The Number Plane THE NUMBER PLANE - Wikispacessks7c.wikispaces.com/file/view/Mathletics L4 Geometry...

The

Num

ber P

lane

THE NUMBER PLANETHE NUMBER PLANETHE NUMBER PLANE

PASSPORT

PASSPORT

www.mathletics.co.nz

2HSERIES TOPIC

1The Number Plane

Mathletics Passport © 3P Learning

Q A company charges $150 to host a party. It also charges extra for each guest. The extra charge amounts to $25 for every 5 guests. Use a graph to find how much it would cost for a party with 35 guests.

0 5 10 15 20 25 30 35 40

350

300

250

200

150

100

50

0

Cost ($)

Number of guests

Fill in the mind-map below with what you already know about number planes. A start has been made for you!

Give this a go!

Uses a coordinate system developed by the French Mathematician René Descartes and its full name is:

The Cartesian Number Plane

Number Plane

Work through the book for a great way to solve this

The Number Plane


The Number PlaneHow does it work?

Grid plane

Grid planes have a vertical axis and a horizontal axis. They can be used to help us find places on maps such as street directories.

The horizontal axis reference is always written first.

A B C D E F G H

7

6

5

4

3

2

1Horizontal axis

A B C D E F G H

7

6

5

4

3

2

1

7

6

5

4

3

2

1

Building is in grid B4

Building is in grid G6

grid E6

Vertical axis

Key

Park

Football field

Building

Lake

Road

A B C D E F G H

Use this map and key to answer the questions below:

2 2HSERIES TOPIC

(ii) Which grid is completely covered by the lake?

(i) Write the grid reference for the two buildings on the map.

2HSERIES TOPIC

3The Number Plane


How does it work? The Number Plane

Main Road

Nor

th A

ve

Pi St

Happy St

Lile Lane

Hap

N

p

N

Main Road

Nor

th A

ve

Pi St

Happy St

Lile Lane

Happy St

Nor

th A

ve

Pi St

Happy St

Li�le Lane

Main Road

Here is another example.

A B C D E F G H

7

6

5

4

3

2

1

Vertical axis

Horizontal axis

North Ave connects to Main Road at B5

Happy St changes direction at E2

7

6

5

4

3

2

1

Use this map and key to answer the questions below:

A B C D E F G H

7

6

5

4

3

2

1

A B C D E F G H

(i) Name the road that connects to Main Road at B5.

(ii) Which road changes direction at E2?

4 The Number Plane


2HSERIES TOPIC

How does it work? The Number PlaneYour Turn

Grid plane 1 Use this map and key to answer the questions below:

a At which grid reference does the walking track join the road?

b Which grid reference is completely filled with dam water?

c What is the grid reference where the creek joins with the dam?

d What three features are located at grid reference C6?

e What is the grid reference where the creek enters the State forest?

f Write down the four grid references that contain a building.

g If you were moving towards the road on the walking track and had just exited the State forest, which grid are you in?

A B C D E F G H I J K L M

11

10

9

8

7

6

5

4

3

2

1


11

10

9

8

7

6

5

4

3

2

1

H I J K L M

Key

State forest

Dam

Creek

Building

Road

Walking track

Bridge

2HSERIES TOPIC

5The Number Plane



GRID PLANE *

GRID PLANE * GRID

PLANE

*

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

Grid plane 2 Use this map and key to answer the questions below:

a Where does Pi St join with Vertical Road?

b Where are the change rooms located?

c Where does Dog Leg Lane come to an end?

d Write down the locations of the Police and Ambulance Stations.

e Locate the pond in the smaller park.

f A new Fire Station is going to be built on the corner of Car Park Road and Oval Road, across the road from the park. What will be the grid reference for the new Fire Station?

g Write down the directions (using grid references only) you would give a driver entering the map from Eff St if they were driving to the Post Office on Horizontal Way.


