Slope as the ratio of vertical rise to horizontal run, the slope formula using subscripted coordinates, and the four cases: positive slope for a line rising left to right, negative for one falling, zero for a horizontal line and undefined for a vertical one. Includes why any two points on a line give the same slope.
Subject: Algebra 1 · 65 slides · symbolic lesson
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Title
Algebra 1 · Chapter 4 — Graphing Linear Equations and Functions
The Slope of a Line
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 4.5 The Slope of a Line §4.5, pp. 229-235 — the lesson these objectives are drawn from
Warm-up
Lesson 4.2 noticed that a line's outputs step by a constant amount. This lesson gives that step a name and a formula.
Discussion prompt
In the table for y equals 3x minus 2, each output was 3 more than the one before it as x rose by 1. If instead x rose by 2 each time, what would the step in y be, and what stays the same?
Hint: Compare the change in y with the change in x rather than looking at y alone.
Answer:
\[ \Delta x = 1: \; \Delta y = 3 \qquad \Delta x = 2: \; \Delta y = 6 \]
The step in y doubles to six, but the ratio of the step in y to the step in x is three in both cases. That ratio is what stays the same, and it is the number this lesson is about — the slope.
Concept
The slope of a line is the ratio of the vertical rise to the horizontal run between any two points on it. A rise means nothing on its own; it is steep or gentle only relative to the run it happens over.
slope — The ratio of the vertical rise to the horizontal run between any two points on a line. It is usually written m.
The ramp investigation on page 228 measures this directly with books and a ruler.
Figure (svg): A ramp resting on a stack of books with its rise and run marked
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.5 The Slope of a Line §4.5, pp. 228-229
Section
Section 1
Concept
Slope is the vertical rise divided by the horizontal run. A ramp rising two inches over five inches has slope two fifths, and doubling the run halves the slope even though the rise has not changed.
\[ m = \dfrac{\text{vertical rise}}{\text{horizontal run}} \]
The ramp investigation varies one of the two at a time to make this visible.
Figure (svg): A ramp resting on a stack of books with its rise and run marked
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.5 The Slope of a Line §4.5, pp. 228-229 — the Developing Concepts ramp investigation and the slope ratio
Picture it
Two measurements, one ratio.
Figure (svg): A ramp resting on a stack of books with its rise and run marked
Keeping the rise fixed and lengthening the run makes the ramp gentler, and the ratio gets smaller to match. That agreement between the picture and the number is what makes the ratio the right definition.
Worked example
This is Example 1 from the textbook.
\[ \text{A hill has a vertical rise of } 40 \text{ feet over a horizontal run of } 200 \text{ feet. Find its slope.} \]
Write the ratio
Why: Slope is rise over run.
\[ m = \frac{40}{200} \]
Substitute the measurements
Why: Forty feet of rise, two hundred feet of run.
\[ \frac{40}{200} \]
Simplify the fraction
Why: Divide both by forty.
\[ \frac{1}{5} \]
State the answer
Why: The slope is one fifth.
\[ m = \frac{1}{5} \]
Figure (svg): The solution to Worked example the slope of a hill shown as a ladder of expressions, one row per algebraic move
\[ m = \dfrac{40}{200} = \dfrac{1}{5} \]
Verify: say what the simplified fraction means
Why: One fifth means one foot of rise for every five feet along, which is a gentle grade — about the steepness of a wheelchair ramp. Reading the simplified ratio back as a sentence is a check that the fraction is the right way up, since five feet up for every one along would be a cliff.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.5 The Slope of a Line §4.5, pp. 229-229
Sorting
Compare each ramp with a slope of one half.
Sort into buckets
Sort each rise-and-run pair by how it compares with a slope of 1/2.
Comparing slopes is comparing fractions, which is a Chapter 2 skill. Nothing new is needed, which is one advantage of defining steepness as a ratio.
