This activity gets students to represent fractions as percentages and to find percentages of money amounts using double number lines and 10 by 10 grids.
Decimals and percentages are special cases of equivalent fractions. For example, 75 percent is equivalent to 75/100, which is equivalent to 3/4 and many other fractions, such as 6/8, 15/20, and 30/40. Percentages can be used either as operators (scalars), to enlarge or reduce an amount by a given proportion, or as proportions in their own right. Just like fractions, they are used as both operators and numbers. For example, “20 percent of 35” treats 20 percent as an operator, and “6/24 is equivalent to 25/100 or 25 percent” treats 25 percent as a number.
The first part of the activity treats percentages as numbers. It uses the hundredths grid to convert common fractions like 1/5, 1/4 , and 1/10 to percentages. The students can draw 10 by 10 grids in their exercise books and shade the appropriate fraction of the grid. Alternatively, you could give them cut-out squares to fold into the fractional parts.
Encourage the students to look for connections between the fractions and the corresponding percentages. Consider these connections:

From their experiences with percentages in contexts such as test marks and sports statistics, many students believe that percentages cannot be larger than 100, since 100 percent represents 1 or the whole. However, there are many real-life contexts, like lambing and calving rates and investment growth, in which percentages greater than 100 do occur. (If a farmer gets twice as many lambs as he has ewes, the lambing percentage is 200 percent.) Encourage the students to generalise this:
1 = 100 percent, so 2 = 200 percent, so 4 = 400 percent, and so on.
Question 2 involves using percentages as operators. The students’ book uses a hundredths grid to show this. If, for example, 30 percent of a grid is shaded, the grid can be used to find 30 percent of any amount:

Note that with the hundredths grid, the value of each small square, or hundredth, of the amount can be calculated if necessary and is equivalent to 1 percent. For example, 1 percent of $65 is 0.65 (65 cents) because 65 ÷ 100 = 0.65. So 32 percent of $65 could be calculated by 0.65 x 32 or by adding 0.65 + 0.65 to 30 percent of $65.
A double number line can also be used to find a percentage of any amount. For example, 30 percent of $48 can be shown as:

Double number lines allow the students to solve difficult percentage problems by connecting and co-ordinating more manageable calculations. The double number line reflects the multiplicative relationship between the fraction and the percentage. For example, consider the problem “35 percent of $56”. The order of calculation might be: 10 percent of 56 is 5.60, so 30 percent of 56 is 3 x 5.60 = 16.80, and 5 percent of 56 is 1/2 of 5.60 = 2.80. 35 percent is 30 percent + 5 percent = $19.60.

For more ideas on how to develop double number lines to solve percentage problems, refer to pages 31–34 of Book 7: Teaching Fractions, Decimals, and Percentages in the Numeracy Project series.

Extension

To challenge the students further, pose problems where the conversions to percentages are not tidy and the students need to approximate the answer. You can do this by making the percentage and the money amount untidy, for example, “find 67 percent of $47.95”. In real-life situations, it is often enough to estimate the answers to such problems. 67 percent is about two-thirds, $47.95 is about
$48.00. 2/3 of 48 is 32. So $32 is a close estimate.
Encourage the students to use rounding and their knowledge of percentage to fraction conversions to help them estimate the answers to percentage problems.
 

Answers to Activity

1. Yvette. 1/5 is 2/10. 1/10 = 10%, so 1/5= 20%.

Leonie
1/2 = 50%, and 1/4 is half of 1/2 , so 1/2 of 50% must be 25%.

Andrew
Yes, you can. 110% would be 1 more than 1.

