THE NUMBER PLANE PASSPORT www.mathletics.co.nz
Fill in the mind-map below with what you already know about number planes. A start has been made for you! Uses a coordinate system developed by the French Mathematician René Descartes and its full name is: The Cartesian Number Plane Number Plane Give this a go! Q A company charges $ to host a party. It also charges extra for each guest. The extra charge amounts to $ for every guests. Use a graph to find how much it would cost for a party with guests. Cost ($) Number of guests Work through the book for a great way to solve this P Learning H SERIES TOPIC
How 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. Use this map and key to answer the questions below: Vertical axis Key Park Football field Building Lake Road A B C D E F G H Horizontal axis (i) Write the grid reference for the two buildings on the map. Building is in grid B A B C D E F G H Building is in grid G (ii) Which grid is completely covered by the lake? A B C D E F G H grid E H SERIES TOPIC P Learning
How does it work? Here is another example. Use this map and key to answer the questions below: Vertical axis North Ave Pi St Main Road Li le Lane Happy St A B C D E F G H Horizontal axis (i) Name the road that connects to Main Road at B. North Ave Pi St Main Road North Ave connects to Main Road at B Happy p St Li le Lane A B C D E F G H (ii) Which road changes direction at E? North Ave Pi St Main Road Li le Lane Happy St A B C D E F G H Happy St changes direction at E P Learning H SERIES TOPIC
How does it work? Your Turn Grid plane Use this map and key to answer the questions below: 9 8 9 8 A A B B C C D E F G H D E F G H I I J J K L M Key State forest Dam Creek Building Road Walking track Bridge 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 C? 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? H SERIES TOPIC P Learning
Oval Road Down St How does it work? Your Turn Grid plane GRID PLANE * GRID PLANE * GRID PLANE * Use this map and key to answer the questions below: 9 8 Ver cal Road Kink St Pi St Car Park Road Dog Leg Lane Eff St Park Circuit P.O. Horizontal Way West Bend Car Park Key Park Sports Oval Pond Ambulance Station P.O. Post Office Change Rooms Police Station.../.../... A B C D E F G H I J K L M 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. P Learning H SERIES TOPIC
How does it work? Axes and coordinates Number planes have two number lines as their axes. Vertical axis (usually labelled ) Horizontal axis (usually labelled ) Coordinates help us locate points and are always written in curved brackets separated by a comma (, ). Write down the coordinates of the points A, B and C shown on the number plane below: B A C 8 A 8 The coordinates of point A = (, ) x-value first Horizontal first y-value second Vertical second B 8 The coordinates of point B = (, ) x-value first Horizontal first y-value second Vertical second C 8 The coordinates of point C = (, ) x-value first Horizontal first y-value second Vertical second H SERIES TOPIC P Learning
How does it work? It is good practice to write the coordinates next to the points when plotting. Plot the points given by the coordinates below on the same number plane: D (, ) E (, ) F (, ) G (8, ) H (, ) Plotting points is done by placing a small dot where the x and y coordinate values cross each other. Remember: The horizontal value is always the first number G (8, ) E (, ) D (, ) H (, ) F (, ) 8 Coordinates can also be called ordered pairs because the order we write the numbers in is important. Ordered pairs can also have decimal values. Plot the points I, J and K on the number plane below: I (,.) J (., ) K (.,.) I (,.) K (.,.) J (., ) 8 P Learning H SERIES TOPIC
How does it work? Your Turn Axes and coordinates Write each letter plotted on the number plane with the matching coordinates below: N C 9 8! T F T I A A S 8 9 *(x, y) AXES AND COORDINATES * AXES AND COORDINATES.../.../... (, ) (, ) (, ) (, ) (, ) (9, ) (, ) (,.) (9., ) (8, 8) Plot and label these points on the number plane below: A (, ) B (, ) C (., ) D (, ) E (9, 8) F (,.) G (, ) H (8, ) 9 8 8 9 8 H SERIES TOPIC P Learning
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. Graph the table below that was completed using the rule: y = x+ x 8 y 9 The x and y values are paired into coordinates to plot. x 8 y 9 (, ) (, ) (, ) (, ) (, 9) (8, ) (8, ) 9 (, 9) 8 (, ) (, ) (, ) (, ) 8 9 The plotted points for the rule above lie on a straight line. When this happens, the rule is called a linear relationship. P Learning H SERIES TOPIC 9
