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Introduction
Slope helps you connect a mathematical rule to a result you can explain and check. Compute rise divided by run and keep subtraction order consistent. Connect the ratio to the x-coefficient after isolating y. This lesson belongs to coordinate geometry and graphing 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 slope.
- 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
Subtract signed numbers and divide fractions.
Previous lesson: Linear Relationships
Key mathematical vocabulary
- coefficient
- A factor multiplying a variable term.
- constant
- A value fixed within a problem.
- equation
- A statement that two expressions are equal.
- slope
- Vertical change divided by nonzero horizontal change.
- ratio
- A comparison by division.
Formulas and rules
- y=mx+b
- m=(y₂−y₁)/(x₂−x₁), x₂≠x₁
- y−y₁=m(x−x₁)
Slope and rate of change
Compute rise divided by run and keep subtraction order consistent. Connect the ratio to the x-coefficient after isolating y.
Slope is the ratio of vertical change to horizontal change: m=(y₂−y₁)/(x₂−x₁), provided the denominator is nonzero. Keep the same point order in numerator and denominator. A linear relationship has constant slope.
Read slope from a table using changes, from a graph using two labeled points, or from an equation by isolating y. In y=mx+b it is m; in Ax+By=C with B≠0 it is −A/B; in point-slope form it is the multiplier of x−x₁. A contextual rate includes output units per input unit.
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. Use the same order for both coordinate differences.
- 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. Rise divided by run matches the coefficient of x in a line with this slope.
- 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
Find the slope through (25, 85) and (30, 145).
- Use the same order for both coordinate differences.
- m=(y₂−y₁)/(x₂−x₁)=(145−85)/(30−25).
- m=60/5=12. It means y increases 12 for each increase of 1 in x.
Answer: 12
Verify: Rise divided by run matches the coefficient of x in a line with this slope.
Avoid this mistake: Dividing horizontal change by vertical change.
Example 2 · Basic
Find the slope through (25, 90) and (30, 155).
- Use the same order for both coordinate differences.
- m=(y₂−y₁)/(x₂−x₁)=(155−90)/(30−25).
- m=65/5=13. It means y increases 13 for each increase of 1 in x.
Answer: 13
Verify: Rise divided by run matches the coefficient of x in a line with this slope.
Avoid this mistake: Changing point order in only one difference.
Example 3 · Intermediate
Find the slope through (25, 95) and (30, 165).
- Use the same order for both coordinate differences.
- m=(y₂−y₁)/(x₂−x₁)=(165−95)/(30−25).
- m=70/5=14. It means y increases 14 for each increase of 1 in x.
Answer: 14
Verify: Rise divided by run matches the coefficient of x in a line with this slope.
Avoid this mistake: Calling a vertical line’s slope zero: its slope is undefined.
Example 4 · Advanced
Find the slope through (26, 36) and (31, 46).
- Use the same order for both coordinate differences.
- m=(y₂−y₁)/(x₂−x₁)=(46−36)/(31−26).
- m=10/5=2. It means y increases 2 for each increase of 1 in x.
Answer: 2
Verify: Rise divided by run matches the coefficient of x in a line with this slope.
Avoid this mistake: Dividing horizontal change by vertical change.
Example 5 · Challenge
Find the slope through (26, 41) and (31, 56).
- Use the same order for both coordinate differences.
- m=(y₂−y₁)/(x₂−x₁)=(56−41)/(31−26).
- m=15/5=3. It means y increases 3 for each increase of 1 in x.
Answer: 3
Verify: Rise divided by run matches the coefficient of x in a line with this slope.
Avoid this mistake: Changing point order in only one difference.
Common mistakes and corrections
- Dividing horizontal change by vertical change.
- Changing point order in only one difference.
- Calling a vertical line’s slope zero: its slope is undefined.
Real-world applications
- Find speed from distance-time data. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Calculate a business cost increase per additional item. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Compute rise divided by run and keep subtraction order consistent. Connect the ratio to the x-coefficient after isolating y.
Slope is the ratio of vertical change to horizontal change: m=(y₂−y₁)/(x₂−x₁), provided the denominator is nonzero. Keep the same point order in numerator and denominator. A linear relationship has constant slope.
Frequently asked questions
What should I know before slope?
Subtract signed numbers and divide fractions.
How can I check my work?
Rise divided by run matches the coefficient of x in a line with this slope. 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?
Dividing horizontal change by vertical change. Changing point order in only one difference. Calling a vertical line’s slope zero: its slope is undefined.
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: Find speed from distance-time data.
- 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
- Compute rise divided by run and keep subtraction order consistent. Connect the ratio to the x-coefficient after isolating y.
- Dividing horizontal change by vertical change.
- Changing point order in only one difference.
- Calling a vertical line’s slope zero: its slope is undefined.
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.
Continue learning
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