The purpose of this unit is to engage the student in applying their knowledge and skills of fractions, decimals and percentages to identify a linear trend.

Students develop their skills and knowledge on the mathematics learning progressions; Multiplicative Thinking, and Patterns and Relationships, in the context of film and television program timing.

Students will apply their understanding of proportion and ability to solve problems involving fractions, decimals and percentages. They will look for and generalise the linear relationships described in the given context, film and television program timing.

### Introductory session

(This activity is intended to motivate students towards the context/integrated learning area and to inform teachers of students' location on the learning progressions):

An expert in filmmaking, Michael Hauge, suggests that most Hollywood movies follow the same structure. He says there are five turning points that occur in the same place in each film. These are:

- The opportunity, occurs when the film is 10% of the way through.
- The change of plans, occurs when the film is 25% of the way through.
- The point of no return, occurs when the film is 50% of the way through.
- The major set back, occurs when the film is 75% of the way through,
- The climax, occurs when the film is 90% of the way through.

A group of students are planning to make a 24 minute film about a very short boy who, wants to go to the school dance, but is too shy to ask a short girl he is keen on outright so leaves a note in her locker, but then finds out they got the wrong locker and a different, very tall girl accepts the invitation. The climax is on the night of the dance, whether the short boy turns up to take his tall date to the dance or not.

- Use the Hauge model to plan the the 24 minute sequence.
- How many sections are there separated by the turning points?
- What is the ratio of the sections of film separated by the turning points?

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 arts learning area. Mathematical discussion that should follow this activity involves:

- How could the structure of a film, with these five turning points be represented graphically.
- If a feature film of two hours followed the same structure, when would each of the five turning points occur?
- In reality, the turning points take time to unfold. Is there room in a short film for the five turning points?
- Sometimes a film is criticised because ‘nothing ever happens’ or because it was ‘long and drawn out’. Discuss how the film may be structured, in terms of its turning points, if such criticism is justified.

### Session two

Focussing on using rates and ratios, reasoning with linear proportions to solve a problem.

**Focus activity**

- The opportunity, occurs when the film is 10% of the way through.
- The change of plans, occurs when the film is 25% of the way through.
- The point of no return, occurs when the film is 50% of the way through.
- The major set back, occurs when the film is 75% of the way through,
- The climax, occurs when the film is 90% of the way through.

- The structure is given as a percentage. How could this be used to show the number of minutes?
- What proportion of the film has run when the change of plan occurs (20 minutes in)?

**Building ideas**

Use the 5 turning point model for the structure of a film to say how long a film will run for in each of the following scenarios:

- the ‘opportunity’ occurs 6 minutes into the film.
- the ‘change of plan’ occurs 15 minutes into the film.
- the ‘point of no return’ occurs 30 minutes into the film.
- the ‘major set back’ occurs 45 minutes into the film.

**Reinforcing ideas**

- the ‘opportunity’ occurs 15 minutes into the film.
- the ‘change of plan’ occurs 21 minutes into the film.
- the ‘point of no return’ occurs 37 minutes into the film.
- the ‘major set back’ occurs one hour into the film.

**Extending ideas**

- the ‘opportunity’ occurs 15 minutes before the ‘change of plan’.
- the ‘change of plan’ occurs 21 minutes before the ‘point of no return’.
- the ‘point of no return’ occurs 37 minutes after the ‘opportunity’.
- the ‘major set back’ occurs one hour after the ‘opportunity’

### Session three

**Focus activity**

One hour of television viewing time typically has only 14 minutes, 15 seconds of commercials. If there are three ad breaks in thirty minutes of television viewing and an average of five commercials per break, what is the average (mean) length of a commercial?

Discussion arising from activity:

- Summarise what we know from the problem.
- What is the best way to go about solving this problem?

**Building ideas**

- How many ads of 15 s can fit into this time?
- Is it possible to fill this time with only ads that are 30 seconds long?

**Reinforcing ideas**

- How many ads of 45 s can fit into this time?
- If some ads are 45 s and some are 20 s, find at least two different ways that a combination of 45 s and 20 s ads could fit into the 14 minutes and 15 seconds of commercial time.

**Extending ideas**

### Session four

**Focus activity**

One hour of tv viewing time typically has 14 minutes, 15 seconds of commercials.

Five years ago, television viewing averaged 13 minutes and 30 seconds of commercial time per hour.

Ten years ago, television viewing averaged 12 minutes and 45 seconds of commercial time per hour.

Describe the rate of change of commercial time, per hour of television viewing over the past ten years.

Discussion arising from activity:

- Is the commercial time increasing or decreasing?
- What is the difference in the time spent on commercials during an hour of TV ten years ago, five years ago, now?
- What is the rate of change in the time spent on commercials during an hour of TV?

**Building ideas**

- now
- five years ago
- ten years ago.

**Reinforcing ideas**

- Find the change in average amount of programme time in an hour of television viewing between now and ten years ago.
- Find the rate of change, per year, of average amount of programme time in an hour of television viewing between now and ten years ago.

**Extending ideas**

Using the data on the typical amount of time over an hour of television time given to commercials, write a rule to find the expected average amount of programme time in an hour of television viewing x number of years from now.

### Session five

**Focus activity**

- Summarise what we know from the problem.
- If a pattern is decreasing, how is this shown in the rate?
- What would this pattern look like on a graph?

**Building ideas**

Years ago | 30 | 25 | 20 | 15 | 10 | 5 | 0 |

Average length of a tv promo ad |

**Reinforcing ideas**

- Five years ago
- today
- In five years’ time.

**Extending ideas**

### Session six

**Focus activity**

Thirty years ago, the average length of a feature film was 108 minutes. The average length of feature films has been increasing at a steady rate so that now it is 130 minutes. At what rate (in minutes per year) is the average length of feature film increasing?

Discussion arising from activity:

- How did you find this rate?
- Is it likely that the length of feature films will always been increasing at this rate? Why/ why not?
- Is it likely that the length of feature films have always been increasing at this rate? Why/ why not?

**Building ideas**

Years ago | 30 | 25 | 20 | 15 | 10 | 5 | 0 |

Average length of a feature film |

**Reinforcing ****ideas**

- Five years ago
- today
- In five years’ time.

**Extending ideas**