Where does it work? Here is another example using different variables. Use the rule b = a + to complete the table of values below: a b 9 a = a = a = a = a = a = b = # + b = # + b = # + b = # + b = # + b = # + = = = = = 9 = After a table of values is completed, we usually plot all the points. Plot the completed table of values on the number plane below: a b 9 (, ) (, ) (, ) (, ) (, 9) (, ) b-axis 9 8 (, ) (, 9) (, ) (, ) (, ) (, ) 8 9 a-axis H SERIES TOPIC P Learning
Where does it work? Your Turn Plotting tables of values Plot each of the completed table of values below: a y = x+ b b = -a x y a b b-axis 8 8 a-axis c m = n d y = x ' n m x y.. m-axis 8 n-axis 8 P Learning H SERIES TOPIC
* PLOTTING TABLES OF VALUES * PLOTTING TABLES OF VALUES Where does it work? Your Turn Plotting tables of values (i) Use the rules to complete the table of values. (ii) Plot the table of values..../.../... a y = x+ x y b q 8 = - p p q 8 q-axis 8 p-axis c n = m - d c = d + m 8 d 8 n c n-axis c-axis 8 m-axis 8 d-axis H SERIES TOPIC P Learning
Where does it work? 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 = + x Input Independent variable Its value is given, so is independent of y y 8 Output Dependent variable Its value depends on the value of x Dependent variable (vertical axis) y x Independent variable (horizontal axis) Remember: We can use different variables such as a, b, m, n, p, q etc Identify the dependent and independent variables in the rules below: (i) y x = + y x = + Axis labelling y Output ` Dependent variable Input Independent variable x (ii) b- = a b- = a Axis labelling a Input ` Independent variable Output Dependent variable b (iii) n = m n = m Axis labelling n Output ` Dependent variable Input Independent variable 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. P Learning H SERIES TOPIC
DEPENDENT AND INDEPENDENT VARIABLES * Where does it work? Your Turn Dependent and independent variables (i) For the rules given below, circle the independent variables. a y = x+ b - b = a c m = n'.../.../... (ii) Circle the correct answer to the statement: Independent variable values are placed onto the horizontal axis. True False (i) For the rules given below, circle the dependent variables. a y - = x b a = b+ c m = n (ii) Circle the correct answer to the statement: Dependent variables are not affected by the values of the independent variables. True False Label each of these axes with the correct variable for the given rule if they were to be graphed. a y = x- b m+ = n c a = b d y = x + e + p = q f x+ = y H SERIES TOPIC P Learning
Where does it work? 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. A bus company charges per kilometre travelled. One passenger is charged $ for travelling km. (i) Graph the bus company charges on the number plane below. Bus charges (,) is called the Origin Cost ($) 8 9 9 8 Kilometres (km) Step Plot the charge for the passenger travelling km Step Draw a line from (,) through the plotted point (ii) Use the graph to find how much it would cost a passenger to travel 8km. Cost ($) Step Draw a horizontal line to the vertical axis to read the cost 8 9 Bus charges 9 8 Kilometres (km) Step Draw a vertical line from 8km up to the graphed line From the graph, the cost for travelling 8km is $. P Learning H SERIES TOPIC
Where does it work? Companies often use a rule to determine how much they charge customers. This rule can be graphed by first completing a table of values. A professional party clown uses this rule to calculate how much he charges for parties:. C n = + C = the amount charged in dollars and n = the number of children (i) Complete the table of values and graph it on the number plane below: n 8 C 8 9 C-axis Clown charges 9 (, 9) 8 Amount charged ($) (8, ) (, ) (, ) n-axis 8 Number of children (, 8) Step Plot the points from the table of values Step Draw a straight line through the plotted points (ii) Use the graph to find how many children can attend the party for $. C-axis 9 Clown charges Step Draw a horizontal line from $ to the graphed line Amount charged ($) 8 Step Draw a vertical line to the horizontal axis to read the value 8 Number of children n-axis From the graph, children could attend the party for $. This example shows that you can start from either axis to read values from the graph. H SERIES TOPIC P Learning
