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Ratios and Data 6-8 / Week 09 / Monday
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Week 09 · Proportional Relationships

Monday

Is it proportional?
// The straight line through zero
⏱ about 20 min

Monday: Is It Proportional?

Rocket carries two seedling pots to the Numbers Desk, each with a paper ruler taped to the side. "Week four. Measuring day."

Raven opens the seedling log. "Seedling A: 2, 4, 6, 8 centimeters at weeks 1 to 4. Seedling B: 3, 4, 5, 6."

"Both grow the same amount each week," Rocket says. "So both are proportional. Done."

"What do you notice when you divide each height by its week?" Raven asks.

Rocket divides. "A gives 2, 2, 2, 2. B gives 3, then 2, then... not 2."

Nova hovers over the log, her light resting on the first row for B. "Where did B start?" she asks.

"At 2 centimeters, before week 1," Rocket says. "A started at zero. That is the difference."

"Proportional means a straight line through zero," Raven says. "This week we test for it."

Rocket, Raven and Nova at the Numbers Desk with two seedling pots. Paper rulers are taped to the pots and the seedling log lies open. Nova hovers with her light on.

The test for proportional

Two quantities are proportional when they grow together in a steady ratio. Double one, and the other doubles too.

The test: divide each y by its x. If every quotient is the same number, the relationship is proportional.

That shared number is the constant of proportionality, written k. It is also the unit rate: the y when x is 1.

Seedling A: 2 ÷ 1 = 2, 4 ÷ 2 = 2, 6 ÷ 3 = 2, 8 ÷ 4 = 2. Every quotient is 2. Proportional, with k = 2.

Seedling B: 3 ÷ 1 = 3, 4 ÷ 2 = 2, 5 ÷ 3 = 5/3, 6 ÷ 4 = 1.5. The quotients differ. Not proportional.

A table, Weeks and Seedling A in cm: 1 and 2, 2 and 4, 3 and 6, 4 and 8
A table, Weeks and Seedling B in cm: 1 and 3, 2 and 4, 3 and 5, 4 and 6

Seedling B grows 1 cm every week, a steady amount. But it started at 2 cm, so height is not proportional to weeks.

Steady growth is not enough. Proportional also means zero of one goes with zero of the other.

In a proportional table, every row is an equivalent ratio. 2:1, 4:2, 6:3 and 8:4 all equal 2.

  1. Write each pair as y ÷ x.
  2. Work out every quotient.
  3. If they are all the same number, it is proportional and that number is k.
  4. If any quotient is different, it is not proportional.
TEST THE TABLES
  • Read the question.
  • Tap your answer.
Seedling A, weeks and cm: (1, 2), (2, 4), (3, 6), (4, 8). Is y proportional to x?
Seedling B, weeks and cm: (1, 3), (2, 4), (3, 5), (4, 6). Is y proportional to x?
Cups of water and seedlings watered: (2, 6), (3, 9), (5, 15). Is y proportional to x?
Minutes and pages read: (10, 6), (20, 11), (30, 18). Is y proportional to x?
FIND K
  • Read the question.
  • Tap your answer.
Seedling A: (1, 2), (2, 4), (3, 6), (4, 8). What is the constant of proportionality k?
Lemon juice and water in cups: (1, 4), (2, 8), (3, 12). What is the constant of proportionality k?
Seconds and meters run: (2, 10), (4, 20), (6, 30). What is the constant of proportionality k?
USE THE RATIO
  • Read the question.
  • Tap your answer.
Seedling A grows 2 cm for every week. How tall is it after 5 weeks?
The lemonade mix uses 4 cups of water for every cup of lemon juice. How much water for 6 cups of lemon juice?
Rocket runs 5 meters every second. How far does he run in 9 seconds?
StatementTrue or false?
Proportional means every y ÷ x is the same number.?
Any table where y grows by the same amount each row is proportional.?
8 ÷ 4 = 2?
6 ÷ 4 = 1.5?
Seedling B is proportional because it grows 1 cm every week.?
WHY THIS EXERCISEThe second and fifth rows are the mistake Rocket made. Naming it is half of fixing it.
Try it
Pour water into a clear cup one spoonful at a time. Mark the level after 2, 4 and 6 spoonfuls.
Measure each mark from the bottom in millimeters. Divide each height by its spoonfuls. Is it proportional?
Pour over a tray or the sink, and wipe any spill right away.
Draw both seedling tables side by side. Beside each row, write y ÷ x. Circle the table where every quotient matches.

Sharp testing. Tomorrow you graph both seedlings and see the straight line through zero.