The Getaway Car — Print Cards

Teacher — unlock codes (cut off before distributing student cards)

Phase 1: RECT01

Phase 2: TRAP02

Phase 3: ZERO03

Phase 4: CASE04

🚔 Case File

Background, read before you begin

A car fled a scene and led police on a 60-second chase before being boxed in. The black box survived and recorded speed every 10 seconds, nothing else. The court wants one number: how far did the car travel? Keep this card, you'll use this data across all four phases.

t (s)	0	10	20	30	40	50	60
v (m/s)	0	12	22	28	26	18	8

Phase 1

Rectangles

Task 1a

Test the obvious approach.

Average all seven readings and multiply by 60 s. Does the result feel trustworthy? What does this method ignore?

Task 1b

Left-hand rectangles.

Split into six 10 s slices. Use the speed at the start of each slice as a constant height. Find the total distance.

Task 1c

Right-hand rectangles.

Repeat using the speed at the end of each slice. Which total is bigger, and why?

Task 1d

What you actually drew.

Sketch both sets of rectangles. What happens to the gap between your totals with 1-second readings instead of 10?

Phase 2

Trapezoids

Task 2a

A smarter height.

Use the average of each slice's two end readings as its height. Recompute the total. What shape have you drawn?

Task 2b

Where does it land?

Compare this total to your two Phase 1 totals. Where does it sit, exactly?

Task 2c

Prove it isn't a coincidence.

Write the left and right sums using \(v_0,\ldots,v_6\). Add and halve them. Show this always equals the Task 2a method.

Task 2d

Name it and generalize it.

This is the Trapezoidal Rule. Write the general formula for \(n\) slices with readings \(v_0,\ldots,v_n\).

Phase 3 · GDC needed

The Limit

Task 3a

Shrink the slices.

Imagine infinitely many, infinitely thin slices. What happens to your estimate's accuracy?

Task 3b

Read the notation.

\(\int_0^{60} v(t)\,dt\): in your own words, what is this instructing you to compute?

Task 3c

A second device.

Run a quadratic regression on your seven \((t,v)\) points on your GDC. Record \(v(t)\) and \(R^2\).

Task 3d

Evaluate the integral.

Use your GDC to evaluate \(\int_0^{60} v(t)\,dt\) with your regression model. Record the value.

Phase 4

The Verdict

Task 4a

Does it clear the threshold?

Police need 1100 m to prove the on-ramp was reached. Which of your four estimates prove that on their own?

Task 4b

How good is the model?

Where does your regression disagree most with the readings? Why might a real car not follow a smooth curve?

Task 4c

When would this fall apart?

If the driver braked and accelerated repeatedly within one 10 s window, could you trust these estimates?

Task 4d, Closing statement

What is the integral doing?

Without saying "area," explain what \(\int_0^{60} v(t)\,dt\) computes, and why the width must shrink to zero.