Session 1

In the first session we introduce the idea of time zones using a globe and by making a phone or internet call. If possible, before teaching Session 1 find an international audience to connect with (e.g. a classroom of students or whānau living in Australia. If possible, arrange to call them during this session.

1. If possible, darken the room. Have someone hold the globe and another student shine the torch on it. An old overhead projector is ideal in place of the torch. You could also find a video showing the movement of the Earth and sun to show. 
   Imagine that the globe is the Earth and the torch is the sun. How do the sun and Earth move in relation to one another?
   Students should suggest that the Earth orbits the sun. 
   How long does that take? (a year or 365¼ days) 
   Ensure that students know that the Earth rotates on its axis. 
   How long does a full turn take? (24 hours or one day)
2. Act out one day. That means little movement of the Earth around the Sun while a full rotation about the axis occurs. Note that the Earth rotates in an anti-clockwise direction (West to East) which explains why the sun appears in the Eastern sky at dawn.
   Discuss what you can see with a classmate. What does this mean for people and animals living on the Earth?
   Out of the discussion students should express the idea that when we are in daylight there are some places in the dark. All places, except the polar regions in summer and winter, undergo day and night in a 24-hour period. You may need to fix a blob of Blu Tack on the globe at the location of your school, and another blob at a location in another country.
3. Rotate the globe, stopping at various points, in the turn.
   What time is it in each place?
   Stop the rotation in a position when the sun is shining directly at New Zealand. 
   What time will this be in Aotearoa? (12 noon unless it is Daylight Savings, the time will be 1:00pm)
   Think about the time in these locations:
   Sydney (two hours earlier)
   Samoa (one hour ahead which becomes 23 hours behind due to the International Date Line)
   Shanghai (5 hours behind)
   London (12 hours behind)
   New York (6 hours ahead which becomes 18 hours behind due to the International Date Line)
   Ask the students if they have whānau living overseas, or if they or someone from their whānau have visited other countries. Think about the time in those locations. What is the time difference between those places and Aotearoa?
4. There are many online videos depicting the movement of the Earth around the Sun. Be sure they represent the axis of the Earth at an angle to the plane of rotation as that explains the seasons.
   
5. Explain that a system was invented to deal with differences. Show Slide One of PowerPoint 1 that shows time zones. 
   Talk to a classmate about what you notice on this map of the world. Be ready to explain what you see to everyone else.
6. In the following discussion try to bring out the following points:
   • In Coordinated Universal Time, Zone Zero is called Greenwich Mean Time (GMT) and is the time zone of countries like Spain, England, Algeria and Mali. Note that these countries are within the same longitudinal vertical strip. Iceland is also in that zone. Since 1968, Iceland has observed GMT all year round. Prior to this, different cities in Iceland observed time zones. Iceland does not have any international borders. Therefore, there was no idea for a time set in relation to neighbouring countries. Daylight Savings was introduced between 1917 and 1921 to allow for more efficient communication (it was around this time that the telephone was introduced to Iceland). Ultimately, GMT was chosen because the moving forward of clocks for Daylight Saving was seen to cause hassle and stress for the Icelandic people. Moving to GMT also meant Iceland was “in sync” with the rest of Europe. Like Iceland, New Zealand had also adopted a local time system. This meant that life was organised, in each city and town, in accordance with its own time zone. By 1868, the use of telegraphs and the post office was becoming more popular. However, having different time zones in different localities meant that different post offices opened and closed at different times. This made it difficult to send and receive any urgent messages. In 1868, and after a lot of arguing between localities, it was decided that New Zealand’s clock time would be 11 hours 30 minutes ahead of GMT. Eventually, in 1946, New Zealand adopted a standard time set 12 hours ahead of GMT, or 180° east of Greenwich. Your students might be engaged by the idea of investigating the time zones of different localities prior to 1868.
   • New Zealand lies in Zone 12 and Eastern Australia in Zone 10.
   • To work out the difference in time between two countries, count how many zones they are apart. For example, New Zealand and Peru are seven zones apart so the time in Lima is 7 hours ahead or 17 hours behind that in Wellington.
7. Ask: Have you ever tried phoning or Skyping someone in Australia? It’s [time] at our place. They live in [location]. What will the time be over there?
8. Let students predict the time in the Australian location using the time zone map.
9. Contact your pre-arranged person/group of people living in Australia. This could be done with Zoom, Skype, Google meets, a cell phone etc. Consider using a video calling application to make the activity more engaging for your students. Be sure to ask them the current time in their location.
10. Discuss: How many time zones do you think Australia has? (three)
    Let students use the map.
    Why does Australia have three time zones? (It is a big country from West to East).
11. Look for Russia on the map. 
    How many time zones does Russia have? (11 zones)
12. Provide the students with copies of Copymaster 1 to work from in pairs. After a suitable time, discuss the answers together as a class. 
    • Question One: This is an open-ended problem so many different answers are possible. From New Zealand any city in South-East Asia will work, such as Kuala Lumpur, Saigon, Bangkok will be five hours different. Mountain State cities in USA such as Denver and Salt Lake City are 5 hours different as well. Any city can be chosen as the reference.
    • Question Two: Mumbai is 7 hours behind New Zealand time. To ring her friend at 8:00am Lucy needs to ring at 3:00pm NZ time. To ring her friend at 6:00pm she would need to ring at 1:00 am NZ time.
    • Question Three: Ireland is 12 hours behind NZ time. The rugby test will start at 7:35am NZ time.
    • Question Four: Korea is four hours behind New Zealand time. The plane takes off at 5:15 am Korean time and lands at 6:15 pm (13 hours later).
    • Question Five: The flight time from Buenos Aires to Auckland is 13 hours and 30 minutes. Buenos Aires is 16 hours behind New Zealand time. Suppose Waimarama leaves Buenos Aires at 10:00pm of 1 January. Her birthday is on 2 January. The time of departure is 2:00pm New Zealand time on the following day, January 2. By the time she flies for 13.5 hours the time she lands is 3:30am on January 3. She misses all of 2 January.
    • Question Six: New Zealand has a special time zone for the Chatham Islands so it has two time zones.

Session 2

In this session students learn to express am and pm times in 24-hour time. They use 24-hour time to make calculations about time difference between two different locations.

1. Use PowerPoint 2 to introduce the lesson. Work through the first five slides setting the context of departing on an International flight. Slide Five has a paper boarding pass and asks students to find important information. The time on the pass is 18-45 which might be written more correctly as 18:45 or 1845.
   What does a time of 18:45 mean?
   Some students may know that the time is given in 24-hour time, sometimes referred to as military time. 
   Why is 6:45pm expressed as 18:45? (It is 18 hours and 45 minutes from the start of the day at Midnight.)
2. You may need to demonstrate counting in hours and minutes until 18:45 is reached, noting the significance of 12:00 noon as the halfway point through a 24-hour day. Proceed through the slides showing other boarding passes to work out the 24-hour times.
3. Copymaster 2 provides a sequence of problems for students to practise reading and writing 24-hour time. Students could work on the problems independently at first, although some may be best supported by encouraging mahi tahi (collaboration) among students. They should share and discuss their thinking and answers. 

Session 3

In this session the class practise working with 24-hour time in the context of the commuter train that runs from Waikanae to Wellington. Use of 24-hour time for public transport is quite rare in New Zealand but is common in Asia and Europe.

1. Look at the first three slides of PowerPoint 3 to introduce the context of two commuters, Henare and Leisha, who work in central Wellington. Commuting can result in long working days. 
   What does it mean to commute to work?
   Why might people use public transport to commute rather than drive their car? (Cost, time, and minimising carbon footprint are important reasons)
   If you were Henare and Leisha what would be the best time to start work? Explain why you would start then.
2. Slide Four has the early morning timetable for the Kapiti Line. It is a simplified timetable compared to the actual timetable but the times are real.
   How much time does it take Henare and Leisha to train from Waikanae to Wellington? (Note that there are two services; normal and express)
   How much longer does it take the normal train than the express train? (3 minutes)
   Which trains are likely to be the most packed? Why? (Trains arriving in Wellington between 7:45am and 9:00am are likely to be popular)
   Which train would you advise Henare and Lisa to take? Why?
3. Encourage the students to consider the ramifications of beginning early or late to avoid the peak periods. An early start means an early return to home, especially if two hours of work can be completed on the train, using the internet.
4. Once the students select a start time show Slide Five which shows the afternoon/evening timetable for Wellington City to Waikanae. Choose several times to see if students recall how to convert from 24-hour time to pm time. For example, the 17:01 train departs at 5:01pm.
5. Provide the students with Copymaster 3 to work from in pairs. Consider pairing up more knowledgeable and less knowledgeable students to encourage tuakana-teina.
   Look to see that your students:
   • read 24-times correctly and transfer those times to am and pm 12-hour times
   • recognise that minute and hour times are based on 60, and calculate times appropriately
   • build with the 24-hour times for existing trains to schedule the new services
   • consider that several services might be rescheduled to accommodate the new trains to best advantage
   • factor in the loss of ‘on train’ working time when calculating the advantage that Henare and Leisha gain from their move
   • support each other, and share and discuss their thinking.

