Tuesday

Raven tapes graph paper to the Numbers Desk. "Weeks across, centimeters up. Plot Seedling A."

Rocket plots (1, 2), (2, 4), (3, 6) and (4, 8). "They line up! A straight line."

"Where does the line go if you follow it back to week 0?" Raven asks.

"Right through (0, 0)," Rocket says. "Zero weeks, zero centimeters. Now Seedling B."

He plots (1, 3), (2, 4), (3, 5), (4, 6). "Also a straight line. But it hits the up axis at 2, not 0."

Nova hovers above the two lines, her light tracing each one. "What do you notice about the point (1, 2) on A?" she asks.

"The x is 1 and the y is k," Rocket says. "One week, 2 centimeters. The unit rate, right on the graph."

"Then the rule for A is y = 2x," Raven says. "Every proportional table has one."

Plot a proportional table and the dots fall on a straight line that passes through the origin, (0, 0).

Seedling B also makes a straight line, but it crosses the vertical axis at 2, not at 0. Not proportional.

Two special points live on every proportional graph. (0, 0) says zero of one means zero of the other.

(1, k) says one unit of x goes with k units of y. That k is the unit rate.

A proportional relationship has the equation y = kx. For Seedling A, y = 2x: height equals 2 times the weeks.

Check it: at x = 3, y = 2 × 3 = 6. The table says 6. At x = 10, y = 2 × 10 = 20.

You can read k from a table, a graph or words. In a table, divide any y by its x. On a graph, read the y at x = 1.

A unit rate can come from fractions too. Raven walks 3/4 kilometer in 1/4 hour. How far in one hour?

Divide: 3/4 ÷ 1/4 = 3. So the unit rate is 3 kilometers per hour. Four quarter hours make an hour, and four 3/4s make 3.

Seedling A grew 1/2 cm in 1/4 week. Divide: 1/2 ÷ 1/4 = 2 cm per week. The same k.

Excellent graphing. Tomorrow you walk, measure and test your own steps for proportionality in the Numbers Lab.