Illinois high school students taking Algebra 1 are assessed by the IAR (Illinois Assessment of Readiness) at the end of the course. The Algebra 1 IAR tests the full range of first-year algebra content, from linear equations and inequalities all the way through quadratic functions and basic statistics. Because it is an end-of-course assessment, it reflects what students are expected to know after a complete school year of instruction.
This guide provides practice problems across each major Algebra 1 topic tested on the IAR, with complete solutions and a structured 4-week study plan.
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Major IAR Algebra 1 Topic Areas
1. Linear Equations and Inequalities
Students solve linear equations in one variable, including equations with variables on both sides and equations that require combining like terms or using the distributive property. They also solve and graph linear inequalities and understand what the solutions represent.
Practice 1: Solve: 4x − 7 = 2x + 9
Solution: 4x − 2x = 9 + 7 → 2x = 16 → x = 8.
Practice 2: Solve and graph: 3x + 5 ≥ 17
Solution: 3x ≥ 12 → x ≥ 4. Graph: closed circle at 4, arrow pointing right.
Practice 3: Solve: 2(3x − 1) = 4x + 8
Solution: 6x − 2 = 4x + 8 → 2x = 10 → x = 5.
2. Linear Functions and Slope
Students interpret slope as rate of change, write equations of lines in slope-intercept form (y = mx + b) and point-slope form, graph lines, and identify parallel and perpendicular relationships.
Practice 4: Write the equation of a line with slope −2 passing through (3, 7).
Solution: y − 7 = −2(x − 3) → y = −2x + 6 + 7 → y = −2x + 13.
Practice 5: A linear function passes through (0, −4) and (5, 6). Find the slope and write the equation.
Solution: Slope = (6 − (−4))/(5 − 0) = 10/5 = 2. Equation: y = 2x − 4.
3. Systems of Linear Equations
Students solve systems using graphing, substitution, and elimination. They interpret the solution as a point of intersection and understand what it means in context.
Practice 6: Solve using substitution:
y = 3x − 2
2x + y = 13
Solution: Substitute: 2x + (3x − 2) = 13 → 5x = 15 → x = 3. Then y = 3(3) − 2 = 7. Solution: (3, 7).
Practice 7: Solve using elimination:
3x + 2y = 12
x − 2y = 4
Solution: Add equations: 4x = 16 → x = 4. Then 3(4) + 2y = 12 → 2y = 0 → y = 0. Solution: (4, 0).
4. Exponential Functions
Students understand exponential growth (y = a · bˣ with b > 1) and exponential decay (b < 1), compare exponential and linear functions, and interpret exponential models in context.
Practice 8: A population of bacteria doubles every hour. Starting with 50 bacteria, write an equation for the population after t hours.
Solution: y = 50 · 2ᵗ.
Practice 9: A car purchased for $24,000 loses 15% of its value each year. Write the equation and find the value after 3 years.
Solution: y = 24,000 · (0.85)³ = 24,000 × 0.6141 ≈ $14,739.
5. Polynomial and Quadratic Expressions
Students add, subtract, and multiply polynomials. They factor quadratics, solve quadratic equations by factoring and using the quadratic formula, and interpret the meaning of solutions.
Practice 10: Factor: x² + 6x − 16
Solution: Find two numbers multiplying to −16 and adding to 6: 8 and −2. Answer: (x + 8)(x − 2).
Practice 11: Solve using the quadratic formula: x² − 4x − 12 = 0
Solution: a = 1, b = −4, c = −12. x = [4 ± √(16 + 48)] / 2 = [4 ± √64] / 2 = [4 ± 8] / 2. x = 6 or x = −2.
Practice 12: Find the vertex of y = x² − 6x + 5.
Solution: Vertex x-coordinate: x = −b/(2a) = 6/2 = 3. y = 9 − 18 + 5 = −4. Vertex: (3, −4).
6. Descriptive Statistics
Students interpret data using measures of center (mean, median) and spread (range, interquartile range, standard deviation at a conceptual level), construct and interpret scatter plots, and identify correlation vs. causation.
Practice 13: A scatter plot shows test scores (y) vs. hours studied (x). The line of best fit is y = 7.5x + 45. What score would you predict for a student who studied 6 hours?
Solution: y = 7.5(6) + 45 = 45 + 45 = 90.
Practice 14: A data set has values: 12, 15, 17, 20, 25, 28, 30. What is the IQR?
Solution: Q1 = 15 (lower middle), Q3 = 28 (upper middle). IQR = 28 − 15 = 13.
Common IAR Algebra 1 Mistakes
- Not distributing the negative correctly: When subtracting a multi-term expression (e.g., in systems via substitution), students often forget to distribute the negative to every term.
- Sign errors in the quadratic formula: When b is negative (like −4), b² is still positive (16), and −b becomes positive. These sign operations trip up many students under pressure.
- Confusing slope of a line with slope of a segment: Slope is a constant property of the entire line, not just the segment shown in a graph. Some students calculate slope only between the visible endpoints and believe other segments would have a different slope.
- Mistaking exponential and linear growth: Students who answer “what type of function is this?” from a table sometimes identify linear growth from an exponential pattern. The key check: linear = constant difference; exponential = constant ratio.
4-Week IAR Algebra 1 Study Plan
Week 1: Linear Equations, Inequalities, and Functions
Solve one- and two-variable linear equations each day. Practice writing equations from slope and a point, or from two points. Cover linear inequalities and compound inequalities. End with a 20-question linear function and equation quiz.
Week 2: Systems of Equations and Exponents
Solve systems by graphing, substitution, and elimination. Identify no solution, one solution, and infinitely many solutions cases. Then cover exponent rules (product, quotient, power, zero, negative). End with a 20-question mixed quiz.
Week 3: Polynomials and Quadratics
Add, subtract, and multiply polynomials. Practice factoring: GCF, trinomials (a = 1 and a ≠ 1), and difference of squares. Solve quadratics by factoring and by the quadratic formula. Graph parabolas and find vertices. End with a 20-question quadratic quiz.
Week 4: Statistics and Full Mixed Review
Cover descriptive statistics, scatter plots, and lines of best fit. Take a 45-question mixed-topic practice test covering all Algebra 1 domains. Identify the two or three areas with the most errors and do a targeted 15-problem review in each area before the exam.
IAR Algebra 1 Resources from ViewMath
ViewMath offers Algebra 1 practice test books and topic-specific workbooks covering all IAR Algebra 1 content areas, with complete answer keys and step-by-step worked solutions. Explore the Algebra 1 collection in the sidebar.
ViewMath is an independent publisher. Our materials are not official IAR or ISBE materials.
