# Algebraic equations: Video and teaching guide

Use this video as a springboard to explore algebraic thinking, and to apply that thinking to a financial context, drawing on reasoning and mathematical modelling.

The video uses the scenario of three young people sharing a taxi ride, each travelling a different distance, and working out a fair way to share the cost. The friends need to decide on a departure time based on taxi fares that change rates after 11:00 pm. We show how to develop algebraic equations to calculate the taxi fare and generate tables of values which are plotted on a Cartesian plane to observe the relationship.

Watch the video below with your students (full screen recommended).

Video duration: 4 min 37 sec

## Suggestions to follow up video

##### Investigating rate

**Objective:** To understand that taxi fares have two components: flag fall and rate per kilometre, and make connections between cost of trip and distance travelled.

In the video, three friends are in a dilemma about when to depart, as the different taxi fare fee structure impacts the overall cost. The two components of a taxi fare are explained. Two different rates are shown that differ depending on time of departure. Travelling after 11:00 pm incurs a more expensive rate but a cheaper flag fall.

**Role of teacher:** Make explicit that there are two components to the taxi fee structure: the flag fall cost and the cost based on the number of kilometres travelled. Ask which of these two values is constant (fixed) and which is variable? The constant is the flag fall (no matter how far you travel you will always pay $10.00/$5.00) and the variable is the number of kilometres travelled – the more kilometres travelled the higher the cost of the trip. Ask which fare they would choose and why.

##### Equations and graphing

**Objective:** To represent the relationship between two variables (cost and distance). Create algebraic equations from word problems involving one or more operations.

To assist in solving the problem and determine the cost of each fare, three different methods are presented to demonstrate the relationship between two variables: the cost and the distance travelled.

**A table of values**

**An equation**

**A graph**

**Role of the teacher: **make explicit that the power of algebra is its ability to generalise the total fare cost (y) for any flag fall value and distance travelled (x). Work through how to write each fare algebraically y=5+3x and the other is y=10+2x. Explain that y is the cost, x is distance travelled.

Summarise and identify costs at the intervals for distance. For example, for a trip of 40km (x = 40) the total cost of each trip would be: before 11:00 pm: 10 + (2 × 40); after 11:00 pm: 5 + (3 × 40).

Present the values in a table to use the functional reasoning.

Table for y = 10 + 2x.

x | y |

0 | 10 |

10 | 30 |

20 | 50 |

30 | 70 |

40 | 90 |

##### A more complex equation (Algebraic equations)

**Objective:** To represent the relationship between two variables (cost) and (distance). Create algebraic equations from word problems involving one or more operations.

In the video, the problem of how to share the ride’s cost fairly is introduced. Each passenger travels a different distance. Ask the students to think about different ways the fare could be shared:

- each person pays a third
- each person pays a portion depending on their fraction of the total ride taken.

The proportional distance each person travels.

The fair split equally.

Another way to split the fare is based on the portion each person travels.

**Role of teacher:** make explicit that the steps to create the equation:

x + 2x + 3x = 90

The video also includes another way to write the equation as an option for investigation:

13 x + 23 x + x = 90.

##### Reflection (Algebraic equations)

Ask students to reflect on their investigations related to taxi fares.

What makes the biggest difference to cost: a fixed cost or a cost that changes depending on a variable? Explain your thinking. When might this not always be the case?

A new taxi driver comes to town and charges $2.00 flag fall and $5.00 per km. What would it cost to travel 3km and 30km?

Is this fee structure good for short trips or long trips? Explain your answer.