Quarry School

Slope: output change divided by input change

Explain it like I am five

Picture a ramp. You walk forward along the floor and climb upward at the same time. To describe its tilt, you compare how much you climbed with how far you walked forward. On a graph, the upward change is called rise, and the forward change is called run. Either change can be negative if you move down or left. The slope is rise divided by run. You are asking how much the output changes for one unit of input change. A climb of 6 over a forward move of 3 means a climb of 2 for each step forward. The total climb is 6, but the slope is 2.

12345678910−5−4−3−2−1123run 2rise 1.2(3, −2)(8, 1)
The verified points give rise three and run five.
Reminder
  • Ordered pairs. (3, −2) means input 3 and output −2.
  • Subtracting negatives. 1 − (−2) = 1 + 2 = 3.
  • Signed division. −7−3 = 73, while −73 is negative.
  • Reducing fractions. 63 = 2 because 6 ÷ 3 = 2.
  • Function notation. f(4) = 9 means the point (4, 9).
  • Absolute value. |−4| = 4 because −4 is four units from zero.
  • Division by zero. 50 is undefined because no number times zero gives five.
  • Units and rounding. 9dollars3items = 3 dollars per item; 103 ≈ 3.33 is rounded.
Why it works. Subscripts 1 and 2 label the first and second points; they are not powers. From input x1 to input x2, the horizontal change is x2 − x1; the vertical change is y2 − y1. Dividing spreads the vertical change across the horizontal units. For y = mx + b, subtracting the outputs gives (mx2 + b) − (mx1 + b) = m(x2 − x1). The starting amounts cancel. Divide by the nonzero input change to get m. That is why every pair of distinct inputs on the same line gives the same slope.
RuleRule: m = riserun = y2−y1x2−x1 = f(x2)−f(x1)x2−x1, where x2 ≠ x1. Units are output units per input unit.
The same idea, five ways
Say it

Say: slope is change in output divided by change in input.

Write it

Write: the output changes by m units per one-unit increase in input.

In math
  • m = riserun
  • m = y2−y1x2−x1
  • m = f(x2)−f(x1)x2−x1
  • Δy = mΔx; Δ means change
  • Graph words: vertical change divided by horizontal change
Like

Compare height gained on a ramp with floor distance covered.

See it
2246run 1rise 2startrise 4, run 2
A ratio of changes describes the tilt.
The same idea, other ways
As a ramp

Walking five feet forward and climbing three gives slope 35. You climb three fifths of a foot for each foot forward.

2461234run 2rise 1.2startfinish
Divide the climb by the forward distance.
As a bill

Four more notebooks add $12. Each notebook adds 124 = $3. Dollars are the output units and notebooks are the input units.

4 more notebooks$3 per notebook$12 moreinputoutput
Input change times rate equals output change.
As repeated steps

Slope −2 means one right and two down. Three right then means six down. Negative tells you the output falls as input increases.

24−2246run 1rise −2startone rightthree right
The size of the output change scales with the input change.
Why either point order works

Going from (3, −2) to (8, 1) gives 35. Going backward gives −3−5, still 35. Both changes reversed, so the ratio stayed the same.

forward: 35
backward: −3−5 = 35
mixed order: 3−5 is wrong
Reverse both differences together.
.1Coordinate slope

Two point addresses tell you the start and finish of a walk. Subtract the y-coordinates to find vertical displacement, the signed upward or downward change. Subtract the x-coordinates to find horizontal displacement. These are rise and run. Use the same start and finish for both.

  • Rule: rise = y2 − y1 and run = x2 − x1, because change is finish minus start.
  • Rule: m = riserun when run ≠ 0, because division gives change per input unit.
2468−4−22run 2rise 1.2startfinish
Rise and run are changes, not the coordinates of a single point.
Reminder
  • Signed subtraction. Subtracting −2 means adding 2.
The same idea, five ways
Say it

Say: finish minus start in both directions.

Write it

Write: divide vertical displacement by horizontal displacement.

In math
  • rise = Δy = y2 − y1
  • run = Δx = x2 − x1
  • m = ΔyΔx
Like

Compare a ramp's climb with its forward distance.

See it
2468−4−22run 2rise 1.2startfinish
The same two addresses supply both changes.
Worked exampleA positive fraction from a negative starting output

Find the slope through (3, −2) and (8, 1). This asks for the change per step even though one output is negative.

2468−4−22run 2rise 1.2(3, −2)(8, 1)
A negative starting height does not make the slope negative.
  1. Rise = 1 − (−2) = 3.Moving upward from −2 to 1 passes through zero.
  2. Run = 8 − 3 = 5.The same walk moves five input units right.
  3. m = 35.Divide rise by run; no common factor remains.
Answer
m = 35.
Check Five times 35 is three, and −2 + 3 = 1.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: use 18 from the second point.
Those are a location's coordinates, not its changes from the first point.
✓ Instead: Subtract both start coordinates first: 1−(−2)8−3 = 35.
Tips and tricks
  • Tip: say finish minus start twice.
.2Slope units and a constant-change assumption

Imagine spreading newly arrived people equally across several years. The population change goes on top and elapsed years go on bottom. That gives people per year. A linear model uses that same yearly amount throughout the interval. Actual populations might change differently within the interval, so state the constant-change assumption.