Session 4

In this session students investigate departure and arrival times of international flights. Students are given two of three conditions and asked to find the third. Departure time, flight time or arrival time might be omitted. In solving the problems students need to compare time zones using 24-hour time and make calculations with time in hours and minutes.

1. PowerPoint 4 shows animations of three flights. Clocks show the times at departure and arrival locations. Ask questions like:
   What is the time difference between Brisbane and Singapore?
   What do you notice about the clocks at each location as the flight proceeds?
   From the departure times in each location how could you calculate the flight time?
   If you are told the flight time how can you calculate the arrival time in the country that you land in?
2. The last question is the most common problem that travellers face. Arriving in an unfamiliar location late at night is alarming and sometimes leads to dangerous situations while travelling to accommodation. The easiest method is to work out the departure time at the arrival location. For example, in summer Adelaide is 2½ hours behind Auckland. An Auckland flight that departs at 2:59pm (1429) is leaving at 12:29pm (1229) Adelaide time. If the flight lands at 5:18pm (1718) then the flight time is calculated as 1718 – 1229. 17 – 12 = 5 hours but the flight is slightly less than that. 18 – 29 = -11 minutes so the flight is 5 hours less 11 minutes or 4 hours and 49 minutes.
3. Model the flight time calculation for each of the three flights shown on PowerPoint 4. Slide Seven has the departure and arrival times of an Auckland to Perth flight. Students might be tempted to calculate the flight time as 2150 – 1925 = 2 hours and 25 minutes. This calculation does not include the time difference. Since Perth is 5 hours behind New Zealand time the flight leaves at 1425 (Perth time) not 1925. The flight time is 2150-1425 = 7 hours and 25 minutes.
4. Give your students copies of Copymaster 4 to work from. Students will need to access a site like www.timeanddate.com/worldclock/ to find the time difference between cities and also use google maps, or a similar programme, to establish where those cities are. Look for students to convert the departure time at the origin to the time at the arrival destination, then find the difference between arrival and departure times. Do students recognise that time operates on a base of 60 so that normal decimal calculation methods need to be adjusted?
5. Finding missing arrival and departure times involves a similar process. Either change the departure time into the time at the arrival location or change the arrival time into the time at the point of departure.

Session 5

In this session, the class practise using timetables and their knowledge of time zones to plan a trip to London from where they live. You could complete this task as a class, or in small groups, and then give pairs of students the opportunity to plan a trip to a destination of their choosing. This could be linked with social sciences learning. 

1. Pose the challenge:
   You are going to have a three-week holiday in the United Kingdom (UK). I want you to plan the trip. Both your parents work, so you cannot leave before December 6th and you cannot return later than December 26th. Plan the transport that will give you the longest possible time in the UK. Allow for travel time, stop overs, and minimalising the overall cost. Use the internet to help you but please don’t book any flights by mistake?
2. Get the students to report back with their results. Who was able to get the longest time in the UK? Who allowed for stopovers in important nexus locations like Singapore or Dubai? Who balanced cost with efficiency? (Cheap flights often take a long time) The students might like to think of important locations to visit in the UK, such as Buckingham Palace, Stonehenge and Strawberry Fields. They could plan an itinerary for the trip.

Session 1: Four Counters

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

1. For this activity students need to work with a partner. Consider pairing more knowledgeable students with less knowledgeable students to encourage tuakana-teina throughout the session. 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 enough time (e.g. a few 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 the majority of students have finished, bring the class together to discuss their methods. Pairs of students could share with other pairs, before presenting their ideas to the class. Expect that students will have devised some reference system. These may include coordinate 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 coordinate system known as Cartesian coordinates 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 coordinate 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 coordinates, a cruiser by three adjacent points, an aircraft carrier by four coordinates in a 2 x 2 square, etc. Players take turns to have shots at each other’s battleships by calling out a coordinate. This is met by the response "hit", where the coordinate is part of a battleship or "miss" if it is not. A battleship is sunk when all of the coordinates 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 coordinate 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. This could be linked to other relevant learning by providing maps from your local area, from areas which your students whakapapa to, or of areas from the past. 
2. In turn, they can set problems for others by giving the coordinates 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 coordinate system superimposed on it and four or five items of treasure (play money) to hide. The coordinate 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 coordinate 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 coordinates, which describe where the pieces of treasure are located. The other group attempts to find the treasure of the first group using the coordinates provided.

Session 3: Dead Centre

Use the Copymaster 2: Dead Centre in this session.

The dead centre of a town is in the middle of all the houses, shops and workplaces. 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. Encourage 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 some 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." The point that minimises distance to the counters (corners of a triangle) is called the Fermat Point, named after another famous mathematician. It is located at the average of the x coordinates and the y coordinates.
   
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 coordinates 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 coordinate 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) coordinates gives (1 + 2 + 4) ÷ 3 = 2.33. The average of the vertical (y) coordinates 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 coordinates. 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 Coordinates

In this lesson we introduce polar coordinates.

1. Polar coordinates use angles and distances to define location. A common example of this is found in nautical navigation where a destination can be described using a compass bearing (given in degrees clockwise from north) and a distance (usually given in nautical miles). For example, you might say "the port is 100 nautical miles away at a bearing of 75 degrees".

2. In mathematics, positions in polar coordinates are defined in a similar way.
   The location of a point Q can be described in relation to point O, the origin. Q is described by the angle between the line OQ and the horizontal (x) axis and by the distance OQ (see diagram).

   So polar coordinates (45°, 5cm) describe the location of the point Q.

   

3. Polar coordinates can be used to play a game of golf. A Copymaster is included as an example of how a polar golf course is designed. Each player begins at the origin marked at the bottom left. They play their first shot by estimating the angle and distance to the first hole (shown at the base of the flag). 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 the landing point is within the black circle at the base of the flag. If any of the hazards are landed on, the given number of penalty points are added. Start the next hole from the origin.
5. The winner is the player who has the lowest score at the end of the game.
6. Students will enjoy making up their own golf holes. A3-size paper gives a more difficult exercise. You may like to use several pages to create a complete golf course of 18 holes. 
7. For an additional challenge, students could have the origin in the middle of the page, meaning that their first shot may have an angle of more than 90°.

Session 5: Coordinate Puzzles

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

1. Coordinate 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 coordinates systems. A coordinate puzzle consists of a set of coordinates that students transfer to a coordinate 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 coordinates: (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 coordinate 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 coordinates 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 coordinates in describing figures with rotational symmetry.
3. Below are three other coordinate puzzles for the students to complete. When they have done so, get students to create their own coordinate 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).

Session 1

In this session we look at the probabilities around flipping coins.

1. Discuss things that happen at a sports game, like tennis, rugby, netball, or cricket, before play starts. (Slide One of PowerPoint 1). This could be framed within the context of current sports events that are relevant to your students (e.g. the Rugby World Cup). A local referee might be able to come to talk to your class about the coin toss. Students could also conduct research into the history of the coin toss, and other game-starting activities (e.g. a whistle, starting clappers). 
   Why is a coin tossed? (A whistle might be hidden in one hand and one captain asked to guess as an alternative).
2. Discuss the advantage that might be gained in winning the toss, especially when playing surface or weather might strongly influence the outcome.
   Is tossing a coin a fair way to decide who kicks off, or the way the team face? Why?
   Students should conclude that there are two possible outcomes, so a coin toss is fair, unless the coin is biased in some way.
3. Imagine a game in which you toss the same coin three times, one after the other. You win if you get two heads and one tail, but lose if any other event occurs. (Slide Two)
   What events might happen that cause you to lose? (HHH, TTT, TTH)
   Does the game seem fair? What does fair mean? (equal chance of winning)
4. Ask students to investigate the game in two ways:

   • Play the game a few times, say ten trials of three consecutive tosses. Insist that they record the trial data in a table. You might want to demonstrate data collection:

