The coordinate plane and its parts: the two axes and the origin, ordered pairs and their two coordinates, plotting a point from the origin, reading coordinates off a graph, the four quadrants and their sign patterns, and scatter plots as pictures of paired data.
Subject: Algebra 1 · 65 slides · symbolic lesson
Open the interactive version of this deck
Title
Algebra 1 · Chapter 4 — Graphing Linear Equations and Functions
The Coordinate Plane
Objectives
Five outcomes, each one you can test yourself on with a pencil and no answer key.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.1 The Coordinate Plane §4.1, pp. 203-208 — the lesson these objectives are drawn from
Warm-up
You have used one number line since Lesson 2.1. This lesson uses two at once.
Discussion prompt
On a single number line, how many numbers do you need to name a point? Now imagine two number lines crossing at right angles — how many do you need then, and does the order matter?
Hint: Try describing where something is in a room using only one number.
Answer:
One number is enough on a line, because there is only one direction to travel. On a plane you need two — one for how far across and one for how far up — and the order matters, since three across and two up is a different place from two across and three up.
That is the whole idea of a coordinate plane: two number lines, and a pair of numbers instead of a single one.
Concept
A coordinate plane is formed by two real number lines that intersect at a right angle at the origin. The horizontal one is the x-axis and the vertical one is the y-axis, and every point in the plane corresponds to an ordered pair of real numbers.
coordinate plane — A plane formed by two real number lines intersecting at right angles at the origin. Every point in it corresponds to exactly one ordered pair of real numbers.
The first number in the pair is the x-coordinate and the second is the y-coordinate.
Figure (svg): A coordinate plane with the x-axis, y-axis and origin labelled, and the point 3 comma 2 plotted
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.1 The Coordinate Plane §4.1, pp. 203-203
Section
Section 1
Concept
The two number lines are called axes and they cross at the origin. Every point corresponds to an ordered pair, whose first number is the x-coordinate and whose second is the y-coordinate.
Figure (svg): A coordinate plane with the x-axis, y-axis and origin labelled, and the point 3 comma 2 plotted
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.1 The Coordinate Plane §4.1, pp. 203-203 — the definition of the coordinate plane and its parts
Picture it
Two axes, one origin, and one plotted point.
Figure (svg): A coordinate plane with the x-axis, y-axis and origin labelled, and the point 3 comma 2 plotted
Each axis is a number line of exactly the kind from Lesson 2.1, with negatives on one side of zero and positives on the other. Nothing about a single axis is new.
Worked example
Vocabulary first, because every later instruction uses it.
\[ \text{Name the horizontal axis, the vertical axis, the crossing point, and the pair naming that point.} \]
The horizontal number line is the x-axis
Why: By convention the first coordinate is measured along it.
The vertical number line is the y-axis
Why: The second coordinate is measured along it.
The point where they cross is the origin
Why: It is the zero of both number lines at once.
Its ordered pair is zero comma zero
Why: Zero units across and zero units up.
\[ (0, 0) \]
Figure (svg): The solution to Worked example name the parts shown as a ladder of expressions, one row per algebraic move
\[ \text{origin} = (0, 0) \]
Verify: check the origin against both axes separately
Why: The origin sits at zero on the horizontal line and at zero on the vertical one, which is exactly what the pair zero comma zero records. Every point's pair is read the same way: its position on each of the two number lines.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.1 The Coordinate Plane §4.1, pp. 203-203
Matching
Four pieces of vocabulary, four descriptions.
Match the pairs
Why: The two axes are ordinary number lines placed at right angles, and the origin is the zero of both at once. The ordered pair is the notation that ties a point to its two measurements, and the word ordered is doing real work — the pair is not a set, and swapping its entries names a different point.
