TSIA2 Mastery Practice Packet Texas Success Initiative Assessment 2.0 Form A
Mathematics Benchmark Practice Packet
College Benchmark CRC 950+ or Diagnostic 6
Student Name
Test Date
Class / Period
Raw Score / 20
TSIA2 Mathematics Formula Reference Calculators strictly permitted on designated items only
Slope & Line: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
Geometry & Right \(\triangle\): \(a^2 + b^2 = c^2\)
Solids & Circles: Cylinder: \(V = \pi r^2 h\)
Section 1: Quantitative Reasoning Questions 1 & 2
1 Unit Rates & Dimensional Analysis
Workspace
A municipal water treatment facility injects fluoridation chemical at a rate of 1.25 milligrams per liter of treated water. If the facility processes \(4.8 \times 10^5\) liters of water each day, how many kilograms of the chemical are required for a 30-day billing cycle? (Note: \(1\text{ kg} = 10^6\text{ mg}\))
A. 1.8 kg
B. 18 kg
C. 60 kg
D. 180 kg
2 Percentages & Successive Price Adjustments
Workspace
An electronics distributor purchased a computer workstation at a wholesale price of \(\$600\). The retail store marked up the wholesale price by \(40\%\). During an annual inventory sale, the marked-up price was discounted by \(25\%\). A customer purchased the workstation and paid an \(8\%\) sales tax on the discounted price. What was the total amount the customer paid?
A. \(\$648.00\)
B. \(\$672.20\)
C. \(\$680.40\)
D. \(\$712.80\)
TSIA2 Mathematics Practice Packet • Form A Page 1 of 7
Sections 1 & 2: Quantitative & Algebraic Foundations Form A
Mixtures, Modeling & Linear Inequalities
Questions 3 – 5
3 Mixture Problems & Concentrations
Workspace
A chemical laboratory technician mixes a \(15\%\) hydrochloric acid solution with a \(40\%\) hydrochloric acid solution to obtain exactly 50 liters of a \(25\%\) hydrochloric acid solution. How many liters of the \(40\%\) acid solution must be used?
A. 15 liters
B. 20 liters
C. 25 liters
D. 30 liters
4 Linear Functions from Tabular Data
Workspace
The table displays the total production cost \(C(n)\), in dollars, for printing \(n\) customized athletic banners. Assuming the cost increases at a constant rate, which equation models \(C(n)\)?
A. \(C(n) = 9.5n\)
B. \(C(n) = 6n + 35\)
C. \(C(n) = 5n + 45\)
D. \(C(n) = 7n + 25\)
5 Multi-Step Linear Inequalities
Workspace
Which inequality represents the complete solution set for \(-\frac{2}{3}(6x - 9) \ge 4x - 14\)?
A. \(x \ge \frac{5}{2}\)
B. \(x \le \frac{5}{2}\)
C. \(x \le -\frac{5}{2}\)
D. \(x \ge -1\)
TSIA2 Mathematics Practice Packet • Form A Page 2 of 7
Section 2: Algebraic Reasoning Form A
Linear Systems, Literal Formulas & Quadratic Roots
Questions 6 – 8
6 Systems of Two Linear Equations
Workspace
A community theater sold 320 total tickets for an evening performance, generating \(\$3,480\) in total ticket revenue. Adult admission tickets were priced at \(\$14\) each, and student admission tickets were priced at \(\$9\) each. Exactly how many adult tickets were sold?
A. 115 adult tickets
B. 120 adult tickets
C. 140 adult tickets
D. 200 adult tickets
7 Rearranging Literal Equations
Workspace
The electric power transmitted through a resistive circuit is given by \(P = \frac{V^2 R}{(R + r)^2}\). If \(P\), \(V\), \(R\), and \(r\) are all positive real numbers, which equation correctly expresses the potential difference \(V\) in terms of \(P\), \(R\), and \(r\)?
A. \(V = \frac{(R + r)\sqrt{P}}{R}\)
B. \(V = (R + r)\sqrt{\frac{P}{R}}\)
C. \(V = \frac{P(R+r)}{\sqrt{R}}\)
D. \(V = \sqrt{P(R + r)^2 - R}\)
8 Solving Quadratic Equations by Factoring
Workspace
What are all the real solutions to the quadratic equation \(3x^2 - 11x - 20 = 0\)?
A. \(x = -\frac{4}{3}\) and \(x = 5\)
B. \(x = \frac{4}{3}\) and \(x = -5\)
C. \(x = -\frac{5}{3}\) and \(x = 4\)
D. \(x = -4\) and \(x = \frac{5}{3}\)
TSIA2 Mathematics Practice Packet • Form A Page 3 of 7
Section 3: Advanced Algebraic Reasoning Form A
Rational Expressions, Function Composition & Vertices
Questions 9 – 11
9 Simplifying Rational Algebraic Expressions
Workspace
Which expression is equivalent to \(\frac{2x^2 + 5x - 12}{4x^2 - 9} \div \frac{x^2 + 4x}{2x^2 - 3x}\) for all real values of \(x\) where the expression is defined?
A. \(\frac{x - 3}{x + 3}\)
B. \(\frac{2x - 3}{2x + 3}\)
C. \(\frac{2x + 3}{2x - 3}\)
D. \(\frac{x + 4}{2x - 3}\)
10 Function Notation & Composition
Workspace
Given the functions \(f(x) = 2x^2 - 3x + 1\) and \(g(x) = 3x - 2\), what is the value of the difference \(f(g(2)) - g(f(2))\)?
A. 8
B. 11
C. 14
D. 19
11 Quadratic Vertex & Maximum Revenue
Workspace
The monthly revenue of an educational publishing firm is modeled by \(R(p) = -5p^2 + 350p - 2400\), where \(p\) is the unit price per book in dollars. What book price \(p\) maximizes the company's monthly revenue, and what is the maximum monthly revenue?
A. Price \(\$30\); Max Revenue \(\$3,600\)
B. Price \(\$35\); Max Revenue \(\$3,725\)
C. Price \(\$40\); Max Revenue \(\$3,600\)
D. Price \(\$35\); Max Revenue \(\$4,150\)
TSIA2 Mathematics Practice Packet • Form A Page 4 of 7
Section 4: Geometric & Spatial Reasoning Form A
Coordinate Geometry, Right Triangles & Composite Area
Questions 12 – 14
12 Perpendicular Lines & Coordinate Intercepts
Workspace
In the \(xy\)-plane, line \(L_1\) passes through points \((-3, 5)\) and \((1, -3)\). Line \(L_2\) is perpendicular to \(L_1\) and passes through the point \((4, -1)\). What is the \(y\)-intercept of line \(L_2\)?
