directed numbers directed numbers directed numbers...
TRANSCRIPT
PASSPORT
PASSPORT
Dire
cted
Num
bers
DIRECTED NUMBERS DIRECTED NUMBERSDIRECTED NUMBERS
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4HSERIES TOPIC
1Directed Numbers
Mathletics Passport © 3P Learning
Work through the book for a great way to solve this
Numbers come in all sizes and forms. They can be positive or negative, whole numbers, fractions or decimals and rational or irrational.
Positive and negative numbers:
Whole numbers, fractions and decimals:
Rational and irrational numbers:
Give this a go!
Calculate the value of: ( 2) 2 2 (2 2)[ ]#- - + +{ }3+Q
Before you start, investigate these terms and write a brief description about them in the boxes.
2 Directed Numbers
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4HSERIES TOPIC
Directed NumbersHow does it work?
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)
Your Turn
1-
21
4
5
36
0.3- o
4
45
8-
13
.0 42
0
2-
.5 0-
7
510-
421
9-
.9 86
303
10
1.19284750219...
31-
83
16-
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.
Natural numbers: the counting rational numbers (not including 0).
-1 Put the special types of rational numbers into their correct group
4HSERIES TOPIC
3Directed 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 C20c , temperature changed C4c+ then C9c-
(ii) What directed number gives the overall change in temperature between the two readings?
Overall change C C
C
4 9
5
c c
c
=+ -
=-
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: g g g575 98 46+ - + Increase = + and decrease = -
` Maria purchased g g g g575 98 46 523+ - + = of olives
rose =+ and falling =-
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 C20c
Temperature by late afternoon C15c
C4c+
C9c-
CentigradeoC
Digital scales need to use directed numbers to calculate changes in mass.
If the first term is positive, we usually don't write in the plus sign.
4 Directed Numbers
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4HSERIES TOPIC
How does it work? Directed NumbersYour Turn
Measuring instruments and integers
1
2
Shade in each thermometer below to accurately display these temperatures:
(i) C11c (ii) C5c- (iii) C6c (iv) C0c (v) C8c above zero (vi) C2c below zero
This thermometer shows a reading of C12c at the start.
Write the new readings that would be displayed after each of these changes in temperature:
(i) Increases by C3c
(ii) Becomes C8c cooler
(iii) Decreases by C13c
(iv) Warms up by C9c
(v) Rose by C22c
(vi) Dropped C28c
(vii) Decrease by C7c then increase by C12c
(viii) Warms by C3c and then a further C9c
(ix) Cools C15c before warming by C19c
(x) Changes to C0c before dropping 4 Cc
Cc
Cc
Cc
Cc
Cc
Cc
Cc
Cc
Cc
Cc
Cc
12
20o
15o
10o
5o
0o
-5o
-10o
CentigradeoC
20o
15o
10o
5o
0o
-5o
-10o
CentigradeoC
20o
15o
10o
5o
0o
-5o
-10o
CentigradeoC
20o
15o
10o
5o
0o
-5o
-10o
CentigradeoC
20o
15o
10o
5o
0o
-5o
-10o
CentigradeoC
20o
15o
10o
5o
0o
-5o
-10o
CentigradeoC
30o
25o
20o
15o
10o
5o
0o
-5o
-10o
CentigradeoC
4HSERIES TOPIC
5Directed Numbers
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How does it work? Directed NumbersYour Turn
Measuring instruments and integers
3 The mass of a glass container with some water in it is shown below:
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
(ii) 212 g of water was spilled out of the glass
(iii) 88 g of flavour syrup was stirred in and then 250 g was drunk
(iv) A similar glass with the same amount of water was placed on the scales next to this one.
(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.
Try this tricky one!
A glass container holding cold water below was placed on the digital scales after being moved from a C24c room into a C32c 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?
4
0402 grams
60o
55o
40o
30o
20o
10o
0o
CentigradeoC
C32c
grams
grams
grams
grams
grams
MEASURIN
G INSTRUMENTS AND INTEGERS
* ..../...../20...
0583 grams
6 Directed Numbers
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4HSERIES TOPIC
How does it work? Directed Numbers
+3-1
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.
• 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 (-)
It is important to first decide what will be the positive direction.
