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Algebra 1 9-12 / Week 12 / Monday
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Week 12 · Data From the Loft and Fair Day

Monday

Dots on a grid
// Scatter plots, lines of best fit and the family fair
⏱ about 20 min

Monday: Dots on a Grid

The Loft is a mess of index cards. Comet sorts them into a pile. "Every launch we logged: pumps and top height."

"Seven launches, seven pairs," Wren says. "Plot each one as a dot. Pumps across, height up. What do you notice?"

Comet plots them. "They climb to the right. More pumps, higher flight. Not a perfect line, but close."

"That is a scatter plot," Wren says. "The pattern is the point, not any single dot."

Nova projects a straight line through the cloud of dots. "Would you like a hint? Try to make the gaps above and below balance."

"A line of best fit," Comet says, sliding her ruler. "Now I can guess the height for ten pumps."

"Predict," Wren says. "Guess is for people without data."

Comet, Wren and Nova at the Maker Fair table with the ramp, the launcher and their data poster.

Two measurements per trial

A scatter plot puts one dot for each trial. The dot's position shows two measurements at once: here, pumps and top height.

The heights are the crew's own estimates from their model and their stopwatch, made up for the Loft. The method is what matters.

Read a scatter plot by asking three things. Which way do the dots go? How tight is the pattern? Is it straight or bent?

Launch1234567
Pumps2345678
Top height (m)2.53.55.577.59.59.5
A scatter plot of pumps across and height up, with seven dots climbing to the right.

A line by eye

  1. Lay a ruler across the dots so about as many sit above the line as below it.
  2. Make the gaps on each side roughly balance. Do not force the line through the first or last dot.
  3. Draw the line and read two points on it to find its slope and intercept.
  4. Use the line to predict a value you did not measure.
The same scatter plot with the least-squares line y = 1.25x + 0.18 drawn through the dots.

Nova's line is the least-squares line, y = 1.25x + 0.18. Tomorrow you learn what least squares means.

Your line by eye should be close to it. Both predict about 12.7 m for 10 pumps.

READ THE SCATTER PLOT
  • Read the question.
  • Tap your answer.
In the pumps and height plot, which way do the dots go?
The fitted line is y = 1.25x + 0.18. What is its slope?
Using y = 1.25x + 0.18, what height does the line predict for 10 pumps? Round to one decimal place.
A scatter plot shows pairs of measurements. What does one dot stand for?
The fitted line is y = 1.25x + 0.18. Predict the height for 10 pumps, to one decimal place. Type the number.
WHY THIS EXERCISEA line of best fit turns a cloud of dots into a rule you can use for values you never measured.
StatementTrue or false?
A scatter plot has one dot for each trial.?
A line of best fit must pass through every dot.?
In the pumps plot, more pumps go with greater height.?
The line predicts a height for 10 pumps even though no 10-pump launch was logged.?
The crew's heights are real measurements of real rockets.?
WHY THIS EXERCISEReading the direction and the tightness of a scatter plot is the first step before any line is drawn.
Try it
Lay a piece of string across your plotted dots and slide it until the gaps above and below balance.
Tape it down and read two points on it. Write the slope you get.
Draw the pumps and height scatter plot on graph paper and lay your own line of best fit through it.

Strong start. Tomorrow you learn how the least-squares line is chosen and what its slope and intercept mean.