Worked example
The ramp experiment, with the rise fixed and the run varied.
\[ \text{A ramp rises } 2 \text{ in. Find its slope over runs of } 4, \; 5 \text{ and } 8 \text{ in.} \]
Take the shortest run
Why: Two over four simplifies to one half.
\[ \frac{1}{2} \]
Take the middle run
Why: Two over five does not simplify.
\[ \frac{2}{5} \]
Take the longest run
Why: Two over eight simplifies to one quarter.
\[ \frac{1}{4} \]
Compare them
Why: The same rise over a longer run gives a smaller slope.
Figure (svg): A ramp resting on a stack of books with its rise and run marked
\[ \tfrac{1}{2} > \tfrac{2}{5} > \tfrac{1}{4} \]
Verify: check the order against the picture
Why: The shortest run gives the steepest ramp and the longest gives the gentlest, which is what pushing the base of a ruler further from a stack of books actually does. This answers the first Think About It question on page 228: with the rise fixed, a longer run means a smaller slope.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.5 The Slope of a Line §4.5, pp. 228-228
Trap
\[ \text{rise } 40 \text{ ft}, \; \text{run } 200 \text{ ft} \]
Write m = 200/40 = 5
Why: The larger number on top looks more natural, and the two words are easy to swap.
A slope of five would mean five feet up for every one along, which is a cliff rather than a hill. The number is the reciprocal of the truth.
\[ m = \dfrac{\text{rise}}{\text{run}} = \dfrac{40}{200} = \dfrac{1}{5} \]
Say the answer as a sentence before accepting it
Why: One up for every five along describes a gentle hill, which matches the measurements.
The word rise comes first in the phrase rise over run, and the sentence test catches it whenever the phrase does not.
Faded example
Rise over run, then simplify.
Fill in the blanks
\text200 40, \; \text5 200: \quad m = \dfrac______} = \dfrac______}
Why: The run goes on the bottom, and dividing both parts by forty gives one fifth. Reading that back as one foot up for every five along confirms the fraction is the right way up.
Prediction
The ramp investigation on page 228 asks this directly.
Predict first
The rise stays at 2 inches and the run is increased from 4 inches to 8 inches. What happens to the slope?
Correct: It halves, from 1/2 to 1/4.
\[ \dfrac{2}{4} = \dfrac{1}{2} \qquad \dfrac{2}{8} = \dfrac{1}{4} \]
Why: Doubling the denominator of a fraction halves it, and the picture agrees: pushing the base of the ruler twice as far from the books makes a noticeably gentler ramp. This is the first Think About It question from the investigation, and it is the reason steepness has to be a ratio rather than just a rise.
Socratic
The investigation could have measured only the rise.
Discussion prompt
Explain why the rise on its own does not describe steepness, using two ramps with the same rise. Then say what a slope of 1 means about the rise and the run.
Hint: Picture a two-inch rise over two inches and over twenty.
Answer:
A rise of two inches over a run of two inches is a very steep ramp, while the same rise over twenty inches is nearly flat. Both have the same rise, so the rise cannot be what steepness means — the comparison with the run is doing all the work.
A slope of one means the rise and the run are equal, so the ramp climbs one unit for every unit along and makes a forty-five degree angle. That is the third Think About It question from the investigation, and it gives a useful landmark: slopes above one are steeper than forty-five degrees and slopes below one are gentler.
Section
Section 2
Concept
For a line through two points, the rise is the difference of their y-coordinates and the run is the difference of their x-coordinates. The slope is the first divided by the second.
\[ m = \dfrac{\text{rise}}{\text{run}} = \dfrac{y_2 - y_1}{x_2 - x_1} \]
Either point may be labelled first, but both subtractions must be done in the same order.
Figure (svg): The slope formula with its numerator and denominator labelled
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.5 The Slope of a Line §4.5, pp. 229-230 — The Slope of a Line box and the note on subtracting in the same order
Picture it
One subtraction on the top and one on the bottom.
Figure (svg): The slope formula with its numerator and denominator labelled
The subscripts are just labels saying which point is which. They are read x sub one and y sub one, and they are not exponents or multiplication.
