Who improved the most?
Can a correct calculation still lead to a misleading recommendation?
The situation
Two fictional teams completed the same fitness trial. Each person recorded how many fewer seconds they needed on a second attempt. Positive change means a faster time; negative change means a slower time.
| Team | Individual improvements |
|---|---|
| A | −2, 1, 1, 2, 3, 19 |
| B | 3, 4, 4, 5 |
The school wants to recognize improvement. It has not defined what “most improved” means.
Estimate an answer and record what you think matters. You do not need a finished method yet.
Your investigation
- Make an initial recommendation. Define your rule before calculating.
- Compare total improvement, mean improvement, and the typical person’s improvement. Where do the recommendations disagree?
- Draw the data on number lines. Explain why the value 19 matters.
- Design a recognition rule that you can defend. Name what your rule rewards and what it ignores.
Support cards
Open one when your teacher invites you to. Use the idea, then return to your own work.
Find a way in
Place each person on a number line. Which team has more people? Does total improvement answer a question about a typical person?
A tool to formalize
The mean is the total divided by the number of values. The median is the middle of the ordered values; with an even number of values, average the two middle values.
Extend the question
A person improves by 5 seconds. If their original time was 100 seconds, what percent change is that? What if it was 20 seconds? Why do we need starting times?
Challenge a claim
FICTIONAL GENERATED-STYLE RESPONSE · NOT LIVE AI“Team A improved by 24 seconds and Team B by 16 seconds, so every member of A improved more than every member of B.”
Identify the exact assumption or step you accept or reject. A confident explanation is not a mathematical check.
Defend and revise
Make a claim in one sentence. Support it with a labeled model and a calculation. State an assumption. Have a partner try to find a case that breaks your claim. Revise in a different color.
Close the support cards. Explain the key idea on your own, then solve the changed case your teacher selects. The group’s answer is not evidence that every person understands.
Make the mathematics portable
Construct a different pair of data sets where one team has the larger mean and the other the larger median. Explain the mechanism.
A new person joins Team B with a change of −6 seconds. Reconsider your original rule. Explain what changed and what stayed the same.