6
-4
-3
-2
-10123456789
Start
Finish
-8
-7
-6
-5
-4
-3
-2
-10 1 2 3 4 56
-6
-5
-4
-3
-2
-10123456789
-6
-5
-4
-3
-2
-10123456789
-4 + 10.5 =
-9 -8
d
-5
(-2.5) - 4.5 =
-12 -11 -10 -9
c
+2 1
2
4
-7
8 1 - 5.5 =
2
-9 -8
-7
Write down the calculation shown on each of these number lines:
a
..../..
*
=2
-8
-7
-6
-5
b
-3
-2
-10 12 34
..
=-3.5
-9 -8
-7
-6
-5
c
-4
-3
-2
-10123456789
-4
-3
-2
-10123456789
-4
-3
-2
-10123456789
=
-9
-8
-7
-6
-5
d
=
-9
16
-4
.../20.
MBER LIN
NU
E
-9
D SUBTRA
AN
C
USING
ON
A
TI
4
ADDITIO
N
Try these trickier ones!
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-7
-6
-5
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Where does it work?
Directed Numbers
The mysterious Dr Thermos!
Dr Thermos controls the temperature of his laboratory using large hot or cold spheres.
+ 1cC
Hot Sphere
- 1cC
Cold Sphere
Count the number of each sphere type to find the overall temperature of his laboratory.
Calculate the temperature of the laboratory for these sphere combinations:
(i)
7 hot spheres and 3 cold spheres
` + 7cC
and
- 3cC
` laboratory is + 4cC
(ii)
+7 and -3 combine to make +4
2 hot spheres and 3 cold spheres
` + 2cC
and
- 3cC
` laboratory is - 1cC
+2 and -3 combine to make -1
To change the temperature, Dr Thermos adds or removes spheres.
Show two ways Dr Thermos can raise the temperature by + 2cC for the combination of spheres below.
9 hot spheres and 6 cold spheres
` + 9cC
and
- 6cC
` laboratory is currently + 3cC
+9 and -6 combine to make +3
(i) Dr Thermos could add two hot spheres
9 hot spheres +2 hot spheres and 6 cold spheres
` + 11cC
and
- 6cC
` laboratory is now + 5cC
` a change of + 2cC
+11 and -6 combine to make +5
(ii) Dr Thermos could remove two cold spheres
9 hot spheres and 6 cold spheres -2 cold spheres
` + 9cC
and
- 4cC
` laboratory is now + 5cC
` a change of + 2cC
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+9 and -4 combine to make +5
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Your Turn
Directed Numbers
1
..../..
Write down the temperature of Dr Thermos’ laboratory for these sphere combinations:
a
.../20.
..
b
Laboratory temperature:
cC
c
Laboratory temperature:
cC
Laboratory temperature:
cC
d
Laboratory temperature:
2
! THE MYST
E
- 1cC
Cold Sphere
OS
+ 1cC
Hot Sphere
cC
Write the number and type of hot or cold spheres that would need to be added to the laboratory to
cause these changes in temperature.
a
+ 3cC
b
- 9cC
d
+ 20cC
3 hot spheres
c
3
18
E
! TH MYST
E
The mysterious Dr Thermos!
OS
S DR.THER
OU
M
I
R
S DR.THE
OU
RM
RI
Where does it work?
- 6cC
Write the number and type of hot or cold spheres that would need to be removed from the
laboratory to cause these changes in temperature.
a
- 7cC
b
+ 2cC
c
- 5cC
d
+ 8cC
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Your Turn
Directed Numbers
The mysterious Dr Thermos!
4
For each of these combinations of hot and cold spheres, explain two different ways Dr Thermos can
change the temperature of the laboratory to equal the amount given in the square brackets.
a
[+4cC]
(i)
First way:
(ii) Second way:
b
[-2cC]
(i)
First way:
(ii) Second way:
Try these trickier ones!
5
If Dr Thermos could only use the spare spheres shown below each given combination, explain
two different ways he could change the temperature of the laboratory to equal the amount in the
square brackets.
You must use all of the spare spheres for at least one of the methods
a
[+6cC]
(i)
First way:
(ii) Second way:
Spare spheres:
b
Using all the spare spheres
[-1cC]
(i)
First way:
(ii) Second way:
Spare spheres:
Using all the spare spheres
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Where does it work?