2. a. Answers and a possible method for each (based on one of the ideas shown) are:
i. $11. 10% or 1/10 of 55 is 5.50. So 20% is 2 x 5.50 = $11.00
ii. $12. 10% of 80 is 8. 5% is 4. 8 + 4 = $12
iii. $52

iv. $21

v. $38. 95% is 100% – 5%. 10% of 40 is 4, so 5% is 2. 40 – 2 = $38.
vi. $60. 100% of 48 is 48. 25% of 48 is 12. 48 + 12 = $60.
b. Discussion and opinions will vary. (All the methods will work.)
c. Answers and a possible method for each (based on one of the ideas shown) are:
i. $14.40. 50% of 36 is 18;
10% of 36 is 3.60; 18 – 3.60 = $14.40
ii. $28. 10% of 35 is 3.50;
3.50 x 8 = $28
or 35 – 3.50 – 3.50 = $28
iii. $15.60. 50% of 24 is 12;
10% of 24 is 2.40; 5% of 24 is 1.20;
12.00 + 2.40 + 1.20 = $15.60
iv. $12

Getting Started

1. Explain to the students that the Board of Trustees/Principal wants a scale model of their school for display in the administration area (or similar). If possible, show students a scale model (it doesn't have to a building).
   Can someone explain what a scale model is?
   How can we create a scale model?
   What will we need to make it?
    
2. Look for students to consider material resources, photographs, and measurement instruments.
    
3. Use Google Maps to get a bird’s eye view of the school layout. Zoom in on the site using the + - symbol and go to photo view. The scale appears in the bottom right of the screen. For example, a 2.5 cm length on screen may represent 20 metres.
    
4. Discuss the meaning of the scale and how it might be used to find the real length and width of each building. As a class, use a ratio table to work out the dimensions of one building. Model the processes involved and ensure a range of students have the chance to contribute. For example:

5. Provide the students with a photocopy of the Google Maps bird’s eye view showing your school buildings. Ask them to work in small groups to work out the length and width of each major building using the scale of 1cm:8m. Model the processes involved, as necessary, and roam to support students as they work.
    
6. After a suitable time bring the class together to discuss strategies and check answers. Review the key calculations involved and any misconceptions.
    
7. Use Google Maps on street view mode to tour the school from a side-on view, as if walking around the school.
    
8. Put the students into small groups. Assign a building to each group. There may be duplications but that is fine.
   I want you to take photographs of the building you have been assigned. You will need enough photographs to construct several views of the building so you can make a scale model. What might you need to consider before you go out?

Students might make a list of considerations such as:

• All four side views are needed. A side might need several shots.
• Use a measurement benchmark so lengths can be estimated. The benchmark might be an object, such as a metre ruler, or a person from the team whose height is known.
• Important details might be focused on, such as doorways, artwork, or objects such as alarm boxes or arches.
   

9. After the discussion, send the groups out with a digital camera to take their photographs. When the groups return save the images to a drive or USB key for safe storage.

Exploring

1. Ask students to estimate the length, width and height of the building they are assigned using the chosen benchmark.
    
2. Discuss the students’ estimates of the height and length of their building and how they arrived at these estimates. What is the height of the tallest building? The shortest? What is the length of the longest building? The shortest?
    
3. Discuss the effect of perspective on the estimates. Structures in the foreground look larger than those at a distance. The students may need to use other visual clues to counter this effect (e.g., vehicles, trees, people).
    
4. Compare the estimates of length and width with the measurement obtained from Google Maps. The images on Google are very accurate.
    
5. Discuss how the actual length and height of the buildings can be measured. The length can be measured using a metre ruler or a tape measure. Elicit some ideas from the students as to how they might measure the height indirectly. Some ways they can measure the height include:
   • One student holds up a metre stick beside the building. Other students walk away from the building until a referent such as a paper clip, or their thumb held at arm’s length, appears to be as long as the metre stick. Using the referent students count the number of paper clips, or thumbs high the building is, then convert this to metres.
   • Take a photo of a metre stick (or other known measure) being held upright beside the building. Use the metre stick as a referent to find the height of the building.
   • Measure the shadow cast by a metre stick. Then measure the building’s shadow at the same time of day. Use the measurements to draw a scale model on grid paper. Read the height of the building from the drawing.
   • Use trigonometry, although that is not expected at this level.

