21 Looking at gradients
The term gradient refers to the rate at which something is changing.
The gradient of an actual slope refers to how steep it is, which in turn is just the rate that the height changes with respect to movements along the slope. It might be easier to think of steps on a staircase. If the height of the steps is large relative to the width/depth of the step, then it will be steep and hard to climb. If the height of the step is small respect to the width/depth of the step, it should be easier to climb and descend the stairs. Incidentally, international building code regulations state that heights should be between 10.2cm and 17.8 cm, while the depth/width (referred to as ‘the tread’) should be at least 27.9 cm.
The concept of the gradient is often taught in the context of relationships between distance, velocity and acceleration. If your distance is changing at a high rate with respect to time, this means your velocity is high, while if your velocity is changing at a high rate with respect to time, it means your acceleration is high.
Let’s makes some predictions
See if you can predict how the gradients will change in the following situations.
Confirm your conjectures
Once you’ve worked through these, take a look at the video to see one way you could confirm your conjectures in Scratch.
You can visit the project page or to further investigate how the height of the changes with time, depending on the shape of the bottle.
Acknowledgement
The problems presented under the heading Let’s make some predictions have been adapted from The Language of Functions and Graphs (1985) by Shell Center for Mathematical Education (this version modified by Deakin University and licensed under CC BY NC 4.0 with permission from Shell Centre for Mathematical Education).