Worked example
The diagram on page 229 uses the points (3, 2) and (8, 4).
\[ \text{Find the slope of the line through } (3, 2) \text{ and } (8, 4). \]
Label the points
Why: Take (3, 2) as the first and (8, 4) as the second.
\[ (x 1, y 1) = (3, 2) \]
Find the rise
Why: Subtract the y-values in that order: four minus two.
\[ \text{rise } 2 \]
Find the run
Why: Subtract the x-values in the same order: eight minus three.
\[ r u n 5 \]
Divide
Why: Rise over run.
\[ m = \frac{2}{5} \]
Figure (svg): Two points on a line with the rise and run between them drawn as a right triangle
\[ m = \dfrac{4 - 2}{8 - 3} = \dfrac{2}{5} \]
Verify: count the triangle on the graph
Why: From (3, 2) the line goes five squares right and two squares up to reach (8, 4), which is the same two over five. Counting squares is an independent check on the subtraction, and it also confirms which of the two numbers belongs on top.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.5 The Slope of a Line §4.5, pp. 229-229
Faded example
Take (1, 0) as the first point.
Fill in the blanks
m = \dfrac0}}1}} = \dfrac2___ = ___
Why: Both blanks come from the same point, which is exactly what the same order rule requires. The result is two, and a sketch confirms it: the line goes two up for every one across, rising to the right.
Worked example
Swapping the labels must not change the answer.
\[ \text{Find the slope through } (1, 0) \text{ and } (3, 4) \text{ both ways round.} \]
Take (1, 0) first
Why: Four minus zero over three minus one.
\[ \frac{4}{2} = 2 \]
Take (3, 4) first
Why: Zero minus four over one minus three.
\[ -4 / - 2 = 2 \]
Compare
Why: Both give two.
Say why
Why: Reversing both subtractions negates the top and the bottom, and the negatives cancel.
Figure (svg): The same two points subtracted in both orders, giving the same slope
\[ \dfrac{4 - 0}{3 - 1} = 2 \qquad \dfrac{0 - 4}{1 - 3} = 2 \]
Verify: try reversing only one subtraction and see what breaks
Why: Four minus zero over one minus three gives four over negative two, which is negative two — the wrong sign. Reversing one subtraction and not the other is the error the same order rule exists to prevent, and it always produces exactly the wrong sign.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.5 The Slope of a Line §4.5, pp. 230-230
Error analysis
The student found the slope through (1, 0) and (3, 4).
Annotate
On: \( \begin{aligned} m &= \frac{y_2 - y_1}{x_2 - x_1} \\ &= \frac{4 - 0}{1 - 3} \\ &= \frac{4}{-2} \\ &= -2 \end{aligned} \)
The fix is to write both subscripted points down first and then fill the formula from them, rather than reading numbers off the page in whatever order they appear.
Elimination
Finding the slope through (2, 1) and (6, 9).
Eliminate the wrong options
Which expression gives the slope?
Survives elimination: A
Why: Both subtractions take the second point first, giving eight over four, which is two. Option D is worth noticing because adding produces a plausible-looking number, ten eighths, with no meaning behind it — only differences describe a change.
Translation
Subtract in the same order top and bottom.
Match the pairs
Why: The first three rise to the right and give positive slopes; the fourth falls, since its y drops from nine to seven while x increases, giving negative one half. Sketching each pair before computing predicts the sign and catches an ordering slip.
Socratic
The rule sounds arbitrary until you see what it protects.
Discussion prompt
Explain what goes wrong if you subtract the y-values in one order and the x-values in the other. Then say why reversing both is safe.
Hint: Think about what happens to a fraction when only one part changes sign.
Answer:
Reversing one subtraction negates it, so the fraction changes sign. The size of the answer stays right and the direction is reported backwards — a line that rises is described as falling, which is worse than a small numerical error because it contradicts the picture.
Reversing both negates the top and the bottom, and a negative divided by a negative is positive, so the two changes cancel exactly. That is why either point may be labelled first: the formula does not care which, as long as it is asked the same question twice.
Section
Section 3
Concept
A line with positive slope rises from left to right. A line with negative slope falls from left to right. The sign can be predicted from a sketch before any arithmetic is done.
Reading left to right is the convention; the line itself has no preferred direction.
Figure (svg): Four lines showing positive, negative, zero and undefined slope
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.5 The Slope of a Line §4.5, pp. 230-231 — Examples 2 and 3, Positive Slope and Negative Slope
Picture it
Every line you can draw is one of these.