6. Get students, in groups, to choose an appropriate method to measure the actual length, width, and height of their building. Once the measurement is complete put the data in a table. You might stop students after a few minutes, and get them to share their thinking with another group, in order to check that all student groups are on track.

   Building	Length	Width	Height
   Office Block
   Senior Block
   Library

   Was there a large difference between the estimated dimensions of your building from your photographs and the actual measurements? Why may that have happened?

7. Explain that to make a scale model of the school, the model of each building will need to be in proportion.
   What does ‘in proportion’ mean? (The corresponding lengths of the real building and the model are in the same ratio)
    
8. Demonstrate with a hypothetical building as an example. Show Slide One of PowerPoint One
   What is the length, width, and height of this building?
   Use the squares in your book to draw a scale drawing.
    
9. Discuss what is different and what is the same between their scale drawing and the original. Lengths change but angles stay the same. Most importantly the ratios of the lengths within the figure stay the same. Show the students Slides Two and Three to show what happens if the ratios are not kept constant.
    
10. Discuss why the scale used by each group needs to be the same or the model of the school will not look proportional.
    Using the table of actual building dimensions, ask:
    Architects usually use ratios of 1:100, 1:150, 1:300, etc.
    Why do you think they do that?
    The scale needs to allow for easy conversion of lengths and the scale model has to be the right size for viewing.
     
11. As a class, choose a suitable scale.
    What scale can we use for our models that will work for these dimensions? 
    If we use a scale of 1cm:1m (1cm:100cm) how long will out model of the Admin Block be?
    1:100 is feasible as a 44m long building will need a model that is 44cm long.
    1cm:2m is also feasible as it means a 44m long building will need a model that is 22cm long.

12. Record the selected ratio symbolically, e.g., 1:200.
    Create a table of referent measures to support students to make their models. For example:

    Actual dimension	Model dimension
    2m	1cm
    10m	5cm
    20m	10cm

13. Ask students to work in their groups to determine the dimensions of their scale model.
     
14. Students will need to develop scale drawings (blueprint) of their building including length, width, and height. Discuss features such as doors, windows, and other features of their buildings.
     
15. Once the blueprint has been completed get the students to reflect on their drawings. 
    Are the proportions correct? 
    Will your building fit with the other buildings in the school model?
     
16. Provide time for the students, in groups, to construct their school buildings. Some students will need support to construct nets for the models. The Unit called Representing 3D models in 2D drawings provides useful lessons on creating nets for models.
     
17. Models can be painted with the colours of the actual buildings.

Sharing

1. Allow the students to share and reflect on the scale models.
   Convince another group that your scale model is in proportion to their model.
   What things would you change if you were going to create the model over again?
    
2. Assemble the scale models of buildings into a complete scale model of the school. Include playing fields, paths etc. This is an exercise about scale as well since the areas and locations of the building must also be in proportion to the scale that is used.
   How can we locate each model in the correct place? (Using a co-ordinate system is useful way to do that.)
   How might we use the Google Map image to guide where we locate each model building?

Session 1

In this session we look at the idea of ‘twice as many’ and its variations. There are several ways to demonstrate the idea of scaling using multiplication. Adapt the contexts reflected in these problems to situations that work with the resources in your classroom, and that appeal to the interests of your students.

1. Pose these problems. Use physical materials or diagrams to represent the objects in each problem.
   1. Here I have 6 tennis balls. Room 6 have asked for twice as many tennis balls as that. How many tennis balls do they want?
   2. Mary has nine coloured pencils. There are twice as many pencils in this desk. How many pencils are in the desk? 
   3. Petra has coloured three squares yellow. If we want to have twice as many red squares as yellow squares, how many squares do we have to colour red?
      
   4. Laslo earns $43. Martine earns twice as much as Laslo. How much does Martine earn?
       
2. Ask: What do these problems have in common?
   Look for students to identify that all four problems involve “twice as many.”
   What operation did you use to solve the problems?
   Record the operations using multiplication:
   2 x 6 = 12        2 x 9 = 18        2 x 3 = 6            2 x 43 = 86
   Record the ratio 1:2
   What does this ratio mean? How is it related to the problems?
   Take an example, For every dollar that Laslo earns, Martine earns two dollars.
    
