Which plan is really cheaper?
When does a recommendation stop being true?
The situation
Two fictional suppliers offer activity kits. Plan A costs $24 plus $2 per kit. Plan B costs $6 plus $3.50 per kit. Shipping and tax are included. The school might need 8–20 kits.
The kit count is a whole number. Begin with the assumption that every kit is purchased under one plan.
Estimate an answer and record what you think matters. You do not need a finished method yet.
Your investigation
- Recommend a plan for 8, 12, and 20 kits. Estimate before calculating.
- Build a table, graph, and pair of equations. Explain how the fixed fee and per-kit cost appear in each representation.
- Find the boundary between the plans. Defend a rule for every whole-number kit count.
- Make a recommendation when the number of kits is uncertain. State what information you would ask for.
Support cards
Open one when your teacher invites you to. Use the idea, then return to your own work.
Find a way in
Calculate the cost of zero kits under each pricing rule, then ask how one more kit changes the cost. What do those two quantities tell you?
A tool to formalize
Let n be the number of kits. Write cost as fixed fee + per-kit cost × n. An equation locates a tie; an inequality describes which plan costs less.
Extend the model
Suppose Plan A applies a 10% discount to its entire bill. Does the crossing point still occur at a whole number? Explain how you would decide for integer kit counts.
Challenge a claim
FICTIONAL GENERATED-STYLE RESPONSE · NOT LIVE AI“Plan A is always cheaper because $2 per kit is less than $3.50 per kit.”
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
Design two new plans that tie at 15 kits. Then design two plans that never tie for any nonnegative number of kits. Justify both designs.
Plan B raises its fixed fee to $12. Find the new decision boundary and explain which feature of the graph changes.
Ask a new question: if attendance is uncertain, do you minimize expected cost, maximum cost, or regret? A model requires a goal as well as numbers.