A. \(-3\)
B. \(-\frac{5}{2}\)
C. \(-1\)
D. \(2\)
13 Pythagorean Theorem & Right Triangles
Workspace
A high-voltage crane support wire is anchored to the level ground at a point 36 feet away from the vertical base of a communication mast. The wire attaches to the mast at an elevation of 48 feet. What is the total length of the taut support wire from the ground anchor to the mast attachment?
A. 52 feet
B. 58 feet
C. 60 feet
D. 64 feet
14 Composite Geometric Area
Workspace
A recreation center outdoor court consists of a rectangle adjoining a semicircle along one of its shorter sides. The rectangle measures 28 meters long and 14 meters wide. The adjoining semicircle has a diameter equal to the 14-meter width. Which value is closest to the total surface area of the court? (Use \(\pi \approx 3.14\))
A. \(446\text{ m}^2\)
B. \(469\text{ m}^2\)
C. \(546\text{ m}^2\)
D. \(699\text{ m}^2\)
TSIA2 Mathematics Practice Packet • Form A Page 5 of 7
Sections 4 & 5: Geometry & Data Analysis Form A
Solid Geometry, Proportionality & Distribution Measures
Questions 15 – 17
15 3D Volume Scaling & Dimensions
Workspace
Right circular Cylinder A has radius \(r\) and height \(h\). Cylinder B has a radius that is \(50\%\) greater than Cylinder A and a height that is \(20\%\) less than Cylinder A. What is the ratio of the volume of Cylinder B to the volume of Cylinder A?
A. \(\frac{6}{5}\)
B. \(\frac{7}{5}\)
C. \(\frac{9}{5}\)
D. \(\frac{11}{5}\)
16 Similar Triangles & Proportional Sides
Workspace
In \(\triangle ABC\), segment \(DE\) is parallel to base \(BC\), with point \(D\) located on side \(AB\) and point \(E\) located on side \(AC\). If \(AD = 8\), \(DB = 6\), and the base \(BC = 21\), what is the length of segment \(DE\)?
A. 10.5
B. 12.0
C. 14.0
D. 16.0
17 Central Tendency & Impact of Outliers
Workspace
A initial sample of 7 diagnostic scores is: \(78, 82, 85, 85, 89, 91, 94\). An 8th participant completes the test and scores a \(42\). Which statement correctly describes the resulting change in the dataset's summary statistics?
A. Both the mean and median decrease by \(\approx 5\) points.
B. The mean decreases by \(> 5\) points; median is unchanged.
C. The median decreases, but the mean resists the outlier.
D. The interquartile range triples while the mean stays constant.
TSIA2 Mathematics Practice Packet • Form A Page 6 of 7
Section 5: Probabilistic & Statistical Reasoning Form A
Tables, Probability, Regression & Final Scoring
Questions 18 – 20
18 Two-Way Table Conditional Probability
Given that a randomly selected student-athlete prefers water, what is the probability that they compete in an individual sport?
A. \(\frac{7}{25}\)
B. \(\frac{7}{10}\)
C. \(\frac{14}{27}\)
D. \(\frac{28}{45}\)
19 Compound Probability without Replacement
An inspection bin contains 15 precision microchips: 10 meet full specifications and 5 have minor surface defects. An automated sensor selects 2 microchips at random one after another without replacement. What is the probability that at least one of the selected microchips has a surface defect?
A. \(\frac{1}{3}\)
B. \(\frac{19}{42}\)
C. \(\frac{4}{7}\)
D. \(\frac{2}{3}\)
20 Linear Regression & Residual Analysis
A least-squares linear regression line modeling daily study hours \(x\) and exam score \(y\) is \(\hat{y} = 6.4x + 51.5\). A student who studied 4.5 hours earned an actual score of 84. What is this student's residual, and how is it interpreted?
A. \(-3.7\); the student scored 3.7 points lower than model.
B. \(+3.7\); the student scored 3.7 points higher than model.
C. \(+4.2\); the student scored 4.2 points higher than model.
D. \(-4.2\); the student scored 4.2 points lower than model.
Official Examination Document • Form A
Student Answer Record & Diagnostic Summary
Fill bubble completely:
(A)(B)(C)(D) 06. (A)(B)(C)(D)
(A)(B)(C)(D) 07. (A)(B)(C)(D)
(A)(B)(C)(D) 08. (A)(B)(C)(D)
(A)(B)(C)(D) 09. (A)(B)(C)(D)
(A)(B)(C)(D) 10. (A)(B)(C)(D)
(A)(B)(C)(D) 16. (A)(B)(C)(D)
(A)(B)(C)(D) 17. (A)(B)(C)(D)
(A)(B)(C)(D) 18. (A)(B)(C)(D)
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Quantitative Q1–Q4: ___ / 4
Algebraic Q5–Q11: ___ / 7
Geometric Q12–Q16: ___ / 5
Statistical Q17–Q20: ___ / 4
TSIA2 Mathematics Practice Packet • Form A Page 7 of 7
TSIA2 Solutions Teacher Guide Instructor Assessment Resource Form A Key
TSIA2 Solutions & Diagnostic Guide (Form A)
College Readiness Target 14+ / 20 (70% Benchmark)
Form A Quick-Scan Master Key 20 Multiple Choice Questions
Q01B
Q02C
Q03B
Q04B
Q05B
Q06B
Q07B
Q08A
Q09B
Q10C
Q11B
Q12A
Q13C
Q14B
Q15C
Q16B
Q17B
Q18C
Q19C
Q20B
Detailed Solutions: Quantitative Reasoning (Questions 1 – 3)
1 Unit Rates & Scientific Notation
Correct: B (18 kg)
Step 1 (Daily chemical in mg): Multiply daily volume by injection rate: \(4.8 \times 10^5\text{ L/day} \times 1.25\text{ mg/L} = 6.0 \times 10^5\text{ mg/day}\).
Step 2 (30-day requirement): Multiply daily mass by 30 days: \(30 \times 6.0 \times 10^5\text{ mg} = 1.8 \times 10^7\text{ mg}\).
Step 3 (Convert to kilograms): Since \(1\text{ kg} = 10^6\text{ mg}\), divide by \(10^6\): \(\frac{1.8 \times 10^7}{10^6} = 18\text{ kg}\).
Common Student Trap: Students choosing A (1.8 kg) typically miscount scientific notation powers during unit conversion (\(10^7 / 10^6 = 10\)). Students choosing C forget the 30-day factor.
2 Percentages & Successive Price Changes
Correct: C ($680.40)
Step 1 (Marked-up retail price): Wholesale \(\$600\) increased by \(40\%\): \(\$600 \times 1.40 = \$840.00\).