3. Get the students to think about ‘three times as many’ situations. Ask them to make up ‘three times as many’ problems for others to solve. You could ask students to frame these within the same context that was reflected in the previous problems. For example, three times as many girls as boys are in the dance troupe. If there are 7 boys, how many girls are there? 
    
4. After a suitable time gather the class and share some of the problems. Physically model examples where the numbers are a reasonable size. The boy to girl problem might be shown like this, with blue cubes for boys and yellow cubes for girls:
    
   There are 7 boys. How many girls are there? (21)
   How many dancers are in the troupe altogether? (28)
   What fraction of the troupe is made up of boys? (7/28)
   What fraction of the troupe is made up of girls? (21/28)
   What ratio is there of boys to girls? (1:3)
    
5. Show how splitting the model can show the 1:3 relationship.
   
   Can you simplify the fractions for boys and girls? (1/4 and 3/4).
    
6. Tell the students that you are going to practice multiplying different numbers. Multiplication is fundamental to working with ratios. Divide the class into pairs to play the ‘times as many’ game. Give each pair of students a dice and ask each student to record the following in a table (or provide students with a copy of the following table):
   
    Players take turns to roll the dice and nominate which cell they will fill in. Suppose a player rolls four and nominates 6 x then they put ‘six times as many as four’, 24, on their score card.
   
   Play continues until both players fill their scoreboard. Once a ‘times as many’ option is used it cannot be used again. The player with the highest total wins.

Session 2

In this session, the concept of ratio is applied in more depth. Continue to adapt the contexts reflected in this session's problems to better suit the interests, cultural backgrounds, and current learning foci that are relevant to your students.

1. Use the ‘times as many’ experience to explain the meaning of ratio.
   If I have 10 blocks and you have 5 blocks I have twice as many as you and we say that the ratio of the number of blocks I have, to the number you have, is 2:1 – two to one.
    
2. Give several examples of situations where objects are in the ratio 2:1 and 3:1
   Examples might be mixing apple juice and orange juice in a 3:1 ratio or a boy to girl ratio of 2:1 in the football team.
    
3. Once you’ve introduced the idea ask students to think of situations where the ratio is 2:1, 3:1, 4:1. It is important that the situation involves many objects not just the base ratio. For example:
   In the punch bowl are 8 litres of apple juice and 4 litres of orange juice.
   What is the ratio of apple to orange?
   Hone kicked over 9 goals but missed 3. What was the ratio of goals to misses?
    
4. Share the situation students create. Ask students how they can tell whether things are in the ratio 2:1 or 3:1. Expect answers like:
   “There is twice/three times as many of one thing than the other.”
    
5. Let students work in their pairs again. Ask them to draw a couple of examples where the ratio of yellow squares to red squares is 2:1 and 3:1.
   Can you use more than ten squares in each ratio picture?
   Secondly, ask students to make up their own ratios and draw yellow and red squares in that ratio.
    
6. Get the class together to discuss this second task. Try to extend their concept of ratio from the simple cases of 2:1 and 3:1.
   Is it possible to have ratios 3:2?
   What does this mean?
   Give me some examples.

7. Suppose that Hare has apples to oranges in the ratio of 2:1 and oranges to bananas in the ratio of 2:3.
   If Hare has 15 bananas, how many apples does he have? 
   Let students attempt the problem in pairs.
   A solution looks like this:
   Look at the ratio of oranges to bananas:
   2:3 = □:15 meaning that 2:3 and □:15 are equivalent ratios.
   □ = 10 because 15 ÷ 3 = 5 and 2 x 5 = 10. Hare has 10 oranges.
   Now consider the ratio of apples to oranges:
   2:1 = □:10 meaning 2:1 and □:10 are equivalent ratios.
   There are twice as many apples as oranges, so the number of apples equals 20.
   A ratio table can be used to track the steps. Model the construction of this, and allow students the opportunity to contribute to the filling out of the table.

