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Expressions and Equations 6-8 / Week 09 / Tuesday
2/6
Week 09 · Linear Functions

Tuesday

Linear or not?
// Rate of change, starting value, and graphs that tell a story
⏱ about 20 min

Tuesday: Linear or Not?

Rocket lays beans in a square on the bench. One bean, then a 2 by 2 square, then 3 by 3.

"Side 1, area 1. Side 2, area 4. Side 3, area 9," he counts. "The rule is area = side squared."

Raven plots (1, 1), (2, 4) and (3, 9). "What do you notice about the points?"

"They do not line up," Rocket says. "The jumps grow: 3, then 5, then 7."

"Now plot the perimeter," Raven says. "Four times the side." Rocket plots (1, 4), (2, 8), (3, 12). "A straight line."

Nova hovers over the beans, her light on the 3 by 3 square. "y = mx + b is a line," she says. "s² is not."

Rocket nods. "Linear and not linear. Same beans, two very different graphs."

What makes a function linear

A function is linear when its rule can be written y = mx + b. Its graph is a straight line.

The test in a table: equal steps in x give equal steps in y. The rate of change never changes.

The perimeter of a square, p = 4s, is linear: each extra 1 of side adds 4. Its graph is a straight line through (0, 0).

A table, side, perimeter and area: 1, 4, 1; 2, 8, 4; 3, 12, 9; 4, 16, 16; 5, 20, 25

A function that bends

The area of a square, A = s², is a function: each side gives exactly one area. But it is not linear.

From side 1 to 2 the area jumps 3. From 2 to 3 it jumps 5. From 3 to 4 it jumps 7. The steps keep growing.

Its points (1, 1), (2, 4) and (3, 9) do not lie on one straight line. The graph curves upward.

A grid from 0 to 20 with the straight line p = 4s through (1, 4), (2, 8) and (3, 12)
  1. Look at the rule. If the letter is only multiplied by a number and added to a number, it is linear.
  2. Look at the table. Check that equal steps in x give equal steps in y.
  3. Look at the graph. A straight line is linear. A curve is not.
LINEAR OR NOT?
  • Read the question.
  • Tap your answer.
The perimeter table gives (1, 4), (2, 8), (3, 12). Do these points lie on one straight line?
The area table gives (1, 1), (2, 4), (3, 9). Do these points lie on one straight line?
Do the points (0, 3), (2, 7), (5, 13) lie on one straight line?
Do the points (1, 2), (2, 5), (3, 10) lie on one straight line?
READ THE RULES
  • Read the question.
  • Tap your answer.
Which of these rules is linear?
Which of these rules is not linear?
A = s². What is the area of a square with side 7?
p = 4s. What is the perimeter of a square with side 7?
RATE OF CHANGE
  • Read the question.
  • Tap your answer.
For the perimeter, from (1, 4) to (3, 12), what is the rate of change?
For the area, from side 1 to side 2 the jump is 3. What is the jump from side 2 to side 3?
Which equation fits the perimeter table (1, 4), (2, 8), (3, 12)?
StatementTrue or false?
Every linear function can be written y = mx + b.?
A = s² is a linear function.?
In a linear table, equal steps in x give equal steps in y.?
The graph of the area of a square is a straight line.?
p = 4s is linear with a rate of change of 4.?
WHY THIS EXERCISEKnowing a function is linear is what lets you predict it from just two points.
Try it
Lay counters in squares of side 1, 2, 3 and 4. Count each area and write the jumps between them.
Then count each perimeter, the counters around the edge, and write those jumps. Which stays the same?
Draw the perimeter points and the area points for sides 1 to 4 on one grid. Join each set. Which one is straight?

Straight or bent, you can tell. Tomorrow is Puzzle Lab: the dripping jug and the timer.

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