# Rates of Change

Students are expected to be able to identify and solve problems with constant rates of change involving distance and speed. This resource package provides a variety of activities to provide students with the opportunity to develop and practise these skills.

Students are also required to interpret the gradient at a point on a curve as the instantaneous rate of change; apply the concepts of instantaneous and average rate of change, using gradients of tangents and chords, in numerical, algebraic and graphical contexts, solve and interpret the answers in growth and decay problems, including compound interest and work with general iterative processes.

Visit the secondary mathematics webpage to access all lists: www.nationalstemcentre.org.uk/secondarymaths

### Around the World *suitable for home teaching*

This video, introducing work on the HS2 rail project, could be used to set the context for work on rates of change and talks about some of the careers that can be considered if studying the STEM subjects.

### F1 in Schools: Mathematics - Calculating Speed

Students are required to apply their knowledge of speed, time and distance to calculate the speed of a Formula 1 car using real race data. Using a circuit map of Monaco showing speeds in mph and kph, the gear used and target times at different points on the circuit and results giving race and qualifying times, students are required to plot graphs of the speeds of the different drivers.

The "Are we there yet?" activity can be used as a reminder to students about how to use the speed, distance, time formulae in preparation for the task.

### Compound Measures and Rates of Change

This resource contains thirteen instant maths ideas ideal for use as starter activities, extension work or as probing questions to assess understanding. Ideas include:

What is meant when an aeroplane travels at mach 2.5?

When is speed measured in knots?

If Phileas Fogg travelled around the world in 80 days what was his average speed?

How fast is the earth travelling?

How dense is a human being?

The resource includes a table of densities of common materials.

### New National Curriculum Mathematics 7

The shape, space and measures section of this textbook contains a chapter entitled **Compound measures**, starting on page 135 of the pdf. The section begins with discussion points exploring compound measures such as speed and strength of solutions of liquids.

The next section contains explanations, examples and exercised exploring the notion of speed and how average speed is calculated and used. Further discussion points compare two speeds measured using different units. Further sections explore the interpretation of distance-time graphs and their use to solve problems.

Further sections explore density and other compound measures such as the rate of flow of water out of a bath.

### Kent Mathematics Project Level Eight

Chapter 14, **Rates of change**, beginning on page 76 of the pdf, explores rates of change in a variety of contexts such as the rate of growth of a runner bean plant, the average rate of ascent of an aircraft and the rate of change of the depth of water as a swimming pool fills. Students are required to use tables of data to determine whether a vehicle is travelling at a constant speed or whether the speed is varying. They also interpret distance-time graphs.

### Exploring gradient functions

In this resource students are presented with a general straight line y=mx+c and are shown that the gradient of the line between any two points that lie on the line is always equal to m. Students should be encouraged to explain the proof to check understanding. This section could be used to review existing knowledge.

Students then move on to curves and need to be clear that the gradient of a curve at a given point is equivalent to the gradient of the tangent to the curve at that point. Students then are required to find the gradient of successive chords between two points on the curve as one point moves closer to a fixed point.

Students are required to find the gradient of the curve y=x^{2} at different values of x leading to a generalisation for the equation of the gradient function. Students then explore families of curves each time finding a generalisation for the equation of the gradient function culminating in a generalisation for the equation of the gradient function for the curves with equations of the form y=ax^{2}+bx+c.

The teacher notes give the proof that the gradient equation of y=mx+c is always equal to m and shows how to find the gradient function of y=ax^{2}+bx+c

### Zero to Infinity

This is a Teachers’ TV programme from the series Painting with Numbers. In this episode Professor Marcus du Sautoy explains how humans have developed numbers for their mathematical needs. It explores some interesting challenges including explanation of the role the digit zero played using place value to represent larger numbers, the problems encountered when attempting to divide by zero and takes us to Hilbert's Hotel to demonstrate the mind-boggling concept of infinity.

Exponential growth is explained by placing one grain of rice on the first square of a chessboard, two grains on the second square, four grains on the third and so on. This clip could be used as a starter to motivate interest before looking in more detail at a problem such as the chessboard problem.

### Compound Interest

This resource is from the Integrating Mathematical Problem Solving project by Mathematics in Education and Industry (MEI), this activity shows how compound interest can be calculated over different intervals. As the intervals get smaller and smaller, the total value approaches a limit. You may want to update some of the source material to reflect current interest rates. Topic areas covered are:

• Use of different time intervals for compounding interest

• Multipliers for percentage change

• Compound interest formula

• The exponential function as a limit of increasingly frequent compounding

• Annual Equivalent Rate (AER)

• Annual Percentage Rate (APR)

• Gross interest

• Net interest

• Use of different time intervals for compounding interest

• Continuous compounding

The resource includes detailed teacher guidance. This resource require use of Geogebra software.