Geometry for RIT under 215: describing the axes and origin, plotting points from ordered pairs, reading positions and shapes off a coordinate grid, and finding distances along grid lines.
Subject: NWEA MAP Growth Math · 60 slides · symbolic lesson
Open the interactive version of this deck · Homework for this lesson
Title
MAP Growth · Geometry · RIT under 215
Describing the grid, plotting points, and reading objects off a coordinate plane
Objectives
Three MAP skills sit here, and they all rest on one convention.
The one convention: x comes first, and it is the one that runs across.
Section
Part 1
Concept
A coordinate plane is built from two number lines set at right angles to each other.
Figure (svg): A coordinate grid with the x axis, y axis and origin labelled
x axis — The horizontal number line, running across the page.
y axis — The vertical number line, running up the page.
Origin — The point where the two axes cross, written (0, 0).
Concept
An ordered pair gives directions from the origin: how far across, then how far up.
Figure (svg): A grid with the point four comma three plotted, showing the move right then up
Ordered pair — Two numbers in brackets, such as (4, 3). The first is the x coordinate and the second is the y.
Warm-up
Retrieve before being taught.
Discussion prompt
Where else have you seen a position given by two numbers or letters?
Hint: Think about games and maps.
Answer:
Chess squares, map grid references, seat numbers in a cinema, spreadsheet cells.
Every one of them uses the same idea: one number across, one number up or down. The coordinate plane is the mathematical version.
Pattern
Three steps, whether you are plotting or reading.
Across the corridor, then up the stairs. That phrase fixes the order for good.
Trap
Asked to plot (2, 5), a student goes 5 across and 2 up.
The first number is always the x, which runs across. So (2, 5) means 2 across and 5 up.
Along the corridor, then up the stairs. The x is also alphabetically first, which is a useful reminder.
Socratic
The convention is arbitrary, but having one is not.
Discussion prompt
Why must everyone agree that x comes first, rather than choosing per question?
Hint: Ask what would happen if two people used opposite conventions.
Answer:
Because (4, 3) and (3, 4) are different points. Without an agreed order, an ordered pair would name two possible places and be useless.
The convention is arbitrary in the same way that driving on one side of the road is arbitrary — what matters is that everyone does the same thing.
Definition probe
Get the vocabulary secure before plotting anything.
Sort into buckets
Sort each description by which axis it belongs to.
Explain it to yourself
Explaining it fixes it faster than repeating it.
Discussion prompt
In your own words, how do you find the point (5, 2) starting from the origin?
Answer:
Start at the origin, move 5 units along the x axis to the right, then move 2 units straight up.
The key phrase is then. The two moves happen in a fixed order, and reversing them lands somewhere else.
Discrimination
The vocabulary is what MAP asks about in the describe-the-plane items.
Sort into buckets
Sort each item by what it names.
Analogy
You use coordinate systems constantly without calling them that.
Match the pairs
Why: Every one of these fixes a position with two values in an agreed order. The coordinate plane replaces the letters with numbers and puts the origin at the bottom left.
Real world
The choice of origin is a decision, not a law.
Discussion prompt
Why is the origin usually drawn at the bottom left of a school grid, when a phone screen puts it at the top left?
Answer:
Because a school grid is usually showing only positive numbers, and the bottom left is where both axes start at zero.
A screen counts downwards from the top because that is how images are stored line by line. Both are valid — what matters is knowing which convention is in use.
Section
Part 2
Worked example
Plot the point (4, 3) on a coordinate grid.
Figure (svg): A grid with the point four comma three plotted, showing the move right then up
Start at the origin
Why: Put your finger on (0, 0).
Move along the x axis
Why: The first number is 4, so move 4 units to the right.
Move up
Why: The second number is 3, so move 3 units up.
\[ (4, 3) \]
Verify: by reading it back
Why: From the plotted point, going straight down meets the x axis at 4 and going straight left meets the y axis at 3 — which is the pair we started with.
