Day 14 — Build Your Bot's Distance Equation

Step 1 of 5

Enter two of your runs.

Take two rows from your Distance Lab worksheet — two different drive times and the distances you measured. The line through them is your bot's equation.

💡 What is a data point?

A data point = one run. It has two numbers: the drive time and the distance you measured.

Example: you drove for 3 seconds and the bot went 150 cm.
That data point is written as (3, 150).
Time goes first. Distance goes second.

1

Run 1

Drive time (seconds)

Distance (cm)

Your Point 1: ( — , — )

2

Run 2

Drive time (seconds)

Distance (cm)

Your Point 2: ( — , — )

📊 Your two runs on a graph

Run 1

Run 2

⚠ Your two drive times are the same. You need two different times to build a line. Change one of them.

Fill in all four numbers first.

Step 2 of 5

How much distance per second?

For every 1 more second of driving, how much further does the bot go? That number is the rate of change — your bot's speed.

📊 Your two runs

Point 1

Point 2

📈 What is rate of change?

Rate of change = how much further the bot goes for each extra second — its speed in cm per second.

Formula: rate of change = distance change ÷ time change

Math calls this the slope. We use the letter m.

1

Calculate the rate of change

Step A — Find the distance change

Distance changed from ? to ? cm

Change = ? − ? = ?

Step B — Find the time change

Time changed from ? to ? s

Change = ? − ? = ?

Step C — Divide: distance change ÷ time change

Rate of change (m) = ? ?

m =

📺 What is the y-intercept (b)?

The y-intercept (b) is where the line starts — the distance at time 0.

For a bot it comes out negative: the bot loses a little distance while it speeds up, so this is your start-up cost.

We calculate it with: b = y − m × x
Use your Run 1 numbers.

2

Calculate the y-intercept (b)

Plug Point 1 into the formula: b = y − (m × x)

b = ? − ( ? × ? ) = ? − ?

b =

Confirm both answers before moving on.

Step 3 of 5

This line is your bot's distance equation.

It passes through both runs you entered. A different pair of runs gives a slightly different line — real measurements are never perfectly on one line.

Note: Any two points define exactly one line. This equation fits your two runs exactly — but a different pair of runs gives a slightly different line. Test it on a third run in Step 4 to see how well it holds.

📊 Your two-run model as a line

Point 1

Point 2

Model line

Your Distance Model

y = —x + —

y = Distance (cm)

m = —

x = Time (s)

b = starting value

What each piece means

y

Distance (cm) — how far the bot goes; the number you want to predict.

m

Speed — how many cm the bot goes per 1 second. Your value: — cm/s

x

Time (s) — the drive time you plug in to make a prediction.

b

start-up (b) — the distance lost while speeding up; it comes out negative. Your value: — cm

What your rate of change means

"For every 1 more second I drive, my bot goes m cm further."

What your slope is:
The slope is your bot's speed in cm per second. A faster bot has a bigger slope.
It should be positive — more time means more distance. A negative or zero slope means a measuring slip; check your two runs.

Important: This equation was built from just two runs. It describes your bot well near those times, but every extra run makes it more trustworthy.

✓ Substitution check: plug in Point 1

Substitute Point 1 into your equation. It will come out exactly right — because Point 1 was used to build the equation. This confirms your arithmetic, not whether the model is good.

y = —

Step 4 of 5

How accurate is your equation?

Use a third run — a drive time you did NOT use to build your line. This tests whether your equation predicts a distance it has never seen.

What is a residual?

Residual = predicted distance − actual distance (in cm)

If the bot went further than predicted, predicted − actual is negative.
If it went less far, it is positive.

• Residual = 0: exact hit for this run.
• Positive residual: equation over-predicted (the bot went shorter than predicted).
• Negative residual: equation under-predicted (the bot went further than predicted).

Real measurements are never exact. A miss of a few cm is normal and good.

Your equation

y = —

1

Pick a third run. Enter its time and distance.

Drive time (seconds)

Actual distance (cm)

Enter the test bot's numbers before continuing.

Step 5 of 5

Share with the class.

Read your report card, then type in another team's numbers to compare the two bots and make your keep/change decision.

🏆 Your Group's Report Card

Your Equation

—

Rate of Change (Slope)

—

y-Intercept

—

Test Residual

—

👥 Compare with another team

Go read another team's poster. Type their speed and start-up here to see both bots on one graph — then decide what to keep and what to change.

Their speed (cm per second)

Their start-up (cm, usually negative)

Your decision — write it on your Model Swap slip

You can't copy their numbers — it's a different bot. The question is: does seeing their equation make you trust one of your numbers, or doubt one?

One number we're keeping (we're sure of it) is our = ,
because .

One number we'll change / re-check is our ,
because next to their bot it looks .

Example: "We're keeping our speed, 38 cm/s — all five of our runs lined up. We'll re-check our start-up: ours is −20 but their bot (and two others) got about −7, so ours is probably a bad measurement." The reason has to name numbers, not "theirs looks better."