     Win	Lose
     THH, HTH, HHT, …	HTT, TTT, TTH, HHH,…

   • Think about all the outcomes that might happen. Possibly come up with a model to show the chances of winning and losing.
5. Gather the class to share ideas.
   What did the experimental results show?
6. Collate the number of wins and losses as a class to show that there seems to be a greater chance of losing the game than winning the game.
   Why is the game unfair?  (Look for attention to the events that might happen – a theoretical idea)
7. Show Slide Three. 
   Do you agree with Brian and Jo? Discuss their ideas with a partner and come up with a justification for your view.
8. Share ideas about the theoretical probability of each event occurring. Look for models to show that the different events Brian suggests do not have equal probabilities.
9. Slide Four builds a tree diagram of possible outcomes with progressive mouse clicks. Let students copy the creation of the diagram in their workbooks.
   Act out simulations of three tosses of the same coin and locate the outcome at the appropriate branch of the tree.
   Which outcomes are wins?
   How can we write the probability of winning using numbers? (Use calculators if needed to convert: 3/8, 0.375, 37.5%)
10. Return to Slide Three to discuss Brian and Jo’s ideas again.
    Distinguish between an event (e.g. Three tails or Two heads and one tail) from an outcome, a way that the event might happen (e.g. HTT, THT, HTT).
    What are the chances of getting… three heads? Three tails? Two tails and one head?
11. Ask students to create a game where they use all four coins all at once. 
    Create rules that make the game fair. That means you have an equal chance to winning and losing.
    Test your game to see that it is fair.
12. Roam the room as students work. Do they…? The most obvious rule for the game is that three of a kind wins, i.e. three heads and one tail, or three tails and one head. Any other outcome loses, i.e. four heads, four tails, two heads and two tails.
    • Recognise that the tossing of four coins simultaneously is essentially the same as tossing a single coin four consecutive times.
    • Model all the outcomes in some way, such as a tree diagram or list.
    • Test the game a significant number of times to check that the probabilities are equal.
13. Slide Five extends the tree diagram to show all the outcomes. There sixteen arms (outcomes), six of which give two heads and two tails. The chances of that event are 6/16 or 3/8 that can be written as 0.375 or 37.5%. Three of a kind events (three heads and one tail or three tails and one head) each have a 4/16 or ¼ chance of occurring. Since ¼ + ¼ = ½ the three of a kind events have a combined probability of ½ or 0.5 or 50%.
14. Go online to this Quincunx Simulation or search for a similar one. Most versions allow the user to vary the number of rows to constrain the problems. Set the tool to four rows. Trial 100 marbles. Ask students to explain the results.
    How is the situation similar to the four-coin toss problem?
15. Roam the room as students investigate. 
    • Do they notice that the marble can bounce in two directions, left and right, in the same way that a single coin toss has two outcomes?
    • Do they notice that the middle bin is the most likely of the five destinations? Why?
    • Can they relate each destination to the events of the four-coin toss? (e.g., the middle bin is equivalent to two heads and two tails as it is the result of two lefts and two rights)
16. Students might investigate the effect of changing the simulation.
    What happens if you add a row?
    What happens if you change the probability of left and right from 50%?
    Why does that happen?

Session 2

In this session, we explore the use of theoretical probabilities to make predictions for different numbers of trials.

1. Slide Six presents the problem of predicting the number of females in the first 20 students who leave school. 
   What information do you need to make a prediction? 
   Expect students to talk about the number of each gender in the school influencing the probability of boys and girls leaving (e.g. which gender tends to leave first, who gets kept after school, etc.) Be sensitive to the gender identities of students in the class.
   If the chances of males and females are 50:50, how many females do you predict?
   Are ten females and ten males a likely outcome?
2. Students may make connections to their previous experiences with coin tosses, and the Dalton Board, to make predictions about variation.
   How might we act out this problem? (Simulate the situation)
   Students might suggest tossing a coin twenty times using heads for girls and tails for boys. You might return to a Dalton Board simulation with a single row and 20 marbles to see what happens. Most times a sample of 20 students will not yield a 10:10 result (The probability is only about 18%, which is less than one fifth).
3. Ask students to work in pairs with a coin (or 20 if you have sufficient) to carry out ten trials of 20 individual coin tosses and record their data.
   How many tosses will come up heads? Why do you predict that number?
4. Gather the class.
   How often did you get an outcome of 10 heads and 10 tails? (About 1 trial in every six)
   How much did your results vary? What was your most unpredictable result?
5. Slide Six contains the pooled data from 200 trials of 20 individual tosses, in a graph. You might decide to create your own dot plot of the data from your class.
   What do you notice about the graph?
   Students should observe that the outcomes vary considerably but the distribution centres on the predicted value of 10 heads.
6. Go online to the Spinner learning object and make a spinner with two equal segments of different colours.
   To save time we will use this tool to simulate a lot of coin tossing. Each trial will be 100 individual coin tosses.
   What is your best prediction for the number of heads and tails?
   Do you expect that the data to look different to the 20 individual coin toss trials?
7. Use Slide 8 to compare the data from trials of 100 individual coin tosses with that from trials of 20 individual coin tosses.
   How are the graphs similar? How are they different?
   Students might notice that the graphs both have a ‘bell’ shape and the distributions centre on the predicted values of ten heads and 50 heads, respectively. They might notice that the shape is smoother for the 100 toss data and more evenly spread among the outcomes. It is unlikely they will notice that the variation is proportionally less for the 100 toss data. The spread for the 20 toss data is from 1 to 19. The spread for the 100 toss data is from 35 to 62. 
   The key point is that experimental data varies a lot from the theoretical predictions.
8. Introduce the game on Copymaster 1: Coin Drop. 
   Get a couple of students to try the game, noting the result.
   Does the game look fair? How do you know?
   How could we work out the chances of winning the game?
9. Provide pairs of students with a 20-cent coin and a copy of the grid. Ask them to trial the game and make a prediction about the chances of winning with a single coin toss.
10. Slide 9 shows 16 coins landing on the grid. Five of the coins land without touching a grid line. 
    If you trusted this trial, what is the chance that a single coin toss gives you a win?
    Look for students to express the experimental probability as a fraction using calculators, 5/16, 0.3125 or 31.25%...
    Would you make money playing this game? Explain.
11. In the animated game the player spends 16 x 20c = $3.20 and gets 5 x 50c = $2.50 back. The return does not look good.
    What fraction of the trials did you win?
12. Gather data from the pairs of students, such as 8/39 (8 out of 39 trials were wins), and 5/18 (5 out of 18 trials were wins). The number of trials being different is helpful.
    Are these fractions close to the same or completely different?
13. You might use a calculator to find the decimal or percentage for each fraction, e.g. 8/39 = 0.205 or 20.5%. Converting the fractions will show that the proportions are around 20-30% with the centre being around 25%.
    Our experimental data show that the chance of winning with a single coin drop is about 25%. 
    With your partner, find a way to work out the chance in a theoretical way. 
    Analyse the game.
    Then change the game in some way so the chance of winning is one half, 50%.
14. Roam the room to see how students find the theoretical probability. The coin drop game is literally about sample space (area). Look for students to:
    • Consider where the coin might land that is considered a win
    • Calculate the win area as a fraction of the total area
    • Recognise that to make the game fairer, the area of each square (or other shape) must be greater, or a smaller coin must be used on the existing grid
15. Slide 10 shows one way to find the probability of winning. Mapping the path of the centre of the coin gives a square shape of the possible winning positions. The win square is one quarter the area of the whole square, so the probability of winning is 1/4, assuming the coin lands randomly.
16. Calculating the size of a square grid that makes the probability of winning with a 20-cent coin exactly one half (0.5) is challenging but your students might approximate the grid size closely. Squares that are about 76mm x 76mm will work. Since a 10-cent coin is about the same diameter of a 20-cent coin, reducing the coin size will not work unless you have a supply of old 5-cent coins.

Session 3

In this session, we work on finding probabilities for two other situations, using theoretical and experimental methods. Two Figure It Out activities are used.

1. Introduce the normal game, Paper, Scissors, Rock, with a pair of students. 
   What are the chances that you win the game?
   Does it matter which option you choose, paper, scissors, or rock?
   How might we create a model of all the outcomes?
2. Slide 11 shows a tree diagram of all the possible outcomes.
   In how many outcomes does Player One win? In how many outcomes does Player Two win?
   Both players win for 3 out of 9 outcomes with the other 3 outcomes being draws. Therefore, the game is fair, with both players having a 1/3 chance of winning each time.
3. Go to the Figure It Out activity, Paper, Scissors, Rock. Either access the Figure It Out books in your school or use the PDF to display the page. Answers are available in the teacher notes.
   Let students work on the problems in pairs. Roam the room and look for students to:
   • Understand the conditions under which each player wins.
   • Recognise that a tree diagram is a useful model and builds on the tree diagram of Slide 11.
   • Identify which outcomes lead to wins for each player.
   • Calculate probabilities for each player winning.
   • Trial the game 30 times and match the data with their original model.
4. Gather the class to discuss their answers.
   What ways did you find to make the game fair?
   Giving different numbers of points is one method. Encourage students to look at the probabilities associated with different options. Possible options might be:
   • One or more Rocks.
   • No Scissors
   • Two Papers.
     