12

11

10

9

8

7

6

5

4

3

2

1

Pi St

Ver�

cal R

oad

Kink St

Dog

Leg

Lan

e

Car Park Road

Eff S

t

Park CircuitO

val R

oad

Horizontal Way

West Bend

Dow

n St

Car Park

P.O.

P.O.

Key

Park

Sports Oval

Pond

Ambulance Station

Post Office

Change Rooms

Police Station

6 The Number Plane


2HSERIES TOPIC


Write down the coordinates of the points A, B and C shown on the number plane below:

Axes and coordinates

Number planes have two number lines as their axes.

Coordinates help us locate points and are always written in curved brackets separated by a comma ( , ).

0 1 2 3 4 5

5

4

3

2

1

0

Vertical axis (usually labelled y-axis)

Horizontal axis (usually labelled x-axis)

7

6

5

4

3

2

1

0

y-axis

x-axis

A

B

C

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8

7

6

5

4

3

2

1

0

y-axis

x-axis

0 1 2 3 4 5 6 7 8

7

6

5

4

3

2

1

0

y-axis

x-axis

0 1 2 3 4 5 6 7 8

7

6

5

4

3

2

1

0

y-axis

x-axis

B

A

C

The coordinates of point B = (4 , 6)

x-value first y-value second

Horizontal first Vertical second

The coordinates of point A = (1 , 3)



The coordinates of point C = (7 , 2)



2HSERIES TOPIC

7The Number Plane



Ordered pairs can also have decimal values. Plot the points I, J and K on the number plane below:

Plot the points given by the coordinates below on the same number plane:

It is good practice to write the coordinates next to the points when plotting.

D (3 , 4) E (0 , 5) F (7 , 0) G (8 , 6) H (2 , 2)

Plotting points is done by placing a small dot where the x and y coordinate values cross each other.

0 1 2 3 4 5 6 7 8

7

6

5

4

3

2

1

0

y-axis

x-axis

I (2 , 5.5) J (4.5 , 1) K (7.5 , 3.5)

Coordinates can also be called ordered pairs because the order we write the numbers in is important.

I (2 , 5.5)

J (4.5 , 1)

0 1 2 3 4 5 6 7 8

7

6

5

4

3

2

1

0

y-axis

x-axis

E (0 , 5)

G (8 , 6)

Remember: The horizontal value is always the first number

D (3 , 4)

H (2 , 2)

F (7 , 0)

K (7.5 , 3.5)

8 The Number Plane


2HSERIES TOPIC


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

(x , y)* AX

ES AND COORDINATES

*

AXES AND COORDI

NATES


1 Write each letter plotted on the number plane with the matching coordinates below:

0 1 2 3 4 5 6 7 8 9 10 11 12 13

11

10

9

8

7

6

5

4

3

2

1

0

y-axis

x-axis

0 1 2 3 4 5 6 7 8 9 10 11 12 13

11

10

9

8

7

6

5

4

3

2

1

0

y-axis

x-axis

(7 , 6) (11 , 3) (3 , 11) (1 , 5) (0 , 1) (9 , 0) (6 , 7) (5 , 3.5) (9.5 , 10) (8 , 8)

2 Plot and label these points on the number plane below:

A (2 , 4) B (0 , 7) C (4.5 , 3) D (7 , 10) E (9 , 8) F (12 , 7.5) G (1 , 6) H (8 , 0)

N

C

!

T

F

T

IA

A

S

The Number Plane


The Number Plane

9H 2SERIES TOPIC

x 0 1 2 4 6 8

y 3 4 5 7 9 11

The x and y values are paired into coordinates to plot.

x 0 1 2 4 6 8

y 3 4 5 7 9 11

(0 , 3) (1 , 4) (2 , 5) (4 , 7) (6 , 9) (8 , 11)

Graph the table below that was completed using the rule: 3y x= +

Where does it work?

Plotting tables of values

Tables of values give us coordinates to plot on a number plane and show the graph of the rule given.

