Linear and Angular Speed: a Free Precalculus Color by Number Activity
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Linear and angular speed is the Precalculus lesson where radians stop being abstract: a bike tire, a record, a ceiling fan. Here is the activity I use with my own students, and it is free: a color by number with 6 circular motion word problems, an answer bank, a coloring page and an answer key.
Below is how I teach the two formulas and how I run the activity. Type your email on the sign-up page and the PDF is yours right away.
How to teach linear and angular speed: two speeds on one wheel
Angular speed is how fast something turns: the angle it sweeps per unit of time, in radians per minute or radians per second. One revolution is 2π radians, so 4 revolutions per minute is 8π radians per minute.
Linear speed is how fast a point on the edge travels along the circle: distance per unit of time. The two are tied together by the radius: linear speed = radius × angular speed. A merry-go-round with a 3 meter radius turning 4 times a minute has an angular speed of 8π, about 25.1 radians per minute, and a rider on the edge moves 3 × 8π = 24π, about 75.4 meters per minute. Every point turns at the same angular speed, but the farther from the center, the faster it moves.
The lesson at a glance
- Grade: Precalculus
- Standard: Common Core HSF-TF.A.1: understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle
- You need: the problem page, the answer bank and the coloring page, one set per student, colored pencils and a calculator, the answer key
- Students will be able to: find the angular speed and the linear speed of a turning object, and convert between units
Get the free color by number and answer key
Before you teach
Print the problem page for each student, and keep the answer bank and the coloring page on your desk for now. Students will need colored pencils for the end.
The problems: 6 circular motion word problems
The six problems are things students have seen spin: a bicycle tire, a vinyl record on a turntable, a ceiling fan and a circular saw blade. Some ask for angular speed in radians per minute; the rest ask for linear speed, in miles per hour, centimeters per second or feet per second.
- Change revolutions to radians: multiply by 2π.
- Use the radius, not the diameter: two of the problems give the diameter.
- Linear speed = radius × angular speed.
- Change the units last: inches to miles or feet, minutes to hours or seconds.
The answer bank: check before coloring
When students finish the problems, I hand them the answer bank to check that each answer is on it. That stops the guessing, and students fix their mistakes, usually a radius or a unit, before they start coloring.
The coloring page
Each answer on the bank has a color. Students find the problem numbers on the picture and color each space to match.
The answer key and the finished picture
The answer key shows every answer with its color, and the finished picture shows what the coloring page should look like.
Here is the free circular motion color by number
Type your name and email on the sign-up page. The PDF downloads on the next page, and a copy comes to your inbox.
- The problem page with 6 circular motion word problems
- The answer bank and the coloring page
- The answer key and the finished picture
This activity is from the trigonometry lessons in my Precalculus course
- Converting Between Radians and Degrees
- Reference Angles
- Trig Functions and Their Exact Values
- Linear and Angular Speed
- Proving Trig Identities
- Sum and Difference Identities
- Double Angle and Half Angle Identities
- Solving Trigonometric Equations
- Law of Sines
- Law of Cosines
- Law of Sines and Law of Cosines
- Area of Oblique Triangles
Questions teachers ask about angular speed
What is angular speed?
How fast something turns: the angle it sweeps divided by the time. The formula is angular speed = angle / time, with the angle in radians.
What is the difference between angular speed and linear speed?
Angular speed measures turning (radians per minute). Linear speed measures distance traveled along the circle (like feet per second). Linear speed = radius × angular speed, so points farther from the center move faster even though they turn at the same angular speed.
What is angular speed measured in?
Radians per unit of time, like radians per minute or radians per second. Revolutions per minute is a speed too: multiply by 2π to change it to radians per minute.
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Get the free circular motion color by number
The 6 word problems, the answer bank, the coloring page, the answer key and the finished picture, in one PDF. Type your email and it is yours.
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