     For a touch of humour about modifying the game search “Big Bang Theory Paper Scissors Rock” for a video to see an adaptation of the game. Lizard and Spock are added options. High achievers might investigate how the added options make the game fair and less likely to result in a draw.
      
5. Introduce the Figure It Out activity, Picking Pocket Money. Students will need either blank wooden cubes or cubes and stickers to create the dice. They will also need a calculator to carry out the calculations. 
6. Let students investigate the situation by experiment as suggested on the student page.
7. Roam the room and look for the following:
   • Systematic recording of the trial outcomes.
   • Compare the total amount for the 52 weeks of rolling the dice with the certain outcome of 52 x 5 = $260.
   • Factor in the concept of risk into their decision about whether to roll the dice or take the $5 each week.
   • Acknowledge that samples vary through comparing the result of their experiment with the results of other groups.
8. Gather the class.
   Which option is best? Why?
   Is there a way to estimate the amount that usually occurs with 52 rolls of the dice?
   The data should indicate that most times 52 rolls gives a result of about $260, or a bit less. Possibly display the trial amounts using a dot plot.
   
9. Model how the average for the dice can be calculated then multiplied by 52 to get an expected value for 52 rolls. (1 + 2 + 4 + 5 + 7 + 10 = 29, 29 ÷ 6 = 4.83) Multiplying 4.83 x 52 = 251.3 so the expected total amount from 52 rolls equals $251.33 which is less than $260. $251 should be about the median of the distribution from trialling.
10. Challenge your students with this task:
    How might we change the numbers on the dice so the expected amount from 52 rolls equals 52 lots of $5?
11. Let students work on the problem together in pairs.
    The easiest solution is to label all the faces $5.
    Challenge yourself. Make the number on each face different.
    Any combination of numbers that average five will work, e.g. 1, 2, 3, 7, 8, 9.

Session 4

In this session students investigate the "Top Drop" game.

This is a game for pairs of students. To play the game the players take turns predicting the outcome then dropping a plastic drink bottle top with a smiley face drawn on inside. The first player predicts which way the top will land ‘face up’, ‘face down’ or ‘face side’.

The top must be held on its ‘side’ when it is dropped from a height of approximately 30 cm onto a flat surface. 
If the player’s prediction is correct, they are awarded points, as below. If the player is incorrect, they are awarded no points.

• Face up 1 point
• Face Down 2 Points
• Face Side 10 points

Each player must keep track of the outcome of each drop, plus the points awarded.
The first player to be awarded 20 points wins the game.

1. Demonstrate the game with a pair of students, showing how to record the outcome of each toss. 
2. Ask students to play as many games as they can before you stop them.
   What is the best strategy to win the game?
3. At the conclusion of the game playing time, assemble the class.
   What did you notice?
   Expect students to say that Up occurs more frequently than Down, with Side being unlikely. 
   Ask: How could we find the probability of each event, Up, Down and Side?
   Since the situation cannot be modelled theoretically the only way to estimate the respective probabilities is by experiment.
4. Ask each student in the class to drop the bottle top ten times and record their results.

5. Collate the results so you have a sample size of between 200 and 300 tosses. You might use a spreadsheet to sum the frequencies.
   The data might look like this (variation will occur from these results):

   Face Up	Face Down	Face Side
   159	78	13

   What do you notice about the samples of ten tosses? (A lot of variation and Face Up most common)
   Can we estimate the probabilities of Up, Down, and Side? How?
   Percentages or decimals might be used to estimate the probabilities, e.g. 159 + 78 + 13 = 250 (total number of tosses), 159/250 = 0.64 = 64%, 78/250 = 0.31 = 31%, 13/250 = 0.05 = 5%.

6. You might create a pie chart that shows the percentages:

   

7. Discuss: Do these probabilities help work out the best strategy for the Top Drop Game?
   Students should say that there is twice as much chance of getting Up as Down. That shows in the points allocated.
   Is it worthwhile to predict 'Side', at all?
   Side only occurs once in twenty tosses, on average, but a win gives 10 points. A lucky person might get two Sides and win the game.

8. You might investigate approaches to playing the game to see which approach is most ‘winning.’ Ideas might be:

   • Always select ‘down’.
   • If ‘down’, select ‘Up’ next time. If ‘Up’ select ‘Down’ next time. Always select the opposite to the last outcome. Never select ‘Side’.
   • If two outcomes are the same in a row, select the other one (not ‘Side’) e.g., ‘Down’, ‘Down’ then select ‘Up’.
   • Drop it with left hand, select ‘Up’; drop it with the right hand, select ‘Down’. Change hands each turn.
   • Every 4th one choose ‘Up’, every 10th one choose ‘Side’, otherwise choose ‘Down’.

   Students will come up with their own approaches. Look for signs they understand that each toss is an independent event, that is, one toss has no influence over the other. The task might also open up discussion on randomness.
   Are the results unpredictable, or is there influence (bias) on what happens?

Session 5

In the final session use a Figure It Out activity, Left to chance to assess students’ understanding of theoretical and experimental probability. Left to chance also involves coin tosses, so students should bring some understanding from the first session.

1. Read the student page (PDF available) so students are clear about what is required. Point out that you will not be carrying out a class trial, but each student will be able to carry out their own personal trial.
   How will you record the results of the trial?
   How will you create a model of the outcomes to find the theoretical probability?
2. Provide each student with a coin and 16 counters. Slide 12 of PowerPoint 1 can be used to confirm that students understand how the game is played.
3. Give students 20-30 minutes to work on the task, recording their results and thinking as they go.
4. Use the work samples as evidence of students’ understanding of probability. Important points are:
   1. Do students record the results of the 16 trials systematically? e.g., In a table or as an organised list.
   2. Do they notice that the game seems to favour a loss, even though the sample of 16 trials is small?
   3. Do they address the claims of Simon and Steve in a rational manner? 
      For example, they might say that four trials are not enough to provide solid evidence (Steve’s assertion). They might analyse the various ways that each destination in the fourth row might be arrived at (Simon’s false claim). To do so, they recognise that an event, e.g., landing on the second from the left ‘lose’ space, can happen through many different outcomes, e.g. HHT, HTH, THH.
      Do students systematically find each set of outcomes for different destinations in the fourth row?
   4. Do they calculate theoretical probabilities from the set of all outcomes, using fractions, and possibly decimals and percentages?
      P (Three heads) = 1/8 = 0.125 = 12.5% (This means that the probability of HHH equals 1/8)
      P (Two heads and one tail) = 3/8 = 0.375 = 37.5%
      P (One head and two tails) = 3/8 = 0.375 = 37.5%
      P (Three tails) = 1/8 = 0.125 = 12.5%
      Combining the two ‘lose’ events gives a probability of 6/8 or 3/4. The game is definitely not fair!
      You might use Slide 13 to discuss all the possible outcomes from three-coin tosses, and the number of those outcomes that lead to each ‘win’ and ‘lose’ destination.
   5. Do they extend the pattern they notice in the green game to the blue game?
      Students might do this by extending the set of possible outcomes to four coin tosses in a systematic way. See Slide 14.
   6. Do they sum the numbers of outcomes that give a ‘win’ or ‘loss’ in the blue game?
   7. Do they conclude the blue game is fair?
      P (Win) = 8/16 = 1/2 = 0.5 =50%
      P (Loss) = 8/16 = 1/2 = 0.5 =50%

All lessons in this unit follow the same sequence of phases as given in the diagram above. A poster of the phases is provided as Copymaster 1 for students to refer to. The notes suggest possible student ideas and teacher reactions to those responses. It is not feasible to anticipate all ideas students might give so you are encouraged to be flexible in how you respond to students rather than ‘teach’ the sample ideas and representations provided.

PowerPoint 1 gives the equation sets that drive the unit. The sets are labelled in the top left corner of each slide for reference. Slide One has the first pattern to look at. The pattern involves the addition of fractions with the same denominator i.e., 1/b + (b-1)/b = 1, 1/3 + 2/3 = 1. These patterns will be, relatively easy for students to represent. Slide One can be used as an example to familiarise students with the approach. Be aware that students’ existing schemes for addition of fractions may be incorrect, e.g., they may simply add across the numerators and denominators:1/4 + 3/4 = 4/8.

