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Introduction
Multi-Step Problem Solving helps you connect a mathematical rule to a result you can explain and check. Make a plan before calculating. Keep intermediate work visible so that a later verification can locate any mistake. This lesson belongs to staar algebra i eoc preparation 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 multi-step problem solving.
- 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
Evaluate linear, quadratic, and exponential functions and read tables.
Previous lesson: Mixed TEKS Practice
Key mathematical vocabulary
- coefficient
- A factor multiplying a variable term.
- constant
- A value fixed within a problem.
- exponent
- A number specifying a power.
- factor
- An expression multiplied by another to form a product.
- quadratic
- A degree-two polynomial relationship.
Formulas and rules
- a(b+c)=ab+ac
- Equivalent expressions agree for all allowed inputs.
- A solution satisfies every original condition.
Choose and fit a model with technology
Make a plan before calculating. Keep intermediate work visible so that a later verification can locate any mistake.
A linear pattern adds roughly a constant amount for equal input changes; a quadratic pattern has roughly constant second differences; a nonzero exponential pattern multiplies by roughly a constant factor. These patterns guide model choice but are not proof that a real-world process follows a rule indefinitely.
Enter paired observations in separate calculator lists and select the appropriate linear, quadratic, or exponential regression. Record the full precision of the coefficients and only round the final prediction. Inspect a scatterplot and residuals, then state units, the observed input interval, and limitations. Four exact sample points can fit a curve perfectly; real measurements usually contain variation.
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. The needed numbers have sum 30 and product 216: 12 and 18.
- 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. Multiplying the factors (or divisor and quotient) recovers the polynomial.
- 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
Factor x² + 30x + 216.
- The needed numbers have sum 30 and product 216: 12 and 18.
- Distribution gives x²+18x+12x+216=x²+30x+216.
- Thus x² + 30x + 216=(x + 12)(x + 18).
Answer: (x + 12)(x + 18)
Verify: Multiplying the factors (or divisor and quotient) recovers the polynomial.
Avoid this mistake: Choosing a curve solely because it passes near a few points.
Example 2 · Basic
Use the data (1, 31), (2, 44), (3, 57), (4, 70) to calculate Pearson r and interpret it.
| x | y |
|---|---|
| 1 | 31 |
| 2 | 44 |
| 3 | 57 |
| 4 | 70 |
- The means are x̄=2.5, ȳ=50.5. Calculate centered sums Sxx=5, Sxy=65, Syy=845.
- Use technology or r=Sxy/√(Sxx·Syy): r=65/√(5·845)=1.
- All points lie exactly on an increasing line. r=1 describes linear association; it does not prove causation. With zero variance in either variable, r is undefined.
Answer: r = 1; perfect positive linear association
Verify: The centered-sum calculation independently returns r=1.
Avoid this mistake: Rounding coefficients too early.
Example 3 · Intermediate
Transform f(x)=x² to g(x)=14(x−18)²+13. Describe the changes and find g(19).
- Replacing x with x−18 shifts the graph right 18.
- Multiplication by 14 stretches vertically; adding 13 shifts up 13. The vertex is (18,13).
- At x=19, g=14(1)²+13=27. For f(14x), the coefficient becomes 196, a horizontal compression by factor 1/14.
Answer: 27
Verify: One unit from the new axis, the vertical increase is a.
Avoid this mistake: Extrapolating without stating the observed domain and physical limitations.
Example 4 · Advanced
A rental costs $2 per hour plus $19. The total is $45. Write an equation and find the hours x.
- Model cost: 2x+19=45.
- Subtract the fixed fee 19: 2x=26.
- Divide by the hourly rate 2: x=13 hours.
Answer: 13
Verify: Substitute x=13 into the ORIGINAL equation: both sides equal 45.
Avoid this mistake: Choosing a curve solely because it passes near a few points.
Example 5 · Challenge
For f(t)=230(0.5)t, find f(4) and identify growth or decay.
- The initial amount is 230; the multiplicative factor is 0.5.
- After 4 periods, multiply by 0.5 a total of 4 times: 230·0.54.
- 0.54=0.0625; the output is 14.375. A factor between 0 and 1 means decay.
Answer: 14.375; decay
Verify: Divide by the accumulated factor to recover the initial amount.
Avoid this mistake: Rounding coefficients too early.
Common mistakes and corrections
- Choosing a curve solely because it passes near a few points.
- Rounding coefficients too early.
- Extrapolating without stating the observed domain and physical limitations.
Real-world applications
- Fit a quadratic trend to measured projectile heights. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Fit an exponential trend to a repeated percentage change. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Make a plan before calculating. Keep intermediate work visible so that a later verification can locate any mistake.
A linear pattern adds roughly a constant amount for equal input changes; a quadratic pattern has roughly constant second differences; a nonzero exponential pattern multiplies by roughly a constant factor. These patterns guide model choice but are not proof that a real-world process follows a rule indefinitely.
Frequently asked questions
What should I know before multi-step problem solving?
Evaluate linear, quadratic, and exponential functions and read tables.
How can I check my work?
Multiplying the factors (or divisor and quotient) recovers the polynomial. 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?
Choosing a curve solely because it passes near a few points. Rounding coefficients too early. Extrapolating without stating the observed domain and physical limitations.
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: Fit a quadratic trend to measured projectile heights.
- 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
- Make a plan before calculating. Keep intermediate work visible so that a later verification can locate any mistake.
- Choosing a curve solely because it passes near a few points.
- Rounding coefficients too early.
- Extrapolating without stating the observed domain and physical limitations.
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