LEARNER JOURNEY Example 1: interpret +210°
Read signPositive means counterclockwise.
Read magnitudeRotate 210°.
Break apart210° = 180° + 30°.
LocateThe terminal side is 30° into Quadrant III.
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
Read an angle’s sign as a direction instruction and preserve that direction throughout a solution.
Evidence of mastery
Construct the reasoning, solve at least 4 of 5 mastery problems, and revise an unfamiliar explanation.
Essential terms
- positive angle
- A counterclockwise rotation. This tells how the angle or coordinate is interpreted in a solution.
- negative angle
- A clockwise rotation. This tells how the angle or coordinate is interpreted in a solution.
- signed measure
- An angle measure whose sign records direction.
- magnitude
- The distance of the measure from zero, without its sign.
Look first, then name the mathematics
Build the visual meaning
- Initial side
- −70°
- Clockwise
- Terminal side
Read the complete visual relationship as text
- The initial side starts on the positive x-axis.
- The curved arrow moves clockwise.
- The terminal side ends below the x-axis.
Definition in plain language
The sign of an angle measure tells direction: positive means counterclockwise and negative means clockwise. The magnitude tells how far to rotate.
Why this matters
Direction affects the actual rotation even when a positive and negative angle later prove coterminal. Keeping the sign is essential in conversions and equations.
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.
Read the sign before doing arithmetic.
Translate + as counterclockwise or − as clockwise.
Use the absolute value for the amount of rotation.
Retain the original sign when reporting the angle measure.
The same method in words
- Read the sign before doing arithmetic.
- Translate + as counterclockwise or − as clockwise.
- Use the absolute value for the amount of rotation.
- Retain the original sign when reporting the angle measure.
Interactive decision model
Build the reasoning, one decision at a time
Build the motion represented by −450°.
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: interpret −210°
Read signNegative means clockwise.
Read magnitudeRotate 210°.
Break apartA clockwise half turn reaches −x; 30° more enters Quadrant II.
LocateThe terminal side is in Quadrant II, not Quadrant III.
Example 3: compare −90° and 270°
Interpret −90°Quarter turn clockwise.
Interpret 270°Three quarter turns counterclockwise.
Compare endsBoth end on the negative y-axis.
Compare measuresThey are coterminal but unequal signed rotations.
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 rotation direction defines a positive angle?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Which rotation direction defines a negative angle?
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Write a prediction before viewing the choices.
Not completed yet.
Partially completed problem
Finish the missing step
A prompt supplies part of the reasoning. Predict the missing mathematical move before choices appear.
1. What directed motion does −135° specify?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
Not completed yet.
Independent practice
Six problems for repetition
Solve all six without opening the worked examples unless feedback shows what to revise.
1. What is the magnitude of −275°?
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Write a prediction before viewing the choices.
2. Where does +90° end?
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3. Where does −90° end?
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4. What signed measure is a 60° clockwise turn?
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5. What signed measure is a 300° counterclockwise turn?
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6. What relationship holds for the terminal sides of +180° and −180°?
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Not completed yet.
Mastery and transfer
Mastery check
Demonstrate understanding
Attempt all five. At least four correct answers are required for mastery.
1. How do the angle measures +180° and −180° compare?
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Write a prediction before viewing the choices.
2. Decompose −810° into full clockwise turns and remainder.
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3. Where does −270° end?
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Write a prediction before viewing the choices.
4. Which is clockwise: 75° or −75°?
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Write a prediction before viewing the choices.
5. Why must the sign be kept during degree-radian conversion?
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Not completed yet.
Unfamiliar transfer
Explain, compare, and revise
Compare the rotations −450° and 270°. Explain what is the same and what is different.
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
−450° is one full clockwise turn plus 90° clockwise, so it ends on the negative y-axis. 270° is three quarter turns counterclockwise and also ends on the negative y-axis. They are coterminal because they differ by 720°, but their signed rotations are different.
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.