LEARNER JOURNEY Example 1: classify 125°
Compare125° lies between 90° and 180°.
Use the planeThat open region is upper left.
NameUpper left is Quadrant II.
CheckIt is not equal to an axis landmark, so it is not quadrantal.
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 coordinate-plane regions and axes to name exactly where a terminal side ends.
Evidence of mastery
Construct the reasoning, solve at least 4 of 5 mastery problems, and revise an unfamiliar explanation.
Essential terms
- quadrant
- One of the four open regions formed by the coordinate axes.
- quadrantal angle
- An angle whose terminal side lies on an x- or y-axis.
- axis angle
- Another name sometimes used for an angle ending on an axis.
- boundary
- An axis separating two quadrants; it belongs to neither quadrant.
Look first, then name the mathematics
Build the visual meaning
- Quadrant I
- Quadrant II
- Quadrant III
- Quadrant IV
- Quadrantal angles lie on an axis
Read the complete visual relationship as text
- Quadrants are numbered counterclockwise starting in the upper right.
- An axis is a boundary, not part of either neighboring quadrant.
- Angles ending on axes are quadrantal angles.
Definition in plain language
Quadrant I is upper right, II upper left, III lower left, and IV lower right. A terminal side exactly on an axis has no quadrant; the angle is quadrantal.
Why this matters
Quadrants determine coordinate signs and trigonometric signs. Recognizing axis cases prevents using a quadrant rule where an exact coordinate is 0 or ±1.
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.
Reduce multi-turn angles to a terminal measure from 0° through 360° if needed.
Compare the measure with 0°, 90°, 180°, 270°, and 360°.
If it equals a landmark, name the axis and call it quadrantal.
If it lies strictly between landmarks, name the corresponding quadrant.
The same method in words
- Reduce multi-turn angles to a terminal measure from 0° through 360° if needed.
- Compare the measure with 0°, 90°, 180°, 270°, and 360°.
- If it equals a landmark, name the axis and call it quadrantal.
- If it lies strictly between landmarks, name the corresponding quadrant.
Interactive decision model
Build the reasoning, one decision at a time
Classify the terminal side of 990°.
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: classify 270°
Compare270° equals an axis landmark.
LocateThree quarter turns end on the negative y-axis.
NameThe angle is quadrantal.
Avoid errorIt is not in Quadrant III or IV.
Example 3: classify 585°
Remove a turn585° − 360° = 225°.
Compare225° lies between 180° and 270°.
NameThe terminal side is in Quadrant III.
Connect585° is not quadrantal because its terminal side is not on an axis.
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 quadrant is upper right?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. How should 180° be classified?
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.
Partially completed problem
Finish the missing step
A prompt supplies part of the reasoning. Predict the missing mathematical move before choices appear.
1. Complete: an angle ending on an axis is called ____.
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. Classify 30°.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Classify 100°.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
3. Classify 200°.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
4. Classify 300°.
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Write a prediction before viewing the choices.
5. Where does 90° end?
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6. Where does 360° end?
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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 should −180° be classified?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Classify 765° after removing full turns.
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Write a prediction before viewing the choices.
3. Classify −90°.
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Write a prediction before viewing the choices.
4. Which two quadrants border the negative x-axis?
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Write a prediction before viewing the choices.
5. Why is 0° not in Quadrant I?
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.
Unfamiliar transfer
Explain, compare, and revise
Classify 1170° completely: number of full turns removed, terminal measure, quadrant or axis, and whether it is quadrantal.
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
1170° − 1080° = 90°, so three complete turns are removed. The terminal measure is 90°, whose terminal side is the positive y-axis. Therefore 1170° is quadrantal and belongs to no quadrant.
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.