Math Games
Speed as Passing
Jean Piaget observed that young children understand speed not as a number, but as a relationship — something is fast because it gets ahead. Before we learn to measure velocity, we perceive it as overtaking: who crossed the line first, who left whom behind. This game strips speed back to that primal intuition. Click in and feel it.
The Race
Lesson 1: Uniform Motion and Passing
Type your first name to get started.
Your responses will be saved to your teacher's dashboard as you work.
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The Race
Three cars. Three lanes. Constant speeds. Press Play and watch carefully.
A
B
C
Set speeds and press Play
Watch the race. What do you notice about how the order changes?
Car speeds
Car A
Car B
Car C
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What did you notice?
Describe the race in your own words.
What changed as the race went on? What stayed the same the whole time?
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The three moments
Tap a label on the left, then tap its matching description on the right.
Labels
T₀
T₁
Tꜰ
Descriptions
Car C is ahead. Car B is middle. Car A is furthest behind.
All three cars are at the exact same position on the track.
Car A is ahead. Car B is middle. Car C is furthest behind.
All matched!
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Same speed or same position?
At T₁ — the moment all three cars are exactly level — which of these is true?
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Predict the tie
Before running the race again — mark on the graph where you think all three cars will first be level.
Click on the graph to place your prediction, then run the race below to check it.
Run the race below and compare it to your prediction.
Was your prediction close? What surprised you?
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What controls the tie?
Set Car C's speed equal to Car A's speed below. Press Play and watch what happens.
If Car C has the same speed as Car A, do they ever tie? Explain why or why not.
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Notice and wonder
You have watched the race, sorted the moments, and tested your predictions. Now reflect.
What is one thing you NOTICE about how the cars move?
What is one thing you WONDER — a question the race made you think of?
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Three snapshots
Draw the positions of Cars A, B, and C in each box. Place a dot or short line for each car and label it.
Before T₀
At T₁
After Tꜰ
What is changing from picture to picture? What is staying the same?
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Position vs time graph
A position vs time graph shows where something is at every moment. Time goes across. Position goes up. Three lines are shown — one per car — but not labeled yet.
The steepest line belongs to which car?
All three lines cross at one point. What moment does that crossing represent?
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Velocity vs time graph
For constant speed, the line on a velocity vs time graph is perfectly horizontal — the speed never changes.
Which horizontal line belongs to which car? Explain how you decided.
Compare this graph to the position vs time graph. What is different? What is the same?
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Zoom in: just two cars
Forget Car B for now. Below is the position vs time graph for just Car A and Car C.
Before T₁, which line is higher? What does that mean about the race?
After T₁, which line is higher? What does that tell you?
Which line is steeper? What does that tell you?
"If a moving object A is behind B at one instant and passes in front of B at a later instant, it is judged to be faster." — Jean Piaget
Do you agree? Use the graph to explain why.
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Saying it in symbols
Mathematicians describe what they see in words, pictures, and symbols. Now let's try symbols.
pos(A) = the position of Car A on the track at a given moment
pos(B) = the position of Car B pos(C) = the position of Car C
> means "greater than" (further ahead) = means "at the same position"
Example: pos(A) > pos(C) means "Car A is further ahead than Car C"
pos(B) = the position of Car B pos(C) = the position of Car C
> means "greater than" (further ahead) = means "at the same position"
Example: pos(A) > pos(C) means "Car A is further ahead than Car C"
At T₀ — write the order of positions.
At T₁ — write what is true about all positions.
At Tꜰ — write the new order.
Write the speed inequality for all three cars.
What pattern do you notice between the position order at Tꜰ and the speed order?
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Lesson 1 complete!
All your responses have been saved to your teacher's dashboard.
In Lesson 2 we will ask: what if one of the cars was speeding up?
In Lesson 2 we will ask: what if one of the cars was speeding up?
Here is what you built today:
Representation 1A sequence of pictures at T₀, T₁, and Tꜰ
Representation 2A position vs time graph — three lines crossing at T₁
Representation 3A velocity vs time graph — three flat horizontal lines
Representation 4Symbolic inequalities for position and speed