Algebraic Reasoning Practice Worksheet
Standards Practice Series
Algebraic Reasoning Practice
2 Questions / Code
Name:
Date:
Period:
Score: / 14
Directions: Complete each problem in the designated workspace. Show all mathematical thinking, setups, and units.
AR.1.A Problem Set 1
2 Questions
- A freelance consultant charges a flat fee of $45 plus $32 per hour. A client has a maximum budget of $250. Write and solve an inequality for the maximum whole hours \(h\) the consultant can work.
Work Space
Answer:
- A courier van travels 180 miles on 7.5 gallons of fuel. If gasoline costs $3.40 per gallon, calculate the exact fuel cost per mile traveled. Show your dimensional setup.
Work Space
Answer:
AR.1.B Problem Set 2
2 Questions
- A student solved \(4(2x - 3) = 5x + 9\) and got \(x = 4\) with Step 1 written as \(8x - 3 = 5x + 9\). Identify the error, explain what should have been done, and find the correct solution.
Work Space
Correct \(x\):
- A student claims a theater with 24 rows of 18 seats will seat 4,320 people. Use estimation to explain why this result is unreasonable, then calculate the actual capacity.
Work Space
Actual Seats:
Algebraic Reasoning Standards Practice Page 1 of 4 • Codes AR.1.A, AR.1.B
Algebraic Reasoning Practice
Problem Codes AR.1.C & AR.1.D
Name: Page 2 of 4
AR.1.C Problem Set 3
2 Questions
- Solve \(\frac{2}{3}x - 5 = \frac{1}{4}x + 2\). State whether you chose algebraic clearing of fractions or a graphical/table tool, and solve for the exact value of \(x\).
Work Space & Technique
Exact \(x\):
- Without paper multiplication or a calculator, use mental math or compatible number sense strategies to estimate \(49.8 \times 19.5\). Record your strategy and estimated interval.
Mental Strategy & Reasoning
Estimated Value:
AR.1.D Problem Set 4
2 Questions
- A reservoir containing 500 gallons drains at 25 gal/min. Represent this relationship in two distinct forms: (a) an algebraic equation for \(V(t)\), and (b) a table for \(t = 0, 5, 10, 15\).
Equation & Table Representations
Equation:
- A line passes through \((2, 7)\) and \((5, 16)\). Write a clear verbal interpretation of the slope (rate of change) and express the relationship in slope-intercept form (\(y = mx + b\)).
Slope Calculation & Verbal Meaning
Form:
Algebraic Reasoning Standards Practice Page 2 of 4 • Codes AR.1.C, AR.1.D
Algebraic Reasoning Practice
Problem Codes AR.1.E & AR.1.F
Name: Page 3 of 4
AR.1.E Problem Set 5
2 Questions
- Plan A costs $20/month plus $4/class. Plan B has no monthly fee but costs $9/class. Create an organized data table to determine the number of classes where both plans cost the same.
Organized Comparison Model / Table
Equal Cost at:
- A tile pattern starts with 5 tiles in Stage 1 and adds 3 tiles in each consecutive stage. Draw an organized representation of Stage 3 and write a general expression for Stage \(n\).
Diagram & Algebraic Expression
Stage \(n\) Rule:
AR.1.F Problem Set 6
2 Questions
- Contrast the growth rates of \(f(x) = 3x + 10\) and \(g(x) = 2^x\) from \(x = 1\) to \(x = 5\). Analyze how their rates of change connect to their mathematical families.
Comparative Analysis Work Space
\(g(x) > f(x)\) at:
- A rectangle has a perimeter of 48 cm. Its length is 6 cm greater than its width. Set up an algebraic equation relating length and width, then find its dimensions and total area.
Relationship Formulation & Solution
Total Area:
Algebraic Reasoning Standards Practice Page 3 of 4 • Codes AR.1.E, AR.1.F
Algebraic Reasoning Practice
Problem Code AR.1.G & Standards Self-Audit
Name: Page 4 of 4
AR.1.G Problem Set 7
2 Questions
- Claim: "The sum of any two rational numbers is always rational." Decide if this is Always, Sometimes, or Never True. Provide a formal justification using algebraic definitions.
Formal Mathematical Justification
Claim is:
- Consider: "If \(a^2 > b^2\), then \(a > b\) for all real numbers \(a\) and \(b\)." Determine whether this statement is true or false. If false, provide a precise counterexample and explain why.
Counterexample & Explanation
Evaluation:
Problem Code Self-Audit & Strategy Review
Student Reflection
| Code | Problems | Confidence Level | Key Strategy Used |
|---|
| AR.1.A | Q1, Q2 | High Med Low | |
| AR.1.B | Q3, Q4 | High Med Low | |
| AR.1.C | Q5, Q6 | High Med Low | |
| AR.1.D | Q7, Q8 | High Med Low | |
| AR.1.E | Q9, Q10 | High Med Low | |
| AR.1.F | Q11, Q12 | High Med Low | |
| AR.1.G | Q13, Q14 | High Med Low | |
Metacognitive Reflection: Which code presented the greatest reasoning challenge today, and what adjustment did you make to resolve it?
Algebraic Reasoning Standards Practice Page 4 of 4 • Code AR.1.G & Self-Audit
