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Week 09 · Proportional Relationships

Friday

Proportional or not?
// The straight line through zero
⏱ about 20 min

Friday: Proportional or Not?

Rocket slaps a list onto the Numbers Desk. "I tested everything at home. Proportional or not."

"Lemonade: 4 cups of water for every cup of lemon juice. Proportional. Double the juice, double the water."

"Our stair timer: 3 seconds a step, but you start with a 5 second head start. Not proportional."

Raven reads. "What do you notice about the ones that failed?"

"They all start somewhere other than zero," Rocket says. "A head start, a plant that was already 2 cm tall."

Nova hovers over the list, her light steady. "And the ones that passed?" she asks.

"Zero goes with zero," Rocket says. "No juice, no water. No steps, no meters."

"Then let us review," Raven says, "because next week we start asking the whole street questions."

Proportional in everyday life

A recipe is proportional. The lemonade mix uses 4 cups of water for every cup of lemon juice, so y = 4x.

Steady speed is proportional. A walker at 0.6 meters per step covers 0.6 × steps, and zero steps is zero meters.

A head start breaks it. A stair timer of 3 seconds per step plus a 5 second start is y = 3x + 5. Not proportional.

Percent change compares growth fairly. Sign-ups from 20 to 25 and from 8 to 10 both grew 25%.

A scatter plot shows whether two measurements rise together, fall together or show no pattern.

The week in five lines

  • Proportional means every y ÷ x is the same number, k, the constant of proportionality.
  • A proportional graph is a straight line through (0, 0), and (1, k) shows the unit rate.
  • The equation is y = kx. Find k from a table, a graph, an equation or words.
  • Percent change is the change divided by the starting amount, written per 100.
  • A scatter plot of two measurements can show a positive association, a negative association or none.
PROPORTIONAL OR NOT?
  • Read the question.
  • Tap your answer.
Lemon juice and water in cups: (1, 4), (2, 8), (3, 12). Is y proportional to x?
Steps and seconds on the stair timer: (1, 8), (2, 11), (3, 14). Is y proportional to x?
Rocket's resting heartbeats, minutes and beats: (1, 70), (2, 140), (3, 210). Is y proportional to x?
Days and seedling height from 2 cm: (1, 3), (2, 4), (3, 5). Is y proportional to x?
USE K
  • Read the question.
  • Tap your answer.
Rocket's resting heartbeats, minutes and beats: (1, 70), (2, 140), (3, 210). What is the constant of proportionality k?
Rocket counted 70 heartbeats a minute at rest. At that rate, how many in 5 minutes?
The mix uses 4 cups of water for every cup of lemon juice. How much water for 7 cups of lemon juice?
Raven bikes 5/2 kilometers in 1/2 hour. How many kilometers per hour is that?
PERCENT CHANGE
  • Read the question.
  • Tap your answer.
A seedling grew from 16 cm to 20 cm. What is the percent increase?
Rocket's stair time fell from 25 seconds to 20 seconds. What is the percent decrease?
Relay sign-ups went from 10 to 12. What is the percent increase?
StatementTrue or false?
y = 3x + 5 is a proportional relationship.?
In y = kx, k is the unit rate.?
4 ÷ 16 = 0.25?
Percent change divides the change by the ending amount.?
Dots rising from left to right show a positive association.?
WHY THIS EXERCISEThese five ideas carry straight into statistics next week.
Try it
Find a recipe at home. Double every amount. Check that each new amount divided by the old one is 2.
Then find something with a head start, like a timer that starts at 5. Say why it is not proportional.
Draw the graphs of y = 4x and y = 3x + 5 from x = 0 to x = 3. Circle the one through (0, 0).

Terrific week. You can spot a proportional relationship in a table, a graph, an equation or a sentence. Tomorrow is Data Day.

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