Your Turn
Directed Numbers
The mysterious Dr Thermos!
Puzzle time!
6
Dr Thermos has a puzzle room at the entrance to his laboratory to keep his experiments secret. The
door will open when the display in the puzzle room reads -2cC . The display currently reads + 3cC .
• Each hot (+1cC ) or cold (-1cC ) step can be used only once.
• Only sideways (
) or forward ( ) steps can be taken.
• Steps in the white section are switched on (added to the puzzle room).
• Steps in the grey section are switched off (removed from the puzzle room).
(i) Draw a pathway that will allow you to enter the laboratory of Dr Thermos.
(ii) See if you can find another path!
Finish anywhere along here
Start anywhere along here
20
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Directed Numbers
Adding and subtracting directed numbers
The exercises in Dr Thermos’ laboratory were examples of adding and subtracting directed numbers.
Here is what happened:
Addition
subtraction
=+1c C
=-1c C
When a
was added, the temperature went up
When a
was added, the temperature went down
When a
was removed (Subtracted), the temperature went down
When a
was removed (Subtracted), the temperature went up
The rules for adding and subtracting represent the same thing.
Use the pattern above to calculate these:
= +1 and
Addition
Subtraction
= -1
5+
= 5 + (+1) = 5 + 1 = 6
+ + = + (Same signs together means plus)
5+
= 5 + (-1) = 5 - 1 = 4
+ - = - (Opposite signs together means minus)
5-
= 5 - (+1) = 5 - 1 = 4
- + = - (Opposite signs together means minus)
5-
= 5 - (-1) = 5 + 1 = 6
- - = + (Same signs together means plus)
These rules work for all kinds of directed additions and subtractions
Calculate these additions and subtractions:
(i)
3 + (- 2) = 3 - 2
=1
+ -=-
(ii) (-2) + 3 =-2 + 3 Same as (i) but in a different order
=1
(iii) 6 - (- 2.5) = 6 + 2.5
= 8.5
- -=+
(iv) (-11) - (-4) =-11 + 4
=- 7
- -=+
(v) 2 1 + (- 3) = 2 1 - 3 + - = 2
2
=- 1
2
(vi) (-12.2) + (-5.2) =-12.2 - 5.2 + - = =-17.4
(vii) 9.5 - (- 8.5) + 6 = 9.5 + 8.5 + 6
= 24
(viii) 4 1 + ^-1h - `-6 2 j = 4 1 - 1 + 6 2
3
3
3
3
= 10
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Directed Numbers
ADDING
A
Adding and subtracting directed numbers
1
2
22
a
-5 + 3
b
-4 + 7
c
-4 - 4
d
-11 - 8
e
1 - 16
f
16 - 22
g
6 + (-3)
h
30 + (-17)
i
0 + (-8)
j
-4 + (-5)
k
-10 + (-19)
l
13 - (-5)
m
20 - (-21)
n
-18 - (-6)
o
-15 - (-26)
a
4-8+2
b
-9 + 6 - 8
c
-3 + 4 - 8 - 7
d
- 7 + (- 4) + 13
e
2 - (-6) - 7
f
5 + (-1) - (-14)
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..../.....
*
Calculate these additions and subtractions of integers without a calculator.
SUBTRAC
/20...
DIREC
NG
T
TI
ND
NUMBERS
Your Turn
ED
Where does it work?
Where does it work?
Your Turn
Directed Numbers
Adding and subtracting directed numbers
Simplify these additions and subtractions:
Use the number line on the side if it helps.
-12 -11 -10 -9
3
5.5 - 8
b
-3 + 2.5
c
-2 - 7.5
d
-6 1 - 19
2
e
9.5 - 16
f
7.5 + (-18)
g
0.5 + (-5.5)
h
-12 1 - (-3 1 )
2
2
i
0 - (+8) - (-6)
a
-2 - 7 + 3.5
b
1 + 0.5 - 6
c
2 + (-3.5) + 1.5
d
- 7 1 + ^-9h - `-2 1 j
2
2
e
-8.5 - `-6 1 j + 18
2
f
-4 + ^-4h - `-4 1 j - 4 1
4
4
-8
a
-7
-6
-5
-4
-3
-2
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-1 0 1 2 3 4 5 6 7 8 9 1011 12
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Where does it work?
