Calculating Instantaneous Rates of Change

When calculating instantaneous rates of change students need to  visualise the properties of the gradient for a straight line graph.   I use the starter activity to see if they can match four graphs with their corresponding equations.

Calculating Instantaneous Rates of Change

The only clue is the direction and steepness of the red lines in relation to the blue line y = x.  All the graphs have an intercept of zero, to allow students to focus on the gradient.  At the end of this activity I feedback to make sure students understand that lines which go downwards have a negative gradient and those that are steeper than y = x has a gradient greater than one.

Calculating Instantaneous Rates of Change

To introduce how to calculate an instantaneous rate of change on a curve we discuss how the steepness of the graph changes depending on the x value.  I like to use the Geogebra applet below to demonstrate how the gradient of the tangent changes along the curve.  The teacher can change the function depending on the point they are trying to make.

I provide the students with a print out of the next few slides for them to write on as we progress through the examples.  This saves time and helps students develop a clear written method.  I encourage the students to use integer coordinate pairs for calculating the change in horizontal and vertical whenever possible.  This video demonstrates the written method I teach the students.

Practical Reasons for Calculating Instantaneous Rates of Change

As we progress through the lesson, I emphasise the practical reasons for calculating instantaneous rates of change.   First, we calculate the rate of leakage from a water tank using a distance-time graph.  Second, we use a velocity-time graph to estimate the acceleration of a ball as it travels through the air.   In this example we discuss gravity as a force slowing the acceleration towards the turning-point then increasing as it begins to fall.

Assessing Progress and Feeding Back

Calculating Instantaneous Rates of Change

I use the plenary to check student’s progress and understanding in two key areas.

  1. How well can students find the gradient of a tangent to estimate an instantaneous rate of change?
  2. Can students interpret the practical meaning of the gradient in the context of the two variables?

Students attempt this on mini-whiteboards and present their working to me for assessment.  Estimating an instantaneous rate of change typically takes two, one hour lessons,  During this time students work through the worksheet and several examination questions.   It is important for students to become confident performing the calculations and understanding them within the context of the problems.

Mr Mathematics members can access this lesson online and download the worksheet by clicking here.

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