Worked example
The pair three comma two, taken apart.
\[ \text{In the pair } (3, 2), \text{ say what each number measures.} \]
The first number is the x-coordinate
Why: It says how far the point is along the horizontal axis from the origin.
\[ 3\text{ across} \]
The second number is the y-coordinate
Why: It says how far the point is along the vertical axis.
\[ 2\text{ up} \]
Both are measured from the origin
Why: The origin is the reference point for both numbers.
\[ \text{from } (0, 0) \]
Note the general form
Why: A pair is always written x first, then y.
\[ (x, y) \]
Figure (svg): The ordered pair 3 comma 2 with its two coordinates labelled and the two moves shown
\[ (3, 2): \; x = 3, \; y = 2 \]
Verify: check the point against each axis
Why: Dropping straight down from the point meets the x-axis at three, and going straight across meets the y-axis at two. Reading a point's coordinates is exactly this pair of projections.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.1 The Coordinate Plane §4.1, pp. 203-203
Trap
\[ (3, 2) \]
Take the first number as the height and the second as the distance across
Why: Nothing in the notation announces which is which, so the order has to be remembered.
That plots the point two across and three up, which is a different place — and every conclusion drawn from the graph afterwards is about the wrong point.
\[ (3, 2): \; 3 \text{ across, then } 2 \text{ up} \]
Read the pair alphabetically: x comes before y in the alphabet and first in the pair
Why: The convention is fixed, and the alphabetical order is the standard way of remembering it.
Saying across before up out loud every time is enough. The two coordinates are never interchangeable unless they happen to be equal.
Sorting
In each pair, decide what the named number measures.
Sort into buckets
Sort each item by which coordinate it names.
The rule never varies: first number across, second number up. Position in the pair decides the meaning, which is why the pair is called ordered.
Elimination
Four statements about the point where the axes cross.
Eliminate the wrong options
Which one is correct?
Survives elimination: A
Why: The origin is where both number lines read zero, so its ordered pair is zero comma zero. Option D is worth noticing: on a graph drawn only for positive quantities the origin does appear at the bottom left, but on a full plane it sits in the middle with all four quadrants around it.
Socratic
One number was enough in Chapter 2.
Discussion prompt
Explain why a point on a line needs one number but a point on a plane needs two. Then predict how many would be needed to name a point in space, and give a real example of such a description.
Hint: Count the independent directions in each case.
Answer:
A line offers only one direction of travel, so a single distance from a chosen zero pins a point down completely. A plane offers two independent directions, and neither one alone can distinguish points that differ in the other, so two numbers are needed.
Space needs three, and everyday descriptions use them: a room number gives a floor, a corridor position and a distance along it, and a flight is tracked by latitude, longitude and altitude. The pattern is that the number of coordinates equals the number of independent directions, which is exactly what is meant by the dimension of the space.
Section
Section 2
Concept
To read a plotted point, count how far it lies to the right or left of the origin, then how far above or below. Those two counts, with their signs, are its coordinates.
The signs come from the same left-right and up-down conventions as the number line in Lesson 2.1.
Figure (svg): The ordered pair 3 comma 2 with its two coordinates labelled and the two moves shown
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.1 The Coordinate Plane §4.1, pp. 203-203 — Example 1, Identify Coordinates
Picture it
Across first, then up.
Figure (svg): The ordered pair 3 comma 2 with its two coordinates labelled and the two moves shown
Reading a point and plotting one are the same two moves in opposite directions. Practising one practises the other.
Worked example
This is Example 1 from the textbook, with the points described in words.
\[ \text{A point is } 3 \text{ right and } 2 \text{ down from the origin; another is } 2 \text{ left and } 1 \text{ down. Write both pairs.} \]
Read the first point's horizontal position
Why: Three units right of the origin, so the x-coordinate is positive three.
\[ x = 3 \]
Read its vertical position
Why: Two units down, so the y-coordinate is negative two.
\[ y = -2 \]
Write the pair
Why: Three comma negative two.
\[ (3, -2) \]
Do the same for the second
Why: Two units left is negative two, and one unit down is negative one.
\[ (-2, -1) \]
Figure (svg): The solution to Worked example read four points shown as a ladder of expressions, one row per algebraic move
\[ (3, -2) \qquad (-2, -1) \]
Verify: check each sign against its direction
Why: Right and up give positives; left and down give negatives. Both points here have a negative y-coordinate because both sit below the horizontal axis, which a glance at the picture confirms without counting.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.1 The Coordinate Plane §4.1, pp. 203-203
Translation
Right and up are positive; left and down are negative.
