The Variable That Wasn't
An eighth-grade class is working on linear functions. The teacher poses: "A phone plan costs $20 per month plus $0.10 per text message. Write an equation for the monthly cost C in terms of the number of texts t." Most students write C = 20 + 0.10t. One student writes C = 20t + 0.10 and argues, "t is the number of texts, and you pay $20 for each one plus ten cents." When the teacher says "the $20 is a one-time monthly fee," the student replies, "But it's still a number, so it's a variable too." The teacher wants to address the underlying confusion about what a variable represents and how the constant and coefficient differ in meaning, without simply giving the correct equation.
1. Diagnose the student's misconception in terms of the roles of coefficient, constant, and variable, and explain how a table of values can distinguish the two equations.
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The student has conflated which quantity varies with which quantity is fixed. In C = 20 + 0.10t, t is the independent variable (number of texts), 0.10 is the coefficient (rate per text), and 20 is the constant (fixed monthly fee). The student's equation C = 20t + 0.10 makes the $20 the per-text rate and the $0.10 a fixed charge - reversing the roles.
A table exposes the error. For t = 0: correct model gives C = $20 (you still pay the monthly fee with zero texts); student's model gives C = $0.10. For t = 10: correct gives $21; student's gives $200.10. The t = 0 case is especially diagnostic: the constant term is exactly the value when the variable is zero, which is why the $20 must be the constant. The student's model predicts an absurd $20 per text, which contradicts the problem statement.
2. Propose two questions the teacher could ask that push the student to reason about meaning rather than just match a form, and explain what each question is designed to reveal.
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Question 1: "If I send zero texts this month, what do I pay?" This targets the constant term. It forces the student to identify the fixed charge and see that the constant in the equation equals the output when the input is zero. If the student answers $20, they can be guided to see that 20 must be the term without a t.
Question 2: "Which number changes the bill every time you send one more text?" This targets the coefficient. It surfaces whether the student understands that the coefficient is a rate per unit of the variable. A student who says "$20 changes each time" reveals they are still treating the fixed fee as per-text.
Together the questions separate the two roles without the teacher supplying the answer. Evidence of understanding: the student can write a new equation from a verbal description (e.g., a gym with a $15 joining fee plus $5 per visit) and correctly assign 15 as the constant and 5 as the coefficient, justifying each choice by referencing the zero-input and per-unit-change meanings.