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Expressions and Equations 6-8 / Week 08 / Friday
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Week 08 · Slope

Friday

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

Friday: Slopes All Around

Rocket stands at the bottom of the porch steps with the tape measure. "Every step rises 18 centimeters and runs 27."

"Eighteen over twenty-seven," Raven says. "What do you notice if you simplify it?"

"Two thirds," Rocket says. "The stairs have a slope of two thirds. Every step, the same."

Nova hovers beside the handrail, her light following it up. "The rail is a line too," she says. "Same slope."

"And the wheelchair ramp at the library is much gentler," Raven says. "A small slope, a long run."

Rocket is already sketching. "Stairs, ramps, handrails, roofs. Slopes everywhere."

Nova hums. "Then review, because next week every line gets a name: a linear function."

Slope in everyday life

Stairs: each step has a rise and a run. The crew's porch steps rise 18 cm and run 27 cm, a slope of 2/3.

Ramps: a gentle ramp has a small slope, a small rise for a long run. A steep ramp has a big slope.

Walking and rolling: on a distance-time graph, slope is speed. The steeper line is the faster mover.

A head start shows up as the y-intercept: where the line crosses the vertical axis.

The week in four lines

  • Slope is rise ÷ run between any two points on a line, and it is the same between every pair.
  • On a line through (0, 0), the slope is the unit rate: y = mx.
  • A line that starts at b is y = mx + b. The b is the y-intercept.
  • To compare two walkers, find each slope, whether it comes from a graph, a table or an equation.
EVERYDAY SLOPES
  • Read the question.
  • Tap your answer.
A step rises 18 cm over a run of 27 cm. The next step reaches (54, 36). What is the slope of the stairs?
A ramp rises 10 cm over a run of 200 cm. What is its slope?
A ramp with slope 0.05 and the porch stairs with slope 2/3. Which is steeper?
A handrail has slope 2/3. How many centimeters does it rise over a run of 90 cm?
WEEK REVIEW
  • Read the question.
  • Tap your answer.
What is the slope of the line through (2, 5) and (6, 13)?
Which equation has slope 3 and crosses the y-axis at 2?
A line passes through (2, 5) and (6, 13). What is y when x = 0?
Rocket's graph passes through (0, 0) and (3, 12). Raven's equation is d = 5t. Who is faster?
Week reviewTrue or false?
Slope is run divided by rise.?
The slope is the same between any two points on a line.?
A gentle ramp has a bigger slope than a steep one.?
In y = mx + b, the b is the y-intercept.?
On a distance-time graph, a steeper line means a faster mover.?
WHY THIS EXERCISEThese four ideas are the whole of next week's linear functions.
OUR WEEK, IN ORDER
  • ?We compared two walkers and named the unit rate as the slope.
  • ?We measured a board ramp and showed its slope was the same all along.
  • ?We read slope and y-intercept from a race with a head start.
  • ?We found slope as rise over run between any two points, with similar triangles.
WHY THIS EXERCISEEach day used the same number, slope, in a new place.
Try it
With a grown-up, measure the rise and run of one stair step at home. Write the slope as a fraction.
Then find a ramp or a sloped path and guess whether its slope is bigger or smaller than the stairs.
Draw a side view of three stair steps with each rise and run labeled. Add the handrail as one line above them.

A whole week of slope. Tomorrow is Puzzle Day: the family measures the stairs and the ramps at home.

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