Match the pairs
Why: The four combinations of directions give the four combinations of signs, and each lands in a different quadrant. The digits are identical in all four, so only the signs distinguish them — which is why a sign error moves a point to a completely different region rather than slightly.
Worked example
A point directly on an axis has a zero in its pair.
\[ \text{A point sits on the y-axis, } 2 \text{ units up. Another sits on the x-axis, } 5 \text{ units right. Write both pairs.} \]
Read the first point's horizontal position
Why: It is on the y-axis, so it is zero units across.
\[ x = 0 \]
Read its vertical position
Why: Two units up.
\[ y = 2 \]
Write the pair and note the zero
Why: Zero comma two — the zero is what puts it on the y-axis.
\[ (0, 2) \]
Do the same for the second
Why: Five across and zero up, so five comma zero, which lies on the x-axis.
\[ (5, 0) \]
Figure (svg): The solution to Worked example points on an axis shown as a ladder of expressions, one row per algebraic move
\[ (0, 2) \text{ on the } y\text{-axis} \qquad (5, 0) \text{ on the } x\text{-axis} \]
Verify: check which zero puts a point on which axis
Why: A zero x-coordinate means no horizontal movement, so the point stays on the vertical axis. A zero y-coordinate keeps it on the horizontal axis. The zero names the axis the point is stuck to, which is the opposite of what many people first guess.
Error analysis
The student read three points from a graph. Two are wrong.
Annotate
On: \( \begin{aligned} \text{3 right, 2 down} &\;\rightarrow\; (-2, 3) \\ \text{2 left, 1 down} &\;\rightarrow\; (-2, -1) \\ \text{on the } y\text{-axis, 2 up} &\;\rightarrow\; (2, 0) \end{aligned} \)
Both errors would be caught by reading the pair back as an instruction — negative two comma three says two left and three up, which is not where the point was.
Faded example
Count across, then up or down.
Fill in the blanks
\text-4 \;\rightarrow\; (3, ___)
Why: Four units left gives an x-coordinate of negative four, and three units up gives a y-coordinate of positive three. Reading the pair back as an instruction — go four left, then three up — confirms it lands where the description said.
Discrimination
A zero coordinate puts a point on an axis.
Sort into buckets
Sort each point by where it lies.
Socratic
The naming feels backwards until you think about the movement.
Discussion prompt
Explain why a point with an x-coordinate of zero lies on the y-axis rather than the x-axis. Then say which axis the point (0, 0) belongs to.
Hint: Think about the movement the coordinate describes.
Answer:
The x-coordinate says how far to move horizontally, and zero means do not move horizontally at all. So the point stays on the vertical line through the origin, which is the y-axis. The coordinate that is zero names the movement not made, and the axis you therefore never leave.
The origin has both coordinates zero, so it makes neither movement and lies on both axes at once. It is the only point in the plane belonging to both, which is why it is the natural reference point for everything else.
Section
Section 3
Concept
To plot a point, start at the origin, move horizontally by the x-coordinate, then vertically by the y-coordinate. The sign of each coordinate chooses the direction.
The two moves may be made in either order and land in the same place, but doing them in the written order keeps the coordinates straight.
Figure (svg): The points 3 comma 4 and negative 2 comma negative 3 plotted from the origin
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.1 The Coordinate Plane §4.1, pp. 204-204 — Example 2, Plot Points in a Coordinate Plane
Picture it
One in the top right and one in the bottom left.
Figure (svg): The points 3 comma 4 and negative 2 comma negative 3 plotted from the origin
The two paths from the origin show the moves explicitly. With practice the path disappears and only the point is drawn, but the two moves are still what determine where it goes.