Intuition
Reading is plotting in reverse: drop straight down for the x, and straight across for the y.
Figure (svg): A grid with the point four comma three plotted, showing the move right then up
Doing both checks confirms the pair without guessing.
Concept
The same two numbers in the other order name a different point.
Figure (svg): Two points plotted showing that four comma three and three comma four are different places
(4, 3) and (3, 4) sit in different places, which is exactly why the pair is called ordered.
Prediction
Apply the convention rather than instinct.
Predict first
Starting at the origin, where is the point (2, 6)?
Correct: 2 right and 6 up
Along the corridor, then up the stairs — 2 along, then 6 up.
Why: The first number is the x coordinate, which measures across, so it is 2 right. The second is the y, which measures up, so it is 6 up.
Faded example
Fill in the two moves.
Fill in the blanks
(7, 2): \text7 2 \text___ ___ \text___
Why: The x coordinate, 7, is the horizontal move and comes first. The y coordinate, 2, is the vertical move and comes second.
Check
Solve it on paper before you click.
Check your understanding
Which describes the location of the point (3, 5)?
Answer: A
Why: The first coordinate is x, giving 3 units along the horizontal axis. The second is y, giving 5 units up. Both moves start at the origin.
Error analysis
A student plotted (0, 4).
Annotate
On: \( (0, 4) \;\Rightarrow\; \text{4 units right along the x axis} \)
A zero coordinate is a strong clue: a zero x puts the point on the y axis, and a zero y puts it on the x axis.
Definition probe
Zero coordinates place a point on an axis.
Sort into buckets
Sort each point by where it lies.
Ranking
Only the x coordinate matters here.
Put in order
Why: How far right a point sits is given entirely by its x coordinate, so the order is 1, 3, 5 and 8. The y coordinates are a deliberate distraction.
Pattern
Each frame moves the point one step along the x axis.
Step through it
Which coordinate changed, and which stayed still?
Horizontal movement changes only x. That is why every point on a horizontal line shares the same y value.
Elimination
A point is plotted 3 units right and 0 units up.
Eliminate the wrong options
What are its coordinates?
Survives elimination: p1
Why: Three units right is the x coordinate and comes first, and no vertical movement makes the y coordinate zero, so the point sits on the x axis at (3, 0).
Section
Part 3
Concept
A shape on a grid is defined by the coordinates of its corners.
Figure (svg): Four points plotted on a grid and joined to make a rectangle
Plot the corners, join them in order, and the shape appears.
Worked example
Four points are plotted at (1, 1), (5, 1), (5, 4) and (1, 4). What shape do they make?
Figure (svg): Four points plotted on a grid and joined to make a rectangle
Plot the four corners
Why: Two sit on the line y equals 1 and two on the line y equals 4.
Look at the side lengths
Why: The bottom runs from x equals 1 to 5, so it is 4 units. The left side runs from y equals 1 to 4, so it is 3 units.
Name the shape
Why: Four sides, all meeting at right angles, with opposite sides equal — a rectangle.
\[ 4 \times 3 \text{ rectangle} \]
Verify: the opposite sides
Why: The top also runs 4 units and the right side also runs 3, so opposite sides are equal, confirming a rectangle rather than a general quadrilateral.
Intuition
When two points share an x or a y, the distance between them can simply be counted.
Figure (svg): Two points on a grid with the horizontal gap counted between them
Subtracting the coordinates that differ gives the same answer without counting.
Worked example
How far apart are the points (1, 2) and (5, 2)?
Figure (svg): Two points on a grid with the horizontal gap counted between them
Check what they share
Why: Both have a y coordinate of 2, so they lie on the same horizontal line.
Subtract the x coordinates
Why: 5 minus 1 is 4.
\[ 5 - 1 = 4 \text{ units} \]
Verify: by counting
Why: Counting the squares between the two points along the line also gives 4, so the subtraction is doing exactly what counting does.
Prediction
Check what the two points share first.
Predict first
How far apart are (3, 1) and (3, 7)?
