The purpose of this unit is to engage the student in applying their knowledge and skills of rates to solve problems involving, distance and time within the context of fuel efficiency. This unit integrates with the technology learning area, technical knowledge.

Students develop their skills and knowledge on the mathematics learning progressions; Multiplicative Thinking, and Patterns and Relationships in the context of fuel efficiency.

Students will apply their understanding of proportion, developing their technical knowledge in the context of fuel efficiency and vehicle running costs.

### Structure

This cross-curricular, context based unit has been built within a framework that has been developed with input from teachers across the curriculum to deliver the mathematics learning area, while meeting the demands of differentiated student-centred learning. The unit has been designed around a six session focus on an aspect of mathematics that is relevant to the integrating curriculum area concerned. For successful delivery of mathematics across the curriculum, the context should be meaningful for the students. With student interest engaged, the mathematical challenges often seem more approachable than when presented in isolation.

The first session is an introductory activity that is aimed to spark the imagination of students, to introduce the need for a particular idea or technique in mathematics that would enable them to explore deeper into that context. It is expected that rich discussion may be had around the context and around the nature of the mathematics involved.

The following five sessions are each based around a model of student-centred differentiated learning.

- There is a starting problem to allow students to settle into the session and to focus on the mathematics within the chosen context. These starting problems might take students around ten minutes to attempt and/or to solve, in groups, pairs or individually.
- It is then expected that the teacher will gather the students together to review the problem and to discuss ideas, issues and mathematical techniques that they noticed during the process. It may be helpful to summarise key outcomes of the discussion at this point.
- The remaining group of activities are designed for differentiating on the basis of individual learning needs. Some students may have managed the focus activity easily and be ready to attempt the
*reinforcing ideas*or even the*extending ideas*activity straight away. These could be attempted individually or in groups or pairs, depending on students’ readiness for the activity concerned. The students remaining with the teacher could begin to work through the*building ideas*activity together, peeling off to complete this activity and/or to attempt the*reinforcing ideas*activity when they feel they have ‘got it’. - It is expected that once all the students have peeled off into independent or group work of the appropriate selection of
*building*,*reinforcing*and*extending*activities, the teacher is freed up to check back with the ‘early peelers’ and to circulate as needed.

### Introductory Session

(To engage student interest in the context and to inform teachers of each student’s location on the learning progressions):

Tom commutes daily, driving 25 km each way on the open road. The full cost of driving a car is quoted as being 75 cents per kilometre. How much does Tom’s weekly commute cost him?

In this activity, the teacher(s) will be able to locate their students on the Multiplicative Thinking learning progression by observing how students manage percentages and how they find and express a ratio. This activity integrates mathematical skills and knowledge with the science learning area. Students are solving problems involving motion, using units that are appropriate to the context.

Mathematical discussion that should follow this activity involve:

- What is meant by the full cost of driving a car?
- How is this different to the fuel cost?
- Is 75 c/km a reasonable estimate of the full cost of driving.
- In what situations would the cost be higher?
- In what situations would the cost be lower?

### Session One

Focussing on describing and using a linear trend.

**Focus activity**

A car manufacturer makes the same car in various engine sizes, ie the body of the car is the same, but the engines are different. The graph below shows the average fuel efficiency in km/L of the different engine sizes (cc).

What conclusions can you draw about the relationship between engine size and fuel efficiency?

Discussion arising from activity:

- Which is the most efficient engine?
- Why might someone choose to purchase the less efficient model?
- If the cars have a 40L fuel tank, how far could they drive between fills?
- These data describe average fuel efficiency. What are some ways that the way a car is used will affect its efficiency?
- Note – these data appear to form a straight line. However, only a small range of engine sizes could be considered. Would it be realistic to have a point on this graph that related to, for example, 500 cc?

**Building ideas**

Using the graph of fuel efficiencies (km/L) for different engine sizes, which is based on the data:

engine size (cc) | fuel efficiency (km/L) |

2500 | 10 |

2000 | 11.5 |

1800 | 12.1 |

2200 | 10.8 |

- How far will the 2000 cc model travel on 10 L of fuel?
- How far will the 2500 cc model travel on 10 L of fuel?
- Describe the overall trend of this relationship.
- How far would you expect a 1500 cc model of the same type of car travel on 10 L of fuel?