Equation Pairs Set One

Noticing Consistency

Use ‘think, pair, share’ by inviting students to look independently at the four examples, work out the missing values, then share their ideas with a partner. In the class discussion expect students to express their observations in ways that are clear to others. Students should re-express their ideas if others do not understand what they are saying. You may need to remind students that the ‘something going on’ relates to all four examples, not just one. Expect responses like:

S: The first fraction always has one on the top line.

T: Can you tell us more? What is the denominator, the bottom number?

S: The bottom numbers, denominators are different in each equation.

Discussion opens the possibility of using correct mathematical terms, like numerator (number above the vinculum), and denominator (number below the vinculum).

S: The sum (answer) is always one.

T: All four equations have a sum of one? What about the top right equation?

S: The sum is five fifths. Five fifths make one whole.

Making a claim

Encourage students to state a claim about what is going on with all four examples in Pattern One. They might do so individually at first then work in a small team to refine their ideas and the way they express those ideas.

Expect ideas like:

• If the denominators of the fractions are the same, and the numerators add to the denominator, the sum is one.
• If the two numerators add to the denominator, then the sum is one.

Your aim is for students to express their claims in clear, minimal terms, using correct mathematical language. For example, ‘the top numbers’ is not as clear as ‘the numerators’, and ‘add to’ is not as precise as ‘the sum of’.

Representation

In this phase, students choose representations to show why the pattern holds consistently. Students might choose physical manipulatives, such as fraction strips or circles, draw diagrams such as strips and circles, and use contexts from everyday life, such as sharing cakes or pizza. Encourage students to begin with the first two examples of equations then consider how the same relationships might generalise to the last, and other similar, equation pairs.

Examples might be:

• I made 1/2 + 1/2 = 1 using fraction strips then I made 1/5 + 4/5 = 5/5 . Five fifths make one, so I put a one strip below.
   
• S: I used fraction circles to show that 1/8 + 7/8 = 1 and 1/3 + 2/3 = 1.
  

Some representations are less helpful than others in terms of understanding the structure of the fractions that add to one. Circles are useful for showing that one is complete. Strips may make the iteration of unit fractions more obvious. Strips are also easier for students to draw, and divide up equally. Ensure your students have experience with a mix of circle and strip fraction models. It is important for students to recognise that the denominator in each example defines the size of the equal parts that one is cut into. For example, eighths are created by the equi-partitioning of one into eight parts, so eight of those parts make one. Your questioning is important to support students to connect the symbols and other representations.

• Explain where the eighths are in your representation.
• How many eighths are needed to make one?
• Where is the seven, in 7/8 represented in your diagram/model?
• How does your representation show that the sum is always one?

Argumentation

In this phase students are asked to formalise their noticing by creating a statement that generalises to all cases. The discussion may start with a specific equation pair but must be amended so that it accurately represents the general pattern.

S: With 1/8 + 7/8 = 1 the numerators add to eight. Eight eighths equal one.

T: So how does that work in the same way with the other equations?

This might lead to expressing the relationship in general terms.

S: The first numerator plus the second numerator add up to the denominator.

T: If we gave names to the numerator and denominator of the first fraction, like a and b, could we express the property more simply?

Some students might experiment with algebraic notation such as a/b + c/b = 1. Note that this statement is incomplete since it misses the necessary condition for a + c = b.

T: Do we need to say something about a and c?

S: We could just write a + c = b.

T: Give us an example of that.

S: If b = 3, the numerators 1 and 2 add to 3.

Though it is difficult for this level, increased sophistication is evident in a + c = b, and writing the generalisation with only two variables, a/b+ (b-1)/b = 1. It is also important to note that b/b = 1, for any whole number value of b, except zero. b can also be any integer, since rational numbers are of the form a/b, a and b∈ I, that is a and b must be integers. High achievers should be challenged to express their generalisations using algebra. You may need to support them to develop the coding strategies to do that.

Equation Pairs Set Two

If time permits in the first lesson, ask the students to approach the second equation set more independently. From this point each equation set is discussed succinctly using the phases of the approach.

Noticing consistency

The four equations apply the addition of fractions with the same denominator. Students need to recognise that the denominators in both addends and the in sum are the same. Watch for students incorrectly applying whole number addition, that is, adding both numerators together and adding both denominators together. The completed equations should be:

Note that ten eighths equals 1  2/8 = 1  1/4

Making a claim

In natural language expect the students to use phrases like “you just add the top numbers”. Introduce important vocabulary such as numerator, denominator, addend, and sum to clarify which numbers are being referred to in the claims. If the claim is restricted to one example, encourage students to see what changes and what stays constant across the other three equations. The aim is to broaden the claim to the equivalent of “If the denominator of both addends is the same, then the numerator of the sum is the sum of the numerators of the addends.”

Representation

Expect both physical and diagrammatic representations to be used. A fraction strip representation might look like this:

Diagrams of a ‘sets’ representation can be used but it is unwise to use sets at this point (See the diagram below). The changing whole can be confusing, since the aim is to generalise a relationship for all examples.

Whatever representation is chosen by students they need to be clear about the referent one, be it a length, area, or set. The sum must be defined in terms of that original one whole.

Argumentation

Look for students to justify that if the denominators of two addends are the same then the sum is the sum of the numerators, over that denominator. Algebraically that is represented as:

a/b + c/b = (a+c)/b

Note that a, b, and c are variables, and the generalised argument becomes clearer, and is confirmed, by translating values from the equations back into the general form.

For example, with 3/8 + 7/8 = 10/8, a = 3, b = 8, and c = 7.

An interesting discussion point is whether this is any different to addition of whole numbers. For example, 3 + 7 = 10 is the combining of 3 ones and 7 ones, making 10 ones. The ‘ones’ are regarded as the same unit. With 3/8 + 7/8 = 10/8, eighths are the units being combined.

Developing the process further

PowerPoint One contains seven equation sets. The sets might form the basis of a week-long unit. The phases for each equation set are described succinctly below.

Equations Set Three

Noticing consistency

The four equations both demonstrate the multiplication of unit fractions, that is fractions with one as the numerator. Understanding in general that 1/b × 1/d = 1/bd, e.g. 1/3 × 1/4 = 1/12, is fundamental for understanding equivalent fractions.

Making a claim

In natural language, expect the students to use phrases like “multiply the two denominators and you get the denominator of the answer”. Expect the use of mathematical vocabulary such as factor and product to clarify which numbers are being referred to in the claims. Encourage clarity by asking questions like:

• Do the denominators of the factors need to be the same?
• What can you say about the numerators?

The aim is to state the claim as something like “The product of two unit-fractions is a unit fraction, with a denominator that is the product of the denominators.”

Representation

Expect both physical and diagrammatic representations to be used. Fraction strips are static representations:

Folding a paper strip or a rectangle is a dynamic representation:

A sets model may be used but the product needs to be known to establish the possible numbers of objects in the set. Therefore, identifying the referent one is more complex.

Argumentation

Look for students to state the condition that both addends are unit fractions. The starting product might be expressed as 1/b × 1/d. The product is 1/bd. Students should explain why the relationship occurs. Their explanation needs not be in formal algebraic terms, with letters as variables.

The second factor gives the first equal partitioning and creates some parts. The denominator gives the number of equal parts. Then each part is split into equal parts. That number of parts is given by the denominator of the first factor. Therefore, each of the d parts, is equally partitioned into b parts, so the total number of parts equals b x d.

Equation Pairs Set Four

Noticing consistency

The four equations apply equivalence of improper fractions. The equivalent number is kept at three to limit variation.

Making a claim

In natural language expect the students to use phrases like “the answer is always three.” Some students might indicate what they see in a diagram.

Expect the use of mathematical vocabulary such as denominator, numerator, fraction, equivalent, and whole number to clarify which numbers are being referred to in the claims. Encourage clarity by asking questions like:

• What does the equals sign indicate? (both numbers represent the same amount)
• How is the numerator related to the denominator? (the numerator is the denominator multiplied by three)
• Why is the number on the right side always three? (For example, with 24 eighths you can make three ones because you need eight eighths to make each one.)

The aim is to state the claim as something like “A fraction that has a denominator that divides into the numerator three times, is equal to three,” or “A fraction that has a numerator that is three times the denominator, is equal to three,”

Representation

Fraction circles are useful in demonstrating the completeness of a one.

Eight eighths make one. With 24 eighths you can make 24 ÷ 8 = 3 ones.