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 degree below the zero:
(ii) 30 degree above the zero:
(iii) $12 extra money:
30
30
c=+
=
$
$
12
12
=+
=
5c=-
The positive sign (+) can be hidden
The magnitude of the temperature is 5 degrees
The magnitude of the temperature is 30 degrees
The magnitude of the money is $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: $ $40 $14 $260 + - = left
` the overall change in money for Matt on this day is +$26 The final directed value = overall change
GROW, FORWARD, ABOVE, FURTHER, MORE, FASTER, EXTRA, ASCEND, HI
GHER
, OVER, INCREASE, WARMER,
NORTH
, EAST,
POSITIVE
REVERSE, BELOW, CLOSER, LESS, SLOWER, REDUCE, DESCEND, LOWER, U
NDER,
DECREASE, COOLER, SOUTH,
WEST,
SHRINK,
NEGATIVE
4HSERIES TOPIC
7Directed Numbers
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How does it work? Directed NumbersYour Turn
Pip was given $24 and then spent $15 on a dress the same day.
Directed number for the overall change in money that day:
The temperature initially increased by 6 degrees and then increased by a further 4 degrees.
Directed number for the overall change in temperature:
Nigel went down 6 rungs on a ladder and then up 2 rungs.
Directed number for the overall change in ladder rungs:
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:
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:
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:
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.
1
2
c
Sean’s bank account balance then changed due to being charged $3 in bank fees.
(iii) Directed number: (iv) Magnitude of the change in Sean's bank balance:
Sean’s bank account balance earned interest and increased by $4.
(i) Directed number: (ii) Magnitude of Sean's bank balance increase:
b 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: (iv) Magnitude of this distance walked by Aki:
a The tree in Jamie’s yard grew 3 metres in one year.
(i) Directed number: (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: (iv) Magnitude of the reduction in height:
Think carefully for each of these statements and write down the directed number that matches the overall change.
a
b
c
d
e
f
+$9
MAGNITUD
E AND DIRECTION OF INTEGERS *
..../...../20...
8 Directed Numbers
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4HSERIES TOPIC
How does it work? Directed Numbers
Ascending and descending order
When comparing directed numbers, it often helps to arrange them into numerical order.
Don’t be tricked by the magnitude of a number.
Which of these two values is higher?
C2c C10c-
C10c- represents a very cold temperature (below/less than C0c ) C2c is a cold temperature, however it is (above/greater than C0c )
C2` c is a higher value than C10c-
-10 is a bigger number in magnitude than 2, however the negative sign makes it lower in value
Let’s look at an ascending and descending order example.
(i) Arrange them into ascending order
(ii) Arrange them into descending order
10 , 6 , 4 , 0 , 3 , 5.2 , 5.4 , 14.5- - -
14.5 , 5.4 , 5.2 , 3 , 0 , 4 , 6 , 10- - -
Lowest negative Highest positive
Highest positive Lowest negative
Order is opposite to ascending
For the numbers 4 , 0 , 14.5 , 6 , 5.2 , 3 , 10 , 5.4- - - :
DescendingAscending
Ascending order = lowest to highest Descending order = highest to lowest
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
-12
CentigradeoC
C2c
C10c-
4HSERIES TOPIC
9Directed Numbers
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How does it work? Directed NumbersYour Turn
Ascending and descending order
1
2
3
Circle the word that represents the order of the values in these statements:
Shortest to Tallest Ascending Descending
Longest to shortest Ascending Descending
Closest to farthest Ascending Descending
Warmest to Coolest Ascending Descending
Heaviest to lightest Ascending Descending Thinnest to widest Ascending Descending
Arrange the following groups of numbers into ascending order (lowest to highest).
, , ,
, , , ,
, , , ,
, , , ,
, , , , ,
, , ,
, , , ,
, , , ,
, , , ,
, , , , ,
Arrange the following groups of numbers into descending order (highest to lowest).
a
a
a
b
b
b
c
c
c
d
d
d
e
e
e
f
14 , 0 , 17 , 3
21 , 25 , 20 , 19 , 22
10 , 8 , 12 , 4 , 16- - - - -
2.2 , 2.4 , 2.6 , 0 , 2- -
1 , 11 , , 1.5 , 1 ,51
41- -
2 , 16 , 5 , 8
40 , 31 , 32 , 38 , 29
19 , 11 , 16 , 18 , 13- - - - -
1.6 , 1.9 , 1.3 , 1.0 , 1.2- -
0 , , 2 , 2.1 , 1.2 , 3041
21- - - -
ASCEND
ING AND DESCENDING ORDER * ..../...../20...
