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Expressions and Equations 6-8 / Week 08 / Tuesday
2/6
Week 08 · Slope

Tuesday

Rise over run
// Rise over run, the steepness of a line
⏱ about 20 min

Tuesday: Rise Over Run

Raven draws a line through (1, 2) and (3, 6) on graph paper. "Find the slope without the origin."

Rocket draws a step from the first point: across 2, then up 4. "Rise 4, run 2. Four over two is 2."

"Now use (2, 4) and (5, 10)," Raven says. Rocket draws the step. "Run 3, rise 6. Still 2!"

"What do you notice about the two triangles?" she asks.

"Same shape, different size," Rocket says. "The big one is a stretched copy of the small one."

Nova hovers over the paper, her light along the line. "Similar triangles," she says. "Same ratio of rise to run, anywhere."

Rocket grins. "So one slope fits the whole line." Nova hums. "That is why one equation does too."

Slope between any two points

Pick any two points on a line. The run is the change across, from the first x to the second.

The rise is the change up, from the first y to the second. Slope = rise ÷ run.

From (1, 2) to (3, 6): run 3 - 1 = 2, rise 6 - 2 = 4. Slope = 4 ÷ 2 = 2.

The line y = 2x with a right triangle from (1, 2) to (3, 6): run 2, rise 4
The same line y = 2x with a bigger triangle from (2, 4) to (5, 10): run 3, rise 6

The two triangles are similar: the same shape, one a stretched copy of the other. So rise ÷ run matches.

That is why the slope is the same between any two points on a line. One number describes the whole line.

From slope to equation

A line through (0, 0) with slope m has the equation y = mx. Every point fits: y is m times x.

A line that crosses the y-axis at b has the equation y = mx + b. The b is the y-intercept, the start.

Start at b when x = 0, then rise m for every 1 across. For y = 2x + 3: start at 3, climb 2 each step.

  1. Pick two points on the line and write them as (x, y).
  2. Run: subtract the first x from the second x.
  3. Rise: subtract the first y from the second y.
  4. Slope m = rise ÷ run.
  5. Find b, the y when x = 0. Then write y = mx + b.
RISE OVER RUN
  • Read the question.
  • Tap your answer.
What is the slope of the line through (1, 2) and (3, 6)?
What is the slope of the line through (2, 4) and (5, 10)?
What is the slope of the line through (1, 1) and (3, 7)?
What is the slope of the line through (0, 3) and (4, 5)?
WHICH EQUATION?
  • Read the question.
  • Tap your answer.
Which equation has slope 2 and crosses the y-axis at 0?
Which equation has slope 2 and crosses the y-axis at 3?
Which equation fits the line through (1, 1) and (3, 7)?
Which equation fits the line through (0, 3) and (4, 5)?
READ Y = MX + B
  • Read the question.
  • Tap your answer.
What is the slope of y = 4x + 1?
A line passes through (0, 3) and (4, 5). Where does it cross the y-axis?
On the line y = 2x + 3, what is y when x = 5?
A line passes through (1, 7) and (3, 11). What is y when x = 0?
StatementTrue or false?
Slope is rise divided by run.?
The slope changes depending on which two points you pick.?
In y = mx + b, the b is where the line crosses the y-axis.?
The line y = 2x passes through (0, 0).?
WHY THIS EXERCISEOne slope for the whole line is what lets one equation describe every point on it.
Draw the line y = 2x + 3. Mark where it crosses the y-axis, then draw a rise-over-run step from there.

Rise over run, from any two points. Tomorrow is Puzzle Lab: the ramp test with a board and tape measures.

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