CLASS 9 MATHS · CHAPTER 1 · STUDY NOTES
Orienting Yourself: The Use of Coordinates
Understand orienting yourself: the use of coordinates through short explanations, labelled models and original practice. Read one topic, explain it in your own words, then check.
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A grid turns a place into an address
Learn why a shared starting point and a grid make directions clear.A position needs two numbers
y (units)
x (units)
- Row y = 2 · Every point here is 2 units above the x-axis.
- Column x = 3 · Every point here is 3 units right of the y-axis.
- 1P = (3, 2) · (3, 2)
The two reference lines are perpendicular. Their intersection locates P. Equal units on both axes.
Read the plotted values
| x | y |
|---|---|
| -5 | 2 |
| 5 | 2 |
| x | y |
|---|---|
| 3 | -5 |
| 3 | 5 |
- Coordinate system
- A way to name a position using numbers and agreed reference lines.
Imagine saying “third column, second row” instead of “over there”. A grid makes the position repeatable: another person can find the same place.
The chapter connects this idea with town planning, maps and the history of mathematical ideas across cultures. The useful idea is a common reference, rather than memorising a list of dates.
Let’s look a little closer.
Go deeper: From maps to a floor plan
The book uses a room plan to show how coordinates can help with furniture placement and clear walking routes. A plan gives floor position; it does not automatically give the height of an object.
Take a moment. What do you think?
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Question
Why must two people use the same origin and scale?
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Plot x first, then y
Place a point without mixing up the two coordinates.A position needs two numbers
y (units)
x (units)
- Row y = 2 · Every point here is 2 units above the x-axis.
- Column x = 3 · Every point here is 3 units right of the y-axis.
- 1P = (3, 2) · (3, 2)
The two reference lines are perpendicular. Their intersection locates P. Equal units on both axes.
Read the plotted values
| x | y |
|---|---|
| -5 | 2 |
| 5 | 2 |
| x | y |
|---|---|
| 3 | -5 |
| 3 | 5 |
- Origin
- The crossing of the axes, written (0, 0).
- Ordered pair
- (x, y): first the horizontal coordinate, then the vertical coordinate.
The x-axis is horizontal and the y-axis is vertical. They meet at right angles. Right and up are positive; left and down are negative.
To plot P(3, 2), move 3 units right from the origin, then 2 units up. A coordinate is signed. The distance of P from the y-axis is |3|, not a negative length.
- 1
Mark the origin and equal unit spacing.
- 2
Move to x = 3 along the x-axis.
- 3
Move vertically to y = 2 and mark P.
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Question
Is (2, 3) the same point as (3, 2)?
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Let the signs tell you the region
Locate a point in one of the four quadrants.Four signs, four regions
y (units)
x (units)
- Example points · Read x first and y second.
- 1I (+,+) · (3, 3)
- 2II (−,+) · (-3, 3)
- 3III (−,−) · (-3, -3)
- 4IV (+,−) · (3, -3)
Axes themselves do not belong to a quadrant. Equal units on both axes.
Read the plotted values
| x | y |
|---|---|
| 3 | 3 |
| -3 | 3 |
| -3 | -3 |
| 3 | -3 |
- Quadrant
- One of the four regions between the coordinate axes.
Start at the top-right and count anticlockwise: I, II, III, IV. Their signs are (+,+), (−,+), (−,−), (+,−).
An axis is a boundary, so a point on either axis is not in a quadrant. The origin is on both axes.
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Question
Where is (−4, 2), and why?
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A zero coordinate puts you on an axis
Recognise axis points from their coordinates.One movement disappears
y (units)
x (units)
- Axis points · The unused coordinate is zero.
- AA · (-4, 0)
- BB · (0, 3)
- OO · (0, 0)
All positions are in abstract coordinate units. Equal units on both axes.
Read the plotted values
| x | y |
|---|---|
| -4 | 0 |
| 0 | 3 |
| 0 | 0 |
- Point on the x-axis
- A point (x, 0), with no vertical shift.
- Point on the y-axis
- A point (0, y), with no horizontal shift.
For (−4, 0), move left 4 units and do not move up or down. For (0, 3), stay at x = 0 and move up 3 units.
Only (0, 0) has both coordinates zero. It belongs to both axes.
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Question
Which axis contains (0, −2)?
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Subtract along the changing coordinate
Find a horizontal or vertical length.A horizontal length of 6
y (units)
x (units)
- AB · Only x changes.
- AA · (-2, 1)
- BB · (4, 1)
Equal units on both axes. Equal units on both axes.
Read the plotted values
| x | y |
|---|---|
| -2 | 1 |
| 4 | 1 |
- Absolute value
- A number’s distance from zero. It is never negative.
If y stays the same, the segment is horizontal: length = |x₂ − x₁|. If x stays the same, it is vertical: length = |y₂ − y₁|.
Distance can be zero when the points coincide. Otherwise it is positive. The order of subtraction does not change the length after taking absolute value.
A vertical segment
|(−3) − 2| = 5 unitsFrom (1, 2) to (1, −3), x is unchanged. Count 2 units to zero and 3 more below it.
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Question
Find the distance from (−2, 1) to (4, 1).
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