Step 2 (Sale discount): \(25\%\) reduction from \(\$840\): \(\$840 \times (1 - 0.25) = \$840 \times 0.75 = \$630.00\).
Step 3 (Sales tax addition): Add \(8\%\) tax to the sale price: \(\$630.00 \times 1.08 = \$680.40\).
Common Student Trap: Incorrectly combining percent percentages additively (\(40\% - 25\% + 8\% = 23\%\) net increase \(\rightarrow \$600 \times 1.23 = \$738\)). Percent changes apply sequentially to changing intermediate base amounts.
3 Mixtures & Concentration Equations
Correct: B (20 liters)
Step 1 (Set up equation): Let \(x\) be liters of \(40\%\) solution; then \((50 - x)\) is liters of \(15\%\) solution.
Step 2 (Balance pure solute): \(0.15(50 - x) + 0.40x = 0.25(50)\).
Step 3 (Solve for \(x\)): \(7.5 - 0.15x + 0.40x = 12.5 \implies 0.25x = 5.0 \implies x = \frac{5.0}{0.25} = 20\text{ liters}\).
Common Student Trap: Students solving for \((50 - x) = 30\) select option D. Emphasize labeling variables before writing equations.
TSIA2 Mathematics Solutions Guide • Form A Page 1 of 5
Teacher Solutions Guide Form A
Linear Models, Inequalities & Literal Formulas
Questions 4 – 7
4 Linear Modeling from Data Tables
Correct: B (\(C(n) = 6n + 35\))
Step 1 (Find constant rate of change): Slope \(m = \frac{185 - 95}{25 - 10} = \frac{90}{15} = 6\text{ dollars per banner}\).
Step 2 (Determine initial/fixed setup cost): Using point \((10, 95)\): \(95 = 6(10) + b \implies b = 95 - 60 = 35\).
Common Student Trap: Choosing A (\(9.5n\)) assumes direct variation through the origin (\(95 / 10 = 9.5\)), ignoring the fixed base charge.
5 Multi-Step Linear Inequalities
Correct: B (\(x \le \frac{5}{2}\))
Step 1 (Distribute fraction): \(-\frac{2}{3}(6x) - (-\frac{2}{3})(9) = -4x + 6\). Inequality is \(-4x + 6 \ge 4x - 14\).
Step 2 (Collect variables and constants): Subtract \(4x\) and subtract 6: \(-8x \ge -20\).
Step 3 (Divide by negative number): Divide by \(-8\) and reverse inequality symbol: \(x \le \frac{-20}{-8} \implies x \le \frac{5}{2}\).
Common Student Trap: Forgetting to reverse the direction of the inequality symbol when dividing by \(-8\) leads directly to distractor A (\(x \ge \frac{5}{2}\)).
6 Linear Systems in Context
Correct: B (120 adult tickets)
Step 1 (System formulation): Total tickets: \(A + S = 320\); Total revenue: \(14A + 9S = 3480\).
Step 2 (Substitution method): Substitute \(S = 320 - A\) into revenue equation: \(14A + 9(320 - A) = 3480\).
Step 3 (Simplify): \(14A + 2880 - 9A = 3480 \implies 5A = 600 \implies A = 120\).
Common Student Trap: Solving for student tickets (\(S = 320 - 120 = 200\)) and selecting D. Remind students to re-verify the specific target variable.
7 Rearranging Literal Formulas
Correct: B (\(V = (R + r)\sqrt{\frac{P}{R}}\))
Step 1 (Isolate \(V^2\)): Multiply by \((R + r)^2\) and divide by \(R\): \(V^2 = \frac{P(R + r)^2}{R}\).
Step 2 (Take square root): \(V = \sqrt{\frac{P(R + r)^2}{R}} = \sqrt{(R + r)^2} \cdot \sqrt{\frac{P}{R}} = (R + r)\sqrt{\frac{P}{R}}\).
Common Student Trap: Distributing radicals across terms incorrectly or forgetting to take the square root of the denominator \(R\) (Distractor A).
TSIA2 Mathematics Solutions Guide • Form A Page 2 of 5
Teacher Solutions Guide Form A
Factoring, Rational Expressions & Functions
Questions 8 – 11
8 Solving Quadratics by Factoring
Correct: A (\(x = -\frac{4}{3}, 5\))
Step 1 (Factor trinomial): For \(3x^2 - 11x - 20 = 0\), find factors of \(3(-20) = -60\) that sum to \(-11\): \(-15\) and \(+4\).
Step 2 (Group and factor): \(3x(x - 5) + 4(x - 5) = (3x + 4)(x - 5) = 0\).
Step 3 (Solve roots): \(3x + 4 = 0 \implies x = -\frac{4}{3}\); \(x - 5 = 0 \implies x = 5\).
Common Student Trap: Sign reversal errors in the zero-product property produce distractor B (\(x = \frac{4}{3}, -5\)).
9 Dividing Rational Algebraic Expressions
Correct: B (\(\frac{2x - 3}{2x + 3}\))
Step 1 (Factor all polynomials): \(2x^2 + 5x - 12 = (2x - 3)(x + 4)\); \(4x^2 - 9 = (2x - 3)(2x + 3)\); \(x^2 + 4x = x(x + 4)\); \(2x^2 - 3x = x(2x - 3)\).
Step 2 (Multiply by reciprocal): \(\frac{(2x - 3)(x + 4)}{(2x - 3)(2x + 3)} \times \frac{x(2x - 3)}{x(x + 4)}\).
Step 3 (Cancel common factors): Canceling \((2x - 3)\), \((x + 4)\), and \(x\) leaves \(\frac{2x - 3}{2x + 3}\).
Common Student Trap: Inverting the wrong fraction or canceling non-factored terms directly across addition signs.
10 Function Notation & Composition
Correct: C (14)
Step 1 (Evaluate \(f(g(2))\)): \(g(2) = 3(2) - 2 = 4\). Then \(f(4) = 2(4)^2 - 3(4) + 1 = 32 - 12 + 1 = 21\).
Step 2 (Evaluate \(g(f(2))\)): \(f(2) = 2(2)^2 - 3(2) + 1 = 8 - 6 + 1 = 3\). Then \(g(3) = 3(3) - 2 = 7\).
Step 3 (Calculate difference): \(21 - 7 = 14\).
Common Student Trap: Assuming function composition is commutative (\(f(g(x)) = g(f(x))\)) or making sign errors in \(f(2)\).
11 Parabola Vertex & Maximum Optimization
Correct: B ($35; $3,725)
Step 1 (Find maximizing price): Vertex \(p = -\frac{b}{2a} = -\frac{350}{2(-5)} = \frac{350}{10} = \$35\).