Worked example
This is Example 2 from the textbook.
\[ \text{Plot } (3, 4) \text{ and } (-2, -3). \]
Start at the origin for the first point
Why: Every plot begins there.
\[ \text{at } (0, 0) \]
Move 3 right and 4 up
Why: Both coordinates are positive, so both moves are in the positive directions.
\[ (3, 4) \]
Return to the origin for the second point
Why: Each point is plotted independently.
\[ \text{back to } (0, 0) \]
Move 2 left and 3 down
Why: Both coordinates are negative, so both moves are in the negative directions.
\[ (-2, -3) \]
Figure (svg): The points 3 comma 4 and negative 2 comma negative 3 plotted from the origin
\[ (3, 4) \text{ and } (-2, -3) \]
Verify: read each plotted point back
Why: Reading the first point's position gives three across and four up, matching the pair. Reading the second gives two left and three down. Reading back is the reverse of plotting and catches a swapped or mis-signed coordinate at once.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.1 The Coordinate Plane §4.1, pp. 204-204
Prediction
The signs decide the region before any counting happens.
Predict first
Without plotting, where does the point (-3, 7) lie?
Correct: Top left, since x is negative and y is positive.
\[ (-3, 7): \; x < 0, \; y > 0 \;\Longrightarrow\; \text{Quadrant II} \]
Why: A negative x moves left and a positive y moves up, so the point lands in the top left region. The sizes of the numbers decide how far, never which region — that is settled entirely by the two signs, which is why the quadrant can be named before anything is drawn.
Worked example
Guided Practice 2 to 5. One of them sits on an axis.
\[ \text{Plot } (2, 5), \; (-3, 7), \; (1, -3), \; (-2, 0). \]
Plot (2, 5)
Why: Two right and five up, in the top right region.
Plot (-3, 7)
Why: Three left and seven up, in the top left region.
Plot (1, -3)
Why: One right and three down, in the bottom right region.
Plot (-2, 0)
Why: Two left and no vertical movement, so it lands on the x-axis.
Figure (svg): The solution to Worked example four from guided practice shown as a ladder of expressions, one row per algebraic move
\[ (2, 5), \; (-3, 7), \; (1, -3) \text{ in quadrants; } (-2, 0) \text{ on an axis} \]
Verify: check the sign patterns against the regions
Why: Two positives put a point top right, a negative then a positive puts it top left, and a positive then a negative puts it bottom right. The fourth has a zero, so it belongs to no quadrant at all — which the sign pattern predicts before anything is drawn.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.1 The Coordinate Plane §4.1, pp. 204-204
Trap
\[ (3, 4) \]
Move three right, then four right again
Why: Both numbers are positive, and the first move went right, so the second follows the same way.
That lands on the x-axis at seven. The second coordinate is measured on the other axis entirely, not further along the first one.
\[ (3, 4): \; 3 \text{ right, then } 4 \text{ up} \]
Change direction between the two moves: the first is horizontal and the second vertical
Why: The two coordinates are measured on two different number lines, which is the entire point of having two axes.
Drawing the two moves as a right-angled path from the origin makes the change of direction visible, and it is worth doing until plotting becomes automatic.
Faded example
Say the direction and the distance for each coordinate.
Fill in the blanks
To plot (-2, -3), start at the origin, move 2 units to the left, then 3 units down.
Why: A negative x-coordinate moves left and a negative y-coordinate moves down, landing the point in the bottom left region. The two moves are always in different directions — one horizontal and one vertical — which is what distinguishes plotting from counting along a single line.
Sorting
The sign of the coordinate chooses the direction.
Sort into buckets
Sort each coordinate by the direction it moves you.
Position in the pair chooses the axis and sign chooses the direction along it. Two independent decisions, and neither one can be read from the other.
Counterexample
One case is enough to refute a general statement.
Discussion prompt
A student claims that the order of the coordinates does not matter, since you end up in the same place either way. Give a counterexample, and then name the only pairs for which the claim happens to be true.
Hint: Look for a pair whose two numbers differ.
Answer:
The pairs three comma two and two comma three are different points: the first is three right and two up, the second is two right and three up. They are reflections of each other in the diagonal line through the origin, and no amount of reordering the moves brings them together.
\[ (a, b) = (b, a) \;\Longleftrightarrow\; a = b \]
The claim is true only when the two coordinates are equal, since then the pair is unchanged by swapping. Those points lie on the diagonal through the origin, and they are the only fixed points of the swap — which is a useful thing to notice because that diagonal reappears in Chapter 4's work on reflections.