  • Rule: output units per input unit follow the same top-over-bottom order as the numbers.
  • The calculation gives an average rate; a constant-change model treats it as the rate throughout.
  • Input zero can mean years after a chosen date, not calendar year zero.
1900people3years
= 19003 people per year
≈ 633 people per year, rounded
The units tell you what each step means.
Reminder
  • Exact versus rounded. 103 is exact; 3.33 is approximate, so use ≈.
The same idea, five ways
Say it

Say: about six hundred thirty-three people per year.

Write it

Write: assuming steady growth, each year adds 19003 people.

In math
  • m = 8100−62002012−2009 = 19003 people per year
  • m ≈ 633 people per year, rounded
  • P(t) = 6200 + 19003t, with t years after 2009
Like

Share the total growth equally among the elapsed years.

See it
26210644066706900713073607590782080508280run 1rise 63320092012
A gain of 1900 is spread across three years.
Worked exampleThe verified town rate

A town grows from 6,200 people in 2009 to 8,100 in 2012. Assuming steady change, find its yearly rate. This asks how much of the total growth belongs to each elapsed year.

26210644066706900713073607590782080508280run 1rise 63320092012
The horizontal axis measures years after 2009.
  1. Population change = 8100 − 6200 = 1900 people.Finish minus start gives the added people.
  2. Time change = 2012 − 2009 = 3 years.Elapsed time is the difference of the years.
  3. m = 19003 = 633.333… people per year.Divide total growth by elapsed time.
  4. m ≈ 633 people per year, rounded to the nearest whole number.The tenths digit is 3, below 5, so rounding to the nearest whole number gives 633.
Answer
  • Exact: 19003 people per year.
  • Rounded: about 633 people per year.
Check 6200 + 19003 × 3 = 8100, the recorded final population.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: the rate is 1900 years per person.
Population is the output and time the input; both units and division were reversed.
✓ Instead: Use 19003 people per year.
Tips and tricks
  • Tip: write units before you reduce.
.3Steepness and zero run

A downhill ramp can be as steep as an uphill ramp. The sign tells direction, while absolute value tells size without direction. Absolute value means distance from zero, so |−4| = 4. With matching scales, slope −4 is steeper than slope 1. A flat walk has no rise; a vertical move has no run.

  • Rule: with matching axis scales and units, greater |m| means a steeper line because one horizontal unit creates more vertical change.
  • Rule: a horizontal line has slope zero because its rise is zero and its run is nonzero.
  • Rule: a vertical line has undefined slope because its run is zero. It is not a function of x because one input has multiple outputs.
−22−8−6−4−22468run 1rise −4
Compare tilt on the same axes.
Reminder
  • Absolute value. |−4| = |4| = 4 because both are four units from zero.
  • Zero multiplication. 0 × 7 = 0 explains why 07 = 0 and why 70 has no value.
The same idea, five ways
Say it

Say: size gives steepness and sign gives direction.

Write it

Write: compare absolute slopes using matching scales and units.

In math
  • |−4| = 4 > |1| = 1
  • Horizontal: y = 4, m = 0
  • Vertical: x = 2, slope undefined
Like

A staircase has the same steepness whether you go up or down.

See it
−22−8−6−4−22468run 1rise −4
The falling line has a larger absolute slope than the dashed rising line.
Worked exampleZero on top versus zero on bottom

Find slopes through (1, 4) and (6, 4), then through (2, 1) and (2, 6). This asks whether each move is flat or has no forward travel.

246246firstsecond
No output change means zero slope.
12 ÷ 3 = 412 ÷ 2 = 612 ÷ 1 = 1212 ÷ 0 = ?0 groups: nowhere to put them, so no answer
Five divided by zero has no value.
  1. First rise = 4 − 4 = 0; run = 6 − 1 = 5.Outputs match while inputs differ.
  2. First slope = 05 = 0, so the line is horizontal.Zero spread across five units remains zero.
  3. Second rise = 6 − 1 = 5; run = 2 − 2 = 0.Inputs match while outputs differ.
  4. Second slope would be 50, so it is undefined; x = 2 is vertical and is not a function of x.No number times zero gives five, and input 2 has multiple outputs.
Answer
  • First: slope 0, horizontal.
  • Second: undefined slope, vertical, not a function of x.
Check For the flat line, 0 × 5 recovers rise zero. For the vertical line, no finite slope times run zero can recover rise five.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: −4 < 1, so slope −4 is less steep.
That comparison includes direction rather than magnitude.
✓ Instead: Compare |−4| = 4 with |1| = 1. The slope −4 is steeper on matching axes.
✗ Not this: Counterexample: a vertical line has zero slope.
Vertical means zero run. Zero slope requires zero rise.
✓ Instead: Horizontal has slope zero; vertical has undefined slope.
Tips and tricks
  • Tip: a flat floor has zero rise; a vertical wall has zero run.
Strategy: step by step
  1. 1. Label the two points so each input stays paired with its output. The subscripts 1 and 2 name points, while raised exponents name powers.
  2. 2. Calculate rise as second output minus first output.
  3. 3. Calculate run as second input minus first input, in matching order.
  4. 4. If run is zero, stop. Otherwise divide rise by run and reduce.
  5. 5. Attach units. Multiply slope by run to check that you recover the rise.
Strategy
Strategy: calculate and interpret slope
1
Is the input change zero?
YesDo not divide. Distinct points with that input form a vertical line with undefined slope.
NoDivide output change by input change.
↓
2
Is a rounded answer requested?
YesKeep the exact fraction first, then label the requested approximation.
NoKeep the reduced exact fraction if its decimal does not end.
  1. 1. Name input and output units.
  2. 2. Subtract in matching order.
  3. 3. Check the run, then divide if it is nonzero.
  4. 4. Reduce and interpret the units and sign.
  5. 5. Multiply slope by run to recover rise.
Worked exampleA small climb spread across two steps