Your Turn
Directed Numbers
Adding and subtracting directed numbers
For trickier questions, the calculator can be used.
Use the ( – ) button to change between positive and negative when entering numbers into the calculator.
Sometimes the
button may look like
Simplify using the rules
-29.3 - (-42.9) + (-19.6) =-29.3 + 42.9 - 19.6
( – ) 2
.
9
3
+
4
.
2
9
–
1
9
.
6
=
Enter into calculator
=-6
Use your calculator (only if you need to) for these:
5
24
a
53 - 89
b
-43 + 94
c
34 + (-51)
d
- 25 + (-94)
e
- 94 - (-28)
f
-16 + (-87) - (-41)
g
-2.6 + 10.4
h
-0.5 - 6.7
i
-10.3 + (-5.6)
j
66 - 34.6 + (-24.4)
k
1.06 + 4.5 - 9.7
l
-0.23 + 1.76
m
(- 71.23) + (-52.38)
n
-1.204 - (-5.093)
o
98.23 - (-13.8) -112.3
p
-29 1 + 51
3
q
-30 1 - `-66 1 j
3
2
r
100 + `-54 1 j - 46 1
2
4
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Directed Numbers
Multiplying and dividing directed numbers
The rules for multiplying and dividing are similar to adding and subtracting:
Multiplication
Division
Positive # Positive # Negative # Negative # Positive = Negative = Positive = Negative = Positive
Negative
Negative
Positive
+
+
-
#
#
#
#
+=+
-=+=-=+
(Same signs mean answer is positive)
(Opposite signs mean answer is negative)
(Opposite signs mean answer is negative)
(Same signs mean answer is positive)
The exact same rules apply because division is the opposite operation to multiplication!
These rules always apply to all multiplication and division calculations.
Calculate these multiplications and divisions:
-4 # 3 =-12
-
#
+=-
(ii) -14 ' 2 =- 7
-'+=-
(iii) 12 # (-1.5) =-18
+
#
-=-
(iv) 24 ' (-4.8) =-5
+'-=-
(v) -8 # (-3) = 24
-
#
-=+
(vi) -46.8 ' (-18) = 2.6
-'-=+
(vii) -3 1
2
-
#
+=-
(viii) -32 ' `-3 1 j = 10
5
-'-=+
(i)
#6
=-21
Be careful squaring (or cubing) directed numbers as parentheses affect the question.
2
(x) -3 =-(3 # 3) =-9
(ix) (-3) 2 = (-3) # (-3) = 9
If more than two terms are multiplied or divided, simplify by calculating in order from left to right.
Simplify these mixed questions:
(i)
6 # (-3) ' 9
(ii) (-2) ' 0.5 # (-3)
(iii) (-1) 3
(iv) 8 4 # 5 ' (-2) ' (-11)
5
=-18 ' 9
=-2
Calculate 6 # (-3) first
=-4 # (-3)
= 12
Calculate (-2) ' 0.5 first
+
#
-=-
-'+=-
-
= (-1) # (-1) # (-1)
= 1 # (-1)
=-1
= 44 ' (-2) ' (-11)
=-22 ' (-11)
=2
-=+
Calculate (-1) # (-1) first
-'-=+
Calculate 1 # (-1) next
+'-=-
Calculate 8 4 # 5 first
5
Calculate 44 ' (-2) next
+
#
+=+
+'-=-
-'-=+
Directed Numbers
Mathletics Passport
#
-'+=-
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Where does it work?