Step 2 (Calculate maximum revenue): \(R(35) = -5(35)^2 + 350(35) - 2400 = -5(1225) + 12250 - 2400 = -6125 + 12250 - 2400 = \$3,725\).
Common Student Trap: Squaring \(-5p\) instead of evaluating \(-5(p^2)\) during manual arithmetic substitution.
TSIA2 Mathematics Solutions Guide • Form A Page 3 of 5
Teacher Solutions Guide Form A
Coordinate Geometry, Pythagorean & 3D Solids
Questions 12 – 15
12 Perpendicular Lines & Coordinate Intercepts
Correct: A (\(-3\))
Step 1 (Find slope of \(L_1\)): \(m_1 = \frac{-3 - 5}{1 - (-3)} = \frac{-8}{4} = -2\).
Step 2 (Determine perpendicular slope): \(m_2 = -\frac{1}{m_1} = -\frac{1}{-2} = \frac{1}{2}\).
Step 3 (Find equation of \(L_2\)): Through \((4, -1)\): \(y - (-1) = \frac{1}{2}(x - 4) \implies y + 1 = \frac{1}{2}x - 2 \implies y = \frac{1}{2}x - 3\). Thus \(y\)-intercept is \(-3\).
Common Student Trap: Using the parallel slope \(-2\) instead of the negative reciprocal slope \(\frac{1}{2}\).
13 Pythagorean Theorem & Right Triangles
Correct: C (60 feet)
Step 1 (Set up right triangle legs): Base \(a = 36\text{ ft}\), vertical mast \(b = 48\text{ ft}\).
Step 2 (Pythagorean calculation): \(c = \sqrt{36^2 + 48^2} = \sqrt{1296 + 2304} = \sqrt{3600} = 60\text{ feet}\).
Note: Recognizing the scaled Pythagorean triple \(12 \times (3, 4, 5) = (36, 48, 60)\) saves valuable test time.
Common Student Trap: Linear addition of legs (\(36 + 48 = 84\)) or subtracting squares under the radical.
14 Composite Two-Dimensional Area
Correct: B (\(469\text{ m}^2\))
Step 1 (Area of rectangle): \(A_{\text{rect}} = 28 \times 14 = 392\text{ m}^2\).
Step 2 (Area of semicircle): Diameter = 14 m \(\implies\) radius \(r = 7\text{ m}\). \(A_{\text{semi}} = \frac{1}{2}\pi r^2 = 0.5 \times 3.14 \times 49 = 76.93\text{ m}^2\).
Step 3 (Total sum): \(392 + 76.93 = 468.93 \approx 469\text{ m}^2\).
Common Student Trap: Forgetting to halve the circle area (yields 546) or using the diameter 14 as radius.
15 3D Volume Scaling & Dimensions
Correct: C (\(\frac{9}{5}\))
Step 1 (Express new dimensions): \(r_B = 1.5r_A\) and \(h_B = 0.8h_A\).
Step 2 (Evaluate volume ratio): \(\frac{V_B}{V_A} = \frac{\pi (1.5r)^2 (0.8h)}{\pi r^2 h} = (1.5)^2 \times 0.8 = 2.25 \times 0.8 = 1.8\).
Step 3 (Convert to fraction): \(1.8 = \frac{18}{10} = \frac{9}{5}\).
Common Student Trap: Failing to square the radius factor (\(1.5 \times 0.8 = 1.2 = \frac{6}{5}\)), which leads directly to distractor A.
TSIA2 Mathematics Solutions Guide • Form A Page 4 of 5
Teacher Solutions Guide Form A
Similarity, Probability, Regression & Diagnostic Action
Questions 16 – 20
Q16: Similar Triangles & Side Ratios Correct: B (12.0)
Total side \(AB = 8 + 6 = 14\). By similarity \(\triangle ADE \sim \triangle ABC\): \(\frac{DE}{BC} = \frac{AD}{AB} \implies \frac{DE}{21} = \frac{8}{14} = \frac{4}{7} \implies DE = 21 \times \frac{4}{7} = 12.0\). Trap: Using \(\frac{8}{6}\) yields distractor D (28 or 16).
Q17: Central Tendency & Outlier Resistance Correct: B (Mean drops > 5 pts; median unchanged)
Original: Mean \(\approx 86.29\), Median = 85. Adding 42: new Mean = \(646 / 8 = 80.75\) (drops \(5.54\) pts); new Median = \(\frac{85 + 85}{2} = 85\) (unchanged). Trap: Believing the median must change when an outlier is added.
Q18: Two-Way Conditional Probability Correct: C (\(\frac{14}{27}\))
Given water preference, denominator is total water drinkers (108). Numerator is individual athletes who prefer water (56): \(P(\text{Ind} \mid \text{Water}) = \frac{56}{108} = \frac{14}{27}\). Trap: Dividing by 200 (grand total) gives distractor A (\(\frac{7}{25}\)).
Q19: Compound Probability without Replacement Correct: C (\(\frac{4}{7}\))
Use the complement rule: \(P(\ge 1\text{ defect}) = 1 - P(\text{both no defect}) = 1 - \left(\frac{10}{15} \times \frac{9}{14}\right) = 1 - \frac{3}{7} = \frac{4}{7}\). Trap: Sampling with replacement gives \(\frac{5}{9}\); calculating exactly 1 defect misses the 2-defect case.
Q20: Linear Regression & Residuals Correct: B (+3.7; scored 3.7 pts higher than model)
Predicted \(\hat{y} = 6.4(4.5) + 51.5 = 28.8 + 51.5 = 80.3\). Residual = \(\text{Actual} - \text{Predicted} = 84 - 80.3 = +3.7\). Positive residual indicates outperforming the model. Trap: Subtracting Actual from Predicted flips the sign to \(-3.7\) (distractor A).
TSIA2 Readiness Placement & Action Rubric
Passing Threshold: Score \(\ge 14\)
Needs Remediation (0–9)
Intensive review required on multi-step inequalities, fractional exponents, and linear equation translation. Focus on basic algebraic manipulation.
Approaching Benchmark (10–13)
Solid arithmetic and linear foundation; target practice on rational expression simplification, conditional probability tables, and quadratic optimization vertices.
College Ready (14–20)
Meets Texas TSI 2.0 College Readiness standard. Ready for credit-bearing college algebra, precalculus, or introductory statistics courses.
