Directed Numbers DIRECTED NUMBERS DIRECTED NUMBERS …slau.weebly.com/uploads/1/3/3/0/13308739/directed_numbers_bookle… · Digital scales need to use directed numbers to calculate

PASSPORT

PASSPORT

Dire

cted

Num

bers

DIRECTED NUMBERS DIRECTED NUMBERSDIRECTED NUMBERS

www.mathletics.com.au

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


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


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


4HSERIES TOPIC

How does it work? Directed NumbersYour Turn


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




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


4HSERIES TOPIC


+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



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


4HSERIES TOPIC


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




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


4HSERIES TOPIC



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



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


4HSERIES TOPIC


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



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


4HSERIES TOPIC


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



1

2


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


4HSERIES TOPIC



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


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 +2 hot spheres and 6 cold spheres

9 hot spheres and 6 cold spheres -2 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




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


4HSERIES TOPIC

Where does it work? Directed NumbersYour Turn


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




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




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


4HSERIES TOPIC



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

4HSERIES TOPIC

21Directed Numbers


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


4HSERIES TOPIC



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...

4HSERIES TOPIC

23Directed 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


4HSERIES TOPIC



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


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)



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

26 Directed Numbers


4HSERIES TOPIC



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



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-


Simplify these using a calculator:3

4 Use your calculator to simplify these:

-343 -1024

MULTIPLYIN

G AND DIVIDING DIRECTED NUMBERS

* ..../...../20...

28 Directed Numbers


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


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...

30 Directed Numbers


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



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:

32 Directed Numbers


4HSERIES TOPIC


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


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 first

Simplify the square brackets


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


4HSERIES TOPIC


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...


4HSERIES TOPIC

35Directed Numbers



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


3 ( ( )) ( ) ( )[ ]3 3 3 3 3# #'- - - - + - -{ }2 2

36 Directed Numbers


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 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)


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





MULTIPLYIN

G AND DIVIDING DIRECTED NUMBERS

* ..../...../20...ADDITION

AND SUBTRACTION USING A NUMBER LIN

E *

..../...../20...

ASCEND

ING AND DESCENDING ORDER * ..../...../20...

MEASURIN

G INSTRUMENTS AND INTEGERS

* ..../...../20...

ORDER OF OPE

RATIONS WITH GROUPING SYMBOLS * ..../.....

/20...

Documents

Directed Numbers DIRECTED NUMBERS DIRECTED NUMBERS …slau.weebly.com/uploads/1/3/3/0/13308739/directed_numbers_bookle… · Digital scales need to use directed numbers to calculate