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Expressions and Equations 6-8 / Week 08 / Puzzle Day
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Week 08 · Slope

Puzzle Day

Puzzle Day: the stair slope
// Rise over run, the steepness of a line
⏱ about 20 min

Puzzle Day: The Stair Slope

Saturday morning the family gathers at the bottom of the porch steps with two tape measures.

"Puzzle Day," Raven says. "Measure the rise and run of one step, then of two steps together."

Rocket measures. "One step: rise 18, run 27. Two steps: rise 36, run 54."

"What do you notice about the two slopes?" Raven asks.

"Both two thirds," Rocket says. "Same line, same slope, any two points."

Nova hovers above the handrail, her light steady. "Now find the gentlest slope in the house," she says. "Predict before you measure."

Rocket is already heading for the garden path. Nova hums. "Walk, do not run."

  1. Slope is the steepness of a line: rise ÷ run between any two of its points.
  2. Similar triangles show the slope is the same between every pair of points, so one slope fits the whole line.
  3. A line through (0, 0) is y = mx, and its slope is the unit rate. A steeper line is a faster walker.
  4. A line that starts at b is y = mx + b. The y-intercept b is the head start.
  5. The crew's ramp had slope 0.25 with two books and 0.3 with three. A bigger slope is steeper.
◇ FAMILY SLOPE HUNT
Measure the rise and run of one stair step. Then measure two steps together. Divide rise by run both times.
Find a ramp, a sloped path or a slide. Measure a rise and a run and work out its slope.
Each person predicts the steepest slope in the house before measuring. Then check.
Draw every slope you found as a line on one graph, run across and rise up. Which line is steepest?
Write an equation y = mx for each one, where x is the run and y is the rise.
For a grown-up
This week your student learned slope as rise over run and compared walkers by the slopes of their lines.
Ask them to explain why one step and two steps give the same slope. The answer is similar triangles.
Hold the tape measure on the stairs yourself. No running on stairs, and nothing is rolled down them.
Safety first
A grown-up holds the tape measure on the stairs. Walk on stairs, never run, and keep one hand free for the rail.
Nothing is rolled down the stairs. Soft balls roll on the floor only, away from doors and roads.
PUZZLE DAY CHECK
  • Read the question.
  • Tap your answer.
One step is (27, 18) and two steps are (54, 36), run then rise. What is the slope of the stairs?
The stairs have slope 2/3. Over a run of 81 cm, how many centimeters do they rise?
The garden path has slope 0.1 and the stairs have slope 2/3. Which is gentler?
Which equation gives the rise y for a run x on the crew's stairs?
Puzzle Day checkTrue or false?
One step and two steps on the same stairs give the same slope.?
A gentle garden path has a bigger slope than steep stairs.?
Slope is rise divided by run.?
WHY THIS EXERCISEMeasuring real stairs shows the slope idea is not only on paper.
Draw the family's slopes on one graph, run across and rise up, and label each line with its slope.

Great measuring, crew. Next week: linear functions, and a dripping jug that keeps perfect time.

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