Correct: 6 units
Shared x means a vertical line; shared y means a horizontal one. Subtract whichever coordinate differs.
Why: Both points have an x coordinate of 3, so they lie on the same vertical line. Subtracting the y coordinates gives 7 minus 1, which is 6 units.
Faded example
Two points on the same horizontal line.
Fill in the blanks
(2, 6) \text7 (9, 6): \quad 9 - 2 = ___ \text___
Why: The y coordinates match at 6, so the points lie on the same horizontal line and only the x coordinates differ. Subtracting gives 7 units.
Check
Solve it on paper before you click.
Check your understanding
A rectangle has corners at (2, 1), (6, 1), (6, 4) and (2, 4). What are its side lengths?
Answer: A
Why: The horizontal sides run from x equals 2 to 6, which is 4 units. The vertical sides run from y equals 1 to 4, which is 3 units.
Real world
Any screen you have ever used is a coordinate plane.
Discussion prompt
How does a phone know where you tapped, and what does that have to do with ordered pairs?
Answer:
The screen reports the tap as a pair of numbers: so many pixels across, so many down.
That is exactly an ordered pair, with the origin usually at a corner of the screen. Every app you use is reading coordinates constantly.
Matching
Read the coordinates carefully.
Match the pairs
Why: A zero in the first position keeps the point on the y axis; a zero in the second keeps it on the x axis. Two zeros put it at the origin.
Sorting
Shared coordinates mean shared lines.
Sort into buckets
Sort each pair by the kind of line the two points share.
Matching y means horizontal, matching x means vertical — which reads backwards until you think about what each coordinate fixes.
Explain it to yourself
This explains the distance shortcut rather than just using it.
Discussion prompt
In your own words, why do two points with the same y coordinate lie on a horizontal line?
Answer:
Because the y coordinate is the height, and if both points are at the same height then neither is above the other.
Two points at the same height can only differ by being further left or right, which is exactly a horizontal line.
Section
Part 4
Concept
Given a plotted point, drop a line straight down for x and straight across for y.
Figure (svg): A grid with the point four comma three plotted, showing the move right then up
Doing both, rather than eyeballing, is what prevents the coordinates being swapped.
Step zero
One habit prevents the swap error entirely.
Discussion prompt
You are asked for the coordinates of a plotted point. What should you do first?
Hint: Do the moves in the same order you would use to plot it.
Answer:
Drop straight down from the point to the x axis and read that number first.
Only then go straight across to the y axis. Doing them in that order means you write them in the right order too.
Notation
The brackets and comma carry meaning.
Annotate
On: \( (4, 3) \)
The notation is compact but rigid — every part of it is load bearing.
Elimination
A point sits 6 across and 2 up.
Eliminate the wrong options
What are its coordinates?
Survives elimination: e1
Why: The horizontal distance of 6 is the x coordinate and comes first; the vertical distance of 2 is the y coordinate and comes second.
Two truths and a lie
Two of these are sound; one is not.
Eliminate the wrong options
Which statement about the coordinate plane is false?
Survives elimination: t2
Why: They are different points. (3, 5) is 3 across and 5 up, while (5, 3) is 5 across and 3 up. That is precisely why the pair is called ordered.
Comparison
Each row describes one point three ways.
Comparison matrix
| point | across | up |
|---|---|---|
| (4, 7) | 4 | 7 |
| (0, 3) | 0 | 3 |
| (6, 0) | 6 | 0 |
The across column always matches the first coordinate, and the up column always matches the second.
Error analysis
A student read a point sitting 2 across and 7 up.
Annotate
On: \( \text{2 across, 7 up} \;\Rightarrow\; (7, 2) \)
The swap error is by far the most common on this topic, and it costs a mark even when the plotting was right.
Edge cases
The grid does not have to stop at zero.
Discussion prompt
What would a point like (-3, 2) mean, and where would it sit?
Hint: Remember that each axis is a number line, and number lines go both ways.