Your Turn
Directed Numbers
Multiplying and dividing directed numbers
1
2
26
Simplify these without using a calculator:
a
3 # (- 7)
b
-6 # 8
c
-36 ' 12
d
48 ' (-4)
e
-16 # (-1)
f
-2 # (-2)
g
-12 ' (-12)
h
21 ' (-21)
i
3 # (-4) ' 6
Simplify these without using a calculator:
a
-24 ' 3 ' (-8)
b
-18 ' (-6) # 4
c
-5 # 2 # (-1) # 2
d
-5 2
e
(-4) 2
f
(-3) 3
g
-23
h
(-2) 2 ' 4
i
(-3) 2 # -3
j
63 ' (-2)
k
24 '-42
l
92 '-32 # (3)
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Directed Numbers
TI
MUL PLYI
3
4
Simplify these using a calculator:
a
27 # (-4)
b
-21 # 13
c
d
-95 ' (-19)
e
125 # (-49)
f
g
-53 # 16
h
(-391) ' (-23)
i
-4 # 29
..../...../20...
*
Multiplying and dividing directed numbers
AND DIVI
DIREC
NG
TE
DI
NG
NUMBERS
Your Turn
D
Where does it work?
-162 ' 18
-15 # 15 ' (-3)
Use your calculator to simplify these:
a
-25 # 13 ' (-65)
b
-500 ' (-5) ' 25
c
-552 ' (-23) # (-8)
d
21
2
e
12.5 # 25 ' (-0.25)
f
-1 2 ' (-14) # 5 # (-2)
5
g
-4 # 3.6 # 22 ' 7.92
h
-3404.7 ' (-3.6) ' 1 1 ' 9.7
2
i
260 ' (-26) # 2 3 ' 0.26
5
j
(- 7) 3
k
-210
l
(-8) 3
o
-(-1.5) 5
# (- 6) ' 5
-1024
- 343
m
(-9) 4
n
-(-6) 3
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What else can you do?
Directed Numbers
Combining the basic operations
Questions that mix multiplying/dividing with adding/subtracting need to be done in a certain order.
Multiplication or division operations are ranked the highest and therefore must be completed first.
Calculate these combined operations questions:
(i)
-2 + 3 # 5
-2 + 3 # 5 =-2 + 15
Do this first
= 13
Add -2 and 15
35 ' (-5) - 1 =- 7 - 1
Do this first
=-8
Subtract 1 from -7
(ii) 35 ' (-5) - 1
If more than one multiply/divide sign, work left to right. Do the same for addition/subtraction.
Calculate these combined operations questions:
(i)
-28 ' 7 # 3 + 6
-28 ' 7 # 3 + 6 =-4 # 3 + 6
Do this first
=-12 + 6
=-6
Multiply -12 by 3
Add -12 and 6
(ii) 17 - 4 -(-10) # 2
17 - 4 -(-10) # 2 = 17 - 4 -(-20)
Do this first
= 13 - (-20)
= 33
Subtract 4 from 17
Subtract -20 from 13 ( = 13 + 20 )
(iii) 24 ' 3 + (-5) # 4
24 ' 3 + (-5) # 4 = 8 + (-5) # 4
Do this first
= 8 + (-20)
=-12
Multiply -5 by 4
Add 8 and -20
(iv) 6 # (-3) - 210 ' 30 # (-2)
6 # (-3) - 210 ' 30 # (-2) =-18 - 210 ' 30 # (-2)
Divide -210 by 30
Do this first
=-18 - 7 # (-2)
Multiply -7 by -2
=-18 + 14
Add -18 and 14
=-4
28
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Your Turn
Directed Numbers
2
3
Simplify these combined operations without a calculator:
a
- 4 + 16 ' 8
b
6 # (-8) -12
c
50 ' (-25) + 10
d
-8 - 5 # 4
e
5 + 7 # (-6)
f
-11 # -2 + 18
g
22 - 39 ' (-3)
h
-10 - 4 # 15
..../...../20...
*
1
SIC OP
BA
Combining the basic operations
COMBIN
G THE
IN
ATIONS
ER
What else can you do?
Simplify these combined operations without a calculator:
a
4 # (-2) + 18 - 11
b
-3 # 6 ' 9 + 3
c
- 9 + 48 ' (-12) + 13
d
32 ' 4 - 3 # (-4)
Use a calculator to simplify these combined operations:
a
28 ' (-4) + 13 # 7
b
120 ' (-15) # 24 - 8
c
- 25 # 6 - 224 ' (-32)
d
30 # (-15) - 35 ' 4 # (-16)
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What else can you do?