Figure (svg): Four lines showing positive, negative, zero and undefined slope
The first two are this section and the last two are the next. Deciding which of the four you are looking at is always the first move.
Worked example
This is Example 2 from the textbook.
\[ \text{Find the slope of the line through } (1, 0) \text{ and } (3, 4). \]
Label the points
Why: Take (1, 0) as the first.
\[ (x 1, y 1) = (1, 0) \]
Subtract the y-values
Why: Four minus zero is four.
\[ \text{rise } 4 \]
Subtract the x-values in the same order
Why: Three minus one is two.
\[ r u n 2 \]
Divide and read the sign
Why: Four over two is two, which is positive, so the line rises.
\[ m = 2,\text{ rising} \]
Figure (svg): The solution to Worked example a positive slope shown as a ladder of expressions, one row per algebraic move
\[ m = \dfrac{4 - 0}{3 - 1} = 2 \]
Verify: check the sign against a sketch
Why: Moving from (1, 0) to (3, 4) goes right and up, so the line rises and the slope must be positive. Predicting the sign from the sketch before dividing turns the arithmetic into a confirmation rather than the only evidence.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.5 The Slope of a Line §4.5, pp. 230-230
Discrimination
Predict the sign from the points before computing.
Sort into buckets
Sort each pair of points by the sign of the slope of the line through them.
Worked example
This is Example 3 from the textbook.
\[ \text{Find the slope of the line through } (0, 3) \text{ and } (6, 1). \]
Label the points
Why: Take (0, 3) as the first.
\[ (x 1, y 1) = (0, 3) \]
Subtract the y-values
Why: One minus three is negative two.
\[ \text{rise } -2 \]
Subtract the x-values in the same order
Why: Six minus zero is six.
\[ r u n 6 \]
Divide and simplify
Why: Negative two over six is negative one third.
\[ m = -\frac{1}{3} \]
Figure (svg): The solution to Worked example a negative slope shown as a ladder of expressions, one row per algebraic move
\[ m = \dfrac{1 - 3}{6 - 0} = \dfrac{-2}{6} = -\dfrac{1}{3} \]
Verify: use two different points on the same line
Why: The textbook's study tip suggests (0, 3) and (3, 2), which give two minus three over three minus zero, or negative one third — the same answer. Any two points on a line give the same slope, so a second pair is a genuinely independent check.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.5 The Slope of a Line §4.5, pp. 231-231
Trap
\[ (0, 3) \text{ and } (6, 1) \]
Subtract the smaller y from the larger to avoid a negative: 3 - 1 = 2
Why: Negative numbers are unwelcome, and taking the larger first keeps everything positive.
\[ m = \dfrac{2}{6} = \dfrac{1}{3} \quad \text{(wrong sign)} \]
The line clearly falls from (0, 3) down to (6, 1), so a positive slope contradicts the picture.
\[ m = \dfrac{1 - 3}{6 - 0} = \dfrac{-2}{6} = -\dfrac{1}{3} \]
Take the y-values in the same order as the x-values, negative or not
Why: The order is fixed by the labelling, not by which number is larger.
Sketching the two points first and predicting the sign makes this error impossible to keep: a falling line and a positive answer cannot both be right.
Faded example
Take (0, 9) as the first point.
Fill in the blanks
m = \dfrac-2-1/2 = \dfrac___}___ = ___
Why: The numerator is negative because y falls from nine to seven while x rises, so the line goes down to the right. Simplifying gives negative one half, meaning it falls one unit for every two along.
Prediction
A line has slope negative three quarters.
Predict first
What does the line do as you read it from left to right?
Correct: It falls, dropping 3 units for every 4 across.
\[ m = -\tfrac{3}{4}: \; \text{right } 4 \Rightarrow \text{ down } 3 \]
Why: The negative sign means the line falls as x increases, and the fraction says the drop is three for every four along. The third option reverses rise and run, giving the reciprocal, and the fourth confuses a small slope with a zero one — a slope of three quarters is gentle but not flat.