**Reinforcing ideas**

Using the graph of fuel efficiencies (km/L) for different engine sizes, which is based on the data:

engine size (cc) | fuel efficiency (km/L) |

2500 | 10 |

2000 | 11.5 |

1800 | 12.1 |

2200 | 10.8 |

- Copy, or use a copy of, the graph and rule a line of best fit to show the trend of fuel efficiency versus, for the range of engine sizes shown.
- Write the rule for the trend in these data.
- Given that fuel costs an average of $1.90 per L, what would you expect a 300 km trip in an 1800 cc car to cost?
- How much more expensive would the same trip be in a 2200 cc car?

**Extending ideas**

engine size (cc) | fuel efficiency (km/L) |

2500 | 10 |

2000 | 11.5 |

1800 | 12.1 |

2200 | 10.8 |

- Describe the overall trend of this relationship, for the range of data shown. (Using terms such as linear or non-linear, increasing, decreasing or constant).
- Would you expect this trend to apply for very small (eg 1000 cc) and very large (3500 cc) engines?
- Write a rule for trend shown in these data, incorporating the domain (range of engine sizes this trend applies to).
- Find the range of estimated costs of a 300 km trip for these cars, given that fuel costs an average of $1.90 per L.

### Session Two

Focussing on using rates and proportions to compare data.

**Focus activity**

A test was carried out on a mid sized car, using the three percentage concentrations of octane available in New Zealand petrol stations; 91, 95, 98. The car was first tuned to the fuel concentration, then with 15 L of that fuel concentration in its tank, the car was driven until the tank was empty. This was repeated for each of the fuels and the car was driven until empty on the same journey at similar speeds each time. The results were:

91 octane: The car travelled 230 km.

95 octane: The car travelled 238 km.

If the relationship between octane rating and distance travelled was linear, how far would the car have travelled when the 98 octane was tested?

Discussion arising from activity:

What conditions/variables were held as constant as possible when comparing these fuels?

The test with the 98 octane fuel resulted in a trip of 240 km. Was this what you might have expected? Why/why not?

**Building ideas**

Testing the performance of different fuels, a car was driven on just 15 L of each concentration of octane:

91 octane: The car travelled 230 km.

95 octane: The car travelled 238 km.

98 octane: The car travelled 240 km.

- Find how many km/L the test car can travel on each of these concentrations of octane.
- If 95 octane costs $2.09 per L, what would the cost be of a 100 km trip in the test car using this fuel?

**Reinforcing ideas**

Testing the performance of different fuels, a car was driven on just 15 L of each concentration of octane:

91 octane: The car travelled 230 km.

95 octane: The car travelled 238 km.

98 octane: The car travelled 240 km.

- Is the relationship between concentration of octane and distance the car will travel linear? Justify your answer.
- If 91 octane costs $1.98 per L, what would the cost be of a 25 km trip in the test car using this fuel?
- How many more litres of 91 octane would be needed to travel 25 km than 98 octane in the test car?

**Extending ideas**

Testing the performance of different fuels, a car was driven on just 15 L of each concentration of octane:

91 octane: The car travelled 230 km.

95 octane: The car travelled 238 km.

98 octane: The car travelled 240 km.

On the test day, 91 Octane cost $1.98 per L, 91 Octane cost $2.09 per L, 91 Octane cost $2.16 per L.

Investigate to find which fuel gave the best value for money.

### Session Three

Focusing on finding and describing a rate of change, for data given from a linear trend.

**Focus activity**

A popular petrol fuelled car has average petrol consumption of 7.6 L per 100 km travelled. A similar sized hybrid (electric and petrol) car uses and average of 3.9 L of petrol per 100 km without any further external charging.

How far would the petrol car car travel on 3.9 L of petrol?

Discussion arising from activity:

- How can the hybrid manage to make such a fuel saving without external charging?
- What are the advantages of petrol vehicles over hybrid?
- What are the advantages of hybrid vehicles over petrol?

**Building ideas**

A hybrid car has a petrol consumption of 3.9 L per 100km and uses 95 octane at $2.09 per litre. A petrol car has a consumption of 7.6 L per 100km and uses 91 octane at $1.98 per litre.

- What is the cost of driving the hybrid car for 100 km?
- What is the cost of driving the hybrid car per km?
- What is the cost of driving the petrol car for 100 km?
- What is the cost of driving the petrol car per km?
- Find the percentage savings in the cost per km of fuelling the hybrid car over the petrol car.