Students may develop more schematic ways to represent the connection between numerator and denominator.

Five fifths make one so 15 fifths must make three, since 3 x 5 =15

Argumentation

Look for students to justify, using words of diagrams, that the denominator of a fraction specifies the number of equal parts that make one. For example, in 12/4 the parts are quarters, four of them are needed to make one. Dividing the numerator by the denominator finds how many ones can be made. With 12/4, 12 ÷ 4 = 3, so three ones can be made. The equation 12/4=3 represents that equivalence.

For all four equations define the denominator of the first fraction as the variable, say b. The observation that the numerator is always the denominator multiplied by three can be represented as 3b/b. The equivalence can be written as 3b/b=3.

Arguments may be supported by graphics or other representations.

Students might also generalise that dividing the numerator by the denominator tells how many ones can be made, and the remainder becomes the fraction part of the mixed number, e.g. 13/5 = 2 3/5.

Equation Pairs Set Five

Noticing consistency

The equation set applies equivalence, that is when two fractions represent the same value. Understanding equivalence is essential to ordering fractions by size and to the addition and subtraction of fractions. The completed sets should be:

Making a claim

In natural language expect the students to use phrases like “If the numerator is three times the other numerator, then the denominator must be three times as well.” Students tend to notice patterns across the equation like this:

Expect the use of mathematical vocabulary associated with fractions, particularly numerator and denominator. Draw students’ attention to the equals sign as representing the same value on both sides. Students are most likely to see equals as a signal to work out the answer.  Encourage clarity by asking questions like:

• What is the same about all four equations?
• What is different?
• How are the numerators related?
• How are the denominators related?

The aim is to state the claim as something like “Two fractions are equivalent, if the one numerator is a multiple of the other, and the denominator is the same multiple of the other.”

Representation

Static representations might involve length, area (circles or rectangles), or sets.

A more dynamic representation is to fold a paper strip or rectangle into thirds, shade two thirds, then fold the thirds in half. How many sixths are shaded?

Argumentation

Look for students to justify, using words of diagrams, that the same multiplication is performed on both the numerator and denominator of a fraction to create an equivalent (same value) fraction. For example, if the numerator and denominator of 2/3 are multiplied by six this gives 12/18, and 2/3 = 12/18. Of interest is that the multiplier 6/6 is another name for one, the identity element for multiplication. 2/3 x 6/6 = 12/18 is the same operation as 2/3 x 1 = 12/18.

However, the multiplication means different things for the numerator and denominator. Multiplying the numerator by a number increases the number of parts by that factor. For example, 12/18 has six times as many parts as 2/3. For the denominator the effect is the inverse. Each part in the new fraction is one sixth that of the first fraction, since six times as many of those parts make one. For example, 12/18 has parts that are one sixth the size of the parts in 2/3.

In general, the transformation a/b × n/n = an/bn, gives n times as many parts but each part is one nth the size. a/b  and an/bn represent the same quantity. Note that n/n is always equal to one.

Equation Pairs Set Six

Noticing consistency

The equation set highlights the difference of unit fractions. Note that the denominator of the first fraction is one less than that of the second. This equation set provides an opportunity to apply equivalence. The completed sets should be:

Making a claim

In natural language expect the students to use phrases like “the difference has a denominator that is the product of the two other denominators.” Ask students to be more specific in their description.

• What can you say about the two fractions on the left of each equation? (They are both unit fractions and there is a difference of one in their denominators)
• What does subtraction mean in these equations? (It means "What is the difference between the two fractions? ")
• Can you make up another equation that would go in this set?

The aim is to state the claim as something like “If two unit-fractions have denominators that are one different, the difference of the fractions is the unit fraction with the product of the denominators as its denominator.”

Representation

Students are likely to use specific examples to convince others about how the relationships work. Fraction strips are probably the most useful representation.

It is important for students to consider why the fraction one twelfth fits exactly.

Argumentation

Students need to apply equivalence to argue why a given unit fraction is the difference. Their argument might begin for a specific example.

Since one half and one third are involved, the difference must involve sixths, as both one half and one third can be renamed as ‘so many’ sixths. 1/2 = 3/6 and 1/3 = 2/6 so there is a difference of one sixth.

Encourage the students to generalise by naming the denominator of the first fraction as a variable again.

• Can you find a relationship that applies to all four equations?

If the first fraction is 1/b, the second is 1/(b+1),  and the product is 1/(b(b+1)). Algebraic notation, like this, is not expected at Level Four, but students may invent their own ways to express what they notice. Whatever notation students create, it can be checked by substituting numbers in place of symbols, and seeing if the equations created belong in the set. This could be done as a paired activity, and could be used to encourage tuakana-teina.

Equation Pairs Set Seven

Use set seven as an opportunity to see how well students engage in the generalisation process independently. Ask them to record their claims, representations, and arguments in ways that work for them. Some students may prefer writing their work while others may prefer to capture their ideas using a digital recording.

Noticing consistency

The equation set highlights the multiplication of fractions, that is a fraction of a fraction. The numerators of the factors are the same, as are the denominators. That is done to reduce variation. The completed sets should be:

Making a claim

In natural language expect the students to use phrases like “It is a fraction multiplied by itself. The product is the two numerators multiplied over the two denominators multiplied.” Ask students to be more specific in their description, through questions like:

• What do you notice about the two factors (fractions being multiplied)? (they are the same)
• Do you know a mathematical term for a number multiplied by itself? (square of the number)
• Can you use the term square to describe the product fraction in a simpler way?

The claim, using mathematical language, might be something like, “If a fraction is squared, the product is the numerator squared over the denominator squared.”

Representation

An area model is useful since it illustrates the relationship between a number multiplied by itself and a square. Folding and shading squares of paper works well.

Students might also use strip models, but the squaring of numerators and denominators is not obvious.

Argumentation

The equations include two variables, that is the value of numerator and denominator in the fraction being squared. Students need to recognise that the denominator names the number of partitions of one. The numerator names the count of those parts.

Students may be able to show how they are thinking with variables (lack of closure), by treating the numerator and denominator as variables in an area diagram.

Algebraically this might be written as a/b × a/b = (a×a)/(b×b) or more succinctly as (a/b)² = a²/b².

Teachers’ Notes

Encourage your students to choose appropriate forms of calculation, depending on the complexity of the operations and numbers involved. Mental calculation, recording in written form, and digital devices are all legitimate options. In your discussions ask the class how they got the answers and what methods they used to do the calculations.

Session 1

1. In this session, students investigate the cost of hiring a taxi from different companies. Most taxi fares are worked out as a combination of a fixed hire fee, called flagfall, and a cost per kilometre of travel. Therefore, a taxi fare is linear relationship between the kilometres travelled, the independent variable, and cost, the dependent variable. In large cities tolls are often imposed for airports, bridges and express roads. These costs are also added to fares, making comparison of route a rich context. Uber fares are a combination of flagfall, time of the journey, kilometres covered, and a fixed booking fee. Therefore, Uber fares are a combination of two fixed costs, and two variable costs.
2. Discuss the hiring of taxis, using PowerPoint 1.
   Who has ridden in a taxi recently? Why did you go in a taxi? (Slide One)
   How did the taxi driver calculate the fare? (Slide Two)
   Why was there a dollar amount on the meter before you even started off?
3. The flagfall usually shows in the fare before the taxi moves. Extras are for items like tolls or extra baggage.
4. Ask students to solve these problems in pairs or threes. Allow students to use calculators if they wish, though the calculations are manageable mentally.
   Tiki Taxis charge $5 flag fall and $3.00 per kilometre. (Slide Three)
   Discuss the meaning of “per kilometre” as a rate “for every kilometre travelled.”
   Use Slide Four to show the problems:
   Work with a partner to solve the problems together.
   
   How much would it cost you to travel 25 kilometres in the cab?
   How much would it cost you to travel 32 kilometres in the cab?
   How far could you go for $41?
   How far could you go for $77?
   
   Share and korero about your thinking with another pair or group.
    