   Apples	Oranges	Bananas
   2	1
   	2	3
   20	10	15

8. Pose a similar problem with other ratio for students to work out. For example:
   Monkey has apples to oranges in a ratio of 1:3 and oranges to bananas in a ratio of 2:5.
   If monkey has 12 oranges, how many apples and bananas does she have?
   A ratio table of the problems is shown below. Continue to model the construction of this, whilst gradually releasing responsibility and asking students to fill out an increasing amount of the table.

   Apples	Oranges	Bananas
   1	3
   	2	5
   4	12	30

9. For homework, challenge the students to find examples of ratios in real life.
   Ask your parents or caregiver when they use ratios.

Session 3

In this session we look at more situations with ratios.

1. Collect in students' examples of ratios from real life. 
    
2. Review the learning from the last session with the following problem:
   Suppose that Hare had apples to oranges in the ratio of 2:1 and oranges to bananas in the ratio of 2:1.
   What is the ratio of apples to bananas?
   How can you convince me of this?
   Is the ratio the same no matter how many apples Hare has? Yes. 
   Show this by example.
   If Hare has 12 apples, how many bananas does he have? (3)
   If Hare has 20 apples, how many bananas does he have? (5)
   If Hare has 16 apples, how many bananas does he have? (4)
   How can you explain this?

3. Use a ratio table to show why the number of bananas is always one quarter of the number of apples. Model some of the construction of this, or ask students to complete in independently or with a partner.

   Apples	Oranges	Bananas
   2	1
   	2	1
   4	2	1

   In general, this can be shown algebraically as:

   Apples	Oranges	Bananas
   a	1/2 a
   	1/2 a	¼ a

   The ratio connecting the number of apples, oranges, and bananas can be written as 4:2:1.

4. Ask the students to work together on the problems on Copymaster 1. 
    
5. After a suitable time gather the class to share solutions. Note that there is more than one answer as the total number of squares can be varied.
    
6. Share the examples of ratios that students found from real life. Use an example of a practical situation to pose a problem (perhaps draw a scale diagram in the ratio of 10:1). Model solving the problem and record the key steps of the process on a class chart/on the board.
   
7. Ask students to use one or more of the contexts that they (and their peers) have contributed and, in pairs, create a ratio problem for another pair of students to solve. Roam and support students to ensure the problems can be solved. Ask students to solve a number of the problems posed by their peers, and then gather as a class to discuss the solutions, strategies used, challenges, and problems that they found particularly engaging.

Session 4

In this session students investigate the link between ratios and part-whole fractions.

1. Get the students to work in pairs to draw and colour squares in two colours in various ratios of their choosing. (2:1, 3:1 and 5:2 would be fine.) Three different ratios are enough.

2. Bring the class together with their drawings.
   Choose an example of a simple ratio (2:1 would be ideal).
   How many squares are red?
   How many squares are blue?
   How many coloured squares are there altogether?
   What fraction of the squares is red?
   What fraction of the squares is blue?
   Note with the last two questions there are different ways to express the part-whole fractions. For example, in the ratio 8:4 the fraction that is red is 8/12 or 4/6 or 2/3 and the fraction that is blue is 4/12 or 2/6 or 1/3.
   Start to draw up a table like the one below.

   ratio	No. red squares	No. blue squares	No. of squares	Fraction of red	Fraction of blue
   2:1	10	5	15	10/15 = 2/3	5/15 = 1/3
   2:1	8	4	12	8/12 = 2/3	4/12 = 1/3

3. Ask students for different red to blue square mixes that are in a ratio of 2:1. Illustrate how the given ratio reduces to 2:1 using diagrams. For example, consider 6:3:
   

4. What patterns do you see? (There are always twice as many red squares as blue squares. The same simplified fractions occur. There can be different total numbers of squares)

   ratio	No. red squares	No. blue squares	No. of squares	Fraction of red	Fraction of blue
   3:2	6	4	10	6/10 = 3/5	4/10 = 2/5
   3:2	15	10	25	15/25 = 3/5	10/25 = 2/5
   3:2	30	20	50	30/50 = 3/5	20/50 = 2/5

   What patterns do you see? 

