# Rate of change and area under the curve.

This set of resources starts with the equation of a straight line in the form y=mx+c and checking the understanding that this represents a linear relationship. This will include understanding that m will determine the slope and c the intercept of a linear graph and interpret the gradient of a straight line graph as a rate of change The first few links below can be used jointly to check students understanding of the gradient (slope) and intercept of a straight line before applying it in a scientific context. Following that are some links to specific examples from the parts of the science courses where students will need to carry out these calaculations.

Some common misconceptions are that students may think that gradient = (change in x) / (change in y). When using straight lines to calculate rate of change some students think that the horizontal section of a distance time graph means an object is travelling at constant speed. Some pupils think that a section of a distance time graph with negative gradient means an object is travelling backwards or downhill.

Whilst many students may be familiar with using an equation in science to divide a variable by time, the concept of a rate of change is known to be a challenging one and so if a student is struggling it can be tricky to identify whether it is the concept of the rate of change or the analysis of the graph itself where the difficulties lie.

When first introducing graphs to students and taking data from them it may be beneficial to explicitly introduce and use the concept of ‘rate of change’ so that students are familiar with it. For example, speed may be calculated as “distance over time” but is perhaps more accurately described as “the rate at which distance is changing with respect to time”.

The list on Interpreting Graphs has a number of ideas to support other aspects of graphs work with students in science as well as a list of some of the common conceptual challenges that some students have with graphs.

The calculation of a rate from a graph of a variable against time may well occur in a number of topics across Biology, Chemistry and Physics. However, the drawing of a tangent to a curve to measure the rate of change may be unfamiliar to some teachers of GCSE and in science is most likely to occur in two topics

- The
**rate and extent of chemical change**topic in chemistry where students may have to calculate a rate of reaction at a particular time. - The
**electricity topic**in physics where the rate of change for a component that behaves in a non-liner fashion may be required. An example might be how voltage changes over time in an AC circuit and how that differs from DC.

Calculating the area between the curve and the X-axis by counting the squares under the graph is a further mathematical technique that is required. This is most likely to be needed in the **Force and Motion** topic in physics where the area under a Velocity/Time graph can be used to calculate the distance travelled. For a constant acceleration, the line will be straight and the area can be calculated geometrically, however for a non-uniform motion the square counting technique will be needed.

### Maths is Fun - Gradient

line-height:115%;font-family:"Arial",sans-serif;mso-hansi-font-family:Calibri;

mso-hansi-theme-font:minor-latin;mso-bidi-font-family:"Times New Roman";

mso-bidi-theme-font:minor-bidi">This link provides a very clear explanation of how to calculate the gradient of a straight line. It is followed by a number of questions with answers that can be used to check understanding.

### Maths is Fun Intercept of a straight line

line-height:115%;font-family:"Arial",sans-serif;mso-hansi-font-family:Calibri;

mso-hansi-theme-font:minor-latin;mso-bidi-font-family:"Times New Roman";

mso-bidi-theme-font:minor-bidi">This link provides a definition of the intercept of a straight line. It is followed by a number of questions with answers that can be used to check understanding.

### Maths is Fun - Gradient of a straight line

line-height:115%;font-family:"Arial",sans-serif;mso-hansi-font-family:Calibri;

mso-hansi-theme-font:minor-latin;mso-bidi-font-family:"Times New Roman";

mso-bidi-theme-font:minor-bidi">This link reminds students of the equation of a straight line and links it to the explanations above. Again the iexplantion is followed by a number of questions with answers that can be used to check understanding.

### Rates of Reaction Graphs (Chemistry)

A really well though through AfL class activity on rates of reaction and the shapes of graphs from the Royal Society of Chemistry. The lsite has student sheets, notes, a lesson plan and the answers.

### Motion Graphs (Area and Gradient)

This set of revision notes work through some examples of finding the gradient of a velocity time graph to find the acceleration and the area under a velocity time graph to find the distance travelled. It does not develop the area under the graph further to consider the square counting method for non-uniform motion but does provide a solid starting point.