0 1 2 3 4 5 6 7 8 9 10 11 12 13

11

10

9

8

7

6

5

4

3

2

1

0

y-axis

x-axis

(8 , 11)

(6 , 9)

(4 , 7)

(2 , 5)

(1 , 4)

(0 , 3)

The plotted points for the rule above lie on a straight line. When this happens, the rule is called a linear relationship.

10 The Number Plane


2HSERIES TOPIC

Where does it work? The Number Plane

a 0 1 2 3 4 5

b 1 3 5 7 9 11

Use the rule 2b a 1= + to complete the table of values below:

Here is another example using different variables.

Plot the completed table of values on the number plane below:

a

b

0

2 0 1

1

#

=

= +

=

a

b

1

2 1 1

3

#

=

= +

=

a

b

2

2 2 1

5

#

=

= +

=

a

b

3

2 3 1

7

#

=

= +

=

a

b

4

2 4 1

9

#

=

= +

=

a 5

b 2 5 1

11

#

=

= +

=

After a table of values is completed, we usually plot all the points.

a 0 1 2 3 4 5

b 1 3 5 7 9 11

(0 , 1) (1 , 3) (2 , 5) (3 , 7) (4 , 9) (5 , 11)

0 1 2 3 4 5 6 7 8 9 10 11 12 13

11

10

9

8

7

6

5

4

3

2

1

0

b-axis

a-axis

(5 , 11)

(4 , 9)

(3 , 7)

(2 , 5)

(1 , 3)

(0 , 1)

2HSERIES TOPIC

11The Number Plane


Where does it work? The Number PlaneYour Turn

a y x 1= +

x 0 1 2 3 4 5

y 1 2 3 4 5 6

b 7 ab = -

a 0 1 2 3 4 5

b 7 6 5 4 3 2


1 Plot each of the completed table of values below:

c m n=

n 0 1 2 4 6 7

m 0 1 2 4 6 7

d 3 2xy '=

x 0 1 2 3 4

y 0 1.5 3 4.5 6

0 1 2 3 4 5 6 7 8

7

6

5

4

3

2

1

0

y-axis

x-axis0 1 2 3 4 5 6 7 8

7

6

5

4

3

2

1

0

b-axis

a-axis

m-axis

n-axis

y-axis

x-axis0 1 2 3 4 5 6 7 8

7

6

5

4

3

2

1

00 1 2 3 4 5 6 7 8

7

6

5

4

3

2

1

0

12 The Number Plane


2HSERIES TOPIC


a y x 2= +

x 0 1 2 3 4 5

y

b q p8= -

p 1 2 4 5 6 7

q


2 (i) Use the rules to complete the table of values. (ii) Plot the table of values.

c mn 1-=

m 1 2 3 4 6 8

n

d dc2

1= +

d 0 2 4 6 7 8

c

y-axis q-axis

p-axis

n-axis c-axis

d-axis

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

* PLOT

TING TABLES OF VALUES

* PLOTTING TABLES

OF VALUES

0 1 2 3 4 5 6 7 8

7

6

5

4

3

2

1

0 x-axis0 1 2 3 4 5 6 7 8

7

6

5

4

3

2

1

0

m-axis0 1 2 3 4 5 6 7 8

7

6

5

4

3

2

1

00 1 2 3 4 5 6 7 8

7

6

5

4

3

2

1

0

2HSERIES TOPIC

13The Number Plane



Identify the dependent and independent variables in the rules below:

y x 5= +

Output Input

` Dependent variable Independent variable

b a7- =

Input Output

` Independent variable Dependent variable

n m2=

Output Input

` Dependent variable Independent variable

Dependent and independent variables

The y-value in the table of values is the answer (or output) after x-values are put into the rule.

y x 4= +

x 0 1 2 4 Input Independent variable Its value is given, so is independent of y

y 4 5 6 8 Output Dependent variable Its value depends on the value of x

y

x

Dependent variable(vertical axis)

Independent variable(horizontal axis)

Axis labelling

Axis labelling

Axis labelling

y

x

a

b

n

m

Did you notice a pattern?