Section
Section 4
Concept
The two axes divide the plane into four regions called quadrants. Which quadrant a point lies in is decided entirely by the signs of its coordinates, and the numbering runs anticlockwise from the top right.
A point with a zero coordinate lies on an axis and is in no quadrant at all.
| Quadrant | Sign of x | Sign of y | Example |
|---|---|---|---|
| I | positive | positive | (4, 3) |
| II | negative | positive | (-2, 3) |
| III | negative | negative | (-2, -3) |
| IV | positive | negative | (4, -2) |
Figure (svg): The four quadrants labelled with the sign pattern of the coordinates in each
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.1 The Coordinate Plane §4.1, pp. 204-204 — the quadrant diagram and Example 3
Picture it
Numbered anticlockwise from the top right.
Figure (svg): The four quadrants labelled with the sign pattern of the coordinates in each
The numbering starts where both coordinates are positive and runs anticlockwise, which is the convention throughout mathematics. Only the sign pattern matters, never the size of the numbers.
Worked example
This is Example 3 from the textbook.
\[ \text{Name the quadrant containing } (-2, 3) \text{ and } (4, -2). \]
Read the signs of the first pair
Why: The x-coordinate is negative and the y-coordinate is positive.
\[ (-, +) \]
Match to a quadrant
Why: Negative then positive is the top left region, Quadrant II.
Read the signs of the second pair
Why: Positive then negative.
\[ (+, -) \]
Match to a quadrant
Why: Positive then negative is the bottom right region, Quadrant IV.
Figure (svg): The solution to Worked example name the quadrant shown as a ladder of expressions, one row per algebraic move
\[ (-2, 3) \text{ in II} \qquad (4, -2) \text{ in IV} \]
Verify: check the directions against the region names
Why: A negative x moves left and a positive y moves up, which is the top left — Quadrant II. A positive x moves right and a negative y moves down, which is the bottom right — Quadrant IV. The signs and the directions agree in both cases.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.1 The Coordinate Plane §4.1, pp. 204-204
Sorting
Read the two signs and nothing else.
Sort into buckets
Sort each point into its quadrant.
Two of these pairs have wildly different magnitudes from their quadrant-mates and land in the same regions anyway. Size affects how far into a quadrant a point sits, never which one.
Worked example
Guided Practice 6 to 9. One of them is on an axis.
\[ \text{Name the quadrant for } (5, 3), \; (-2, 0), \; (4, -1), \; (-3, -6). \]
(5, 3) has two positives
Why: Top right, Quadrant I.
(-2, 0) has a zero y-coordinate
Why: It lies on the x-axis and is in no quadrant.
(4, -1) is positive then negative
Why: Bottom right, Quadrant IV.
(-3, -6) has two negatives
Why: Bottom left, Quadrant III.
Figure (svg): The solution to Worked example four from guided practice shown as a ladder of expressions, one row per algebraic move
\[ \text{I}, \; \text{on the } x\text{-axis}, \; \text{IV}, \; \text{III} \]
Verify: confirm that the axis point really has no quadrant
Why: The quadrants are the four open regions strictly between the axes, and a point on an axis is on the boundary rather than inside any of them. Saying it is in no quadrant is the complete and correct answer.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.1 The Coordinate Plane §4.1, pp. 204-204
Trap
\[ (-2, 0) \]
Put it in Quadrant III, since the x-coordinate is negative
Why: The point is on the left, and two of the four quadrants are on the left, so one of them looks like the answer.
The point sits exactly on the x-axis, on the boundary between Quadrants II and III. It belongs to neither.
(-2, 0) lies on the x-axis and is in no quadrant.
Check for a zero coordinate before naming a quadrant
Why: The four quadrants are the open regions between the axes, so a point on an axis is excluded by definition.
This is the same distinction as zero being neither positive nor negative in Lesson 2.1. Boundaries need their own answer rather than being forced into one of the regions.
Matching
Four patterns, four regions.
