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

Friday

Graphs that tell a story
// Rate of change, starting value, and graphs that tell a story
⏱ about 20 min

Friday: Graphs That Tell a Story

Rocket pulls the pinhole plug from the full jug and lets it drain into the tray. "Watch the level."

Raven marks it each minute: 12, 10, 8, 6. "What do you notice about this graph?"

"It goes down," Rocket says. "Two centimeters a minute. The rate of change is negative 2."

"And still a straight line," Raven says. "Linear, but decreasing. Height = 12 - 2 × minutes."

Nova hovers over the tray, her light on the falling water. "Describe the graph in words," she says. "Then draw one from words."

"Falling, straight, empty at 6 minutes," Rocket says.

Nova hums. "Now sketch this: a walker who speeds up, stops, then walks home." Rocket grabs a pencil.

Describing a graph in words

Read a graph from left to right. If it climbs, the function is increasing. If it falls, it is decreasing.

If it is a straight line, the function is linear and the rate of change is constant. A curve means nonlinear.

The draining jug: h = 12 - 2t. Rate of change -2, so decreasing, and linear. It reaches 0 at 6 minutes.

A grid from 0 to 12 with the falling line h = 12 - 2m through (0, 12) and (4, 4)

Sketching a graph from words

A walker speeds up: the line gets steeper. A walker who stops: the line goes flat, because distance stays the same.

A walker who comes home: the line comes back down to zero.

A sketch needs no exact numbers. Show the shape: up, flat or down, and straight or curved.

WordsShape of the graph
Filling at a steady rateA straight line going up
Draining at a steady rateA straight line going down
Stopped, nothing changingA flat line
Speeding upA curve that gets steeper
INCREASING OR DECREASING?
  • Read the question.
  • Tap your answer.
The draining jug follows h = 12 - 2t. Is the height increasing or decreasing?
The filling cup follows h = 2m + 3. Is the height increasing or decreasing?
A line passes through (1, 9) and (4, 3). Is it increasing or decreasing?
A line passes through (0, 1) and (5, 11). Is it increasing or decreasing?
READ THE DRAINING JUG
  • Read the question.
  • Tap your answer.
The draining jug was at (0, 12) and (4, 4), minutes then cm. What is the rate of change?
The jug follows h = 12 - 2t. After how many minutes is it empty?
h = 12 - 2t. How many cm are left after 3 minutes?
The draining jug gave (0, 12), (2, 8), (4, 4). Do these points lie on one straight line?
Week reviewTrue or false?
A graph that falls from left to right shows a decreasing function.?
A negative rate of change means the graph goes up.?
A flat line means the quantity is not changing.?
A = s² is linear.?
The initial value is where the graph meets the vertical axis.?
WHY THIS EXERCISEReading a graph in words is how a picture becomes a story anyone can follow.
OUR WEEK, IN ORDER
  • ?We timed a dripping jug and wrote its rule from two marks.
  • ?We compared two drip cups, one as a table and one as a rule, by rate of change and initial value.
  • ?We found that the perimeter of a square is linear and its area is not.
  • ?We read the card box in three forms: a story, a table and two points.
WHY THIS EXERCISEEach day used the same two numbers, rate and start, in a new form.
Try it
Walk across a room slowly, stop for five seconds, then walk back faster. Sketch your distance from the start.
Label the three parts of your graph: increasing, flat, decreasing.
Sketch a graph of a walker who speeds up, stops for a while, then walks home. Label each part in words.

A whole week of linear functions. Tomorrow is Puzzle Day: graph stories with the family.

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