Unit 3 · Linear RelationshipsF2 · Slope & Lines · Guidance

Slope from Points, Graphs & Tables

Read rate of change in every representation and keep rise over run in order.

29 min
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7 steps · about 29 minutes

Start with the warm-up.

Start

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Warm-up

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Key terms

Slope: Change in output divided by change in input.
Rise: Vertical change between two points.
Run: Horizontal change between the same two points.

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Try it first

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Take a shot before the lesson. Write one relationship from the stem, even if you do not finish.

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The big idea

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See the structure

Slope is change in $y$ divided by change in $x$ . Use the same point order in numerator and denominator so both signs change together.

Key idea

Name quantities before calculating

Write what each number measures. A correct calculation with the wrong quantity or unit still misses the question.

Build a reliable method

In $y=mx+b$ , $m$ is slope and $b$ is the value at $x=0$ . A table is linear when equal input changes produce equal output changes.

Explore in Desmos

See rise over run between two points

Plot the two points, then read 3 from the slope calculation. Moving right 2 and up 6 confirms rise over run

6/2=3

The calculator is live: change anything and see what happens.

See rise over run between two points

Degree mode

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Stretch to a 36

Slope has contextual units, such as dollars per month. Its sign tells direction; its magnitude tells the change for one input unit.

Trap

A believable answer to the wrong question

The choices often include a correct intermediate number. Before selecting it, reread the highlighted ask and check the unit, sign, and domain.

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Worked examples

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Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1: Keep subtraction order consistent

Problem

What is the slope of the line through

(3,2)

and

(11,6)

Use rise over run:

(6-2)/(11-3)

Which calculation represents rise over run?

m=4/8=1/2

Which movement between the two points confirms the slope?

Worked example

Example 2: Interpret slope with units

Problem

A tank model passes through (2 min, 18 L) and (7 min, 43 L). What does its slope mean?

Slope is

(43-18)/(7-2)=25/5=5

Which calculation represents rise over run?

The tank gains 5 liters per minute over this interval.

Which movement between the two points confirms the slope?

Faded example · you finish it

Example 3: Complete the setup and result

Problem

What is the slope of the line through

(1,7)

and

(5,19)

Choose the setup that follows the stated relationship.

Choose the resulting value.

Desmos way

Verify slope with points and a line

Plot (1,7) and (5,19), then graph y=3x+4. Confirm that both points lie on the line.

Type (19-7)/(5-1) and read 3. The numerator is the rise 12 and the denominator is the run 4.

Verify slope with points and a line

Degree mode

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.