TSIA2 Mathematics Solutions Guide • Form A Page 5 of 5
TSIA2 Foundations Practice Packet Texas Success Initiative Assessment 2.0
Mathematics Foundations Practice Packet
Target Placement Level Score Range 920 – 945
Student Name
Date
Period / Instructor
Score / 20
Foundations Formula Guide Diagnostic Levels 4 & 5 Preparation
Rectangle: \(P = 2l + 2w\)
Triangle & Prism: \(A = \frac{1}{2}bh\)
Linear Slope-Intercept: \(y = mx + b\)
Angles & Triangles: Supp.: \(A + B = 180^\circ\)
Part I: Quantitative Reasoning Questions 1 & 2
1 Unit Price & Comparison
Workspace
A grocery store sells breakfast cereal in two package sizes: a 16-ounce box for \(\$4.48\) and a 24-ounce box for \(\$6.00\). What is the unit cost per ounce of the larger box, and how much cheaper per ounce is it compared to the smaller box?
A. \(\$0.25\) per oz; \(\$0.03\) cheaper per oz
B. \(\$0.25\) per oz; \(\$0.05\) cheaper per oz
C. \(\$0.28\) per oz; \(\$0.03\) cheaper per oz
D. \(\$0.30\) per oz; \(\$0.02\) cheaper per oz
2 Percent Discount & Sales Tax
Workspace
A winter fleece jacket has an original price tag of \(\$80.00\). During a seasonal sale, the store offers a \(20\%\) discount on the jacket. If a sales tax of \(5\%\) is calculated on the discounted sale price, what is the final price a customer pays?
A. \(\$60.80\)
B. \(\$64.00\)
C. \(\$67.20\)
D. \(\$68.00\)
TSIA2 Foundations Practice Packet (Score 920–945) Page 1 of 7
Parts I & II: Quantitative & Algebra
Proportions, Fraction Operations & Linear Equations
Questions 3 – 5
3 Scale Ratios & Proportional Distance
Workspace
On an official state roadmap, a distance of \(0.5\) inches represents an actual driving distance of \(18\) miles. If two county courthouses are \(3.5\) inches apart on the map, what is the actual highway distance between them?
A. 98 miles
B. 108 miles
C. 126 miles
D. 144 miles
4 Fractions & Order of Operations
Workspace
What is the exact value of the numerical expression \(\frac{3}{4} + \frac{2}{3} \times \frac{9}{8} - \frac{1}{2}\)?
A. \(\frac{3}{4}\)
B. \(1\)
C. \(\frac{5}{4}\)
D. \(\frac{3}{2}\)
5 Solving Two-Step Linear Equations
Workspace
If \(4x - 13 = 35\), what is the value of \(x\)?
A. \(x = 5.5\)
B. \(x = 8\)
C. \(x = 11\)
D. \(x = 12\)
TSIA2 Foundations Practice Packet (Score 920–945) Page 2 of 7
Part II: Algebraic Foundations
Variable Equations, Inequalities & Verbal Models
Questions 6 – 8
6 Equations with Variables on Both Sides
Workspace
What value of \(x\) satisfies the equation \(5(x - 2) = 2x + 11\)?
A. \(x = 3\)
B. \(x = 5\)
C. \(x = 7\)
D. \(x = 9\)
7 Solving One-Variable Inequalities
Workspace
Which inequality represents the complete solution set for \(3x + 7 < 22\)?
A. \(x < 5\)
B. \(x > 5\)
C. \(x < 9\)
D. \(x > 9\)
8 Translating Words into Algebraic Equations
Workspace
"Seven less than three times a number \(n\) is equal to forty-one." Which equation correctly represents this verbal statement, and what is the value of \(n\)?
A. \(7 - 3n = 41;\quad n = -11\)
B. \(3n - 7 = 41;\quad n = 16\)
C. \(3(n - 7) = 41;\quad n = 20\)
D. \(3n + 7 = 41;\quad n = 11\)
TSIA2 Foundations Practice Packet (Score 920–945) Page 3 of 7
Part II: Algebraic Foundations
Lines, Function Inputs & Quadratic Roots
Questions 9 – 11
9 Slope-Intercept Form of a Line
Workspace
A straight line in the coordinate plane has a slope of \(-\frac{3}{2}\) and crosses the vertical \(y\)-axis at the point \((0, 4)\). Which equation represents this line?
A. \(y = \frac{3}{2}x + 4\)
B. \(y = -\frac{3}{2}x + 4\)
C. \(y = 4x - \frac{3}{2}\)
D. \(y = -\frac{2}{3}x + 4\)
10 Evaluating Linear Functions
Workspace
A linear function is defined by \(f(x) = -3x + 8\). What is the value of \(f(-4)\)?
A. \(-20\)
B. \(-4\)
C. \(4\)
D. \(20\)
11 Factoring Simple Quadratic Equations
Workspace
What are all the real solutions to the quadratic equation \(x^2 - 5x - 14 = 0\)?
A. \(x = -7\) and \(x = 2\)
B. \(x = 7\) and \(x = -2\)
C. \(x = -14\) and \(x = 1\)
D. \(x = 14\) and \(x = -1\)
TSIA2 Foundations Practice Packet (Score 920–945) Page 4 of 7
Part III: Geometric Reasoning
Perimeter, Area, Right Triangles & 3D Volume
Questions 12 – 14
12 Rectangle Perimeter & Area
Workspace
A rectangular backyard garden has a length of \(15\) feet and a width of \(8\) feet. What is the total square footage (area) of the garden, and how many linear feet of border edging are required to enclose its full perimeter?
A. Area = \(120\text{ sq ft}\); Perimeter = \(46\text{ ft}\)
B. Area = \(120\text{ sq ft}\); Perimeter = \(23\text{ ft}\)
C. Area = \(46\text{ sq ft}\); Perimeter = \(120\text{ ft}\)
D. Area = \(60\text{ sq ft}\); Perimeter = \(46\text{ ft}\)
13 Pythagorean Theorem in Real Context
Workspace
A \(13\)-foot safety ladder leans against the exterior wall of a building. If the bottom base of the ladder is placed \(5\) feet away from the base of the wall on level ground, how high up the wall does the top of the ladder reach?
A. 8 feet
B. 10 feet
C. 12 feet
D. 14 feet
14 Volume of Rectangular Solids
Workspace
A warehouse storage container has a rectangular base measuring \(6\) feet long by \(4\) feet wide, and a vertical height of \(3.5\) feet. What is the total storage volume of the container in cubic feet?
A. 48 cubic feet
B. 72 cubic feet
C. 84 cubic feet
D. 96 cubic feet
TSIA2 Foundations Practice Packet (Score 920–945) Page 5 of 7
Parts III & IV: Geometry & Statistics
Angle Relationships, Triangles & Data Measures
Questions 15 – 17
15 Supplementary Angle Relationships
Workspace
Two angles, \(\angle A\) and \(\angle B\), are supplementary angles. If the measure of \(\angle A\) is \(65^\circ\), what is the measure of angle \(\angle B\)?
