# X Marks the Spot

Purpose

This unit uses games and puzzles to explore co-ordinate systems and their uses. The students are given the opportunity to invent their own systems by locating counters on a sheet of paper and by finding the dead centre of a set of objects. The Cartesian and polar co-ordinate systems are used to locate position on maps. Polar co-ordinates are also used to describe certain loci (sets of points with a common property).

Achievement Objectives
GM4-7: Communicate and interpret locations and directions, using compass directions, distances, and grid references.
Specific Learning Outcomes
• Use Cartesian and polar co-ordinates to describe the position of an object.
• Find the location of an object using Cartesian or polar co-ordinates.
Description of Mathematics

This unit emphasises the value of co-ordinates to locate the position of objects in the plane. Two co-ordinate systems are presented. One is the Cartesian system that uses horizontal and vertical distances from a fixed point (the origin). The other is the polar co-ordinate system that uses angles about a fixed line and distances from a fixed point on that line.

Students may well have seen the Cartesian system before as it is the one commonly used on road maps, and maps generally. It was invented by a French mathematician called René Descartes. You’ll read his story on the way through the unit. The origin of the polar co-ordinate system is less clear. Both systems describe some complicated curves simply though the unit does not go into functions at higher levels of the Curriculum. Students are likely to see polar co-ordinates as an interesting curiosity at this point.However, Cartesian co-ordinates provide the basis of a great deal of work that is undertaken in the senior secondary school. Here algebra and geometry are brought together to form a powerful combination that enable some interesting problems to be solved. Most of the calculus in school is built around the application of algebra to geometry via the Cartesian co-ordinate system.

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:

• Act out the finding of locations with Cartesian and Polar co-ordinates by walking on a large grid. You might use tape or chalk to mark out a grid on the carpet of your classroom, or create the grid outside on a concrete surface.
• Directly model the locating of points for your students, using “I do then you do” approaches. Also, model finding an angle with a protractor using the concept of an angle as a turn.
• Limit the complexity of the co-ordinates you use. Beginning learners might use only whole number values and angles that are easy divisions, such as 45⁰. More proficient students might use decimals, such as (4.7, 5.6), and angles that are complex divisions of 360⁰, such as 27⁰ or 135⁰ (going beyond 90⁰).
• Ask students to work collaboratively to support each other. In that way you can share the expertise of proficient students in the class, and allow students the opportunity to be both learners and teachers.

Differentiation involves changing the context of activities to suit the interests of your students. The unit uses quite traditional contexts such as battleships and pirates. Students will find using locations in their local area engaging, but also be prepared to use locations they are less familiar with, such as other towns/cities in Aotearoa. Students might become more engaged in sports they prefer such as basketball, hockey and netball, than golf. Professional sports coaches gather a lot of detail about the locations of their players during games. Traditional settings also provide contexts for study. Māori call the North Island “Te Ika-a-Māui” (the fish of Māui). There were no aeroplanes then. How did they know the island was shaped like a fish? What reference system did pre-European Māori use to navigate?

Required Resource Materials
• Rulers
• Copies of a Cartesian co-ordinate system (Copymaster 1)
• Counters
• Large sheets of paper (newsprint is good)
• Copies of the "dead" centre for some objects (Copymaster 2)
• Street maps of the local area
• Copies of the golf hole (Copymaster 3)
Activity

#### Session 1: Four Counters

This lesson introduces the Cartesian co-ordinate system using a game with four counters and "Battleships".

