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Week 09 · Proportional Relationships

Data Day

Data Day: the proportion hunt
// The straight line through zero
⏱ about 20 min

Data Day: The Proportion Hunt

Saturday morning the crew gathers the family at the Numbers Desk with a tape measure and a jug.

"A proportion hunt," Raven says. "Find three things in this house that are proportional and one that is not."

Rocket is already pouring. "Our scale shows the same grams for every cup of water I add. Proportional."

"What do you notice about the scale before the first cup?" Raven asks.

"It reads zero," Rocket says. "Zero cups, zero grams. Through the origin."

Nova hovers over the scale, her light on the display. "Now try it with the jug on the scale first," she says.

Rocket sets the empty jug down. The scale reads 300. "A head start! Now it is grams = k × cups + 300. Not proportional."

"Everyone finds one," Raven says. "Then we graph them all."

  1. Two quantities are proportional when every y ÷ x is the same number, the constant of proportionality k.
  2. A proportional graph is a straight line through (0, 0). The point (1, k) shows the unit rate.
  3. The equation is y = kx. A head start, like y = 3x + 5, breaks proportionality.
  4. Unit rates can come from fractions: 3/4 kilometer in 1/4 hour is 3 kilometers per hour.
  5. Percent change is the change divided by the starting amount. A scatter plot shows whether two measurements rise together.
◇ FAMILY PROPORTION HUNT
Each person finds one pair of quantities at home and measures three pairs of values. Steps and meters, cups and grams, pages and minutes.
Divide each y by its x. If the three quotients match, it is proportional. Write its k and its equation y = kx.
Find at least one pair with a head start, like a jug on the scale, and explain why it fails the test.
Graph every proportional pair on one sheet, each as a line through (0, 0). Which has the biggest k?
Pick one family result from last month and this month, like books read. Work out its percent change together.
For a grown-up
The hunt uses the whole week: the y ÷ x test, the line through (0, 0), y = kx and percent change.
Pouring happens over a tray or the sink, with spills wiped right away. The tape measure rolls back slowly.
If a result is disputed, divide y by x together for each row rather than giving the answer.
Safety first
Pour water over a tray or the sink and wipe spills so no one slips.
Walk, never run, when measuring steps. Keep the tape measure away from faces as it rolls back.
HUNT CHECKS
  • Read the question.
  • Tap your answer.
The crew's scale, cups of water and grams, starting empty: (1, 240), (2, 480), (3, 720). Is y proportional to x?
The crew's scale, cups of water and grams, with the jug on it: (1, 540), (2, 780), (3, 1020). Is y proportional to x?
The crew's scale, cups and grams: (1, 240), (2, 480), (3, 720). What is the constant of proportionality k?
The crew's scale showed 240 grams for each cup of water. How many grams for 5 cups?
Hunt checkTrue or false?
A scale that reads 300 before the first cup gives a proportional table.?
720 ÷ 3 = 240?
Every proportional relationship can be written as y = kx.?
A head start of 5 seconds changes y = 3x into y = 3x + 5.?
WHY THIS EXERCISETelling proportional from almost proportional is the skill that makes every table honest.
Draw the family's proportional pairs as lines on one graph through (0, 0). Label each line with its k.
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