10 Directed Numbers
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4HSERIES TOPIC
How does it work? Directed NumbersYour Turn
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:
12th floor
8th floor
Ground floor
2nd floor
Ground floor
15th floor
4th floor
10th floor
5th floor
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.
During the first 15 minutes, did the operator mostly ascend or descend in the elevator? Explain you answer.
Arrange the directed movements into descending order.
, , , , , , , ,
, , , , , , , ,
13 14 15
10 11 12
7 8 9
4 5 6
1 2 3
G >|< >|<
a
b
c
COMBO TIME *
COMBO TIME * COMBO TIME *..../...../
20...
4HSERIES TOPIC
11Directed Numbers
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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.
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
-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 11 12 13 14 15
Lower numbers are further left and higher numbers are further right.
Display the numbers 4, 1, 7- - on a number line.
Display the numbers , , , . , ,1 2 421 5 5 9 0- - on a number line.
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, 1 7` 1-
Read the numbers from right to left ( )
Read the numbers from left to right ( )
This example requests numbers to be plotted using a given rule.
Values increase (ascending)Values decrease (descending)
Display all the even positive integers between -5 and 13 on a number line.
Write these numbers in descending order.
Write these numbers in ascending order.
` The numbers in descending order are: 12 , 10 , 8 , 6 , 4 , 2
` The numbers in ascending order are: 5.5 , 2 , 0 , 1 , 4 , 921- -
Half values are plotted by placing the dot half-way between the integers on either side.
42
1 is half way between 4 and 5-5.5 is half way between -5 and -6
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13
Remember:1 means 'less than' and 2means 'greater than'
Even and positive
12 Directed Numbers
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4HSERIES TOPIC
How does it work? Directed NumbersYour Turn
a
d
g
j
a
b
a
b
b
e
h
k
c
f
i
l
The number line
Insert the correct symbol 1 (less than) or 2 (greater than) for each of these.1
6 4 3 -12 -5 4
8 -8 721- -7
921 9.9 -5.1 -5.8
-12.5 -21.5 1121 11
31
-11 -2
21-
0
.1 8- 1.2
List the numbers displayed below in ascending order
List the numbers displayed below in descending order
2
3
-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 11 12
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
, , , ,
, , , ,
, , , ,
, , , ,
-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 11 12 13 14 15
THE NUMB
ER LINE * THE NUMBER LINE *
..../...../20...
4HSERIES TOPIC
13Directed Numbers
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How does it work? Directed NumbersYour Turn
a
d
a
b
c
d
c
b
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
The number line
Display the following sets of numbers on a number line.4
5
5, 8, 1, 3, 6.- -
8.5, 1.5, 2, 7.5, 5.- -
4 , 4, 6 , , 5.21
21
21- -
7, 7.5, 3 , 0, 2 , 2.21
21-
Display the following sets of numbers on a number line.
The first six odd integers above -8.
All the negative even integers less than 7 but greater than negative 8.
All the multiples of 3 higher than 7 but lower than 18.
All the integers that are 3 or 7 whole numbers away from -2.
0.521 =
14 Directed Numbers
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4HSERIES TOPIC
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.
Positive directionNegative direction
Remember
Use a number line to calculate these:
(i) 8 + 4
Starting at +8, 4 to the right finishes at 12 right (+12)8 4 12` + =
Starting at -6, then 7 to the left finishes at 13 left (-13)16 7 3`- - =-
(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 11 12
(iii) (iv)3 + (-2) (-2) + 3
-2
-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3
Negative numbers are often written inside parentheses
( )3 2 1` + - = ( 2) 3 1` - + =
Same as (iii) but in different order
+3
Starting at +3 then 2 to the left finishes at 1 right (+1) Starting at -2, 3 to the right finishes at 1 right (+1)
(v) 10 - 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 11 12
10 26 4 12` - + =-
-26
+4
Starting at +10, then 26 to the left followed by 4 to the right finishes at 12 left (-12)
Always start from 0 and move using each directed number one at a time.