Directed Numbers
Order of operations
For more complex calculations involving parentheses and indices, we use a similar convention:
P
Parentheses
I
Indices
Do these first
D
M
Division
A
Multiplication
S
Addition
Then do these
Subtraction
Do these last
Simplify these directed number calculations using the correct order of operations:
(i)
(12 - 8) # 7
Subtract 8 from 12 inside the parentheses
(12 - 8) # 7 = 4 # 7
= 28
Multiply 4 and 7
(ii) -21 ' (2 + 5) + 3
-21 ' (2 + 5) + 3 =-21 ' 7 + 3
=-3 + 3
=0
Add 2 and 5 inside the parentheses
Divide -21 by 7
Add -3 and 3
(iii) (-4) 2 ' (8 - 6)
(-4) 2 ' (8 - 6) = (-4) 2 ' 2
= 16 ' 2
=8
Subtract 6 from 8 inside the parentheses
Square -4
Divide 16 by 2
(iv) 15 # (-5 + 3) 3 - 20
15 # (-5 + 3) 3 - 20 = 15 # (-2) 3 - 20
= 15 # (-8) - 20
Add -3 and 5 inside the parentheses
Cube -2
Multiply 15 by -8
=-120 - 20
=-140
Subtract 20 from -120
(v) 29 - (13 - 4 # 5) + 32
29 - (13 - 4 # 5) + 32 = 29 - (13 - 20) + 32
= 29 - (- 7) + 32
= 29 - (- 7) + 9
= 36 + 9
= 45
30
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Multiply -4 by 5 inside the parentheses
Subtract 20 from 13 inside the parentheses
Square 3
Subtract -7 from 29
Add 36 and 9
What else can you do?
Your Turn
Directed Numbers
Order of operations
1
2
Simplify these directed number calculations using the correct order of operations without a calculator:
a
(16 - 10) # 4
b
34 ' (6 + (- 8))
c
-5 # (27 ' 9)
d
(-13 - (-19)) # 3
e
-5 # (14 - 9) - 5
f
4 # (36 ' 2) - (-4)
g
18 + (16 ' 4 + 10)
h
(18 ' 2 - (-4)) - 7
Simplify these calculations containing indices using the correct order of operations without a calculator:
a
(-2) 2 ' (-2)
b
(- 7 - 5) # (-1) 3
c
(8 - 14) 2 - 16
d
(33 - 43) ' 8
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Your Turn
Directed Numbers
TI
ERA ONS
P
O
..../...../20...
a
26 ' (14 + 2 # 6) + (-1) 2
b
-100 # (12 - 32 ' (-4)) ' 40
c
(11 # 23 - 10) ' 39 + 8
d
9 # ((-8) - 8 ' 4 + 16) - 43
e
200 ' (-4) # (-3) ' (16 - 14 # (-1) 3)
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ERA ONS
OP
32
Try simplify these trickier ones using the correct order of operations:
ER OF
ORD
3
*
Order of operations
ER OF
ORD
What else can you do?
Directed Numbers
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What else can you do?
Directed Numbers
Order of operations with grouping symbols
For questions containing multiple pairs of grouping symbols, complete each pair separately.
Grouping symbol names:
( ) = parentheses
[ ] = square brackets
{ } = braces
Simplify these combined operations questions containing multiple brackets:
(i)
(10 # 2) ' (- 7 - 3)
Simplify the parentheses
(10 # 2) ' (- 7 - 3) = 20 ' (-10)
=-2
Divide 20 by -10
10 # 2 = (10 # 2) ' (- 7 - 3)
-7 - 3
The same question
can be written as a
fraction.
(ii)
numerator = (numerator) ' (denominator)
denominator
34 - 3 # 4
22 ' (-2)
34 - 3 # 4 = (34 - 3 # 4) ' (22 ' (-2))
22 ' (-2)
= 22 ' (-11)
=-2
Re-write with parentheses
Simplify the parentheses
Divide 22 by -11
When questions contain brackets within other brackets, work from the inside to the outside.