Socratic
The connection between a minus sign and a direction deserves a reason.
Discussion prompt
Explain why a negative slope means the line falls from left to right, starting from what the numerator and denominator measure. Then say what happens if you read the line from right to left instead.
Hint: Ask what makes a fraction negative.
Answer:
A fraction is negative when exactly one of its parts is. Reading left to right means the run is positive, since x increases, so the only way the slope can be negative is for the rise to be negative — and a negative rise is a fall.
Reading right to left makes both parts negative, and the slope comes out the same. That is the same cancellation as the order rule from the last section, and it is why the convention of reading left to right is a convention rather than a fact about the line.
Section
Section 4
Concept
A horizontal line has no rise, so its slope is zero divided by something, which is zero. A vertical line has no run, so its slope would be something divided by zero, which has no meaning at all.
\[ \dfrac{0}{4} = 0 \qquad \dfrac{4}{0} \text{ is undefined} \]
Zero slope is a perfectly good number; undefined slope is the absence of one.
Figure (svg): Two columns contrasting a zero slope with an undefined slope
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.5 The Slope of a Line §4.5, pp. 231-232 — Examples 4 and 5, Zero Slope and Undefined Slope
Picture it
Same digit, opposite consequences.
Figure (svg): Two columns contrasting a zero slope with an undefined slope
The two cases connect straight back to Lesson 4.3: zero slope belongs to the family y equals b, and undefined slope to the family x equals a.
Worked example
This is Example 4 from the textbook.
\[ \text{Find the slope of the line through } (-1, 2) \text{ and } (5, 2). \]
Subtract the y-values
Why: Two minus two is zero.
\[ \text{rise } 0 \]
Subtract the x-values in the same order
Why: Five minus negative one is six.
\[ r u n 6 \]
Divide
Why: Zero divided by six is zero.
\[ m = 0 \]
Name the line
Why: Both points have the same y-coordinate, so the line is horizontal.
Figure (svg): The solution to Worked example a zero slope shown as a ladder of expressions, one row per algebraic move
\[ m = \dfrac{2 - 2}{5 - (-1)} = \dfrac{0}{6} = 0 \]
Verify: check against Lesson 4.3
Why: Two points sharing a y-coordinate lie on the horizontal line y equals 2, which that lesson showed has every point at the same height. No height is gained anywhere along it, so a slope of zero is exactly what the picture requires.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.5 The Slope of a Line §4.5, pp. 231-231
Sorting
Look for a shared coordinate.
Sort into buckets
Sort each pair of points by the slope of the line through them.
A shared coordinate is the whole test, and which coordinate is shared decides which of the two special cases you have.
Worked example
This is Example 5 from the textbook.
\[ \text{Find the slope of the line through } (5, -1) \text{ and } (5, 3). \]
Subtract the y-values
Why: Three minus negative one is four.
\[ \text{rise } 4 \]
Subtract the x-values in the same order
Why: Five minus five is zero.
\[ r u n 0 \]
Try to divide
Why: Four divided by zero has no meaning.
Name the line
Why: Both points have the same x-coordinate, so the line is vertical.
Figure (svg): The solution to Worked example an undefined slope shown as a ladder of expressions, one row per algebraic move
\[ m = \dfrac{3 - (-1)}{5 - 5} = \dfrac{4}{0} \quad \text{undefined} \]
Verify: say what the division by zero is reporting
Why: Slope asks how much the line climbs per unit across, and a vertical line goes no distance across at all — so the question has no answer rather than an answer of zero. Reporting undefined is the complete and correct answer here, not a failure to finish.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.5 The Slope of a Line §4.5, pp. 232-232
Trap
\[ (5, -1) \text{ and } (5, 3): \; m = \dfrac{4}{0} \]
Write m = 0, since there is a zero in the fraction
Why: A zero appears in the working, and zero is the nearest available answer.
A slope of zero would mean a horizontal line, and these two points are one directly above the other. The two answers describe perpendicular lines.
\[ \dfrac{4}{0} \text{ is undefined; the line is vertical} \]
Ask which position the zero is in before answering
Why: Zero on the top gives zero; zero on the bottom gives no value at all.