-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 11 12 13 14 15
+4
Start
Start
Start
Start
Start
Finish
Finish
Finish Finish
Finish
4HSERIES TOPIC
15Directed Numbers
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How does it work? Directed NumbersYour Turn
1
2
Addition and subtraction using a number line
Show the calculation for each of these on a number line and write down the answer.
a
a
b
c
b
c
d
e
3 5+ =
12=-
3=-
4=
( )4 8- + =
( )3 1- - =
7 15- =
14 6 3- + =
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Write down the calculation to get the given answer shown on each of these number lines:
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
-15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3
16 Directed Numbers
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4HSERIES TOPIC
How does it work? Directed NumbersYour Turn
Addition and subtraction using a number line
Show the calculation for each of these on a number line and state the answer.3
4
a
a
b
c
d
1.5 221+ =
2=
.4 10 5- + =
.821 5 5- =
4
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
Write down the calculation shown on each of these number lines:
Try these trickier ones!
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
( . ) .2 5 4 5- - =
221+
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
c =
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
ADDITION
AND SUBTRACTION USING A NUMBER LIN
E *
..../...../20...
b
.3 5=-
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
d =
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
Start Finish
4HSERIES TOPIC
17Directed Numbers
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Directed Numbers
The mysterious Dr Thermos!
Dr Thermos controls the temperature of his laboratory using large hot or cold spheres.
Count the number of each sphere type to find the overall temperature of his laboratory.
Hot Sphere Cold SphereC1c+ C1c-
Calculate the temperature of the laboratory for these sphere combinations:
(i)
(ii)
7 hot spheres and 3 cold spheres
9 hot spheres and 6 cold spheres
9 hot spheres +2 hot spheres and 6 cold spheres
9 hot spheres and 6 cold spheres -2 cold spheres
2 hot spheres and 3 cold spheres
C7` c+
C2` c+
C9` c+
C11` c+
C9` c+
C3c-
C3c-
C6c-
C6c-
C4c-
and
and
and
and
and
` laboratory is C4c+
` laboratory is C1c-
` laboratory is currently C3c+
` laboratory is now C5c+
` laboratory is now C5c+
` a change of C2c+
` a change of C2c+
+7 and -3 combine to make +4
+2 and -3 combine to make -1
+9 and -6 combine to make +3
+11 and -6 combine to make +5
+9 and -4 combine to make +5
To change the temperature, Dr Thermos adds or removes spheres.
Show two ways Dr Thermos can raise the temperature by C2c+ for the combination of spheres below.
(i) Dr Thermos could add two hot spheres
(ii) Dr Thermos could remove two cold spheres
Where does it work?
18 Directed Numbers
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4HSERIES TOPIC
Where does it work? Directed NumbersYour Turn
The mysterious Dr Thermos!
Hot Sphere Cold SphereC1c+ C1c-
Write down the temperature of Dr Thermos’ laboratory for these sphere combinations:1
a
c
b
d
Laboratory temperature: Cc
Laboratory temperature: Cc
Laboratory temperature: Cc
Laboratory temperature: Cc
2
3
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
a
b
b
c
c
d
d
C3c+
C7c-
C9c-
C2c+
C6c-
C5c-
C20c+
C8c+
3 hot spheres
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.
THE MYSTER
IOUS DR.THERMOS ! THE MYSTERIOUS DR.THE
RM
OS !..../...../20...
4HSERIES TOPIC
19Directed Numbers
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Where does it work? Directed NumbersYour Turn
The mysterious Dr Thermos!
4
5
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
b
a
b
C[ 4 ]c+
C[ ]2c-
C[ ]6c+
C[ ]1c-
First way:
First way:
First way:
First way:
Second way:
Second way:
Second way:
Second way:
(i)
(i)
(i)
(i)
(ii)
(ii)
(ii)
(ii)
Try these trickier ones!
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
Using all the spare spheres
Using all the spare spheres
Spare spheres:
Spare spheres:
20 Directed Numbers
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4HSERIES TOPIC
Where does it work? Directed NumbersYour Turn
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 C2c- . The display currently reads C3c+ .
• Each hot ( C1c+ ) or cold ( C1c- ) 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
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21Directed Numbers
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Where does it work? Directed Numbers
. ( . ) . .9 5 8 5 6 9 5 8 5 6
24
- - + = + +
=
Adding and subtracting directed numbers
The exercises in Dr Thermos’ laboratory were examples of adding and subtracting directed numbers.