**Reinforcing ideas**

A hybrid car has a petrol consumption of 3.9 L per 100km and uses 95 octane at $2.09 per litre. A petrol car has a consumption of 7.6 L per 100km and uses 91 octane at $1.98 per litre. The average driving distance of a single car is 14 000 per km per year.

- What is the cost of driving the hybrid car for 14 000 km?
- What is the cost of driving the petrol car for 14 000 km?
- Calculate the expected average fuel savings for running the hybrid over the petrol car in this example.

**Extending ideas**

Tom has a petrol car, but wants to change to a hybrid. Use the following information to estimate the time it would take for Tom to recover the cost of upgrading from his petrol car to a hybrid.

- A hybrid car has a petrol consumption of 3.9 L per 100km and uses 95 octane at $2.09 per litre. A petrol car has a consumption of 7.6 L per 100km and uses 91 octane at $1.98 per litre.
- The average driving distance of a single car is 14 000 per km per year.
- To trade his petrol car in for a hybrid, it will cost Tom $7 500.

### Session Four

Focusing on finding and describing a linear trend.

**Focus activity**

Tom commutes 25 km each way to work. He currently drives a car that runs on 91 octane ($1.98 per L) and uses 7.6 L of fuel to drive 100 km. What is the cost of his weekly commute?

Discussion arising from activity:

- Tom earns a net weekly pay of $700. Do you consider Tom’s commute to be a reasonable expense?
- What costs other than fuel will he have in order to drive himself to work each day?
- How could he be saving some of the cost of his commute?

**Building ideas**

Tom commutes 25 km each way to work. He currently drives his car spending $38 per week on fuel to commute.

- How far does Tom travel going to and from work in a week?

The full running cost of his car (tyres, insurance, registration, maintenance, etc) is 75c per km.

2. What is the full running cost of Tom’s weekly commute?

3. What proportion of Tom’s $700 per week pay goes to fund his commute?

**Reinforcing ideas**

Tom commutes 25 km each way to work. He currently drives his car spending $38 per week on fuel to commute.

- How does Tom’s weekly commuting fuel costs compare with the estimated running costs of a car at 75c per km?
- What percentage of Tom’s $700 per week pay goes to fund his commute?

**Extending ideas**

Tom decides to reduce the cost of his commute by giving a ride to a colleague who lives nearby. His colleague has offered to pay half of the petrol costs. Use the information below to work out the proportion of the actual cost of the commute his colleague is paying for.

- Tom commutes 25 km each way to work.
- He currently spends $38 per week on fuel to commute.

The full running cost of his car (tyres, insurance, registration, maintenance, etc) is 75c per km.

### Session Five

Focusing on finding and describing a linear trend.

**Focus activity**

Tom currently drives to work. The running cost of his car is 75c/km. If he changed to a hybrid car, the cost of driving would drop to 40c/km. If he changed to an electric car, the cost would be 30c/km. If he rode a bicycle, the cost would be 20c/km.

Show this information on a suitable graph.

Discussion arising from activity:

- Which is the most sensible option for Tom’s commute?
- What factors other than running costs should Tom consider if he is thinking about changing how he gets to work?

**Building ideas**

Tom drives a total of 250 km each week to get to work. The running cost of his current car is 75c/km. If he changed to a hybrid car, the cost of driving would drop to 40c/km.

- Tom works 48 weeks in a year. How far does he travel in his annual commute?
- What is the annual cost of driving his current commute?

What would his annual cost of commuting be if he drove a hybrid car.

How much could Tom save, annually, by switching to a hybrid car?

**Reinforcing ideas**

Tom currently drives 25 km each day each way to get to work. The full cost of his commute is $188 per week. He has decided to get fit and save money by cycling to work. Consider the following points to work out the time it will take for Tom to "break even" if he starts commuting by bike.

- Tom wants to buy a $2000 road bike.
- The average maintenance and repair costs of commuting by bike is 20c/km

**Extending ideas**

Tom drives a total of 250 km each week to get to work. The running cost of his car is 75c/km. If he changed to an electric car, the cost would be 30c/km. and he would need to find $10 000 to fund the replacement and to fit a charging station in his garage.

- Tom works 48 weeks in a year. How much would he save on the annual cost of commuting if he drove an electric car.

Tom would need to find $10 000 to fund the replacement of his current car to an electric one and to fit a charging station in his garage.

2. How many years would it take for Tom to pay the cost of changing from a petrol to an electric car for commuting?