Mathematics 7–10 · Year 9
Stacking cups: a linear relationship with a measured gradient and intercept
Number and algebra
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The idea
The height of a stack of identical cups grows by the same amount for every cup added, so height against number of cups is a straight line whose gradient is the lip height and whose intercept is the cup body below the lip.
What you need
- 12 identical rigid plastic cups per group (one style per group, different styles across groups)
- 30 cm ruler
- Grid paper
How to do it
- Measure the height of one cup and the height of the lip (the part of a cup that shows above the cup it sits in) to the nearest millimetre.
- Predict the height of a stack of 10 cups from those two measurements.
- Stack 1, 2, 3, up to 8 cups, measuring the total height each time; plot height against number of cups.
- Draw a line of best fit by eye through the points; read the gradient (cm per cup) and the vertical intercept.
- Write the rule h = a + b n and use it to predict the height of 20 cups and the number of cups that reach a 60 cm shelf; then test the 12-cup prediction by measuring.
- Compare rules between groups with different cup styles and explain the different gradients.
What you should see
For a cup 12.0 cm tall with a 1.5 cm lip the heights are 12.0, 13.5, 15.0, 16.5 cm for 1 to 4 cups, the gradient is 1.5 cm per cup and the intercept is 10.5 cm (the body below the lip), so 10 cups measure 25.5 cm and the rule predicts 12 cups at 28.5 cm. Other cup styles give different gradients and intercepts but the same straight-line shape. The learner knows it worked when the 12-cup measurement lands within 3 mm of the prediction made from the 1 to 8 data.
What changes
- What you change
- number of cups
- What you measure
- height of the stack
- What you keep the same
- identical cups
- stack pushed fully down each time
- measured from the desk to the top rim
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- The height of n cups is n times the height of one cup.
- The intercept of a real relationship must be zero.
- The gradient is the height of a cup.
Safety card
Hazards
No hazard is listed.
Controls
No control is listed.
Note
No chemicals or 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 K–10 Syllabus (2022), Stage 5 Linear relationships B; page read 2026-09-22MA5-LIN-C-02
- Mathematics K–10 Syllabus (2022), Stage 5 Data analysis B; page read 2026-09-22MA5-DAT-C-02
- Australian Curriculum v9AC9M9A03AC9M9A05
Sources
The pages the author read to write this activity.