5. Ask students to take one of the equivalent ratios to 2:3 and show how it simplifies by drawing pictures.
    
6. Finish the session with this puzzle:
   I have a mixture of black and white marbles in a ratio of 3:5.
   There are 64 marbles in total. How many marbles are there of each colour?

Session 5

In this session students consider how part-whole fractions can be expressed as ratios.

1. Remind the class of what has happened so far in this unit.
   If someone was sick this week and needed to catch up, what would you tell them about ratios?
    
2. Pose this problem for students to solve in pairs then share as a class.
   Leah takes a packet of mixed nuts from the pantry.
   She notices that one fifth of the nuts are brazil nuts, one third are hazelnuts, and seven fifteenths are cashews.
   There are 30 nuts in the packet. What is the ratio of brazil nuts to hazelnuts to cashews?
   Make sure the word version of the problem is visible and let students work in small teams to solve the problem. Provide materials, like counters if needed.
    
3. Roam the room as students work. Look for the following:
   • Do students use trial and error? If so, direct them to address the clues.
   • Do students approach the task systematically and record their attempts? If not, suggest they use a table or ratio symbols.
   • Do students recognise that if one fifth of the nuts are brazils then the total number must be a multiple of five?
   • Do students recognise that if one third of the nuts are hazels then the total number must be a multiple of three?
   • Do students recognise that the fraction “seven fifteenths” gives a strong clue about the total number of cashew nuts? (7/15 = 14/30)
4. After a suitable time ask groups to prepare to share their ideas. Preparation might include writing solutions, or showing working, including diagrams. Gather the class to share ideas and address the questions above.
   The solution is a ratio of Brazil:Hazelnut:Cashew of 6:10:14 that can be simplified to 3:5:7.

5. Show the students a table containing important information in the problem but with some omissions to complete.

   Total number of nuts	Fraction of brazil nuts	Fraction of hazelnuts	Fraction of cashews	Number of brazil nuts	Number of hazelnuts	Number of cashews
   30	1/3		7/15		10

   If you had this information about Leah’s bag of nuts, how would you complete the cells?

   Students should comment that 1/3 of 30 equals 10 and 7/15 of 30 equals 14. They should also notice that 10/30 = 1/3.

6. Tell students that you are going to "go nuts" and give them another problem. They will need a blank table to keep track of the clues and find the answer.

   Total number of nuts	Fraction of brazil nuts	Fraction of hazelnuts	Fraction of cashews	Number of brazil nuts	Number of hazelnuts	Number of cashews

   Three fifths of the nuts are hazelsnut.
   There are 15 brazil nuts.
   The total bag of nuts holds 100.
   What is the ratio of brazils:hazelnuts:cashews?

   Total number of nuts	Fraction of brazil nuts	Fraction of hazelnuts	Fraction of cashews	Number of brazil nuts	Number of hazelnuts	Number of cashews
   100		3/5		15

7. Let students work in groups to solve the problem. Roam the room to see if the students are explaining ideas to each other and justifying those ideas. For example:

   The number of hazelnuts is 3/5 of 100. That is 60 hazelnuts because 1/5 of 100 equals 20.

   Total number of nuts	Fraction of brazil nuts	Fraction of hazelnuts	Fraction of cashews	Number of brazil nuts	Number of hazelnuts	Number of cashews
   100		3/5		15	60

   That means the number of cashews equals 100 – 15 – 60 = 25.

   Total number of nuts	Fraction of brazil nuts	Fraction of hazelnuts	Fraction of cashews	Number of brazil nuts	Number of hazelnuts	Number of cashews
   100		3/5		15	60	25

   The fractions are 15/100 = 3/20 (brazil nuts) and 25/100 = 1/4 (Cashews).