The check is to look at the two points: a shared y-coordinate means horizontal and zero slope, and a shared x-coordinate means vertical and undefined slope.
Faded example
Compute both and name each line.
Fill in the blanks
Through (-1, 2) and (5, 2): m = 0/6 = 0. Through (5, -1) and (5, 3): m = 4/0 is undefined.
Why: Zero on top divides perfectly well and gives zero, so the horizontal line has a slope. Zero on the bottom gives nothing at all, so the vertical line has none. The two look similar on the page and describe perpendicular lines.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Zero slope | Undefined slope | |
|---|---|---|
| Which part is zero | the rise | the run |
| The line is | horizontal | vertical |
| Its equation looks like | y = b | x = a |
| Is it a function of x? | Yes | No |
Every row is the mirror image of the other column, and the last row is the one from Lesson 4.3 — the vertical line fails the definition of a function as well as having no slope.
Socratic
It would be convenient if it just gave zero or infinity.
Discussion prompt
Explain why four divided by zero cannot be given a value, using what division means. Then say why zero divided by four is perfectly fine.
Hint: Division asks what number times the divisor gives the dividend.
Answer:
Four divided by zero asks what number multiplied by zero gives four. Everything multiplied by zero gives zero, so no such number exists and no value can be assigned. Choosing one anyway would break the rule that multiplying by zero gives zero, which is more valuable than filling in the gap.
Zero divided by four asks what number multiplied by four gives zero, and the answer is zero — a single, unambiguous value. The two expressions look similar and are completely different questions, which is why one has an answer and the other does not. This is the same rule Lesson 2.8 stated about dividing by zero, applied to a new situation.
Section
Section 5
Concept
The slope may be computed from any two points on a line, and every choice gives the same answer. That is why it is meaningful to speak of the slope of the line rather than the slope between two points.
\[ \dfrac{1}{2} = \dfrac{2}{4} = \dfrac{3}{6} \]
The rise-and-run triangles for different pairs of points have different sizes and the same shape.
Figure (svg): A line with three different rise-over-run triangles all giving the same ratio
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.5 The Slope of a Line §4.5, pp. 231-231 — the Study Tip that any two points on a line give the same slope
Picture it
Different sizes, same shape.
Figure (svg): A line with three different rise-over-run triangles all giving the same ratio
The textbook notes that the proof of this belongs to geometry. What you can do now is check it on examples, which is worth doing at least once.
Worked example
The line through (1, 1), (3, 2) and (7, 4).
\[ \text{Compute the slope from each pair of points on this line.} \]
Use the first two points
Why: One over two.
\[ \frac{1}{2} \]
Use the last two points
Why: Two over four, which simplifies to one half.
\[ \frac{1}{2} \]
Use the outer two points
Why: Three over six, which also simplifies to one half.
\[ \frac{1}{2} \]
Conclude
Why: All three agree, so the slope belongs to the line.
\[ m = \frac{1}{2} \]
Figure (svg): A line with three different rise-over-run triangles all giving the same ratio
\[ \dfrac{1}{2} = \dfrac{2}{4} = \dfrac{3}{6} \]
Verify: notice that the three fractions are equivalent rather than equal as written
Why: One over two, two over four and three over six are three names for the same number, exactly as in Chapter 2. Slope is well defined precisely because the different triangles produce equivalent fractions rather than different values.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.5 The Slope of a Line §4.5, pp. 231-231
Pattern
Each frame uses a wider pair of points on the same line.
Step through it
What do all four pairs have in common, and what does that let you say about the line?
Every pair simplifies to one half. The triangles grow but the ratio does not, which is what allows slope to be called a property of the line.
Worked example
A second pair is an independent check on the first.
\[ \text{A student says the line through } (0, 3) \text{ and } (6, 1) \text{ has slope } \tfrac{1}{3}. \text{ Check it.} \]
Recompute from the given points
Why: One minus three over six minus zero is negative two over six.
\[ -\frac{1}{3} \]
Compare with the claim
Why: The size matches and the sign does not.