A. \(25^\circ\)
B. \(35^\circ\)
C. \(115^\circ\)
D. \(125^\circ\)
16 Area of a Triangle
Workspace
A collegiate sports pennant has the shape of a triangle with a horizontal base of \(12\) inches and a perpendicular height of \(7\) inches. What is the surface area of the pennant in square inches?
A. 24 sq inches
B. 42 sq inches
C. 84 sq inches
D. 96 sq inches
17 Median and Range of a Dataset
Workspace
The list below shows the number of books read by six high school students during a summer reading program: \(4,\; 7,\; 3,\; 9,\; 7,\; 6\). What is the median number of books read, and what is the range of the dataset?
A. Median = 6.0; Range = 6
B. Median = 6.5; Range = 6
C. Median = 6.5; Range = 7
D. Median = 7.0; Range = 5
TSIA2 Foundations Practice Packet (Score 920–945) Page 6 of 7
Part IV: Statistics & Probability
Basic Probability, Charts & Final Answer Record
Questions 18 – 20
18 Simple Event Probability
An opaque draw bag contains 8 blue marbles, 5 red marbles, and 7 green marbles. If one marble is drawn at random from the bag, what is the probability that the drawn marble is NOT red?
A. \(\frac{1}{4}\)
B. \(\frac{2}{5}\)
C. \(\frac{3}{5}\)
D. \(\frac{3}{4}\)
19 Interpreting Survey Frequencies & Percents
Fifty incoming college students were surveyed regarding their transportation to campus: 12 walk, 8 ride a bicycle, 18 take the bus, and 12 drive. What percent use either a bicycle or public bus?
Bicycle (8) + Bus (18)
A. \(36\%\)
B. \(40\%\)
C. \(52\%\)
D. \(60\%\)
20 Proportional Sampling & Population Estimates
A wildlife biologist tags 40 fish in a conservation lake. Later, a random sample of 120 fish is captured, and 15 are found to have tags. Based on this proportional sample, what is the estimated total fish population in the lake?
A. 240 fish
B. 300 fish
C. 320 fish
D. 450 fish
TSIA2 Foundations Examination
Student Answer Sheet & Diagnostic Tracker
Fill bubble:
(A)(B)(C)(D) 06. (A)(B)(C)(D)
(A)(B)(C)(D) 07. (A)(B)(C)(D)
(A)(B)(C)(D) 08. (A)(B)(C)(D)
(A)(B)(C)(D) 09. (A)(B)(C)(D)
(A)(B)(C)(D) 10. (A)(B)(C)(D)
(A)(B)(C)(D) 16. (A)(B)(C)(D)
(A)(B)(C)(D) 17. (A)(B)(C)(D)
(A)(B)(C)(D) 18. (A)(B)(C)(D)
(A)(B)(C)(D) 19. (A)(B)(C)(D)
(A)(B)(C)(D) 20. (A)(B)(C)(D)
Quantitative Q1–Q4: ___ / 4
Algebraic Q5–Q11: ___ / 7
Geometric Q12–Q16: ___ / 5
Statistical Q17–Q20: ___ / 4
TSIA2 Foundations Practice Packet (Score 920–945) Page 7 of 7
TSIA2 Foundations Teacher Guide Teacher Instructional Resource
TSIA2 Foundations Solutions & Diagnostic Guide
Foundations Benchmark Score Range 920 – 945
Foundations Quick-Scan Master Key 20 Multiple Choice Questions
Q01A
Q02C
Q03C
Q04B
Q05D
Q06C
Q07A
Q08B
Q09B
Q10D
Q11B
Q12A
Q13C
Q14C
Q15C
Q16B
Q17B
Q18D
Q19C
Q20C
Solutions: Quantitative Reasoning (Questions 1 – 3)
1 Unit Price & Difference Comparison
Correct: A (\$0.25/oz; \$0.03 cheaper)
Step 1 (Smaller box rate): \(\frac{\$4.48}{16\text{ oz}} = \$0.28\text{ per ounce}\).
Step 2 (Larger box rate): \(\frac{\$6.00}{24\text{ oz}} = \$0.25\text{ per ounce}\).
Step 3 (Price difference): \(\$0.28 - \$0.25 = \$0.03\text{ per ounce savings}\).
Common Student Pitfall: Identifying the smaller box unit price as the target answer (distractor C) or subtracting total package prices instead of per-ounce rates.
2 Discount & Sales Tax Calculation
Correct: C (\$67.20)
Step 1 (Sale price after 20% discount): Discount = \(\$80 \times 0.20 = \$16.00\). Sale price = \(\$80.00 - \$16.00 = \$64.00\) (or \(\$80 \times 0.80 = \$64.00\)).
Step 2 (Apply 5% sales tax): Tax = \(\$64.00 \times 0.05 = \$3.20\).
Step 3 (Final total): \(\$64.00 + \$3.20 = \$67.20\).
Common Student Pitfall: Stopping after calculating the discounted price without adding tax (\$64.00, distractor B) or subtracting the tax instead of adding it (\$60.80, distractor A).
3 Proportional Map Scales
Correct: C (126 miles)
Step 1 (Set up proportion): \(\frac{0.5\text{ in}}{18\text{ mi}} = \frac{3.5\text{ in}}{x\text{ mi}}\).
Step 2 (Cross-multiply): \(0.5x = 3.5 \times 18 = 63\).
Step 3 (Solve for distance): \(x = \frac{63}{0.5} = 126\text{ miles}\). (Alternatively: \(3.5 / 0.5 = 7\); \(7 \times 18 = 126\text{ miles}\)).
Common Student Pitfall: Multiplying \(3.5 \times 18 = 63\) and neglecting to divide by the \(0.5\) scale factor.
TSIA2 Foundations Solutions Guide (Score 920–945) Page 1 of 5
Teacher Solutions Guide
Arithmetic Operations, Two-Step Equations & Inequalities
Questions 4 – 7
4 Order of Operations with Fractions
Correct: B (1)
Step 1 (Multiplication precedes addition/subtraction): \(\frac{2}{3} \times \frac{9}{8} = \frac{18}{24} = \frac{3}{4}\).
Step 2 (Substitute into expression): \(\frac{3}{4} + \frac{3}{4} - \frac{1}{2}\).
Step 3 (Add and subtract with common denominator): \(\frac{6}{4} - \frac{2}{4} = \frac{4}{4} = 1\).
Common Student Pitfall: Working left-to-right without obeying PEMDAS (\(\frac{3}{4} + \frac{2}{3} = \frac{17}{12}\), then multiplying by \(\frac{9}{8}\)), yielding erroneous fractional values.
