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### Introduction to slope

In this animated presentation students are introduced to the mathematical concept of slope (or gradient) of a straight line. The mathematical definition of slope is explained in terms of, and derived from, two practical contexts.

### EagleCat: linear graph

Explore the graphs of linear equations in the form y = mx + c. Observe changes to the gradient and y-intercept under various transformations. Alternately, change the equation and observe changes in the y-intercept or change the y-intercept and see how the equation changes.

### Triathlon: elite triathletes

Compile a news article reporting results of a major triathlon. Compare graphs of four triathletes' performance (distance against time) for the swimming, running and cycling legs of the course. Interpret the graphs to answer questions about each of the race legs and the overall performance of the four triathletes. Insert ...

### Triathlon: the course

Compile a news article reporting results of a major triathlon. Look closely at graphs of a triathlete's performance (distance against time and distance against altitude) for the cycling leg of a course. Interpret the graphs to answer questions about the triathlete's peformance and the altitude of the course. Insert your ...

### Triathlon: distance-time graphs

Watch triathletes performing in swimming, cycling and running legs of a triathlon. See how distance-time graphs are used to represent and compare race performances. Examine axis units, scales and gradients to compare course and performance variables. For example, notice that a flat line on a graph shows that an athlete ...

### Triathlon: triathlete

Compile a news article reporting results of a major triathlon. Look closely at graphs of a triathlete's performance (distance against time) for the swimming, running and cycling legs of the course. Interpret the graphs to answer questions about each of the race legs and the overall performance of the triathlete. Insert ...

### Differential calculus: linear graphs

Observe the linear distance–time graph of a rocket travelling at a constant velocity. Calculate the average and instantaneous velocity of the rocket over different time intervals. Notice how as each time interval becomes smaller the rocket's average velocity is equivalent to its instantaneous velocity. Work out how the ...

### Differential calculus: non-linear graphs

Observe the non-linear distance–time graph of a rocket travelling at a changing velocity. Calculate the average and instantaneous velocity of the rocket over different time intervals. Notice how as each time interval becomes smaller, the rocket's average velocity approaches its instantaneous velocity. Use the slider to ...

### Differential calculus: the derivative

Observe the non-linear distance–time graph of a rocket travelling at a changing velocity. The distance, s, travelled by the rocket after t seconds is determined by the formula s(t) = t². Calculate the average velocity of the rocket over time intervals that become progressively shorter. Tabulate the results and look for ...

### Differential calculus: power functions

Observe the non-linear graphs of various power functions (such as f(x) = x², or f(x) = x³) and select the expressions for finding the gradient of the secant between small changes in x represented by Δx. Tabulate the values of f'(x) and plot the derivative of each function. Determine the pattern between the graphs for each ...

### Differential calculus: cubic function

Observe the non-linear distance–time graph of a rocket travelling at a changing velocity. The distance, s, travelled by the rocket after t seconds is determined by the formula s(t) = t³ – 2. Calculate the average velocity of the rocket over time intervals that become progressively shorter. Tabulate the results and look ...

### Differential calculus: quartic function

Observe the non-linear distance–time graph of a rocket travelling at a changing velocity. The distance, s, travelled by the rocket after t seconds is determined by the formula s(t) = t⁴ + t². Calculate the average velocity of the rocket over time intervals that become progressively shorter. Tabulate the results and look ...

### Differential calculus: derivatives graphing tool

Use this revision tool to examine the relationship between functions and their derivatives by observing their graphs. Select a polynomial function (for example, f(x) = x³ – 3x²) from the menu. Estimate the gradients of the given tangents and position them on the graph of the function. Draw the graph of the derivative function. ...

### Differential calculus: linear and non-linear graphs

Observe the linear and non-linear distance–time graphs of a rocket travelling at both constant and changing velocities. Calculate the average and instantaneous velocities of the rocket over different time intervals. Notice what happens to the average and instantaneous velocities as the time intervals become smaller. Work ...

### Differential calculus: derivatives and power functions

Observe the non-linear distance-time graph of a rocket travelling at a changing velocity. Calculate the average velocity of the rocket over time intervals that become progressively shorter. Tabulate the results and derive a formula for finding the instantaneous velocity at a given point. In the second activity, observe ...

### Differential calculus: cubic and quartic functions

Observe the non-linear time graph of a rocket travelling at a changing velocity. The distance, s, travelled by the rocket after t seconds is determined by the formulas: s(t) = t³ – 2 and s(t) = t⁴ + t². Calculate the average velocity of the rocket over time intervals that become progressively shorter. Tabulate the results ...

### Triathlon: assessment

Test your understanding of distance–time graphs. For example, look closely at graphs of a triathlete's performance (distance against time) for the swimming, running and cycling legs of a triathlon. Interpret the graphs to answer questions about each of the race legs and the overall performance of that triathlete. Compare ...

### Constant acceleration

Learn to use your knowledge of graphs and calculus to analyse the way things move. Follow the derivation of equations to describe the position, velocity and acceleration of a moving object. Deal conceptually with both positive and negative accelerations. This resource consists of a video in three sections with animations ...

### Inclines - mathematics activities

This photograph of a road sign suggests investigations of the different indicators of slope used in our environment. It may lead to explorations of trigonometric relationships in relation to gradient. Teachers are encouraged to scan all the ideas suggested here as relevant to the various year level groupings, as there is ...

### Triathlon: assessment: teacher guide

Use this resource with 'Triathlon: assessment' (L8274). First, see a profile of that assessment object, which gives you some helpful background information. Then follow clear guidelines on the interpretation of student's responses to questions based on interpreting information on a distance–time graph in the assessment object.