Find the slope through (1, 2) and (3, 6). This asks how much the output changes for each one-unit increase in input.

242468run 1rise 2startone stepfinish
Two right and four up means two up per one right.
  1. Rise = 6 − 2 = 4.The output climbs from 2 to 6, four units upward.
  2. Run = 3 − 1 = 2.That climb occurs across two input units.
  3. m = 42 = 2.Divide the total climb by the number of input units.
Answer
m = 2.
Check Move one right and two up twice: (1, 2) to (2, 4) to (3, 6). This reaches the supplied second point.
Ladder: from easy to exam-hard. Press Try it first on any rung to hide its steps and use them as hints.
Rung 1Rung 1: one step right

Find the slope through (0, 0) and (1, 2). This asks how high one step right takes you.

21234run 1rise 2startfinish
When run is one, rise is slope.
  1. Rise = 2 − 0 = 2.The output increases two.
  2. Run = 1 − 0 = 1.The input increases one.
  3. m = 21 = 2.Divide output change by input change.
Answer
m = 2.
Check One step of slope two takes output zero to two.
Rung 2Rung 2: a downward move

Find the slope through (4, 7) and (6, 1). This asks how much the output falls per step right.

46−2246810run 1rise −3startfinish
Moving right while falling gives negative slope.
  1. Rise = 1 − 7 = −6.The output falls six.
  2. Run = 6 − 4 = 2.The input increases two.
  3. m = −62 = −3.Six down across two units is three down per unit.
Answer
m = −3.
Check −3 × 2 = −6, and 7 − 6 = 1.
Rung 3Rung 3: signs and a fractional slope

Find the slope through (3, −2) and (8, 1), then repeat backward. This asks whether reversing your walk changes the slope.

2468−4−22run 2rise 1.2firstsecond
Use either direction in both differences.
  1. Forward: 1−(−2)8−3 = 35.Both changes use finish minus start.
  2. Backward: −2−13−8 = −3−5 = 35.Reversing both differences changes both signs.
Answer
Both orders give m = 35.
Check 35 × 5 = 3 forward, and 35 × (−5) = −3 backward.
Rung 4Rung 4: an exact rate with units

A town grows from 6,200 people in 2009 to 8,100 in 2012. Assume constant change. Find the yearly rate exactly and rounded. This asks for people gained per year.

26210644066706900713073607590782080508280run 1rise 63320092012
Measure time from the first observation.
  1. Population change = 8100 − 6200 = 1900.Population is output.
  2. Time change = 2012 − 2009 = 3.Time is input.
  3. m = 19003 people per year.Divide output change by input change.
  4. m ≈ 633 people per year, rounded to the nearest whole number.The exact rate is 633.333…; its tenths digit is 3, below 5, so rounding to the nearest whole number gives 633.
Answer
  • Exact: 19003 people per year.
  • Rounded: about 633 people per year.
Check 6200 + 19003 × 3 = 8100.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: 1−(−2)3−8 gives the slope through (3, −2) and (8, 1).
The point order matches neither direction of travel; only the denominator was reversed.
✓ Instead: Use 1−(−2)8−3 = 35, or reverse both differences.
✗ Not this: Counterexample: a climb of four across two units gives slope four.
Four is total rise. Slope gives rise per input unit.
✓ Instead: m = 42 = 2. Check: 2 × 2 = 4.
✗ Not this: Counterexample: two population counts prove a town grows steadily every year.
They determine an average over the interval. Growth between observations might vary.
✓ Instead: Treat that rate as a constant slope only under a stated or justified constant-change assumption.
Tips and tricks
  • Tip: remember rise over run. Output changes vertically, so it goes on top.
  • Tip: keep coordinate pairs together before subtracting.
  • Tip: 19003 is exact; 633 is rounded and will not reproduce the endpoint exactly.
Trap. Trap: subtracting outputs in one order and inputs in the other reverses the sign. Use the same starting point for both differences.