Check with a third point on the line
Why: The point (3, 2) is on it, and two minus three over three minus zero is negative one third.
\[ -\frac{1}{3}\text{ again} \]
State the verdict
Why: The slope is negative one third; the claim dropped the sign.
Figure (svg): The solution to Worked example check a claimed slope with a third point shown as a ladder of expressions, one row per algebraic move
\[ m = -\dfrac{1}{3} \text{, not } \dfrac{1}{3} \]
Verify: confirm with the picture
Why: From (0, 3) the line goes right and down to (6, 1), so it falls and the slope must be negative. Two independent computations and the sketch all agree, which is as sure as this kind of check gets.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.5 The Slope of a Line §4.5, pp. 231-231
Trap
Two lines are drawn on one plane, and a student picks (1, 1) from one and (7, 4) from the other.
Apply the slope formula to the two points
Why: Both points are visible on the page and the formula only asks for two of them.
The formula returns a number, but it is the slope of a third line through those two points — not of either line drawn.
Check that both points lie on the same line before computing
Why: Slope is a property of one line, and the formula cannot tell whether the points belong to it.
Substituting each point into the line's equation confirms membership, which is the check from Lesson 4.2 doing useful work again.
Elimination
The line has slope 2 and passes through (1, 0).
Eliminate the wrong options
Which point is NOT on that line?
Survives elimination: A
Why: From (1, 0) the point (2, 3) is one right and three up, a ratio of three rather than two. The other three all give exactly two, which is what being on the line requires — and this test is a quick way to check membership without ever writing the equation down.
Two truths and a lie
Three statements about slope. Two are true and one is not.
Eliminate the wrong options
Which statement is false?
Survives elimination: A
Why: The false statement is the first. A line falling steeply has a large negative slope, and as a number that is smaller than a gentle positive one — negative five is less than one half, though the first line is far steeper. Comparing steepness means comparing how far the slopes are from zero, which is the absolute value idea from Lesson 2.2, while the slope itself also carries a direction.
Socratic
The textbook says the proof belongs to geometry, but the idea can be seen now.
Discussion prompt
Explain informally why widening the run between two points on a line increases the rise in the same proportion. Then say what would happen to this argument if the graph were a curve instead.
Hint: Think about the constant step from Lesson 4.2.
Answer:
A linear equation adds the same amount to y for each unit added to x, as the constant step in the table showed. So doubling the run doubles the rise exactly, tripling triples it, and the ratio between them never moves. The triangles are scaled copies of one another.
For a curve the step is not constant — Lesson 4.2's table for y equals x squared stepped by three then five then seven — so widening the run changes the rise by a different proportion and the ratio moves. That is precisely why a curve has no single slope, and why finding a slope at a point on a curve turns out to need calculus.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Slope | The line | How to spot it from two points |
|---|---|---|
| Positive | rises left to right | y increases as x increases |
| Negative | falls left to right | y decreases as x increases |
| Zero | horizontal | the y-coordinates match |
| Undefined | vertical | the x-coordinates match |
The last column can be checked by eye before any subtraction, which makes it the fastest way to catch a slip in the arithmetic.
Pattern
Whether the slope is asked for from points, a graph or a description, the same five moves cover it.
Step two costs a few seconds and catches the ordering error, which otherwise produces a perfectly plausible answer with the wrong sign.
OpenStax Elementary Algebra 2e, §4.4 Understand Slope of a Line §4.4
Check
Subtract in the same order top and bottom.
Check your understanding
What is the slope of the line through (2, 1) and (6, 9)?
Answer: A
Why: The rise is nine minus one, which is eight, and the run is six minus two, which is four. Eight over four is two, and the positive sign matches a line rising from left to right.
Check
Look for a shared coordinate.
Check your understanding
What is the slope of the line through (4, -2) and (4, 6)?
Answer: A
Why: Both points have an x-coordinate of four, so the run is zero and the division has no value. The line is vertical, which by Lesson 4.3 is the graph of x equals 4.
Check
The sign carries a direction.
Check your understanding
A line has slope -2/3. What does it do from left to right?
Answer: A
Why: The negative sign means the line falls as x increases, and the fraction says two down for every three along. The rise is on top and the run underneath, so the two numbers are not interchangeable.
