LEARNER JOURNEY Example 1: convert 60°
Set up60°·(π rad/180°).
CancelThe degree units cancel.
Reduce60/180=1/3.
Conclude60°=π/3 radians.
Start here
Build the concept from the beginning
Starting point
No prior trigonometry is assumed. Every required term and decision is taught on this page.
Learning target
Use units to choose the correct conversion factor, cancel degrees, and simplify exact π fractions.
Evidence of mastery
Construct the reasoning, solve at least 4 of 5 mastery problems, and revise an unfamiliar explanation.
Essential terms
- conversion factor
- A ratio of equivalent quantities, so its value is 1.
- cancel units
- Remove a unit appearing once in a numerator and once in a denominator.
- simplify
- Reduce numerical factors without changing value.
- exact radian form
- A reduced multiple or fraction of π.
Look first, then name the mathematics
Build the visual meaning
- 150°
- Degrees cancel
- 150π/180 radians
- 5π/6 radians
Read the complete visual relationship as text
- The conversion factor equals 1 because π radians = 180°.
- Degrees cancel diagonally.
- The answer keeps radians and π when exact.
Definition in plain language
For a degree measure d, radians = d(π/180). Preserve the sign, cancel degrees, and reduce the fraction.
Why this matters
Dimensional cancellation prevents using the reciprocal factor and provides a built-in reasonableness check.
Reasoning path
See the method as a sequence
Move from left to right. Each decision prepares the next one; on a phone, follow the numbered cards from top to bottom.
Write the signed degree measure.
Multiply by π radians/180°.
Cancel the degree unit.
Reduce the numerical fraction and keep π; check the angle size and sign.
The same method in words
- Write the signed degree measure.
- Multiply by π radians/180°.
- Cancel the degree unit.
- Reduce the numerical fraction and keep π; check the angle size and sign.
Interactive decision model
Build the reasoning, one decision at a time
Construct the conversion of 150° to radians.
Choose this step, then check the construction.
Choose this step, then check the construction.
Choose this step, then check the construction.
The reasoning path changes as each decision is selected.
Choose every step, then check the construction.
Model before practice
Three fully explained examples
Each example names the decision, carries it out, and interprets the result.
Example 2: convert −225°
Preserve sign−225°·(π/180°).
CancelDegrees cancel.
Reduce−225/180=−5/4.
Conclude−225°=−5π/4 radians.
Example 3: convert 765°
Set up765°·(π/180°).
Find common factorgcd(765,180)=45.
Reduce765/45=17 and 180/45=4.
Conclude765°=17π/4 radians; the large value preserves its extra turns.
Practice with decreasing support
Guided practice
Practice immediately after the model
Predict first. Then use the visual relationship and reasoned feedback to check two decisions.
1. Which factor converts degrees to radians?
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2. Convert 180°.
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Partially completed problem
Finish the missing step
A prompt supplies part of the reasoning. Predict the missing mathematical move before choices appear.
1. Finish: 90°·π/180° = ____.
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Independent practice
Six problems for repetition
Solve all six without opening the worked examples unless feedback shows what to revise.
1. Convert 30°.
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2. Convert 45°.
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3. Convert 135°.
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4. Convert 240°.
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5. Convert 330°.
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6. Convert −120°.
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Mastery and transfer
Mastery check
Demonstrate understanding
Attempt all five. At least four correct answers are required for mastery.
1. Convert 720°.
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2. Convert 15°.
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3. Convert 270°.
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4. Convert −450°.
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5. Why is 150°·180/π not a valid degrees-to-radians setup?
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Not completed yet.
Unfamiliar transfer
Explain, compare, and revise
Convert 1125° to radians exactly. Show the conversion factor, cancellation, reduction, and a turn-count check.
Guided self-check
Judge the locked first attempt against every criterion. Each judgment gives revision guidance.
Choose Met or Needs revision.
Choose Met or Needs revision.
Choose Met or Needs revision.
Choose Met or Needs revision.
Model reasoning
1125°·(π rad/180°) = 1125π/180 rad. Divide numerator and denominator by 45 to get 25π/4 rad. Since 1125° is three full turns plus 45°, the result should be 6π+π/4=25π/4, which confirms the conversion.
Write a first attempt before opening the self-check.
The record reveals exact answers and model reasoning only after the corresponding attempt or self-check has occurred.