Match the pairs
Why: Reading down the list of patterns traces the four quadrants anticlockwise from the top right, which is exactly the numbering convention. Learning the order as a rotation rather than as four separate facts makes it much harder to forget which is which.
Elimination
Three of these lie strictly inside a region.
Eliminate the wrong options
Which point is in no quadrant?
Survives elimination: A
Why: A zero x-coordinate places the point on the y-axis, which is a boundary rather than part of any quadrant. Option B is worth noticing: a coordinate of 0.1 is very small but not zero, so that point genuinely is inside a quadrant however close to the axis it looks.
Socratic
The order is a convention, and it is not arbitrary.
Discussion prompt
Describe the path traced by the quadrant numbers one to four, and say why starting where both coordinates are positive is a natural choice. Then predict which quadrant most real-world graphs use.
Hint: Follow the numbers around the diagram.
Answer:
The numbers run anticlockwise starting from the top right, which is the region where both coordinates are positive. Starting there is natural because it is the quadrant that behaves most like ordinary arithmetic — everything is positive, and it is the region you would draw if you had never met negative numbers.
Most real-world graphs use Quadrant I alone, because most measured quantities cannot be negative: time, distance, mass, price, population. That is why graphs in newspapers usually show only the top right corner, with the origin at the bottom left of the page rather than the middle.
Section
Section 5
Concept
A scatter plot displays paired data by plotting one point for each pair. The pattern the points make shows the relationship between the two quantities, if there is one.
scatter plot — A graph in which each pair of measurements is plotted as one point, used to reveal a relationship between two quantities.
Figure (svg): A scatter plot of wing length against wing beat rate for several birds
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.1 The Coordinate Plane §4.1, pp. 207-207 — the wing-length exercises the lesson opens with
Picture it
Six birds, six points.
Figure (svg): A scatter plot of wing length against wing beat rate for several birds
The points fall from left to right, so longer wings go with slower beating. No calculation produced that conclusion — the picture did.
Worked example
Six birds, with wing length in centimetres and wing beats per second.
\[ \text{Plot the pairs } (5, 14), (10, 11), (18, 8), (26, 5), (38, 3), (52, 2). \]
Choose which quantity goes on each axis
Why: Wing length across and beat rate up, so each bird's pair is read as length then rate.
Choose scales that fit the data
Why: Lengths run to about fifty and rates to about fifteen, so the axes need different scales.
Plot one point per bird
Why: Each pair becomes a single dot, and the birds are not joined up.
Describe the pattern
Why: The points fall steadily from left to right.
Figure (svg): A scatter plot of wing length against wing beat rate for several birds
\[ \text{as wing length rises, beat rate falls} \]
Verify: check the description against the extreme points
Why: The shortest wing at five centimetres beats fourteen times a second, and the longest at fifty-two beats twice. The two ends confirm the direction, and every point in between sits consistently along that trend.
Sorting
Describe each scatter in words.
Sort into buckets
Sort each description by the kind of pattern it describes.
A scatter plot's job is to distinguish these three cases, and it does so before any calculation. Chapter 5 will fit a line to the first two kinds.
Worked example
Reading is the more useful direction, since most scatter plots arrive already drawn.
\[ \text{Using the plot, estimate the beat rate for a bird with } 30 \text{ cm wings.} \]
Find 30 on the horizontal axis
Why: It falls between the plotted lengths of 26 and 38.
Read the rates of the neighbouring points
Why: Five beats a second at 26 centimetres and three at 38.
\[ 5\text{ and } 3 \]
Estimate between them
Why: Thirty is about a third of the way from 26 to 38, so the rate is a little below five.
\[ \text{about } 4.5 \]
State the answer as an estimate
Why: About four and a half beats a second.
Figure (svg): The solution to Worked example read a scatter plot shown as a ladder of expressions, one row per algebraic move
\[ \text{about } 4.5 \text{ beats per second} \]
Verify: check that the estimate sits between its neighbours
Why: Four and a half lies between the three and the five of the neighbouring birds, which it must for a falling trend. An estimate outside that range would contradict the pattern the plot shows.
Trap
Six birds are plotted, and the student draws a line from each point to the next.