In these cases, the dependent variable is the one by itself on one side of the equal sign. When a variable is by itself in a rule, it is called the subject of the rule.

(i) y x 5= +

(ii) b a7- =

(iii) n m2=

Remember: We can use different variables such as a, b, m, n, p, q etc

14 The Number Plane


2HSERIES TOPIC



1 (i) For the rules given below, circle the independent variables.

a y x 5= + b b a12- = c m n 4'=

(ii) Circle the correct answer to the statement:

“Independent variable values are placed onto the horizontal axis.”

True False

2 (i) For the rules given below, circle the dependent variables.

a 5 xy =- b 12a b= + c m n4 =

(ii) Circle the correct answer to the statement:

“Dependent variables are not affected by the values of the independent variables.”

True False

3 Label each of these axes with the correct variable for the given rule if they were to be graphed.

a y x2 1= - b m n3+ = c ba2

=

d xy3

1= + e p q3+ = f 3x y2 + =

DEP

ENDENT

AND

INDEPENDENT VARIABLES *

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

2HSERIES TOPIC

15The Number Plane



DEP

ENDENT

AND

INDEPENDENT VARIABLES *

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

A bus company charges per kilometre travelled. One passenger is charged $36 for travelling 24km.

Fees and charges

Fees charged by companies are often graphed on number planes to help you quickly read how much it would cost you for different situations.

The cost is usually dependent on how far you travel, how long you use the service etc. Therefore cost is always put on the vertical axis.

(i) Graph the bus company charges on the number plane below.

0 3 6 9 12 15 18 21 24

63

54

45

36

27

18

9

0

Cost ($)

(ii) Use the graph to find how much it would cost a passenger to travel 18km.

Bus charges

Kilometres (km)

Step 1Plot the charge for the passenger travelling 24km

Step 2Draw a line from (0,0) through the plotted point

0 3 6 9 12 15 18 21 24

63

54

45

36

27

18

9

0

Cost ($)

Bus charges

Kilometres (km)

Step 1Draw a vertical line from 18km up to the graphed lineStep 2

Draw a horizontal line to the vertical axis to read the cost

From the graph, the cost for travelling 18km is $27.

(0,0) is called the Origin

16 The Number Plane


2HSERIES TOPIC


Companies often use a rule to determine how much they charge customers.This rule can be graphed by first completing a table of values.

C = the amount charged in dollars and n = the number of children

(i) Complete the table of values and graph it on the number plane below:

(ii) Use the graph to find how many children can attend the party for $75.

Clown charges

Number of children

From the graph, 10 children could attend the party for $75.

n 0 4 8 12 16

C 50 60 70 80 90

C-axis

n-axis

(0 , 50)

(4 , 60)

(8 , 70)

(12 , 80)

(16 , 90)

0 2 4 6 8 10 12 14 16

110

100

90

80

70

60

50

0

Amount charged ($)

Clown charges

Number of children

Step 1Draw a horizontal line from $75 to the graphed line

Step 2Draw a vertical line to the horizontal axis to read the value

n-axis

This example shows that you can start from either axis to read values from the graph.

0 2 4 6 8 10 12 14 16

110

100

90

80

70

60

50

0

Amount charged ($)

Step 1Plot the points from the table of values

Step 2Draw a straight line through the plotted points

C-axis

A professional party clown uses this rule to calculate how much he charges for parties: 2.5C n 50= +

2HSERIES TOPIC

17The Number Plane



0 15 30 45 60 75 90 105 120 135

42

36

30

24

18

12

6

0

Extra marks

Increase in marks from studying

Graph showing cost of hosting party per guest

Fees and charges

1 A recent survey on studying found that marks achieved increased by an average of 6 marks from a previous test for every 30 minutes of study.

(i) Graph this relationship between study time and marks below: (psst: plot the extra marks after 30 minutes study first)

(ii) Use the graph to find how many more marks a student could get after studying for 105 minutes.