5 Solving Two-Step Linear Equations
Correct: D (\(x = 12\))
Step 1 (Isolate the variable term): Add 13 to both sides: \(4x - 13 + 13 = 35 + 13 \implies 4x = 48\).
Step 2 (Divide by the coefficient): \(x = \frac{48}{4} = 12\).
Common Student Pitfall: Subtracting 13 instead of adding 13: \(4x = 35 - 13 = 22 \implies x = 5.5\) (distractor A).
6 Equations with Variables on Both Sides
Correct: C (\(x = 7\))
Step 1 (Distribute on the left side): \(5(x - 2) = 5x - 10\). Equation: \(5x - 10 = 2x + 11\).
Step 2 (Collect variables on one side): Subtract \(2x\) from both sides: \(3x - 10 = 11\).
Step 3 (Solve): Add 10: \(3x = 21 \implies x = \frac{21}{3} = 7\).
Common Student Pitfall: Incomplete distribution (\(5x - 2 = 2x + 11\)) or combining terms with opposite sign errors.
7 Solving One-Variable Inequalities
Correct: A (\(x < 5\))
Step 1 (Subtract constant): \(3x + 7 < 22 \implies 3x < 22 - 7 \implies 3x < 15\).
Step 2 (Divide by positive coefficient): \(x < \frac{15}{3} \implies x < 5\).
Common Student Pitfall: Flipping the inequality symbol when dividing by a positive number (\(x > 5\), distractor B). Reinforce that the sign only reverses when multiplying or dividing by a negative number.
TSIA2 Foundations Solutions Guide (Score 920–945) Page 2 of 5
Teacher Solutions Guide
Verbal Translations, Coordinate Lines & Quadratic Roots
Questions 8 – 11
8 Translating Words to Algebraic Equations
Correct: B (\(3n - 7 = 41;\; n = 16\))
Step 1 (Parse syntax): "Three times a number \(n\)" = \(3n\). "Seven less than..." subtracts 7 from \(3n\): \(3n - 7\).
Step 2 (Set up equation and solve): \(3n - 7 = 41 \implies 3n = 48 \implies n = 16\).
Common Student Pitfall: Translating "seven less than" literally in reading order as \(7 - 3n = 41\) (distractor A). Remind students that "less than" reverses the operand order.
9 Slope-Intercept Form (\(y = mx + b\))
Correct: B (\(y = -\frac{3}{2}x + 4\))
Step 1 (Identify parameters): Slope \(m = -\frac{3}{2}\). Point \((0, 4)\) is the \(y\)-intercept, so \(b = 4\).
Step 2 (Formulate equation): \(y = mx + b \implies y = -\frac{3}{2}x + 4\).
Common Student Pitfall: Swapping the slope and the intercept (\(y = 4x - \frac{3}{2}\), distractor C) or omitting the negative sign on the slope.
10 Evaluating Linear Functions with Negatives
Correct: D (20)
Step 1 (Substitute \(x = -4\)): \(f(-4) = -3(-4) + 8\).
Step 2 (Multiply negative numbers): \(-3 \times (-4) = +12\).
Step 3 (Add constant): \(12 + 8 = 20\).
Common Student Pitfall: Multiplying \(-3\) by \(-4\) and retaining a negative sign (\(-12 + 8 = -4\), distractor B). Emphasize that negative times negative equals positive.
11 Solving Quadratics by Factoring (\(a = 1\))
Correct: B (\(x = 7\) and \(x = -2\))
Step 1 (Find factors of \(-14\) that sum to \(-5\)): The factors are \(-7\) and \(+2\), since \((-7)(2) = -14\) and \(-7 + 2 = -5\).
Step 2 (Factored form): \((x - 7)(x + 2) = 0\).
Step 3 (Zero product property): \(x - 7 = 0 \implies x = 7\); \(x + 2 = 0 \implies x = -2\).
Common Student Pitfall: Choosing the factor numbers without inverting their signs for the actual roots (yielding distractor A: \(x = -7\) and \(x = 2\)).
TSIA2 Foundations Solutions Guide (Score 920–945) Page 3 of 5
Teacher Solutions Guide
Perimeter, Area, Pythagorean Theorem & Prism Volume
Questions 12 – 15
12 Perimeter and Area of Rectangles
Correct: A (Area = 120 sq ft; Perim = 46 ft)
Step 1 (Calculate area): \(\text{Area} = \text{length} \times \text{width} = 15 \times 8 = 120\text{ square feet}\).
Step 2 (Calculate perimeter): \(\text{Perimeter} = 2(15) + 2(8) = 30 + 16 = 46\text{ linear feet}\).
Common Student Pitfall: Adding only two adjacent sides for perimeter (\(15 + 8 = 23\), distractor B) or reversing the numerical answers for area and perimeter.
13 Pythagorean Theorem for Missing Leg
Correct: C (12 feet)
Step 1 (Identify triangle components): Hypotenuse \(c = 13\text{ ft}\), base leg \(a = 5\text{ ft}\), vertical leg \(h\).
Step 2 (Apply theorem): \(h^2 + 5^2 = 13^2 \implies h^2 + 25 = 169\).
Step 3 (Solve): \(h^2 = 169 - 25 = 144 \implies h = \sqrt{144} = 12\text{ feet}\).
Common Student Pitfall: Adding the squares (\(169 + 25 = 194\)) as if solving for the hypotenuse rather than a missing leg.
14 Volume of Rectangular Prisms
Correct: C (84 cubic feet)
Step 1 (Formula): \(V = \text{length} \times \text{width} \times \text{height}\).
Step 2 (Substitute values): \(V = 6 \times 4 \times 3.5 = 24 \times 3.5\).
Step 3 (Arithmetic): \(24 \times 3 = 72\); \(24 \times 0.5 = 12\); \(72 + 12 = 84\text{ cubic feet}\).
Common Student Pitfall: Decimal multiplication errors, such as dropping the \(0.5\) factor to get \(24 \times 3 = 72\) (distractor B).
15 Supplementary Angle Relationships
Correct: C (\(115^\circ\))
Step 1 (Definition): Supplementary angles sum to \(180^\circ\).
Step 2 (Calculate): \(\angle B = 180^\circ - 65^\circ = 115^\circ\).
Common Student Pitfall: Confusing supplementary with complementary angles (\(90^\circ - 65^\circ = 25^\circ\), distractor A). Mnemonic reminder: "C" comes before "S"; 90 comes before 180.
