The Instantaneous Rate Lab
How fast is something changing at a single instant? Draw a tangent, measure its gradient, and use the same rate-of-change thinking to grow and shrink money with compound multipliers.
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How fast, right now? ⏱️
Average speed over a journey is easy. But how fast are you going at **one instant**? The **instantaneous rate of change** answers that. On a curved graph it is the steepness at a single point, and the same idea powers compound growth in money.
Words to know 🔑
Four terms run through this module:
Average or instantaneous? ⚖️
The gradient of a straight line on a curve tells you a **rate**, but which one depends on the line.
Gradient of a tangent 🔢
An interactive activity.
Match each graph to what the gradient means 🔗
- A distance-time graph
- A velocity-time graph
- A tangent at a point
- A chord between two points
- The gradient is the speed
- The gradient is the acceleration
- The instantaneous rate at that point
- The average rate between the points
Draw, pick, divide 💡
To find the instantaneous rate at a point on a curve: **draw the tangent** at that point, **pick two points far apart** on it, then work out **rise / run**. At GCSE this is done by drawing and measuring, not by calculus.
Plot the distance 📈
An interactive activity.
Compound growth 🏦
An interactive activity.
Why compound wins 🎯
Why does compound interest give more than simple interest over time?
- Interest is earned on interest already added
- The interest rate is always higher
- The money is taxed less
- The bank adds a fixed bonus
Spot the true statements 🖍️
An interactive activity.
The method in order 🪜
An interactive activity.
Rates in words 🧩
The instantaneous rate of change at a point is the gradient of the _____ drawn there. To find it, you work out the rise divided by the _____. On a distance-time graph this gradient is the _____. When money grows by 4% a year it is multiplied by 1.04 each year, which is called _____ interest. To find a value after a percentage decrease, you multiply by a number less than _____.
Pick the right tool 🧭
An interactive activity.
Explain the rate ✍️
An interactive activity.