Expected extra marks achieved by studying for 105 minutes = marks.

2 A company charges $150 to host a party. It also charges extra for each guest. The extra charge amounts to $25 for every 5 guests.

(i) Graph this relationship between charges and guests below: (psst: plot the charge for 0 guests and the extra charge for 10 guests.)

(ii) Use the graph to find how much it would cost for a party with 35 guests.

A party with 35 guests will cost: $

Minutes spent studying

Number of guests

Remember me?350

300

250

200

150

100

50

00 5 10 15 20 25 30 35 40

Cost ($)

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

* F

EES AND CHARGES

* FEES AND C

HARGES

18 The Number Plane


2HSERIES TOPIC


Fees and charges

3 Acabcompanyusesthefollowingruletodeterminethefaresforpassengers: $ .F k0 50 4#= +

F = The total fare in dollars and k = the number of kilometres travelled

(i) Completethetableofvaluesandgraphitonthenumberplanebelow:

(ii) Usethegraphtofindhowfarsomeonecouldtravelbycabiftheyhad$7.

Apersonwith$7couldtravel:km

4 Matthewispaidbyhisfavouriterugbyunionclubtoencouragefansupportateveryhomegame accordingtothefollowingrule: 75 150P g= +

P = The total paid and g = the number of home games

(i) Completethetableofvaluesandgraphitonthenumberplanebelow:

(ii) Usethegraphtofindhowmanygameshemustattendtoearn$1200.

Matthewmustattendgamestoearn$1200.

0 1 2 3 4 5 6 7 8 9 10 11 12

14

12

10

8

6

4

2

0

Fare ($)

Cab fare per kilometre travelled

Kilometres travelled

k 0 4 8 12

F

k-axis

F-axis

$ .F k0 50 4#= +

g 0 2 6 10

P

0 2 4 6 8 10 12 14 16

1400

1200

1000

800

600

400

200

0

Pay ($)

Graph showing payment for Matthew

Number of home games

g-axis

P-axis

75 150P g= +

The Number Plane

19H 2SERIES TOPIC

What else can you do?

More axes and coordinates

We can extend the number plane to include all integers by introducing negative numbers.

The axes of a Cartesian number plane splits the grid into four sections called quadrants.

The quadrants are numbered as shown above. The coordinates in each quadrant have different combinations of positive and negative values.

Coordinate combinations

2nd quadrant: negative, positive (- , +) 1st quadrant: positive, positive (+ , +)

3rd quadrant: negative, negative (- , -) 4th quadrant: positive, negative (+ , -)

Points on the horizontal axis ( ) have a second coordinate value of 0 eg: (3 , 0).

Points on the vertical axis ( ) have a first coordinate value of 0 eg: (0 , 2).

x-axis

y-axis

x-values decrease x-values increase

y-values increase

y-values decrease

1 2 3 4 5 6 7 8 -8 -7 -6 -5 -4 -3 -2 -1

-1

-2

-3

-4

-5

-6

-7

7

6

5

4

3

2

1

Remember: (0, 0) is calledThe Origin. Usually labelled ‘0’

2nd 1st

3rd 4th

0

20 The Number Plane


2HSERIES TOPIC

What else can you do? The Number Plane

Write down the coordinates of the points A, B, C and D shown on the number plane below:

x-axis

y-axis

1 2 3 4 5 6 7 -7 -6 -5 -4 -3 -2 -1

-1

-2

-3

-4

-5

-6

-7

7

6

5

4

3

2

1

2nd 1st

3rd 4th

x-axis

y-axis

A

D

B

C

A

D

B

C

2nd quadrant: x negative, y positivex-value first y-value second

Point B: (-4 , 5)

1st quadrant: x positive, y positivex-value first y-value second

Point A: (5 , 3)