Here is what happened:C1c=+ C1c=-
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
5 + = 5 + (+1) = 5 + 1 = 6
5 + = 5 + (-1) = 5 - 1 = 4
5 - = 5 - (+1) = 5 - 1 = 4
5 - = 5 - (-1) = 5 + 1 = 6
Addition
Subtraction
Addition
subtraction
The rules for adding and subtracting represent the same thing.
Use the pattern above to calculate these:
+ + = + (Same signs together means plus)
+ - = - (Opposite signs together means minus)
+ - = -
- - = + - - = +
+ - = - + - = -
- + = - (Opposite signs together means minus)
- - = + (Same signs together means plus)
= +1 and = -1
These rules work for all kinds of directed additions and subtractions
Calculate these additions and subtractions:
(i)
(iii)
(v)
(vii)
(ii)
(iv)
(vi)
(viii)
3 ( 2) 3 2
1
+ - = -
=
( . ) .
.
6 2 5 6 2 5
8 5
- - = +
=
( )221 3 2
21 3
21
+ - = -
=-
( 2) 3 2 3
1
- + =- +
=
( 11) ( 4) 11 4
7
- - - =- +
=-
( 12.2) ( 5.2) 12.2 5.2
17.4
- + - =- -
=-
4 4 1 631 1 6
32
31
32
10
+ - - - = - +
=
^ `h j
Same as (i) but in a different order
22 Directed Numbers
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Where does it work? Directed NumbersYour Turn
Adding and subtracting directed numbers
Calculate these additions and subtractions of integers without a calculator.
1
2
a
a
c
e
b
b
d
f
e
h
k
n
c
f
i
l
o
d
g
j
m
5 3- + 4 7- +
1 16-
30 ( 17)+ -
10 ( 19)- + -
18 ( 6)- - -
4 4- -
16 22-
( 8)0 + -
13 ( )5- -
( )15 26- - -
9 6 8- + -
7 ( 4) 13- + - +
( ) ( )5 1 14+ - - -
11 8- -
6 ( 3)+ -
4 ( 5)- + -
( )20 21- -
4 8 2- +
3 4 8 7- + - -
( )2 6 7- - -
ADDING A
ND SUBTRACTING DIRECTED NUMBERS
*
..../...../20...
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23Directed Numbers
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Where does it work? Directed NumbersYour Turn
Adding and subtracting directed numbers
Simplify these additions and subtractions:Use the number line on the side if it helps.
3
4
a
d
g
a
c
e
b
d
f
b
e
h
c
f
i
.5 5 8-
621 19- -
. ( . )0 5 5 5+ -
.2 7 3 5- - +
( . ) .2 3 5 1 5+ - +
.8 5 621 18- - - +` j
.1 0 5 6+ -
721 9 2
21- + - - -^ `h j
4 4 441 4
41- + - - - -^ `h j
.3 2 5- +
.9 5 16-
12 ( 3 )21
21- - -
.2 7 5- -
. ( )7 5 18+ -
( ) ( )0 8 6- + - -
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
24 Directed Numbers
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Where does it work? Directed NumbersYour Turn
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
2 3. 2 –( – ) .9 + 4 . 9 1 69 =
Simplify using the rules
Enter into calculator
. ( . ) ( . ) . . .29 3 42 9 19 6 29 3 42 9 19 6- - - + - =- + -
6=-
Use your calculator (only if you need to) for these:
5 a
d
g
j
m
p
b
e
h
k
n
q
c
f
i
l
o
r
53 89-
( )25 94- + -
. .2 6 10 4- +
. ( . )66 34 6 24 4- + -
( . ) ( . )71 23 52 38- + -
2931 51- +
43 94- +
( )94 28- - -
. .0 5 6 7- -
. . .1 06 4 5 9 7+ -
. ( . )1 204 5 093- - -
30 6621
31- - -` j
( )34 51+ -
( ) ( )16 87 41- + - - -
. ( . )10 3 5 6- + -
. .0 23 1 76- +
. ( . ) .98 23 13 8 112 3- - -
100 5421 46
41+ - -` j
4HSERIES TOPIC
25Directed Numbers
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Where does it work? Directed Numbers
Multiplying and dividing directed numbers
The rules for multiplying and dividing are similar to adding and subtracting:
Positive # Positive = PositivePositive # Negative = NegativeNegative # Positive = NegativeNegative # 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)
Multiplication
Division 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:
(i)
(i)
(ii)
(iii)
(iv)
(i)
(iii) (iv)
(v) (vi)
(vii)
(ix)
(viii)
(x)
4 3 12#- =-
6 ( 3) 9# '-
( 2) 0.5 ( )3#'- -
( )1 3-
8 5 ( 2) ( )54 11# ' '- -
14 2 7'- =-
12 ( 1.5) 18# - =- ( . )24 4 8 5' - =-
8 ( 3) 24#- - = 46.8 ( 18) 2.6'- - =
3 6 2121#- =-
( 3) ( 3) ( 3) 92#- = - - =
32 351 10'- - =` j
3 (3 3) 92#- =- =-
Be careful squaring (or cubing) directed numbers as parentheses affect the question.