Simplify these combined operations questions containing multiple brackets:
(i)
3 # 6(1 - 6) ' 5 @
3 # 6(1 - 6) ' 5 @ = 3 # 6- 5 ' 5 @
= 3 # -1
=-3
Simplify the parentheses first
Simplify the square brackets
Multiply 3 by -1
(ii) 30 - "18 - 626 ' (2 2 - 17) @,
30 - "18 - 626 ' (2 2 - 17) @, = 30 - "18 + 626 ' (4 - 17) @,
Simplify the parentheses
= 30 - "18 + 626 ' (-13) @,
= 30 - "18 + - 2 ,
Simplify the square brackets
= 30 - 16
Simplify the braces
= 14
Subtract 16 from 30
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Directed Numbers
Simplify these questions:
a
(16 + 9) + (14 - 7)
b
(1 - 9) - (3 - 8)
c
(7 - 11) # (4 + 2)
d
(18 - 3) ' (-12 - (- 7))
e
46 - 19
5-2
f
-2 # 10
48 ' 12
h
8 # 5 - 12
42 ' (-6)
= (46 - 19) ' (5 - 2)
= 27 ' 3
=9
g
2
34
(50 + 54 ' 9) ' (-24 ' 3)
Simplify these questions with multiple grouping symbols:
a
-12 ' 6(7 + 11) ' 9 @
b
610 + (-14 ' 2) @ # 8
c
6-6 # (13 - (- 7)) @ ' 4
d
5 # 6(22 - 9) # 3 @
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UP
GRO ING
1
H
Order of operations with grouping symbols
NS WI
TIO
T
A
R
OF OP
ER
E
D
R
Your Turn
20...
/
..../.....
LS *
MBO
O
SY
What else can you do?
What else can you do?
Your Turn
Directed Numbers
Order of operations with grouping symbols
a
{[12 + (6 # 8)] ' 3} -25
b
(-2) 3 + {- 2 + [2 # (2 + 2)]}
e?
er m
Rememb
Give these three tricky ones a go to earn an awesome stamp.
(14 - 8) # (-1)
15 - (-3)
* AWE
3
ME *
SO
a
2
E*
OM
4
Simplify these:
/20...
..../.....
* AWES
3
b
-4 # {- 72 + (-6) 2 ' [(4 # (-5)) - (-2)]}
c
-3 # {[(-3 - (-3)) 2 + (-3) 2] ' 3 # (-3)}
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Cheat Sheet
Directed Numbers
Here is a summary of the important things to remember for directed numbers
Types of numbers
• Positive and negative numbers are numbers that have both size and direction.
Eg: +2 or -3
• Rational numbers can be written as a fraction. They include whole numbers and terminating or
recurring decimals. Eg: 8, 2.3, 4.5o , 1 , 4 .
2
• Irrational numbers cannot be written as a fraction. These include decimals that keep going
without following a pattern, and roots of numbers that are not perfect squares/cubes etc.
Eg: 3 , 1.2938164928..., r.
• Integers are whole numbers only. They are rational and have no decimal or fraction parts.
Eg: 3, -4, 100.
Magnitude and direction of numbers
• Directed numbers are positive (+) or negative (-) whole numbers.
• They show both the size (magnitude) and direction of values associated with change.
• It is important to first decide what will be the positive direction.
Ascending and descending order
Ascending order = lowest to highest
Descending order = highest to lowest
Adding and subtracting directed numbers
Addition
Subtraction
+
+
-
+ = +
- = -
+ = -
- = +
(Same signs together means plus)
(Opposite signs together means minus)
(Opposite signs together means minus)
(Same signs together means plus)
Multiplying and dividing direct numbers
Positive # Positive # Negative # Negative # Multiplication
Division
Positive = Negative = Positive = Negative = Positive
Negative
Negative
Positive
+
+
-
#
#
#
#
+=+
-=+=-=+
(Same signs mean answer is positive)
(Opposite signs mean answer is negative)
(Opposite signs mean answer is negative)
(Same signs mean answer is positive)
The exact same rules apply because division is the opposite operation to multiplication!
Order of operations
The word PIDMAS helps to remember the order in which we need to perform calculations.
P
Parentheses
Indices
Do these first
36
H
4
SERIES
TOPIC
I
D
Division
M
Multiplication
Then do these
Directed Numbers
Mathletics Passport
© 3P Learning
A
S
Addition
Subtraction
Do these last
Directed Numbers
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