Point C: (-6 , -4)

x-value first y-value second3rd quadrant: x negative, y negative

Point D: (3, -2)

x-value first y-value second4th quadrant: x positive, y negative

7

6

5

4

3

2

1

1 2 3 4 5 6 7 -7 -6 -5 -4 -3 -2 -1 0

-1

-2

-3

-4

-5

-6

-7

0

2HSERIES TOPIC

21The Number Plane



Plot and label these points on the same number plane:

Remember, when plotting points the first number is always the x-value (or horizontal value).

x-axis

y-axis

1 2 3 4 5 6 7 8 -8 -7 -6 -5 -4 -3 -2 -1

-1

-2

-3

-4

-5

-6

-7

7

6

5

4

3

2

1

E (0 , 5)

H (2.5 , -5)

F (-2 , 1.5)

I (-4 , -6)

E (0 , 5) F (-2 , 1.5) G (-7 , 0)

H (2.5 , -5) I (-4 , -6) O (0 , 0)

G (-7 , 0) O (0 , 0)0

22 The Number Plane


2HSERIES TOPIC

What else can you do? The Number PlaneYour Turn


1 Put the letter for the coordinates below into the correct group where they would be found if plotted.

There may be one or two that fit into more than one group.

A (-2 , 1) B (8 , 0) C (5 , 5) D (-2 , 0) E (4 , -5) F (0 , -1) G (-3 , -6)

H (2 , 7) I (-4 , 0) J (4 , 1) K (-5 , 1) L (-1 , -1) M (3 , 9) N (7 , -3)

O (0 , 0) P (-6 , -1) Q (9 , -8) R (0 , 4) S (9 , 2) T (6 , -7) U (0 , -8)

V (-2.5 , -1.5) W (-1.5 , 3.5) X (0 , -4.5) Y (-8 , 0) Z (-0.5 , 0.5)

3rd quadrant

2nd quadrant

1st quadrant

x-axis

Coordinates4th quadrant

y-axis

2HSERIES TOPIC

23The Number Plane




2 What does a Penrose Triangle look like? Follow the instructions below to find out!

Plot and join the following sets of coordinates together with straight lines using a ruler.

1st Line: start (0 , -4) (2 , 2) (-4 , 0) (0 , -4) stop

2nd Line: start (-6 , 2) (9 , 7) (3 , -11) (0 , -12) (5 , 3) (-4 , 0) (-6 , 2) stop

Colour in the “7” shape formed by the second line.

3rd Line: start (0 , -4) (-1 , -7) (-11 , 3) (-12 , 0) (0 , -12) stop

Colour in the “V” shape formed by the third line a different colour.

4th Line: start (-11 , 3) (7 , 9) (9 , 7) stopy-axis

x-axis0

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

-11

-12

-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

24 The Number Plane


2HSERIES TOPIC


(i) What coordinates must she enter for the grid below if the rule is: “Any multiple of 7 or a number with a 7 in it ”?

Blank screens Screens lit according to rule

` The coordinates to be lit up are:

(1 , 1) (1 , 6) (2 , 5) (3 , 3) (3, 4) (4 , 3) (5 , 2) (5 , 5) (6 , 1)

(ii) If the rule is changed to “Any multiple of 3”, which coordinates will remain lit from the previous rule?

(3 , 4) (3 , 3) (6 , 1)

(iii) What rule did the artist use if she entered the following coordinates?

(3 , 6) (6 , 5) (3 , 3) (6 , 2)

An electronic artist lights up displays of squares from a large grid of numbered LCD screens according to a rule. She must enter the coordinates into the lighting unit so the correct squares are lit up.

Interesting applications

Let’s look at some applications of number planes requiring a little more thought.