If more than two terms are multiplied or divided, simplify by calculating in order from left to right.
Simplify these mixed questions:
18 9
2
'=-
=-
( )4 3
12
#=- -
=
( 1) ( 1) ( 1)
1 ( 1)
1
# #
#
= - - -
= -
=-
44 ( 2) ( 11)
22 ( 11)
2
' '
'
= - -
=- -
=
Calculate 6 ( 3)# - first
Calculate ( ) .2 0 5'- first
Calculate ( 1) ( 1)#- - first
Calculate ( 1)1 # - next
Calculate 8 554# first
Calculate 44 ( 2)' - next
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Where does it work? Directed NumbersYour Turn
Multiplying and dividing directed numbers
1 Simplify these without using a calculator:
a
d
g
b
b
e
e
h
h
k
c
c
f
f
i
i
l
3 ( 7)# -
( 4)48 ' -
( )12 12'- -
6 8#-
18 ( 6) 4#'- -
16 ( 1)#- -
( )4 2-
21 ( 21)' -
( )2 42 '-
2 44 2'-
36 12'-
5 2 ( 1) 2# # #- -
2 ( 2)#- -
( 3)3-
3 ( 4) 6# '-
( 3) 32#- -
9 3 (3)2 2#'-
Simplify these without using a calculator:
a
d
g
j
( )24 3 8' '- -
52-
23-
( )6 23 ' -
2
4HSERIES TOPIC
27Directed Numbers
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Where does it work? Directed NumbersYour Turn
a
d
g
b
e
h
k
n
c
c
f
f
i
i
l
o
27 ( 4)# -
( )95 19'- -
53 16#-
( )500 5 25' '- -
12.5 25 ( 0.25)# ' -
b
e
h
21 13#-
125 ( 49)# -
( ) ( )391 23'- -
. ( . ) .3404 7 3 6 121 9 7' ' '- -
210-
( )6 3- -
4 29#-
552 ( 23) ( 8)#'- - -
162 18'-
1 ( 14) 5 ( 2)52 ' # #- - -
15 15 ( 3)'#- -
260 ( 26) 2 0.2653
#' '-
( )8 3-
( . )1 5 5- -
a
d
g
j
m
25 13 ( 65)# '- -
2 ( 6) 521 '# -
4 3.6 22 7.92# # '-
( )7 3-
( )9 4-
Multiplying and dividing directed numbers
Simplify these using a calculator:3
4 Use your calculator to simplify these:
-343 -1024
MULTIPLYIN
G AND DIVIDING DIRECTED NUMBERS
* ..../...../20...
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4HSERIES TOPIC
Directed NumbersWhat else can you do?
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)
(ii)
(i)
(ii)
(iii)
(iv)
2 3 5#- +
35 ( 5) 1' - -
28 7 3 6#'- +
17 4 ( 10) 2#- - -
24 3 ( 5) 4#' + -
6 ( 3) 210 30 ( 2)#'# - - -
( ) ( ) ( )
( )
6 3 210 30 2 18 210 30 2
18 7 2
18 14
4
# # #
#
' '- - - =- - -
=- - -
=- +
=-
2 3 5 2 15
13
#- + =- +
=
( )35 5 1 7 1
8
' - - =- -
=-
28 7 3 6 4 3 6
12 6
6
#' #- + =- +
=- +
=-
17 4 ( 10) 2 17 4 ( 20)
( )13 20
33
#- - - = - - -
= - -
=
24 3 ( 5) 4 8 ( 5) 4
( )8 20
12
#' #+ - = + -
= + -
=-
Add -2 and 15
Subtract of 1 from -7
Multiply -12 by 3
Add -12 and 6
Subtract 4 from 17
Subtract -20 from 13 ( = 13 + 20 )
Multiply -5 by 4
Divide -210 by 30
Multiply -7 by -2
Add -18 and 14
Add 8 and -20
If more than one multiply/divide sign, work left to right. Do the same for addition/subtraction.
