Mathematics 11–12 · Year 12

Drop height against bounce height: scatterplot, correlation and least-squares line

Bivariate data analysis: Scatter plots and lines of best fit (Mathematics Standard 2, Year 12; Mathematics Standard 1, Year 12 for the scatter plot, a line of best fit by eye or with digital tools, and interpolation and extrapolation only, because Pearson's r and the least-squares line are Standard 2 content)

Practical, model not builtLow risk

This site has no interactive model of its own. Where a step or a material names a Concept Studio model, simulation or tool, it has not been built; an external simulation a step names (for example PhET) is not part of this site.

The idea

Two measured variables with a physical link give a scatterplot whose strength and direction a correlation coefficient measures and whose trend a least-squares line predicts, within the measured range only.

What you need

A bouncy ball, a 2 m tape fixed to a wall, a phone filming at 60 frames per second
A scientific calculator with linear regression, or a spreadsheet

How to do it

Drop the ball from rest at 0.40, 0.60, 0.80, 1.00, 1.20, 1.40 and 1.60 m; read the first bounce height from the video three times at each height and keep the mean.
Plot bounce height (dependent) against drop height (independent) and describe the form, direction and strength.
Standard 2: compute Pearson's r and the least-squares line y = a + b x on the calculator and interpret b as the fraction of height returned. Standard 1: draw a line of best fit by eye or with digital tools and describe the strength of the association as strong, moderate or weak.
Predict the bounce from 1.10 m (interpolation) and from 5 m (extrapolation) and explain which prediction is trustworthy.

What you should see

If each bounce returns the same fraction of the drop height, the points lie on a line through the origin, so r is close to 1, the intercept a is close to 0 and the gradient b equals the height ratio r from the bouncing-ball entry. For the model's test data (0.4, 0.26), (0.8, 0.51), (1.2, 0.77), (1.6, 1.02) the line is y = 0.005 + 0.635 x with r = 0.99997, predicting 0.704 m from 1.10 m. A 5 m prediction extrapolates beyond the data, where air resistance and ball deformation can change the ratio. The learner knows it worked when r exceeds 0.95 (Standard 1: when the points lie close to the line drawn by eye) and a 1.10 m check drop lands close to the interpolated value.

What changes

What you change: drop height
What you measure: first bounce height
What you keep the same: ball
floor surface
release from rest
peak read from the same camera position

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

A high r proves that drop height causes the bounce height; here physics supplies the cause, r only measures the linear association.
The line can be used for any drop height; predictions outside 0.4 to 1.6 m are extrapolations.

Safety card

Low riskLearners carry it out

Hazards

standing on a chair to reach 1.6 m

Controls

use a step stool with a spotter

Note

No chemicals and no heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.

Curriculum references

The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.

Mathematics Standard 11–12 Syllabus (2024), Year 12 Standard 1 focus area Bivariate data analysis; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MST-12-S1-06
Mathematics Standard 11–12 Syllabus (2024), Year 12 Standard 2 focus area Bivariate data analysis; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MST-12-S2-08
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

All Concept Studio activities