1. For this activity students need to work with a partner. Each student will require a large sheet of paper (A3 or bigger), coloured crayons, and four counters of different colours.
2. Before students start the activity, explain the rules below and allow them two minutes to think about how they might want to prepare their sheets of paper to help them with the challenge. This gives them a chance to develop some form of reference system for location.
3. The students take turns to place four counters on their sheet of paper. One counter must be in each quadrant (quarter) of the paper and no counters can be on the edge. The counters do not have to be placed symmetrically though the actual position of the counters is up to the individual student. Once a student has put the counters in place they must give their partner instructions for locating the counters. The nature of these instructions is up to the student themselves. The other student must then mark on their own paper the location of the counters using an X in the appropriate coloured crayon. At no stage is the second student allowed to see their partner’s paper while placing their X, nor do the players have access to rulers or protractors.
4. Once the Xs are marked the players compare the locations of the counters and the corresponding Xs by overlaying the marked paper. Measurements of difference can be taken if you wish.
5. When each pair of students is finished, bring the class together to discuss their methods. Expect that students will have devised some reference system. These may include co-ordinate grids, specific reference points or conventions involving length and direction. Inform them that they have re-invented systems for location which were created centuries ago in response to similar problems involving finding positions of objects, particularly in navigation and warfare.
6. Share the story of René Descartes, a famous French mathematician and philosopher. Descartes enjoyed sleeping in and while doing so noticed a fly on the ceiling. Not wishing to get out of bed himself he wondered how he could tell a servant the exact location of the fly. Thus, he invented the co-ordinate system known as Cartesian co-ordinates after René himself. Students will enjoy the story more if they can act it out. Get a student to be the servant who goes to a corner of the room and follows these instructions: "Go three steps across and two steps up. Now splat the fly!" Emphasise the importance of knowing which direction is across and which is up. After all, the spot represented by (3,2) is not the same spot as (2,3).
7. Students may wish to play the game "Battleships" to reinforce the convention that the horizontal value is given first and the vertical value second. That is, (5,6) refers to the location 5 units across and 6 units up. "Battleships" is played on a Cartesian co-ordinate grid. Each player has their own copy of this grid, which is masked from their opponent. They nominate and shade some points on the plane to represent battleships. A destroyer might be represented by two adjacent co-ordinates, a cruiser by three adjacent points, an aircraft carrier by four co-ordinates in a 2 x 2 square, etc. Players take turns to have shots at each other’s battleships by calling out a co-ordinate. This is met by the response "hit", where the co-ordinate is part of a battleship or "miss" if it is not. A battleship is sunk when all of the co-ordinates representing that battleship are discovered. The first player to sink all their opponent’s battleships is the winner.

#### Session 2: Buried Treasure

X can represent the position of a counter. More traditionally it represents the spot where pirate treasure is buried. We now follow that theme.

1. Maps invariably have a co-ordinate system of some kind so that the location of streets, landforms or other features can be found quickly. This idea can be reinforced easily by giving students a copy of the street directory and the map of a local town or city and asking them to locate certain landmarks.
2. In turn they can set problems for others by giving the co-ordinates of landmarks and requiring others to find them.
3. An interesting activity is to play "Pirate’s Treasure". Each group of students receives a map of the school with a co-ordinate system superimposed on it and four or five items of treasure (play money) to hide. The co-ordinate system should be numerical with two-digit numbers marking the lines on the map (eg. 00 for zero, 01 for one, 12 for twelve etc).
4. Discuss with the students how a location can be described very precisely by using a six-digit reference. The third and sixth digits are like a decimal reference, which more accurately define the horizontal and vertical distances. For example the co-ordinate below could be given as 015 008.

5. Each group goes outside and "buries their treasure" marking its location on their map of the school. They return to the classroom and provide another group the five co-ordinates, which describe where the pieces of treasure are located. The other group attempts to find the treasure of the first group using the co-ordinates provided.

Use the Copymaster Dead Centre in this session.

There is an old joke about the dead centre of a town being the cemetery. Here we get the class to think about the location of the real "dead" centre of a set of objects. In this situation we mean the point that is in some sense in the middle of these objects.

1. Discuss what the term "dead centre of town" means. Expect ideas like, "Exactly in the middle", "The oldest part of town", and "Where all the shops are." Suggest that towns and cities do not always grow symmetrically, especially where waterways and landforms intervene, so the centre of town is not always the oldest part or where the main shopping centre is located.
2. Pose the following scenario. Each picture shows ten dots. These dots represent where ten people from the same club live. They want to meet in the place that is in the "dead centre" of all their houses. Find a way to work out where that centre is. Point out that the dead centre will be the place that minimises (makes smallest) the total travel distance that the club members need to get there. (You might say that they are looking for a new place to locate their clubhouse.)
3. Using spatial estimation supported by measurement is a cumbersome way to find the centre. It is, however, likely to be the preferred option of most groups. Some guidance may be necessary. Since the houses in Scenario One are almost co-linear this is the easiest arrangement to work out by measurement methods. However, since the houses are not the same distance apart the process is more complicated than might at first be thought. An idea of average is likely to emerge. This might take several forms such as, "If we get the centre exactly the distances of points on the left will add to the same as the distances on the right," or, "We measured the distances of each house from the left-hand house and found the average distance. That told us how far to the right the centre was."
4. This concept of averages is potentially powerful when faced with the more difficult Scenarios Two and Three. Remind the students that they have used co-ordinates previously to describe location and that these number pairs might have some use in the dead centre problems. Consider a simple case.The points have a co-ordinate system superimposed on them, so they have number pairs of (1,3), (2,1), and (4,2), respectively. Finding the average of the horizontal (x) co-ordinates gives (1 + 2 + 4) ÷ 3 = 2.33. The average of the vertical (y) co-ordinates gives (3 + 1 + 2) ÷ 3 = 2. So the point (2.33, 2) gives the centre of the three houses.
5. Get the students to apply the average method to Scenarios Two and Three. This method can also be used to establish whether the school is in the "dead centre" of where students in your class live. Using a street reference the location of each student’s house can be described using co-ordinates. These can be loaded into a spreadsheet, x-values in one column, y-values in the other, and each column averaged using a function formula. The location given by the averages will define the dead centre of where students in your class live.