Calculate these combined operations questions:
Combining the basic operations
Do this first
Do this first
Do this first
Do this first
Do this first
Do this first
4HSERIES TOPIC
29Directed Numbers
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What else can you do? Directed NumbersYour Turn
Combining the basic operations
1 Simplify these combined operations without a calculator:
4 16 8'- +
( )50 25 10' - +
5 7 ( 6)#+ -
( )22 39 3'- -
a
c
e
g
b
d
f
h
6 ( ) 128# - -
8 5 4#- -
11 2 18#- - +
10 4 15#- -
2
3
Simplify these combined operations without a calculator:
a
c
b
d
b
d
4 ( 2) 18 11# - + -
( )9 48 12 13'- + - +
28 ( 4) 13 7#' - +
25 6 224 ( 32)# '- - -
3 6 9 3# '- +
32 4 3 ( 4)#' - -
120 ( 15) 24 8#' - -
30 ( 15) 35 4 ( 16)# #'- - -
Use a calculator to simplify these combined operations:
a
c
COMBIN
ING THE BASIC OPERATIONS
*..../...../20...
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4HSERIES TOPIC
What else can you do? Directed Numbers
Order of operations
For more complex calculations involving parentheses and indices, we use a similar convention:
P I D M A S
Parentheses Indices Division Multiplication Addition Subtraction
Do these first Then do these Do these last
Simplify these directed number calculations using the correct order of operations:
(i)
(ii)
(iii)
(iv)
(v)
(12 8) 7#-
( )21 2 5 3'- + +
(12 8) 7 4 7
28
# #- =
=
21 (2 5) 3 21 7 3
3 3
0
' '- + + =- +
=- +
=
( 4) (8 6) ( 4) 2
16 2
8
2 2' '
'
- - = -
=
=
( ) ( )4 8 62 '- -
15 ( ) 205 3 3# - + -
29 (13 4 5) 32#- - +
29 (13 4 5) 3 29 (13 20) 3
29 ( 7) 3
29 ( 7) 9
36 9
45
2 2
2
#- - + = - - +
= - - +
= - - +
= +
=
( ) ( )
( )
15 5 3 20 15 2 20
15 8 20
120 20
140
3 3# #
#
- + - = - -
= - -
=- -
=-
Subtract of 8 from 12 inside the parentheses
Multiply 4 and 7
Add 2 and 5 inside the parentheses
Subtract of 6 from 8 inside the parentheses
Add -3 and 5 inside the parentheses
Multiply -4 by 5 inside the parentheses
Divide -21 by 7
Square -4
Cube -2
Subtract 20 from 13 inside the parentheses
Add -3 and 3
Divide 16 by 2
Multiply 15 by -8
Square 3
Subtract 20 from -120
Subtract -7 from 29
Add 36 and 9
4HSERIES TOPIC
31Directed Numbers
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What else can you do? Directed NumbersYour Turn
Order of operations
Simplify these directed number calculations using the correct order of operations without a calculator:1
a
c
e
g
a
c
b
d
f
h
b
d
(16 10) 4#-
5 (27 9)# '-
5 (14 9) 5#- - -
( )18 16 4 10'+ +
( ) ( )2 22 '- -
( )8 14 162- -
( 7 5) ( 1)3#- - -
( )3 43 83 '-
( ( ))34 6 8' + -
( 13 ( 19)) 3#- - -
4 (36 2) ( 4)# ' - -
( ( ))18 2 4 7' - - -
2 Simplify these calculations containing indices using the correct order of operations without a calculator:
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4HSERIES TOPIC
What else can you do? Directed NumbersYour Turn
Order of operations
3 Try simplify these trickier ones using the correct order of operations:
a
b
c
d
e
( ) ( )26 14 2 6 1 2' #+ + -
100 (12 32 ( 4)) 40' '#- - -
(11 2 10) 39 83# '- +
9 (( 8) 8 4 16) 43# '- - + -
200 ( 4) ( 3) (16 14 ( 1) )3' '# #- - - -
ORDER OF O
PERATIONS * ORDER OF OPERATIONS
* ..../...../20...