Join the points, as with the line graphs of Lesson 1.7
Why: Joining made the trend clearer there, so it looks like an improvement here too.
The line claims that a bird exists at every length in between with exactly that beat rate, which the data never measured. Each point is one bird, not a stage in a continuous change.
Leave the points separate, and describe the trend in words or with a single straight line of best fit.
Join points only when the horizontal axis is an ordered sequence such as time
Why: Lesson 1.7's rule applies unchanged: joining asserts that values exist between the plotted ones.
Six birds are six separate creatures, not one bird changing over time. The scatter shows a trend without claiming any particular bird between the measured ones.
Elimination
Wing length and beat rate for six different birds.
Eliminate the wrong options
Which representation is appropriate?
Survives elimination: A
Why: A scatter plot shows both measurements for every bird at once, which is what makes the relationship visible. The other three each throw away one of the two quantities or assert something the data does not support — and option D is the most common real mistake, since summarising to an average feels like progress.
Estimation
The trend supports an estimate between measured values.
Predict first
The plot shows 5 beats per second at 26 cm and 3 at 38 cm. Roughly what would you expect at 32 cm?
Correct: About 4.
\[ \tfrac{5 + 3}{2} = 4 \text{ beats per second} \]
Why: Thirty-two is halfway between twenty-six and thirty-eight, so the rate should be about halfway between five and three, which is four. Estimating between measured points is reasonable when the trend is steady, though it remains an estimate rather than a measurement.
Socratic
A visible relationship is not the same as an explanation.
Discussion prompt
The wing data shows that longer wings go with slower beating. Give one thing this establishes and one thing it does not. Then suggest why the relationship might exist.
Hint: Think about what the plot could look like for two quantities that are unrelated but both change over time.
Answer:
It establishes that the two quantities vary together in the data collected — knowing a bird's wing length lets you estimate its beat rate. It does not establish that long wings cause slow beating, and it says nothing about birds outside the range measured, from five to fifty-two centimetres.
A plausible reason is that a longer wing moves more air per beat and has more mass to accelerate, so fewer beats are needed and each one takes longer. That is a physical explanation the data is consistent with, but the plot alone cannot confirm it — distinguishing a relationship from an explanation is a habit worth building now, since Chapter 5 will fit lines to data of exactly this kind.
Comparison
Fill the blanks from memory before you scroll back. Only the signs matter.
Comparison matrix
| Quadrant | Sign of x | Sign of y | Example |
|---|---|---|---|
| I | positive | positive | (4, 3) |
| II | negative | positive | (-2, 3) |
| III | negative | negative | (-2, -3) |
| IV | positive | negative | (4, -2) |
Reading down the table traces the quadrants anticlockwise from the top right, which is the whole of the numbering convention.
Pattern
Whether you are plotting a point, reading one, or building a scatter plot, the same five moves cover it.
Step four says the origin every time. Plotting a second point from where the first one landed is the commonest cause of a whole set of misplaced points.
OpenStax Elementary Algebra 2e, §4.1 Use the Rectangular Coordinate System §4.1
Check
Reading a point. Count across before up.
Check your understanding
A point lies 4 units left of the origin and 5 units up. What is its ordered pair?
Answer: A
Why: Four units left gives an x-coordinate of negative four, and five units up gives a y-coordinate of positive five, in that order. Reading the pair back gives the original description, which is the check.
Check
Naming a quadrant. Read the signs only.
Check your understanding
In which quadrant does the point (-7, -1) lie?
Answer: A
Why: Both coordinates are negative, so the point is to the left of the origin and below it — the bottom left region, which is Quadrant III. The sizes of the two numbers play no part in the decision.
Check
A point on an axis. Look for a zero.
Check your understanding
Where does the point (0, -6) lie?
Answer: A
Why: The x-coordinate is zero, so the point makes no horizontal movement and stays on the vertical axis. The y-coordinate of negative six carries it six units below the origin.
Real world
A weather station records the temperature and the number of ice creams sold on twelve days.
Discussion prompt
Describe how you would display this data so that a relationship between the two quantities becomes visible, saying which quantity goes on each axis and why. Then say what pattern you would expect, and one thing the graph could not establish even if the pattern appeared clearly.