Answer:
A negative x means moving left from the origin rather than right, so (-3, 2) is 3 units left and 2 units up.
The axes are full number lines, so they extend in both directions. MAP below RIT 215 stays in the positive quadrant, but the convention itself never changes.
Faded example
Three corners of a rectangle are given.
Fill in the blanks
(2, 2), \; (7, 2), \; (7, 5), \; (2, 5)
Why: The fourth corner must share its x with the first corner and its y with the third, so it sits at (2, 5). The rectangle is then 5 units by 3 units.
Section
Part 5
Concept
Describing, plotting and reading are three views of the same convention.
Figure (svg): Two points plotted showing that four comma three and three comma four are different places
If you can say what x and y mean, and in what order, all three follow.
Worked example
A square has corners at (1, 1), (4, 1) and (4, 4). Where is the fourth corner, and what is the side length?
Figure (svg): Four points plotted on a grid and joined to make a rectangle
Look at what the given corners share
Why: Two share y equals 1, and two share x equals 4.
Find the missing pattern
Why: The fourth corner must share x equals 1 with the first, and y equals 4 with the third.
Name the missing corner
Why: It is at (1, 4).
Find the side length
Why: From x equals 1 to 4 is 3 units.
\[ (1, 4), \text{ side } 3 \text{ units} \]
Verify: all four sides
Why: Bottom, top, left and right all measure 3 units, so the shape really is a square and the missing corner is right.
Sorting
Many items only need one of the two numbers.
Sort into buckets
Sort each question by which coordinate answers it.
Notice the last two: a point is on the y axis when its x is zero, which reads backwards at first.
Explain it
A classmate keeps swapping the coordinates.
Discussion prompt
What phrase or physical habit would fix this for them permanently?
Answer:
Along the corridor, then up the stairs. You cannot climb the stairs before walking down the corridor.
Pair it with the physical habit of tracing the horizontal move with a finger before the vertical one, and the order stops being a thing to remember.
Missing information
Not every question can be answered.
Discussion prompt
A question says: a point is 5 units from the origin. What are its coordinates?
Answer:
There is not enough information — many points are 5 units from the origin, including (5, 0), (0, 5) and (3, 4).
A single distance does not fix a position on a plane; you need two numbers, which is exactly why coordinates come in pairs.
Estimation
Reading approximately is often enough to pick an option.
Predict first
A point sits just past the middle of a grid running 0 to 10 on both axes, slightly above centre. Which is the best estimate?
Correct: (6, 6)
Estimating against the centre of the grid is enough to eliminate every other option here.
Why: The middle of a 0 to 10 grid is (5, 5). Just past the middle and slightly above centre puts both coordinates a little above 5, which matches (6, 6).
Invariant
Moving along one axis leaves the other coordinate alone.
Step through it
What is identical in all three, and what has changed?
The y coordinate is invariant under horizontal movement. That is why points on a horizontal line all share a y value.
Commit first
Decide your answer and your confidence before revealing.
Predict first
Which point lies on the x axis?
Correct: (7, 0)
A zero in the second position keeps the point on the x axis; a zero in the first keeps it on the y axis.
Why: A point lies on the x axis when it has not moved up or down at all, which means its y coordinate is 0. That is the second number, so (7, 0) is the one.
Exit ticket
One item that tells you whether the deck landed.
Predict first
Starting at the origin, you move 3 right and 8 up. What point are you at?
Correct: (3, 8)
Why: The horizontal move of 3 is the x coordinate and is written first. The vertical move of 8 is the y coordinate and is written second, giving (3, 8).
Connect it up
Your handwriting, one page, from memory.
Draw it
Draw a coordinate grid with both axes labelled and the origin marked. Plot and label (4, 3) and (3, 4) to show that order matters. Beside the grid write what x and y each measure, and the phrase you use to remember the order.
If you can draw this from memory, all three skills in this deck are covered.
Recap
Three skills, one convention.
Almost every lost mark on this topic is a swapped pair, and doing the two moves in a fixed order prevents it.
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