Real world
Building codes limit a wheelchair ramp to a slope of one twelfth. A doorway sits 30 inches above the pavement.
Discussion prompt
Work out the shortest run the ramp may have, and say what the slope one twelfth means in words. Then say what happens to the required run if the doorway is twice as high.
Hint: Rise over run equals one twelfth.
Answer:
\[ \dfrac{30}{\text{run}} = \dfrac{1}{12} \;\Longrightarrow\; \text{run} = 360 \text{ in} = 30 \text{ ft} \]
A slope of one twelfth means one inch of climb for every twelve inches along, which is why the ramp needs thirty feet of pavement for a thirty-inch step. The number sounds small and the consequence is large, which is exactly the point of the code.
Doubling the height doubles the run to sixty feet, because the ratio is fixed. The rise and the run are in direct proportion whenever the slope is held constant, which is the same reasoning as the equivalent fractions in the last section — and it is why accessible entrances often need switchbacks.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
What is the slope of the line through (3, 5) and (3, -2)?
Correct: Undefined, since the run is zero.
\[ m = \dfrac{-2 - 5}{3 - 3} = \dfrac{-7}{0} \quad \text{undefined} \]
\[ \text{compare } (3, 5), (8, 5): \; m = \dfrac{0}{5} = 0 \]
Why: Both points have an x-coordinate of three, so the denominator of the slope formula is zero and the division has no value. The line is vertical. The first option is the trap worth naming: a zero appears in the working, but it is in the denominator, and a zero there means no answer rather than an answer of zero. Zero slope belongs to horizontal lines, which are perpendicular to this one.
Explain it
They can plot points and have never met slope.
Discussion prompt
In no more than four sentences, explain what slope measures and why it has to be a ratio rather than a single measurement. Then give them the one thing to check before trusting an answer.
Hint: Start from a ramp rather than from a formula.
Answer:
A usable answer: slope measures how steeply a line climbs, by comparing how far it goes up with how far it goes across. A rise of two inches is steep over three inches and almost flat over thirty, so the rise alone cannot describe steepness — only the ratio can.
The thing to check is the sign against a sketch. If the two points go up as you read to the right, the slope has to be positive, and any negative answer means the subtractions were done in mismatched orders. That check catches the commonest error in the whole lesson and takes about five seconds.
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: The order is fixed by writing both labelled points down before touching the formula. Rise over run is fixed by saying the answer as a sentence and checking it against the picture. The two zeros are fixed by asking which position the zero is in — top gives zero, bottom gives nothing. Predicting the sign is fixed by asking whether y goes up or down as x goes up, which needs no arithmetic 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 coordinate plane and on it draw four lines: one rising, one falling, one horizontal and one vertical. Label each with its slope, writing undefined rather than a number for the vertical one. On the rising line, mark three different pairs of points and draw the rise-and-run triangle for each pair, writing the fraction beside each triangle and showing that all three simplify to the same number. Underneath the plane, write the slope formula with both subtractions labelled, and beside it write out one full computation with the two points labelled first. In the lower corner, write the two fractions zero over four and four over zero, and beside each say what it equals and which kind of line it belongs to. Finally, in the margin, write one sentence saying why the rise alone cannot describe steepness.
Your three triangles on the rising line should give three different-looking fractions that simplify to the same number. If any of them does not, recheck which coordinate went on top.
Recap
Five things, and the fourth is the distinction that costs the most marks.
| If the question says | Your first move is |
|---|---|
| Find the slope through two points | Label the points, then fill the formula |
| Is the slope positive or negative | Ask whether y rises or falls as x rises |
| The x-coordinates are the same | The slope is undefined; the line is vertical |
| The y-coordinates are the same | The slope is zero; the line is horizontal |
| How steep is it | Compute rise over run and read it as a sentence |
Lesson 4.6 looks at the simplest family of lines of all — the ones through the origin, where y is a constant multiple of x. Their slope turns out to be that constant, which makes them the natural model for any two quantities in fixed proportion.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.5 The Slope of a Line §4.5, pp. 229-235 — everything on these slides traces back here
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