Week 12 · Data From the Loft and Fair Day

Friday

When a line is wrong

// Scatter plots, lines of best fit and the family fair

⏱ about 20 min

Friday: When a Line Is Wrong

Comet plots one flight: height every second from launch to landing. "0, 15, 20, 15, 0. Line of best fit, Nova?"

Nova projects a flat line across the middle. "The least-squares line. Would you like a hint about the residuals?"

"Below, above, above, above, below," Wren says. "The residuals make a pattern. The line is wrong here."

"Of course it is," Comet says. "That is our parabola from last week. No line fits a hill."

Wren pins up the folded strip data. "Folds against layers: 2, 4, 8, 16, 32. What do you notice?"

"Doubling. Exponential," Comet says. "A line would miss low and high and hit the middle."

"So before you fit a line, look at the shape," Wren says. "The dots tell you which model to try."

Residuals with a pattern

If a line fits, the residuals look random: some above, some below, no pattern. If they curve, the model is wrong.

For the one-flight data, the least-squares line is flat: slope 0, height 10 all the way across. Its residuals run below, above, above, above, below.

That pattern says the data bend. The parabola h = -5t² + 20t from week 11 passes through every dot exactly.

$A scatter plot of one flight, five dots in an arch, with a flat least-squares line through the middle.$

$The parabola h = -5t² + 20t, which passes through all five flight dots.$

Three shapes to know

Shape of the dots	Model to try	Check	Loft example
straight band	linear, y = mx + b	equal differences in a table	pumps and height
an arch or a U	quadratic, y = ax² + bx + c	differences change by a constant	height during one flight
a curve that climbs faster and faster	exponential, y = a × rˣ	equal ratios in a table	folds and layers

Folds	1	2	3	4	5
Layers	2	4	8	16	32
Line predicts	-2	5.2	12.4	19.6	26.8

The folds line y = 7.2x - 9.2 has a high r of 0.93. Yet it predicts -2 layers for one fold and misses at both ends.

A high r does not mean the right model. The shape of the dots does.

WHICH MODEL?

Read the question.
Tap your answer.

Layers after 1, 2, 3, 4, 5 folds: 2, 4, 8, 16, 32. Is the pattern linear, exponential or neither?

Heights for 2, 3, 4, 5, 6 pumps on a smooth day: 3, 4.5, 6, 7.5, 9. Linear, exponential or neither?

Heights at 0, 1, 2, 3, 4 seconds: 0, 15, 20, 15, 0. Linear, exponential or neither?

The residuals of a fitted line run below, above, above, above, below. What does that tell you?

MIXED REVIEW

Read the question.
Tap your answer.

The one-flight dots follow h = -5t² + 20t. What is the vertex, the top of the flight?

Back to pumps and height, y = 1.25x + 0.18. What is the slope?

For the fins data, what is r to two decimal places?

One flight: at t = 2 the height is 20 and the flat line predicts 10. What is the residual?

Statement	True or false?
Residuals with no pattern mean the line fits well.	?
A high r always means a line is the right model.	?
Equal ratios in a table point to an exponential model.	?
Height during one flight is best modeled by a line.	?
Look at the shape of the dots before choosing a model.	?

WHY THIS EXERCISEThis is the week, and the course: pick the model the data ask for, then read it in the build.

Try it

Fold a strip of paper five times and count the layers each time. Plot folds across and layers up.
Try to lay a straight line through the dots. Where does it miss?

Draw the folds and layers dots with the straight line through them. Shade where the line misses high and low.

Loft Log: write the three shapes and the model each one calls for.Tomorrow is Fair Day. List the three tables you will run and the data each one collects.Look back at week 1. Write one thing about expressions or units that helped you this week.

Terrific week, and a terrific course. Tomorrow is Fair Day, and the family is your crowd.

Grown-up checked my answers

← Thursday Loft Day →