   Total number of nuts	Fraction of brazil nuts	Fraction of hazelnuts	Fraction of cashews	Number of brazil nuts	Number of hazelnuts	Number of cashews
   100	3/20	3/5	1/4	15	60	25

   The ratio of Brazil: Hazelnut: Cashew is 15:60: 25 = 3: 12: 5.

8. Ask different students if they have noticed any connection between fractions and ratios. Let them use a specific example, like the ratio of Brazils: Hazelnuts: Cashews of 2:3:5. What fractions are in that ratio?
   
   Drawing a diagram will illustrate that the total of the numbers forms the denominator, and the fractions are formed by using each number as a numerator.
   In the 2:3:5 ratio the denominators are 2 + 3 + 5 = 10.
   The fractions are 2/10, 3/10, and 5/10. 2/10 and 5/10 can be simplified to 1/5 and 1/2.
   
   
9. Ask students to create their own mix of Brazils: Hazelnuts: Cashews and record their mix as fractions and as a ratio.

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.

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.

Using Materials

In Mixing Colours,  a sequence was provided for introducing the part-whole relationships in ratios and expressing them as fractions and percentages.  The lesson for Extending Mixing Colours should initially follow the same sequence.  The notes below show how the Number Properties phase of the lesson can be made  more complex.

It is important for students at the Advanced Proportional stage to have multiplicative strategies to compare ratios without needing to replicate the ratios until a common part or whole is found, e.g., at earlier stages, to compare 2:4 with 3:5, both are duplicated (2:4 = 4:8 = 6:12 … and 3:5 = 6:10 …) until a common part of 6   measures of A is found.

Using Imaging

Follow the steps on the next page to create a computer spreadsheet that allows ratios to be displayed simultaneously as a table and connected graph. This spreadsheet can be accessed online by downloading Material Master 7–10.   Create a table like this:

2. For each mixture, create a pie graph and embed it in the spreadsheet rather than save it as a separate file. This means that any changes made to the cell entries are transferred to the pie chart.

The spreadsheet can then be used to compare mixtures visually and numerically. In the example above, the proportion of blue changes while the actual amount of it stays constant (two units).

Ask the students how the percentage of each colour was calculated and why this made the comparison easier. Percentages provide a common denominator of 100 for the equivalent fractions. In Mixture A, there are 2 units of blue out of a total number of 2 + 5 = 7 units.  2/7 = 2  ÷ 7 = 0.2857 = 28.57%.

Pose other problems requiring students to compare ratios. Ask the students to anticipate which mixture gives the darkest shade of green before the spreadsheet is used to confirm. (A blue and yellow mixture produces green. The greater the proportion of blue, the darker the green will be.)

Good examples are (blue:yellow):

Using Number Properties

Provide similar problems but allow students access only to their calculators rather than to any visual display. If confusions arise, the spreadsheet can be used to check  solutions. Expect students to justify their calculation steps and encourage estimation and use of leveraging from known benchmark fractions, e.g., 5/11 > is slightly less than one-half or 50%.

Further problems are:

Using Number Properties

Problem: “Mrs Norris hires students to work on her garden. She agrees to pay them $400 total.

Diane, Sonya, and Mere work 13 hours, 10 hours, and 17 hours respectively on the garden. How much pay will Sonya receive?”

Discuss why the ratio of hours 13 : 10 : 17 leads to Sonya receiving $100. 
(Answer: 13 + 10 + 17 = 40. 10/40 x 400 = 100.)

Problem: “Two friends buy a house that costs $193,500. Bernie contributes $89,500,  and Bronwyn contributes $104,000. Five years later, they sell it for $234,000. How  much money should each of them receive?
(Answer: Bernie should receive 89500/193500 x 234 000 = $108,233 (to the nearest  dollar), so Bronwyn should receive 234 000 – 108 233 = $125,767.)

Examples: Worksheet (Material Master 8–29).

Understanding Number Properties:

John and June buy a house together. John contributes $g, and June contributes $h. When they sell the house, they receive $t altogether. How would you work out how much John should receive as his share? (Answer: $(g/g+h?) x t.)