* FEES AND CHARGES * FEES AND CHARGES Where does it work? Your Turn Fees and charges A recent survey on studying found that marks achieved increased by an average of marks from a previous test for every minutes of study. (i) Graph this relationship between study time and marks below: (psst: plot the extra marks after minutes study first) Increase in marks from studying.../.../... Extra marks 8 9 Minutes spent studying (ii) Use the graph to find how many more marks a student could get after studying for minutes. Expected extra marks achieved by studying for minutes = marks. A company charges $ to host a party. It also charges extra for each guest. The extra charge amounts to $ for every guests. (i) Graph this relationship between charges and guests below: (psst: plot the charge for guests and the extra charge for guests.) Graph showing cost of hosting party per guest Cost ($) Remember me? Number of guests (ii) Use the graph to find how much it would cost for a party with guests. A party with guests will cost: $ P Learning H SERIES TOPIC
Where does it work? Your Turn Fees and charges A cab company uses the following rule to determine the fares for passengers: F = $. # k+ F = The total fare in dollars and k = the number of kilometres travelled (i) Complete the table of values and graph it on the number plane below: F = $. # k+ k 8 F F-axis Cab fare per kilometre travelled 8 Fare ($) 8 9 Kilometres travelled k-axis (ii) Use the graph to find how far someone could travel by cab if they had $. A person with $ could travel: km Matthew is paid by his favourite rugby union club to encourage fan support at every home game according to the following rule: P = g+ P = The total paid and g = the number of home games (i) Complete the table of values and graph it on the number plane below: P = g+ g P P-axis Graph showing payment for Matthew 8 Pay ($) 8 Number of home games g-axis (ii) Use the graph to find how many games he must attend to earn $. Matthew must attend games to earn $. 8 H SERIES TOPIC P Learning
What else can you do? More axes and coordinates We can extend the number plane to include all integers by introducing negative numbers. x-values decrease -8 - - - - - - - - y-values increase nd st - rd - - th - - - x-values increase 8 y-values decrease Remember: (, ) is called The Origin. Usually labelled 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 nd quadrant: negative, positive (-, +) st quadrant: positive, positive (+, +) rd quadrant: negative, negative (-, -) th quadrant: positive, negative (+, -) Points on the horizontal axis ( ) have a second coordinate value of eg: (, ). Points on the vertical axis ( ) have a first coordinate value of eg: (, ). H SERIES TOPIC 9
What else can you do? Write down the coordinates of the points A, B, C and D shown on the number plane below: B nd st A - - - - - - - - - D C rd - - - - - th nd quadrant: x negative, y positive x-value first y-value second Point B: (-, ) - - - - - - - - - - C - Point C: (-, -) - - x-value first y-value second - rd quadrant: x negative, y negative B D A st quadrant: x positive, y positive x-value first y-value second Point A: (, ) Point D: (, -) x-value first y-value second th quadrant: x positive, y negative H SERIES TOPIC P Learning
What else can you do? Remember, when plotting points the first number is always the x-value (or horizontal value). Plot and label these points on the same number plane: E (, ) F (-,.) G (-, ) H (., -) I (-, -) O (, ) E (, ) F (-,.) G (-, ) O (, ) -8 - - - - - - - 8 - - - I (-, -) - - - - H (., -) P Learning H SERIES TOPIC
What else can you do? Your Turn More axes and coordinates 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 (-, ) B (8, ) C (, ) D (-, ) E (, -) F (, -) G (-, -) H (, ) I (-, ) J (, ) K (-, ) L (-, -) M (, 9) N (, -) O (, ) P (-, -) Q (9, -8) R (, ) S (9, ) T (, -) U (, -8) V (-., -.) W (-.,.) X (, -.) Y (-8, ) Z (-.,.) nd quadrant rd quadrant st quadrant th quadrant Coordinates H SERIES TOPIC P Learning
What else can you do? Your Turn More axes and coordinates 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. st Line: start (, -) (, ) (-, ) (, -) stop nd Line: start (-, ) (9, ) (, -) (, -) (, ) (-, ) (-, ) stop Colour in the shape formed by the second line. rd Line: start (, -) (-, -) (-, ) (-, ) (, -) stop Colour in the V shape formed by the third line a different colour. th Line: start (-, ) (, 9) (9, ) stop 9 8 - - - -9-8 - - - - - - - - 8 9 - - - - - - -8-9 - - - P Learning H SERIES TOPIC