Identify the numbered LCDscreens at the given coordinates

` Rule = “Any multiple of 9”

37 38 39 40 41 42

31 32 33 34 35 36

25 26 27 28 29 30

19 20 21 22 23 24

13 14 15 16 17 18

7 8 9 10 11 12

1 2 3 4 5 6

37 38 39 40 41 42

31 32 33 34 35 36

25 26 27 28 29 30

19 20 21 22 23 24

13 14 15 16 17 18

7 8 9 10 11 12

1 2 3 4 5 6

37 38 39 40 41 42

31 32 33 34 35 36

25 26 27 28 29 30

19 20 21 22 23 24

13 14 15 16 17 18

7 8 9 10 11 12

1 2 3 4 5 6

1 2 3 4 5 6

7

6

5

4

3

2

1

1 2 3 4 5 6

7

6

5

4

3

2

1

1 2 3 4 5 6

7

6

5

4

3

2

1

2HSERIES TOPIC

25The Number Plane



The skate park Jill is going to is four units East and one unit North of the Post Office located at (-1 , -2). Jill needs to post a letter for her mother before going to the skate park. Her house is five units North from the Post Office:

Distances between points on the number plane are measured in units.

Directions can be given as East (right), West (left), North (up) and South (down).

(i) Plot the coordinates of her house, the skate park and Post Office below.

(ii) Jill made a new friend at the skate park and gave the following directions to reach her house:

“I live four vertical and four horizontal units away from here.”

Explain why these directions are not enough information.

From the skate park, there are four different coordinates that match. Jill’s instructions:

(-1 , 3), (7 , 3) , (-1 , -5) and (7 , -5).

Her new friend could go to the wrong place.

(iii) What directions should Jill have given to be more accurate?

From the skate park, I live four units West then four units North.

x-axis

y-axis

1 2 3 4 5 6 7 8 -8 -7 -6 -5 -4 -3 -2 -1 0

-1

-2

-3

-4

-5

-6

-7

7

6

5

4

3

2

1

(3 , -1) skate park

Post Office (-1 , -2)

Jill's home (-1 , 3)

NORTH

EAST

SOUTH

WEST

26 The Number Plane


2HSERIES TOPIC



1 The electronic artist in the example lights up the numbered LCD screens using the following rule:

“Any multiple of 5 or a number with a 6 in it.”

(i) Lightly colour in the LCD screens that suit this rule and write down their coordinates.

(ii) The artist changes the rule to: “Any number divisible by 6 and including numbers with a 6 in it.” List the coordinates of the LCD screens that will be turned off from the previous rule.

( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )

2 Mark and Fabio are using portable radios in a bush-walking exercise. Mark's location is at the Look Out, four units West and three units North from the Base Camp located at (0 , 0). Fabio’s location is a Waterfall which is five units East and six units South from the Base Camp.

(i) Plot the locations of the Look Out, Base Camp and Waterfall.

37 38 39 40 41 42

31 32 33 34 35 36

25 26 27 28 29 30

19 20 21 22 23 24

13 14 15 16 17 18

7 8 9 10 11 12

1 2 3 4 5 6

x-axis

y-axis

1 2 3 4 5 6 7 8 -5 -4 -3 -2 -1

-1

-2

-3

-4

-5

-6

4

3

2

1

1 2 3 4 5 6

7

6

5

4

3

2

1

* AWESOME * * AWESOM

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

(ii) Do these locations form a straight line?

Yes No

(iii) What directions should Fabio radio to Mark to guide him to the Waterfall?

0

4 Which of the following ticket locations are closest to the stage for the concert venue pictured here?

Location 1: 1st quadrant, two units away from both axes.

Location 2: (2 , -2)

Location 3: Four units along the horizontal axis between the 1st and 4th quadrants.

Location 4: (-3 , 0)

Location 5: Three units along the horizontal axis and two units directly below the 2nd quadrant.

Location 6: The centre of the venue (0 , 0).

Closest Location:

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27The Number Plane




3 The coordinates (4 , 0) and (0 , 3) are two vertices of a square. Write down two different sets of coordinates that could be the other two vertices of this square.