5. Check the students’ work as you roam the class. Look to see whether they:
   • create, and enact, a correct sequence of operations, for example, 5 + (25 x 3) = $80 for the 25 kilometre fare
   • recognise that the order of operations is important, and how a digital device processes that order
   • use inverse thinking to calculate the number of kilometres for a given fare, for example (77 – 5) ÷ 3 = 24km.
6. After an appropriate time, share answers as a class. Try to generalise the methods to calculate either a fare, when given the distance, or a distance, when given the fare.
   You might use a flowchart like this:
   
   Ask:
   Why is the order that we carry out the operations important?
   How do we find the distance if we are given a fare?
   You might decide to record equations for the flowchart processes. Define the variables; f represents the fare in number of dollars, and d represents the distance in number of kilometres.
   If we start an equation, f=, what do we include to show how the fare is calculated, what fare is equal to? (f = 3d + 5)
   If we start an equation, d=, what do we include to show how the distance is calculated, what distance is equal to? (d = (f – 5)/3). In this case pay special attention to the vinculum in a fraction also being a symbol for division.
7. Ask students to go online and locate a graphing tool or use Excel. Set up a graph of the relation between distance and fare. Since distance is the independent (explanatory) variable, it needs to be measured on the horizontal (x) axis. Fare is the dependent (responding) variable and needs to be measured on the vertical (y) axis. Demonstrate graphing a single ordered pair (25, 80) to show the fare for 25 kilometres (Slide Five).
   
8. Ask your students to graph ordered pairs for the other three answers they have about Tiki Taxis, i.e. (32, 101), (12, 41) and (24, 77). Note that the order of the variables in the ordered pair is important.
9. After students complete their graph (Slide Six) discuss the relation as a class.
   Do you see any pattern? (Student might notice that the graph is linear)
   Why do you think the points lie on a line? (Same rate of $3 per kilometre)
10. Ask students to find other ordered pairs of distance and fare to add to their graph. The task is significant as you want students to recognise that the equation of the graph represents an infinite set of possible discrete points.
11. After a suitable time ask your students to enter the equation f = 3d + 5 (Slide 7).
    What do you notice?
    Can we work out fares and distances using the graph? How?
    What would the fare for 30 kilometres be? ($95)
    What is the distance of a fare that costs $65? (20 km)
    Animating Slide 7 shows how the answers can be read off the axes.
12. Use Slide Eight to introduce this problem:
    Tiki Taxis find they are not making enough money to cover their costs, and to give their drivers a decent wage.
    Which option is better, if they want to make more money?
    Raise flagfall to $10.00
    or
    Increase the rate to $3.50 per kilometre
13. Let students work on the problem in their teams. Roam the room to observe their approach.
    • Do they recognise that increasing cost in any way may mean that Tiki Taxis get fewer customers?
    • Do they understand increasing flagfall by $5 increases all fares by $5, irrespective of the number of kilometres travelled? This makes short trips relatively more expensive. Is that a good thing for the company?
    • Do they recognise that increasing the rate by $0.50 has a more positive effect on income, as more kilometres are travelled?
    • Do they use tables, graphs, or equations to represent the two increase scenarios?
    • Do all students have a chance to korero about their group's thinking?
    • Do all students have a chance to fill both tuakana and teina roles in their group?
14. After a suitable time gather the class to share ideas. Slide Nine shows a graph of the two increase scenarios.
    What does the graph show? (An increase in flagfall of $5 gives more income than an increase in rate of $0.50 per kilometre up to 10 kilometres)
    How far do you think most taxi trips are? (Create some make-believe trips online using Google Maps and use the distance function)
    What recommendations would you make to the owners of Tiki Taxis?

Session 2

1. In this session students look at the time required to roast poultry. They are encouraged to connect the situations of hiring a taxi and roasting a chicken or turkey. Both situations involve linear relations between two variables.
2. Use Slide One of PowerPoint 2 to introduce the scenario of chicken consumption.
   What does 43 kilograms per capita mean? (An average of 43 kilograms for every person)
   Discuss the fact that this might include bones as well as meat, but every person includes little children and elderly people who eat relatively little.
   How many whole chickens make up 43 kilograms? (invite predictions then move to Slide Two)
3. Slide Two shows how the sizes of processed chickens are worked out. Sizes go up in two as weight increases by 0.2 kilograms or 200 grams. Use the information to work how many whole chickens each kiwi eats on average.
   What operation should we perform? (43 ÷ 1.2 = 35.83)
   Therefore, kiwis average nearly 36 whole size 12 chickens per year. That’s three chickens every month!
4. Slide Three shows the way to calculate correct cooking time for stuffed roast chicken.
   How long should you cook a size 12 chicken?
   What calculation should you perform? (1.2 x 55 + 20 = 86)
5. Use an online graphing tool or Excel to create axes of weight (in kilograms) and cooking time (in minutes). Plot the point (1.2, 86).
6. Slide Four shows the graph and plotted ordered pair. Animating the Slide shows how to read off both measures from the point. Check students know what each number refers to, i.e. 1.2 kg and 86 minutes.
7. Ask students to graph other ordered pairs, such as (Slide Five):
   How long should you roast a size 18 chicken for? 1.8 x 55 + 20 = 119
   If a chicken should be roasted for 97 minutes, what mass is the chicken? (97 – 20) ÷ 55 = 1.4
   If a chicken should be roasted for 75 minutes, what mass is the chicken? (75 – 20) ÷ 55 = 1.0
8. Each answer provides another ordered pair to add to the graph. Plot (1.8, 119), (1.4, 97) and (1.0, 75). Slide Six has a completed graph.
   What pattern do you notice? (Linear)
   Why does this pattern occur? (Constant rate of 55 minutes per kilogram)
   How is this situation the same as the rental car situation? (A constant rate and fixed time. The fixed time has a similar effect as flagfall in the rental car situation). Share and korero about your thinking with another pair or group.
   
   Students may be able to relate roasting a chicken to their knowledge of cooking chicken in a hāngi. How long should chicken be cooked for in a hāngi? Does cooking time increase if more chickens are being cooked? Why or why not?
    
9. Slides Seven and Eight introduce this problem:
   In the Aotearoa New Zealand, it is becoming more common to cook a turkey to celebrate Christmas with whānau. Commercial turkeys are much larger than chickens and have a mass of between 3.6 and 10.8 kilograms, etc.
   Let students work on the problems collaboratively with support from digital graphing tools and calculators. Solutions are below:
   • Is the graph of mass and cooking time linear? (Slide Nine)
     The relationship is close to linear.
     
   • Find a rule to work out the roasting time for any mass of turkey. 
     The amount of cooking time goes up by a little over 15 minutes for each extra kilo. For example, between the point (8, 210) and (10, 240), mass increases by 2 kilograms and cooking time increases by 30 minutes. That is 15 minutes per extra kilogram. Students may recognise that their rule should begin as t = 15m but a fixed amount of time needs to be added. They might take any point, say (8, 210), and use it to work out the extra fixed amount.
     8 x 15 = 120
     120 + 90 = 210
     The rule is t = 15m + 90 and can be tested using other known masses and times. For example, 15 x 10 + 90 = 240 minutes.
   • Use your rule to calculate the roasting time for a 7 kilogram turkey.
     15 x 7 + 90 = 195 minutes
   • Do turkeys take less time, per kilogram, to roast than chickens?
     Why might that be?
     Chickens take an extra 55 minutes per kilogram compared to 15 minutes extra for turkeys. Mostly that is because a much larger bird takes way longer in the oven to reach a cooking temperature (90 minutes vs 20 minutes) but having reached that temperature cooks more quickly. There is little difference in the density of chicken and turkey though turkey is a little leaner, has less fat proportionally.
10. Finish the session by posing this problem (Slide 10). Students might work independently on the problem if you want to use the opportunity to assess their understanding of linear relations.
    Kiwis eat about 24 kilograms of pork, per capita, each year.
    Here are the recommended roasting time details for a shoulder roast.
    Put in the oven for 30 minutes at a temperature of 220⁰C.
    Turn the oven down to 180⁰C. 
    Cook the roast for 20 minutes per 500 grams.
    Create a graph to help people work out the roasting time for pork.
    Slide Eleven provides a model graph.
    Ask students to write an equation for the relation between mass and cooking time. (t = 40m + 30)
    
    Have students share and korero about their thinking with each other.

Session Three

1. In this session students compare the costs of two different hire car firms. They make decisions about which firm is best for given distances.
2. Have a general discussion about hire cars.
   Has anyone’s family or friends ever hired a car?
   Why did you need to hire a car rather than use your own?
   How many days did you hire the car for?
   How far did you travel?
   Do you know how much it might have cost?
   What different things did you have to pay for?
3. Share Copymaster 1 with the students. Introduce Question 1 then let students work on the problems in pairs.
   Question 1: Suppose that Rent-A-Bomb charges $25 a day and 30c per kilometre.
   How much does it cost for a two-day hire if you travel 300 kilometres?
   2 x 25 + 300 x 0.3 = $160
   If you use the car for three days, and the hire costs $291 in total, how many kilometres do you travel?
   (291 – 3 x 25) ÷ 0.3 = 920km
   If you travel 900 kilometres, in total, and the hire costs $650, how many days do you hire the car for?
   (645 – 900 x 0.3) ÷ 25 = 15 days

4. After suitable time, gather the class to discuss the answers.
   What different ways did you use to solve these problems?
   Which strategies are the most efficient? Have students share and korero about their thinking with each other. 
   Note that students are likely to use trial and improvement strategies, particularly to problems b) and c). These strategies can be made more efficient by organising data systematically in tables.  For example, b might be approximated by choosing likely numbers of kilometres.