#### Session 4: Polar Co-ordinates

In this lesson we introduce polar co-ordinates.

1. Polar co-ordinates use angles and distances to define location. The most common example of this method is found in orienteering where competitors are given compass bearings (eg. 168° South, 400 metres) to tell them where the next point on the course is located.
2. In mathematics the angles in polar co-ordinates are defined in this way. First a line is drawn starting from the point O, the origin. Then any point P can be described by first joining P to O. Then P is described by the angle θ between OP and the initial line and by the distance OP.

So polar co-ordinates (45°, 5cm) translate to the location of the point Q.

3. Polar co-ordinates can be used to play a game of golf. A Copymaster is included as an example of how a polar golf hole is designed. Each player begins on the tee that is shown as the "crosshairs" with the polar angles. They play their first shot by estimating the angle and distance to the hole (shown at the base of the flag). This shot is recorded in a table. The landing point of the shot is found by measuring using a protractor and ruler. The next shot is taken from the landing point and all references for angle and distance are taken from that new point.
4. A player gets the ball in the hole if they hit the base of the flag exactly. If any of the hazards are reached, then the given number of penalty points is deducted. The winner is the player who takes the least number of shots to get to the hole.
5. Students will enjoy making up their own golf holes. A3 sized paper gives a more difficult exercise. Make up a complete golf course of 18 holes. (The golf game has an interesting extra application, that of integers, where scores above and below par are added to get a total for a round.)

#### Session 5: Co-ordinate Puzzles

This lesson provides an application of polar co-ordinates to revise and strengthen the work of the previous session.

1. Co-ordinate puzzles are both a good way to reinforce the finding of locations described by number pairs and also to highlight the relative strengths of rectangular and polar co-ordinates systems. A co-ordinate puzzle consists of a set of co-ordinates that students transfer to a co-ordinate system and join the points in the order that they are given. The result should be a figure that they recognise.
2. Consider for example, the set of polar co-ordinates: (0°, 4cm), (45°, 4cm), (90°, 4cm), (135°, 4cm), (180°, 4cm), (225°, 4cm), (270°, 4cm), (315°, 4cm), (360°,4cm). Get students to plot these points on a polar co-ordinate system. This results in a set of points. If connected by straight lines these points form the corners of a regular octagon or if connected by smooth curves form a circle. Taking the circle scenario, tell the students to give the co-ordinates of other points that also lie on the circle. Ask them how we might have anticipated that the points given before would form a circle. (They are the same distance from a fixed point). This puzzle highlights the power of polar co-ordinates in describing figures with rotational symmetry.
3. Below are three other co-ordinate puzzles for the students to complete. When they have done so get them to create their own co-ordinate puzzles for a partner to solve.

Puzzle One: (5,10), (3,7),(0,5), (3,3), (5,0), (7,3), (10,5), (7,7).

Puzzle Two: (0°, 0cm), (45°, 1cm), (90°, 2cm), (135°, 3cm), (180°, 4cm), (225°, 5cm), (270°, 6cm), (315°, 7cm), (360°, 8cm), (0°, 9cm), (45°, 10cm), (90°, 11cm), (135°, 12cm), (180°, 13cm), (225°, 14cm), (270°, 15cm), (315°, 16cm), (360°, 17cm).

Puzzle Three: (5, 6), (6, 5.9), (7, 5.7), (8, 5.4), (9, 4.8), (10, 3), (9, 1.2), (8, 0.6), (7, 0.3), (6, 0.1), (5, 0), (4, 0.1), (3, 0.3), (2, 0.6), (1, 1.2), (0, 3), (1, 4.8), (2, 5.4), (3, 5.7), (4, 5.9), (5, 6).

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