What else can you do? Interesting applications Let s look at some applications of number planes requiring a little more thought. 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. (i) What coordinates must she enter for the grid below if the rule is: Any multiple of or a number with a in it? 8 9 8 9 8 8 9 9 8 9 8 9 8 9 Blank screens 8 9 Screens lit according to rule ` The coordinates to be lit up are: (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (ii) If the rule is changed to Any multiple of, which coordinates will remain lit from the previous rule? (, ) (, ) (, ) (iii) What rule did the artist use if she entered the following coordinates? (, ) (, ) (, ) (, ) 8 9 8 9 8 9 8 9 ` Rule = Any multiple of 9 Identify the numbered LCD screens at the given coordinates H SERIES TOPIC P Learning
What else can you do? Distances between points on the number plane are measured in units. NORTH WEST EAST SOUTH Directions can be given as East (right), West (left), North (up) and South (down). The skate park Jill is going to is four units East and one unit North of the Post Office located at (-, -). 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: (i) Plot the coordinates of her house, the skate park and Post Office below. Jill's home (-, ) -8 - - - - - - - - Post Office (-, -) - - - - - - 8 (, -) skate park (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: (-, ), (, ), (-, -) and (, -). 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. P Learning H SERIES TOPIC
What else can you do? Your Turn Interesting applications The electronic artist in the example lights up the numbered LCD screens using the following rule: Any multiple of or a number with a in it. (i) Lightly colour in the LCD screens that suit this rule and write down their coordinates. 8 9 8 9 8 9 8 9 * AWESOME *.../.../... * AWESOME * (ii) The artist changes the rule to: Any number divisible by and including numbers with a in it. List the coordinates of the LCD screens that will be turned off from the previous rule. (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) 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 (, ). 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. (ii) Do these locations form a straight line? Yes No - - - - - - 8 - - - - - (iii) What directions should Fabio radio to Mark to guide him to the Waterfall? H SERIES TOPIC P Learning
What else can you do? Your Turn Interesting applications The coordinates (, ) and (, ) are two vertices of a square. Write down two different sets of coordinates that could be the other two vertices of this square. - - - - - - - - - 8 First set: (, ) and (, ) Second set: (, ) and (, ) Vertices are the corner points Which of the following ticket locations are closest to the stage for the concert venue pictured here? Location : st quadrant, two units away from both axes. Location : (, -) Location : Four units along the horizontal axis between the st and th quadrants. Location : (-, ) Location : Three units along the horizontal axis and two units directly below the nd quadrant. - - - - - - Location : The centre of the venue (, ). Closest Location: - - Stage - - - P Learning H SERIES TOPIC
What else can you do? Your Turn Interesting applications 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: Application icon 9 8 Calculator Music Documents Shares Gymnas cs 8 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? (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) 8 H SERIES TOPIC P Learning
What else can you do? Your Turn 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. O Y 9 A D R C 8 O O F F U Y F - - -9-8 - - - - - - - N A O - - - R G 8 9 N H I - D - - O P - -8 P -9 Order and coordinates are: Hint:!rewsna ruoy kcehc ot sdrawkcab em daer (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) P Learning H SERIES TOPIC 9
Cheat Sheet 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. Axes and coordinates Number planes have two number lines as their axes. The Origin (, ) nd Quadrant (negative, positive) Vertical axis () st Quadrant (positive, positive) -8 - - - - - - - 8 - - - rd Quadrant th Quadrant - (negative, negative) (positive, negative) - - - Coordinates help us locate points and are always written in curved brackets with a comma in between. (horizontal value, vertical value) Horizontal axis () Horizontal value is always written first. E.g. (-, ) means move two units horizontally to the left, then three units vertically up from the origin. Dependent and independent variables y = x+ x y 9 (, ) (, ) 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. H SERIES TOPIC P Learning
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AXES AND COORDINATES * AXES AND COORDINATES GRID PLANE * GRID PLANE * GRID PLANE * * FEES AND CHARGES * FEES AND CHARGES * PLOTTING TABLES OF VALUES * PLOTTING TABLES OF VALUES *(x, y).../.../....../.../....../.../....../.../....../.../... DEPENDENT AND INDEPENDENT VARIABLES *