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Algebra 1 9-12 / Week 04 / Monday
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Week 04 · Linear Graphs and Rates of Change

Monday

The cart draws a line
// The cart on the ramp, one second at a time
⏱ about 20 min

Monday: The Cart Draws a Line

Comet sets the plank on two sawhorses, barely tilted. "Gentle slope. The cart should roll nice and steady."

Wren has taped marks along the plank every 10 centimeters. "I call out the mark each second. You write."

Comet lets go. "One mark. Four. Seven. Ten. Thirteen." The cart rolls off the end into a towel.

"What do you notice?" Wren asks, reading the list.

"It goes up by three every second," Comet says. "Three marks a second. But it started at mark one, not zero."

Nova hovers above the notebook and projects the numbers as dots. They sit in a perfect straight line.

"Would you like a hint?" she asks. "Three per second and a start of one. You wrote a rule like that last week."

"d = 3t + 1," Comet says. "Our engineer, help us read it as a graph."

Comet, Wren and Nova in the Loft timing the small wooden cart as it rolls down the taped plank.

From readings to a rule

The crew's readings are their own Monday timings on the gentle plank. Each second the cart passes 3 more marks.

At t seconds the cart is at d = 3t + 1 marks. A mark is 10 centimeters, so in centimeters d = 30t + 10.

The 3 is the rate of change: 3 marks per second. The 1 is the starting value, the mark at t = 0.

Seconds (t)Marks (d = 3t + 1)Centimeters (30t + 10)
0110
1440
2770
310100
413130
516160
619190
A grid from 0 to 20 with the line d = 3t + 1 and the crew's seven readings marked as dots.

A solved problem to study

Wren wants the rate of change from two readings only: (1, 4) and (3, 10). Here is his work.

Change in output: 10 - 4 = 6 marks. Change in input: 3 - 1 = 2 seconds. Rate: 6 ÷ 2 = 3 marks per second.

Then the start: going back one second from (1, 4) takes away 3, so d = 1 at t = 0. The rule is d = 3t + 1.

Why does this work? On a line the change per second is the same everywhere, so any two points give the same rate.

The line through (1, 4) and (3, 10) with a right triangle showing a run of 2 and a rise of 6.
READ THE CART'S LINE
  • Read the question.
  • Tap your answer.
The cart passed mark 4 at 1 second and mark 10 at 3 seconds. What is the rate of change, in marks per second?
The cart passed (1, 4) and (3, 10). Which mark was it at when t = 0?
The rule is d = 3t + 1. Which mark does the cart pass at t = 6 seconds?
WRITE THE RULE
  • Read the question.
  • Tap your answer.
A cart passes 3 marks a second and starts at mark 1. Which equation is its line?
A second cart passed (2, 7) and (4, 13). Which equation fits its line?
In centimeters the cart moves 30 a second from 10. Which equation is its line?
StatementTrue or false?
The rate of change of d = 3t + 1 is 3 marks per second.?
The y-intercept of d = 3t + 1 is 3.?
The point (4, 13) is on the graph of y = 3t + 1.?
The point (5, 15) is on the graph of y = 3t + 1.?
10 - 4 = 3 × 2?
WHY THIS EXERCISEReading slope and intercept straight from the rule is the fastest way to picture the line.
The cart follows d = 3t + 1. Which mark does it pass at t = 4? Type the number.
In centimeters, d = 30t + 10. How far is the cart at t = 2? Type the number.
What is the rate of change of d = 3t + 1, in marks per second? Type the number.
Wren's two points were (1, 4) and (3, 10). How many marks did the cart gain between them? Type the number.
WHY THIS EXERCISEThe change in output is the top of the rate of change. The change in input is the bottom.
Try it
On graph paper, draw axes with seconds across and marks up. Label both and write the scale: one square is one mark.
Plot the crew's seven readings from the table and lay a ruler along them. Do they make one straight line?

Strong start, engineer. Tomorrow you graph rules by hand, find both intercepts and meet a line that goes down.