CHAPTER 1 · RESOURCES
AIMS Quick Revision
Think first. Use AI purposefully. Verify always.
Reconnect the ideas, check a diagram, then try five familiar questions. This is revision, not an examination.
PDF version planned
Concept Map
Key Ideas
The origin O is (0, 0), where the perpendicular axes meet. On the x-axis, y = 0; on the y-axis, x = 0.
- LOCATE: Position is described relative to an agreed reference.
- BUILD: The x-axis is horizontal; the y-axis is vertical.
- NAVIGATE: x-coordinate: signed horizontal position from the origin.
- ORIENT: I: (+, +). II: (−, +). III: (−, −). IV: (+, −).
- MEASURE: Same y? Compare x. Same x? Compare y.
- CONNECT: H = (x₂, y₁) shares a row with A and a column with B.
- HALFWAY: Average x-coordinates together and y-coordinates together.
- APPLY: Plot → Observe → Measure → Connect → Verify.
Formula Bank
For A(x₁, y₁) and B(x₂, y₂), with the same unit scale on both axes:
Axis-parallel distance
Same y: |x₂ − x₁|. Same x: |y₂ − y₁|.
General distance
Midpoint
Visual Recall
P(−2, 3): left 2, up 3. The sign pair (−, +) places P in Quadrant II. Points on an axis belong to no quadrant.
Important Connections
- A shared reference, scale and order turn two location clues into one point.
- Coordinate differences describe movement; their magnitudes describe lengths.
- Perpendicular horizontal and vertical changes connect the grid to the Baudhāyana–Pythagoras theorem.
- Half of each coordinate change gives the midpoint. Measurements support a geometric claim only when its conditions are checked.
Common Mistakes
- Giving a column twice does not supply the missing row. Agree on the reference, scale and order.
- Do not swap the axis labels or use unequal spacing for equal units.
- Do not reverse the coordinates: horizontal comes first, vertical second.
- A point with a zero coordinate is on an axis, not in a quadrant.
- A signed change can be negative; a distance cannot.
- Adding the two leg lengths measures the route through the helper point, not the straight segment.
- Average corresponding coordinates. Equal distances alone do not establish that a point lies on the segment.
- A sketch or equal lengths alone cannot justify every property of a shape. State what your calculation proves.
Mini Examples
NAVIGATE · Plot P(−2, 3), then check it by reading back from the diagram.
Read back: horizontal −2, vertical 3. So P = (−2, 3).
Read the worked example →MEASURE · Find the distance from A(−3, 2) to B(2, 2).
AB = |2 − (−3)| = 5 units. Reversing gives |−3 − 2| = 5.
Read the worked example →CONNECT · Find AB for A(−2, −1) and B(1, 3).
AB = 5 units. It exceeds either leg but is less than their sum, 7.
Read the worked example →HALFWAY · Find the midpoint of A(−2, −1) and B(2, 3).
The average y-coordinate is (−1 + 3) ÷ 2 = 1, so M(0, 1).
Read the worked example →APPLY · A(−2, −1), B(2, −1), C(2, 3), D(−2, 3) mark a layout. Verify its shape and centre.
All four side lengths are 4. The layout is a square (and therefore also a rectangle). Both diagonal midpoints are (0, 1). Each diagonal has length √32. These checks agree with the plotted shape.
Read the worked example →5-Question Quick Check
Selected from existing AIMS practice. Try independently, use a hint if needed, and explain each correction.
Choose an answer or enter a number, then check. A retry hint appears after an incorrect attempt. You can change your choice and try again.
Feedback is based on reviewed answer keys, not AI. Completion records correct responses after any retries; it is not an exam mark.