Hint: Only one of the two quantities is plausibly the cause of the other.
Answer:
Plot a scatter of twelve points, with temperature across and sales up. Temperature goes on the horizontal axis because it is the quantity you would use to predict the other — sales do not change the weather.
You would expect a rising pattern: warmer days, more ice creams. Every point is one day, and the days are not joined, since there is no continuous change from one day's dot to the next.
What the graph could not establish is that temperature causes the sales. The pattern is consistent with that explanation and also with others — school holidays fall in warm months, and so does tourist traffic. A scatter plot shows that two quantities move together, and deciding why they do is a separate question the picture cannot answer.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
Is the point (0, 5) in Quadrant I?
Correct: No, it lies on the y-axis and is in no quadrant.
\[ (0, 5): \; x = 0 \;\Longrightarrow\; \text{on the } y\text{-axis} \]
\[ (0.1, 5) \text{ would be in Quadrant I} \]
Why: The quadrants are the four open regions strictly between the axes, and a point with a zero coordinate sits on a boundary rather than inside any of them. This is the same distinction as zero being neither positive nor negative on the number line of Lesson 2.1: a boundary case needs its own answer rather than being forced into one of the neighbouring categories.
Explain it
They can use a number line confidently and have never seen two at once.
Discussion prompt
In no more than four sentences, explain how a coordinate plane names a point, using a real place they would recognise. Then tell them the one thing about the notation they must not get wrong, and how to remember it.
Hint: Street grids and cinema seats both work as examples.
Answer:
A usable answer: a coordinate plane is two number lines crossing at right angles, and a point is named by how far across it is and how far up. It is like a cinema seat named by its row and its number, or a street address given as a block east and a block north — one number is never enough, because two places can share it.
The thing not to get wrong is the order: the first number is always across and the second is always up. The letters help, since x comes before y in the alphabet and across comes before up in the pair — and saying across, then up out loud each time is enough to make it stick.
Exit ticket
Name the weakest spot before you close the deck. That is the one worth ten minutes tonight.
Predict first
Which of these would you least want to be handed cold on a quiz tomorrow?
Correct: Whichever you picked is the right answer — and each one has a specific fix.
Why: Order is fixed by saying across, then up aloud every time, and by reading the plotted point back as a pair. Signs are fixed by naming the direction before counting the distance. Quadrant naming is fixed by learning the four sign patterns as one anticlockwise rotation rather than four separate facts. Axis points are fixed by checking for a zero before naming any quadrant at all. Pick yours and do five of that kind tonight rather than twenty mixed ones.
Connect it up
Do this on paper. It is worth more than rereading the slides.
Draw it
Draw one large coordinate plane in the middle of a page, labelling both axes, the origin and all four quadrants with their sign patterns. Plot one point in each quadrant and label it with its ordered pair, then plot one point on each axis and write beside it why it belongs to no quadrant. In one corner, draw the two moves from the origin for a point with two negative coordinates, showing the change of direction between them. Below the plane, plot a small scatter of five or six points from paired data of your own choosing, label both axes with units, and write one sentence describing the pattern. Finally, in the margin, write two ordered pairs that are swaps of each other and mark both on your plane to show they are different points.
Your two swapped points should be reflections of each other across the diagonal through the origin. If they landed on top of each other, check whether the two coordinates you chose happened to be equal.
Recap
Five things, and the second one is the convention everything else depends on.
| If the question says | Your first move is |
|---|---|
| Write the ordered pair | Count across from the origin first |
| Plot the point | Start at the origin, whichever point it is |
| Name the quadrant | Read the two signs and ignore the sizes |
| The point (0, -4) | Check for a zero — it is on an axis |
| Make a scatter plot | One point per pair, and do not join them |
Lesson 4.2 puts equations on this plane: an equation in two variables has solutions that are ordered pairs, and plotting all of them turns out to give a straight line.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.1 The Coordinate Plane §4.1, pp. 203-208 — everything on these slides traces back here
Want this taught 1-on-1? Alexander tutors Algebra 1 — $55/session, free consultation.