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BetterGrades Precalculus · Unit 2 · Lesson

Quantities, variables, and dependency

Identify the input, output, parameters, and units in a quantitative relationship and explain how the quantities depend on one another.

Opening

Start with the situation

An input is supplied to a relationship, an output is determined by it, and parameters remain fixed for one selected model.

The same dependency may arrive as a story, table, graph, or formula. Learning to preserve the inputs, outputs, units, and domain while moving among those forms is the central language of the course.

Before you begin

Prerequisite check

  • Use function notation from the algebra and function readiness unit.
  • Read ordered pairs and interval notation.
  • Attach units to contextual quantities.
Core explanation

Explanation

Define every quantity and unit, identify the direction of dependency, and check that terms combined in a formula have compatible units.

The same symbol can play different roles in different models; the context determines whether it is input, output, or parameter.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through quantity dependency map, parameter family graph, or another equivalent representation.

Conceptual reading

What the idea is really doing

A function is a dependency, not merely an equation. Inputs, outputs, units, and domain must agree in words, tables, graphs, and formulas; each representation should tell the same mathematical story.

This lesson narrows that lens to one goal: identify the input, output, parameters, and units in a quantitative relationship and explain how the quantities depend on one another. The point is not to memorize an isolated trick; it is to know what evidence makes the conclusion valid and how a second representation can check it.

Reusable method

A reliable route through the problem

  1. Define every quantity.
  2. Unit.
  3. Identify the direction of dependency.
  4. Check that terms combined in a formula have compatible units.

Verification: Choose two representations and make them verify one another. A table can test a formula, a graph can expose a domain or range claim, and units can reveal a model that is algebraically neat but conceptually wrong.

Foundation walkthrough

Plan before calculating

Problem

A rental costs $18\$18 plus $7\$7 per hour.

Plan
Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Define every quantity and unit, identify the direction of dependency, and check that terms combined in a formula have compatible units.
Conclusion
C(h)=18+7hC(h)=18+7h dollars.
Why the check works
The fixed fee and hourly rate have different parameter meanings.
Worked examples

See the idea in three forms

foundation example

A rental costs $18\$18 plus $7\$7 per hour.

SolutionC(h)=18+7hC(h)=18+7h dollars.

The fixed fee and hourly rate have different parameter meanings.

representation example

Units of 6060 in d(t)=60td(t)=60t miles.

SolutionMiles per hour.

This example expresses quantities, variables, and dependency in a second form.

transfer example

Perimeter with width 55 and length L.

Solution2L+102L+10

The same symbol can play different roles in different models; the context determines whether it is input, output, or parameter.

Quantity dependency map. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The fixed fee and hourly rate have different parameter meanings.
Read this graph as text

Quantities, variables, and dependency · Quantity dependency map. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The fixed fee and hourly rate have different parameter meanings. The figure uses concrete points, curves, arrows, intervals, or matrix structure instead of relying on color alone.

Labels, point shapes, line styles, arrows, and position carry the mathematical meaning; color is supplementary.

Why it matters: Use the mathematical objects in this figure to support the lesson outcome: Identify the input, output, parameters, and units in a quantitative relationship and explain how the quantities depend on one another.

Anchor figure · Quantity dependency map

Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The fixed fee and hourly rate have different parameter meanings.

Parameter family graph. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for quantities, variables, and dependency.
Read this graph as text

Quantities, variables, and dependency · Parameter family graph. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for quantities, variables, and dependency. The figure uses concrete points, curves, arrows, intervals, or matrix structure instead of relying on color alone.

Labels, point shapes, line styles, arrows, and position carry the mathematical meaning; color is supplementary.

Why it matters: Use the mathematical objects in this figure to support the lesson outcome: Identify the input, output, parameters, and units in a quantitative relationship and explain how the quantities depend on one another.

Mechanism figure · Parameter family graph

Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for quantities, variables, and dependency.

Unit consistency ledger. Compare the valid path with the tempting shortcut. The figure shows why manipulating symbols before defining what they measure leads to a false conclusion.
Read this graph as text

Quantities, variables, and dependency · Unit consistency ledger. Compare the valid path with the tempting shortcut. The figure shows why manipulating symbols before defining what they measure leads to a false conclusion. The figure uses concrete points, curves, arrows, intervals, or matrix structure instead of relying on color alone.

Labels, point shapes, line styles, arrows, and position carry the mathematical meaning; color is supplementary.

Why it matters: Use the mathematical objects in this figure to support the lesson outcome: Identify the input, output, parameters, and units in a quantitative relationship and explain how the quantities depend on one another.

Comparison and error figure · Unit consistency ledger

Compare the valid path with the tempting shortcut. The figure shows why manipulating symbols before defining what they measure leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is manipulating symbols before defining what they measure.

Check yourself

Write cost for $9\$9 plus $2\$2 per movie.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice

Ten concrete questions

Practice 101

Write cost for $9\$9 plus $2\$2 per movie.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 202

Units of 6060 in d(t)=60td(t)=60t miles.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 303

Perimeter with width 55 and length L.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 404

Horizontal axis for cost as function of hours.

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Practice 505

Explain why this conclusion is valid: C(h)=18+7hC(h)=18+7h dollars. Use the foundation problem as evidence: A rental costs $18\$18 plus $7\$7 per hour.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 606

Solve the representation example, then name the feature of quantities, variables, and dependency that it illustrates: Units of 6060 in d(t)=60td(t)=60t miles.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 707

Correct this reasoning and identify the first unsafe assumption: A frequent error is manipulating symbols before defining what they measure.

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Practice 808

Connect two representations for this example: A rental costs $18\$18 plus $7\$7 per hour. Describe what a graph, table, mapping, or algebraic form would have to show.

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Practice 909

Create a nearby example by changing one number or condition in this prompt: Perimeter with width 55 and length L. Predict the effect, solve your new example, and compare it with the original.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

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Practice 1010

Write a short verification checklist for quantities, variables, and dependency, then apply it to one worked example from this lesson.

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Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Lesson close

Connect forward

The next lesson, Relations and functions, uses this result as part of a larger structure.

Source record

Original BetterGrades manuscript, rights-separated references.

  • Yoshiwara, Modeling, Functions, and Graphs
  • Lippman and Rasmussen, Precalculus Volume 1
  • Stitz and Zeager, Precalculus

No long source passage is reproduced.