   Number of kilometres	Kilometre cost	3-day hire	Total cost
   300	300 x 0.3 = $90	$75	$165
   1000	1000 x 0.3 = $300	$75	$375
   800	800 x 0.3 = $240	$75	$315
   …	…	…	…

   The equations given represent an efficient solution that comes from recognising a rule for working out the number of kilometres from the cost or the inverse of that.
   A flowchart for a 3-day hire might support students to see the relationships:
   
   To solve problem 1b put $291 into the cost box and work backwards to find the distance.

5. Introduce Rent-a-dent, a firm that charges $40 per day and 15c per kilometre. Discuss the parts of problem 2 then ask students to work in their teams. Allow suitable time for students to work on the problems. Then gather the class to discuss their strategies and solutions. Answers are below:
   • How much does it cost for a four-day hire if you travel 400 kilometres in total?
     4 x 40 + 400 x 0.15 = $220   
   • If you use the car for two days and the total cost is $197 how many kilometres do you travel?
     {197 – (2 x 40)} ÷ 0.15 = 780km
   • If you travel 800 kilometres, and the total cost is $480, how many days do you hire the car for?
     {480 – (800 x 0.15)} ÷ 40 = 9 days
     You might use tables again to organise data and flowcharts to suggest the most efficient ways to calculate the answers.
6. Let’s compare the two firms, Rent-a-bomb and Rent-a-dent.
   To simplify things let’s hire the car for one day.
   Which company is the cheapest to use?
   
   Can you explain your thinking?
   Students might offer ideas like:
   Rent-a-bomb charges less to hire the car but Rent-a-dent charges less per kilometre.
    
7. Go to an online graphing tool or Excel.
   How might we graph the deals from both companies on the same graph?
   What are the variables? (Distance and cost)
   What rule connects kilometres and cost for each company?
   What is the equation for that rule?
   Collectively develop a graph that looks like this (PowerPoint 3, slide 1):
   
8. Send students away in pairs to create their own graph and answer these questions (animate Slide One for questions to appear):
   • Which company belongs to each line? How do you know? Label the lines.
   • Which company is cheapest if you only want to travel only 80 kilometres? How do you know?
   • What does the intersection of the lines mean?
   • Which company is the cheapest to hire from, for one day?

Session 4

In this session students look at mobile phone plans. For many people mobile phones are also their source of emails, music and internet. Plans are developed by companies to attract consumers who meet patterns of phone use. The best plan for one person is not necessarily the best for another. Variables to consider are time in phone calls, number of texts, and amount of data.

1. Have a general discussion about mobile phones.
   Who has their own mobile phone? What do you use your phone for?
   How many times a week would you use it to phone someone?
   How many text messages would you send in a day?
   What does it cost to use your phone?
   Who pays for it?
   The discussion should show that use of mobile phones is variable and that they cost money.
   How do you select the best deal for your phone?
2. Search online for a tool that compares companies in Aotearoa. Browsing through such a site shows how complex the choice of plans and companies is. Slide One of PowerPoint 4 asks:
   What details do you usually look for in mobile phone plans?
   Generally, plans include the amount of data, minutes of calls, and number of texts. Sometimes they include special deals like extra data, memberships to entertainment providers or music channels, and interest free purchase of phones.
3. Use Slide Two to discuss these features. Hotspot is the ability to run data through another device, such as a laptop, in locations where there is coverage.
   Mobile phone plans contain a lot of variables. Let’s simplify the situation by looking at just one variable, data volume.
   What is meant by data? (Pieces of information)
   Slide Three shows the commonly used file sizes.
   A gigabyte equals 1000 x 1000 x 1000 = 1 000 000 000 = 1 billion bytes. To put things into perspective:
   1 GB of data is equivalent to about 210 000 text only emails, or 250 3-minute MP3 videos, or 600 high resolution images. A standard movie DVD contains about 4.5 GB of data.
   Terabytes (TB) are another common unit of data. For example, if you buy an external hard drive for your computer it is likely to be several terabytes in size. One terabyte equals 1 000 GB, which equals 1 000 000 000 000 bytes (1 trillion).
4. Slide Four introduces three different plans.
   What is the same and what is different about these plans?
   Students should notice that everything is the same except the volume of data.
   Is there a relationship between the amount of data and the price of the plan?
   If a plan is exactly the same, but allows 8GB of data, how much should you pay per month?
5. Let students work on the problem. Allow access to online graphing tools. Encourage students to develop rules connecting the amount of data and the cost. Suggest that a graph may be useful. It may be helpful to remind students of the rules developed in Session 1 and Session 2.
6. After a suitable time gather the class and discuss their findings. In general Flight has a base charge for the package of unlimited calls and texts. It charges an extra $8per month for each 1GB of data. A rule might be c = 8d + 13, if c represents cost in dollars, and d represents amount of data in GB. The graph is linear.
7. Slide Five shows three plans from a different company, Bolt, that trades off data for call minutes.
   All of these plans cost the same, but which one is best?
   Justify your choice of plan.
   Let students discuss the plans. They should notice that the relationship between data and call time is negative. That is, when one goes up the other goes down. Choosing one of the plans is likely to be dependent on usual phone usage, preference for internet browsing versus making calls to friends and family.
8. Show Slide Six which has a graph of the relationship between data and call time.
   Is the relationship linear?
   How is the relationship different from the others we have seen?
   Imagine a plan with same cost that has only call time. How many minutes would you get?
   Imagine a plan with same cost that has only data. How many gigabytes would you get?
9. Animating Slide Six shows a trendline that answers the last two questions.
   Find a rule that relates call time to data. Let t represent call time, in minutes, and d represent amount of data, in gigabytes. (t = 240 – 40d)
10. Slide Seven contains an investigation in which students consider their own mobile phone preferences. They choose one plan and justify their choice to a partner.
11. After a suitable time gather the class to discuss their choices.
    Is there a best deal? Explain.

Session 5

In this session we give the students a chance to operate a hire car company.

1. Play the hire car game as given below.
2. Get the class to report back at the end of the session.

The Hire Car Game: Students work in teams of three to operate their own hire car company. Each team has overheads of $20 a day per small car, $25 a day for a medium car, and $35 a day for a large car, whether the car is used or not. Overheads cover the cost of the vehicle, insurance, building rental, cleaning staff, etc. Costs occur each day irrespective of whether the car is hired or not.

1. Each team sets their price for hiring a small, medium, or large car to the rest of the class. For example, they might set the costs of cars as follows:
   Small Car              $35 per day + 0.25c per kilometre
   Medium Car          $40 per day + 0.30c per kilometre
   Large Car              $45 per day + 0.35c per kilometre
   Each company choses its stock level of cars from a minimum of 4 cars to a maximum of 10 cars. The company nominates the size of each car and assigns each car an identification code, e.g. Car 1.
2. Hire time consists of enough rounds that each hire company has a chance to get customers. Three teams of three players become the competing hire companies. Each other player is a customer who rolls two dice, then adds the two numbers that come up. If a customer gets a total of 2, 3 or 4 they must try to hire a small car; for a total of 5, 6, 7, 8 or 9 they hire a medium car; and for 10, 11 or 12 they hire a large car. To work out the number of kilometres they will travel the customer rolls the dice again, totals the two numbers, then multiplies by 15km. For example, a customer might roll 4 and 2 first. 4 + 2 = 6 so they hire a medium car. Next, they roll 1 and 4. (1 + 4) x 15 = 75km which is the distance of their trip.
3. Each customer must hire from the company that gives them the cheapest price for the size of car and kilometres they travel. No friends here – this is business!
4. The company owners keep track of their receipts and bills. Financial records can be audited by IRD (teacher) at any time. A fine of $200 applies to poor financial records. Companies might use a spreadsheet to track income and expenses.
5. If a company is out of cars for any reason, they can (i) give the customer a larger car at the cost of the original sized car; (ii) borrow a car from a competitor at an extra $10 cost; or (iii) pass the client on to another company.
6. Play the game each day for one or two weeks.
7. The winner is the company that make the largest profit.