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Introduction
Proportional Relationships helps you connect a mathematical rule to a result you can explain and check. A proportional rule has a constant output-input ratio and passes through the origin. Contrast it with a linear rule containing a fixed fee. This lesson belongs to linear equations and builds toward independent Algebra I problem solving.
Alignment uses 19 TAC §111.39(c). Review lessons sample their listed standards. Readiness and supporting labels describe STAAR emphasis, not whether a topic is required.
Learning objectives
- Identify the quantities and representations used in proportional relationships.
- Apply the lesson's rule to solve a numerical or contextual problem.
- Explain each mathematical step using the correct vocabulary.
- Verify an answer and correct a likely error.
Prerequisite knowledge
Use ratios and solve one-step equations.
Previous lesson: Rearranging Formulas
Key mathematical vocabulary
- constant
- A value fixed within a problem.
- ratio
- A comparison by division.
- input
- A value supplied to a rule.
- output
- The result of a rule at an allowed input.
- expression
- A mathematical phrase representing a value.
Formulas and rules
- y=mx+b
- m=(y₂−y₁)/(x₂−x₁), x₂≠x₁
- y−y₁=m(x−x₁)
Direct variation
A proportional rule has a constant output-input ratio and passes through the origin. Contrast it with a linear rule containing a fixed fee.
Direct variation is the relationship y=kx, where k is a constant. The ratio y/x is fixed for every nonzero allowed input, and the graph passes through (0,0). The constant k gives output units per input unit.
Given a nonzero point, divide output by input to find k, then write the whole model. To solve for an unknown input, divide the output by nonzero k. A line with a nonzero starting fee is linear but not a direct variation. If k=0, y=0 and division by k is unavailable.
A method you can explain
- Read the question and identify the known information, unknown quantities, and any domain restrictions. Select a suitable representation and tool.
- Apply the relationship explained above. Find the constant of variation: k=36/3=12.
- Write each calculation or logical inference on a separate line. Keep units and signs visible. When two methods are possible, compare which one uses the given information most directly.
- Check the result in the ORIGINAL conditions, not merely the last simplified line. The supplied nonzero point satisfies the rule; the slope is unchanged.
- Interpret what the answer means. Decide whether its size, sign, and precision make sense; identify a restriction or assumption that would change the conclusion.
Five worked examples
Example 1 · Beginner
3 kilograms cost $36 at a constant price. Write the direct variation model C=kd.
- Find the constant of variation: k=36/3=12.
- At input zero the output is 0, so the intercept is 0.
- The equation is C = 12d. Preserve equality when changing forms; all forms describe the same points.
Answer: C = 12d
Verify: The supplied nonzero point satisfies the rule; the slope is unchanged.
Avoid this mistake: Finding k by multiplying the coordinates instead of dividing.
Example 2 · Basic
3 kilograms cost $39 at a constant price. Write the direct variation model C=kd.
- Find the constant of variation: k=39/3=13.
- At input zero the output is 0, so the intercept is 0.
- The equation is C = 13d. Preserve equality when changing forms; all forms describe the same points.
Answer: C = 13d
Verify: The supplied nonzero point satisfies the rule; the slope is unchanged.
Avoid this mistake: Adding an intercept that the situation does not have.
Example 3 · Intermediate
3 kilograms cost $42 at a constant price. Write the direct variation model C=kd.
- Find the constant of variation: k=42/3=14.
- At input zero the output is 0, so the intercept is 0.
- The equation is C = 14d. Preserve equality when changing forms; all forms describe the same points.
Answer: C = 14d
Verify: The supplied nonzero point satisfies the rule; the slope is unchanged.
Avoid this mistake: Assuming direct variation only applies to whole-number ratios.
Example 4 · Advanced
3 kilograms cost $6 at a constant price. Write the direct variation model C=kd.
- Find the constant of variation: k=6/3=2.
- At input zero the output is 0, so the intercept is 0.
- The equation is C = 2d. Preserve equality when changing forms; all forms describe the same points.
Answer: C = 2d
Verify: The supplied nonzero point satisfies the rule; the slope is unchanged.
Avoid this mistake: Finding k by multiplying the coordinates instead of dividing.
Example 5 · Challenge
3 kilograms cost $12 at a constant price. Write the direct variation model C=kd.
- Find the constant of variation: k=12/3=4.
- At input zero the output is 0, so the intercept is 0.
- The equation is C = 4d. Preserve equality when changing forms; all forms describe the same points.
Answer: C = 4d
Verify: The supplied nonzero point satisfies the rule; the slope is unchanged.
Avoid this mistake: Adding an intercept that the situation does not have.
Common mistakes and corrections
- Finding k by multiplying the coordinates instead of dividing.
- Adding an intercept that the situation does not have.
- Assuming direct variation only applies to whole-number ratios.
Real-world applications
- Price bulk food sold by weight without a fixed charge. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Relate distance to time at constant speed starting from zero. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
A proportional rule has a constant output-input ratio and passes through the origin. Contrast it with a linear rule containing a fixed fee.
Direct variation is the relationship y=kx, where k is a constant. The ratio y/x is fixed for every nonzero allowed input, and the graph passes through (0,0). The constant k gives output units per input unit.
Frequently asked questions
What should I know before proportional relationships?
Use ratios and solve one-step equations.
How can I check my work?
The supplied nonzero point satisfies the rule; the slope is unchanged. A numerical check can detect an error; when claiming an identity for all inputs, also explain the algebraic property that makes it true.
Which mistakes should I watch for?
Finding k by multiplying the coordinates instead of dividing. Adding an intercept that the situation does not have. Assuming direct variation only applies to whole-number ratios.
Can I use another valid method?
Yes. Explain your assumptions, show enough intermediate work to justify the result, and verify it in the original problem. Equivalent expressions and valid alternative methods receive credit.
Mathematical process practice · A.1A–A.1G
- A.1A: Price bulk food sold by weight without a fixed charge.
- A.1B: Plan, solve, check, and interpret the written response.
- A.1C: Select paper, a calculator, a table, or a graph; explain the choice.
- A.1D: Connect symbols with a verbal explanation and another representation.
- A.1E: Organize the solution with a table, diagram, or labeled equations.
- A.1F: Explain how the representation and the algebraic rule express the same relationship.
- A.1G: Justify each operation and the reasonableness of the conclusion.
Quick revision notes
- A proportional rule has a constant output-input ratio and passes through the origin. Contrast it with a linear rule containing a fixed fee.
- Finding k by multiplying the coordinates instead of dividing.
- Adding an intercept that the situation does not have.
- Assuming direct variation only applies to whole-number ratios.
Practice and explain
Try all four multiple-choice questions, then show your reasoning in the written response. Objective scores and AI feedback assess different skills.
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Original Cambridge Veritas practice. These are not official STAAR items. Cambridge Veritas is not affiliated with or endorsed by TEA.
Official Algebra I TEKS (§111.39) · TEA assessed curriculum · STAAR blueprint · TEA instructional materials