First set: ( , ) and ( , )

Second set: ( , ) and ( , )

Vertices are the corner points

Stage

x-axis

y-axis

-1

-2

-3

-4

-5

-6

1 2 3 4 5 -5 -4 -3 -2 -1

7

6

5

4

3

2

1

x-axis

y-axis

1 2 3 4 5 6 7 8 -5 -4 -3 -2 -1

-1

-2

-3

-4

7

6

5

4

3

2

1

0

0

28 The Number Plane


2HSERIES TOPIC


Calculator

Shares

Documents

Gymnas�cs

8

9

1

2

3

4

5

6

7

5 6 7 1 2 3 4 8 0

Music


5 Akido’s intelligent touchpad has the five application icons on the front screen as shown in the diagram below. This touchpad uses coordinates to recognise where Akido has touched the screen so the correct application is started.

If any of these 9 points on the application icon are pressed, the application will start:

a Write down all the coordinates that will open the Gymnastics application if a touch was detected there.

( , ) ( , ) ( , ) ( , ) ( , )

( , ) ( , ) ( , ) ( , )

b Akido moves all his picture icons to new locations. He moves the Music application icon down five units and to the left two units. What are the new coordinates that will open his Music application?

( , ) ( , ) ( , ) ( , ) ( , )

( , ) ( , ) ( , ) ( , )

x-axis

y-axis

..../...../20...///Application

icon

2HSERIES TOPIC

29The Number Plane



A bit of fun at the end

Coordy the Frog likes to sit on every lily pad in the pond at different times of the day. Coordy cannot adjust how far he jumps very well, so he always jumps the same distance. Starting from his current lily pad his first jump is to the closest Y. Write down the letter order and coordinates of the lily pads he jumps to during each day.

Order and coordinates are: Hint: !rewsna ruoy kcehc ot sdrawkcab em daeR

( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )

( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )

( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )

..../...../20...///y-axis

x-axis

12

11

10

9

8

7

6

5

4

3

2

1

1 2 3 4 5 6 7 8 9 10 11

-1

-2

-3

-4

-5

-6

-7

-8

-9

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

Y

O

R

G

O

O

D

N

U

F

F

H

P

I

N

P

D

R O

O

A

A

C

Y F0

30 The Number Plane


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Cheat Sheet The Number Plane

Here is a summary of the things you need to remember for the number plane

Grid plane

• Grid planes have a vertical axis (usually marked with numbers) and a horizontal axis (usually marked with letters).

• A letter paired with a number is used to locate a grid area on maps.


• Number planes have two number lines as their axes.

• Coordinates help us locate points and are always written in curved brackets with a comma in between.(horizontal value , vertical value)

Horizontal value is always written first. E.g. (-2 , 3) means move two units horizontally to the left, then three units vertically up from the origin.


Input Independent variable Its value is given, so is independent of y

Output Dependent variable Its value depends on the value of x

Graphing tables of values• Coordinates can be formed by pairing dependent and independent variables from a table of values.• Plotting these points on a number plane will graph the rule used to complete the table of values.• When plotting graphs of fees for services, the cost is placed on the vertical axis.

x 1 2 3 4

y 3 5 7 9

(1 , 3) (3 , 7)

y x2 1= +

x-axis

y-axis

0 1 2 3 4 5 6 7 8 -8 -7 -6 -5 -4 -3 -2 -1

The Origin (0 , 0)

Vertical axis (y-axis)

Horizontal axis (x-axis)

2nd Quadrant(negative , positive)

1st Quadrant(positive , positive)

3rd Quadrant(negative , negative)

4th Quadrant(positive , negative)

7

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

-7

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31The Number Plane


The Number Plane Notes

32 The Number Plane


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The Number Plane Notes

DEPENDENT

AND INDEPENDENT

VARIABLES *

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

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

(x , y)* AX

ES AND COORDINATES

*

AXES AND COORDI

NATES

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

FEES A

ND CHARGES *

FEES AND CHARG

ES

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

* PLOT

TING TABLES OF VALUES

* PLOTTING TABLES

OF VALUES

GRID

PLANE * GRID PLANE * GRID P

LANE

*

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

The Number Plane THE NUMBER PLANE - Wikispacessks7c.wikispaces.com/file/view/Mathletics L4 Geometry...

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