4HSERIES TOPIC
33Directed 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)
(i)
(ii)
(ii)
(10 2) ( 7 3)# ' - -
( ) ( ) ( )10 2 7 3 20 10
2
# ' '- - = -
=-
(10 2) ( 7 3)7 3
10 2## '
- -= - -
denominatornumerator (numerator) (denominator)'=
( )34 3 422 2
#
' --
( )34 3 4 (34 3 4) (22 ( 2))
( )22 2
22 11
2
##
'' '
'-
- = - -
= -
=-
The same question can be written as a fraction.
Simplify the parentheses
Re-write with parentheses
Simplify the parentheses
Simplify the parentheses first
Simplify the square brackets
Simplify the parentheses
Simplify the square brackets
Simplify the braces
Subtract 16 from 30
Multiply 3 by -1
Divide 22 by -11
Divide 20 by -10
When questions contain brackets within other brackets, work from the inside to the outside.
Simplify these combined operations questions containing multiple brackets:
3 ( )1 6 5'# -6 @
30 26 ( )18 2 172'- - -6 @" ,
( ) ( )
( )
30 18 26 30 18 26
30 18
30
30 16
14
2 17 4 17
26 13
18 2
2' '
'
- - = - +
= - +
= - +
= -
=
- -
-
-
6 6
6
@ @
@
" "
"
"
, ,
,
,
3 ( ) 5 3
3 1
3
1 6 5 5# #
#
' '=
= -
=-
- -6 6@ @
34 Directed Numbers
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4HSERIES TOPIC
What else can you do? Directed NumbersYour Turn
1
2
Simplify these questions:
a
c
e
g
a
c
b
d
b
d
f
h
(16 9) (14 7)+ + -
(7 11) (4 2)#- +
5 246 19
--
( ) ( )50 54 9 24 3' ' '+ -
( ) ( )1 9 3 8- - -
( ) ( ( ))18 3 12 7'- - - -
2 1048 12#
'-
( )42 68 5 12#
' --
Simplify these questions with multiple grouping symbols:
12 ( )7 11 9' '- +6 @
6 (13 ( 7)) 4# '- - -6 @
( ) 810 14 2' #+ -6 @
5 (2 9) 32## -6 @
( ) ( )
27 3
46 19 5 2
9
'
'
= - -
=
=
ORDER OF OPE
RATIONS WITH GROUPING SYMBOLS * ..../.....
/20...
Order of operations with grouping symbols
4HSERIES TOPIC
35Directed Numbers
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What else can you do? Directed NumbersYour Turn
3 Simplify these:
a
a
b
c
b
4 Give these three tricky ones a go to earn an awesome stamp.
( )(14 8) ( 1)
15 3
2 3#
- -- -
Remember me?
* AWESOME *
* AWESOME
*..../...../20.
..
( 2) 2 2 (2 2)[ ]#- - + +{ }3+
( )[ ]12 6 8 3 25# '+ -{ }
4 72 ( 6) (4 ( 5)) ( 2)[ ]#'#- - + - - - - }{ 2
Order of operations with grouping symbols
3 ( ( )) ( ) ( )[ ]3 3 3 3 3# #'- - - - + - -{ }2 2
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4HSERIES TOPIC
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: or2 3+ -
• Rational numbers can be written as a fraction. They include whole numbers and terminating or
recurring decimals. Eg: , , , ,8 2.3 4.5 .2
1 4o
• 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: , ,1.2938164928... .3 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
Positive # Positive = PositivePositive # Negative = NegativeNegative # Positive = NegativeNegative # Negative = Positive
#+ + = +
#+ - = -
#- + = -
#- - = +
Multiplication
Addition
Subtraction
+ + = + (Same signs together means plus)
+ - = - (Opposite signs together means minus)
- + = - (Opposite signs together means minus)
- - = + (Same signs together means plus)
Adding and subtracting directed numbers
Multiplying and dividing direct numbers
Division 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 I D M A SParentheses Indices Division Multiplication Addition Subtraction
Do these first Then do these Do these last
(Same signs mean answer is positive)
(Opposite signs mean answer is negative)
(Opposite signs mean answer is negative)
(Same signs mean answer is positive)