TSIA2 Foundations Solutions Guide (Score 920–945) Page 4 of 5
Teacher Solutions Guide
Triangle Area, Statistics, Probability & Placement
Questions 16 – 20
Q16: Area of a Triangle Correct: B (42 sq inches)
\(\text{Area} = \frac{1}{2}bh = \frac{1}{2}(12)(7) = 6 \times 7 = 42\text{ sq in}\). Pitfall: Omitting the \(\frac{1}{2}\) factor produces \(12 \times 7 = 84\) (distractor C).
Q17: Median and Range of a List Correct: B (Median = 6.5; Range = 6)
Order values: \(3, 4, 6, 7, 7, 9\). Median is the average of 3rd and 4th terms: \(\frac{6 + 7}{2} = 6.5\). Range = \(9 - 3 = 6\). Pitfall: Finding the median without sorting the list first gives incorrect values.
Q18: Complementary Event Probability Correct: D (\(\frac{3}{4}\))
Total marbles = \(8 + 5 + 7 = 20\). Non-red marbles = \(8 + 7 = 15\). Probability = \(\frac{15}{20} = \frac{3}{4}\). Pitfall: Calculating the probability of drawing red (\(\frac{5}{20} = \frac{1}{4}\), distractor A).
Q19: Calculating Percent from Survey Frequency Correct: C (52%)
Combined count = \(8\text{ (bike)} + 18\text{ (bus)} = 26\). Total = \(50\). Percent = \(\frac{26}{50} = \frac{52}{100} = 52\%\). Pitfall: Calculating only bus percentage (\(\frac{18}{50} = 36\%\), distractor A).
Q20: Proportional Sample Population Estimate Correct: C (320 fish)
\(\frac{15\text{ tagged in sample}}{120\text{ total sample}} = \frac{40\text{ tagged in lake}}{N\text{ total population}}\). Simplifying: \(\frac{1}{8} = \frac{40}{N} \implies N = 40 \times 8 = 320\text{ fish}\).
TSIA2 Foundations Placement & Advancement Rubric
Foundational Mastery: Score \(\ge 14/20\)
Below 920 Band (< 10)
Needs intensive review in basic integer arithmetic, fraction multiplication, order of operations, and two-step inverse operations before retaking.
Secure 920–945 Band (10–13)
Demonstrates command of foundational algebra and geometry. Ready for targeted transition into rational factoring and system substitution.
Ready for 950+ Packet (14–20)
Has mastered developmental benchmarks. Advance immediately to the TSIA2 Mastery Practice Packet (CRC 950+ college-ready assessment).
TSIA2 Foundations Solutions Guide (Score 920–945) Page 5 of 5
TSIA2 Core Practice Packet Texas Success Initiative Assessment 2.0
Mathematics Core Skills Practice Packet
Target Placement Level Score Range 900 – 920
Student Name
Date
Class / Period
Score / 20
Core Skills Reference Guide Diagnostic Levels 1, 2 & 3 Preparation
Perimeter: Rect.: \(P = 2l + 2w\)
Area: Rect.: \(A = l \times w\)
Circle Basics: Radius: \(r = \frac{d}{2}\)
Triangle Angles: Sum \(= 180^\circ\)
Part I: Quantitative Arithmetic Questions 1 & 2
1 Operations with Signed Integers
Workspace
What is the value of the arithmetic expression \(-18 - (-7) + (-12)\)?
A. \(-37\)
B. \(-23\)
C. \(-1\)
D. \(13\)
2 Percent of a Number
Workspace
A restaurant meal check totals \(\$45.00\). If a customer decides to leave a \(15\%\) tip for the food server, what is the exact amount of the tip?
A. \(\$3.00\)
B. \(\$6.75\)
C. \(\$7.50\)
D. \(\$9.00\)
TSIA2 Core Skills Practice Packet (Score 900–920) Page 1 of 7
Part I: Quantitative Arithmetic
Fraction Addition, Decimals & Uniform Rates
Questions 3 – 5
3 Adding Unlike Fractions
Workspace
What is the sum of \(\frac{3}{8} + \frac{1}{4}\), expressed in simplest form?
A. \(\frac{1}{3}\)
B. \(\frac{4}{12}\)
C. \(\frac{5}{8}\)
D. \(\frac{7}{8}\)
4 Decimal Arithmetic with Money
Workspace
Marcus has a \(\$50.00\) bill. At a school supply store, he purchases three spiral notebooks for \(\$4.25\) each and a backpack for \(\$24.50\). How much cash change should Marcus receive?
A. \(\$11.25\)
B. \(\$12.75\)
C. \(\$13.50\)
D. \(\$21.25\)
5 Constant Speed & Distance
Workspace
A motorist drives along an interstate highway at a constant speed of \(55\) miles per hour. How many total miles will the motorist travel in \(3.5\) hours?
A. 165.0 miles
B. 182.5 miles
C. 192.5 miles
D. 220.0 miles
TSIA2 Core Skills Practice Packet (Score 900–920) Page 2 of 7
Part II: Algebraic Foundations
One-Step Equations & Combining Like Terms
Questions 6 – 8
6 One-Step Linear Equations (Subtraction)
Workspace
Solve for the variable \(y\): \(y - 17 = 42\).
A. \(y = 25\)
B. \(y = 35\)
C. \(y = 59\)
D. \(y = 69\)
7 One-Step Linear Equations (Division)
Workspace
Solve for \(k\): \(\frac{k}{6} = 14\).
A. \(k = 2.33\)
B. \(k = 20\)
C. \(k = 74\)
D. \(k = 84\)
8 Combining Like Terms
Workspace
Which algebraic expression is completely simplified and equivalent to \(5x + 8 - 2x + 3\)?
A. \(3x + 11\)
B. \(7x + 11\)
C. \(3x + 5\)
D. \(14x\)
TSIA2 Core Skills Practice Packet (Score 900–920) Page 3 of 7
Part II: Algebraic Foundations
Distributive Property, Expressions & Coordinates
Questions 9 – 11
9 Distributive Property
Workspace
Which expression represents the expanded form of \(4(2a - 5)\)?
A. \(6a - 9\)
B. \(8a - 5\)
C. \(8a - 20\)
D. \(8a + 20\)
10 Evaluating Multi-Variable Expressions
Workspace
If \(x = 5\) and \(y = 3\), what is the value of the algebraic expression \(4x - 2y + 7\)?
A. 19
B. 21
C. 27
D. 33
11 Coordinate Plane Quadrants
Workspace
In which quadrant of the standard coordinate plane is the ordered point \((-4, 6)\) located?
A. Quadrant I
B. Quadrant II
C. Quadrant III
D. Quadrant IV
TSIA2 Core Skills Practice Packet (Score 900–920) Page 4 of 7
Part III: Geometric Reasoning
Perimeter of Polygons, Square Area & Circle Parts
Questions 12 – 14
12 Perimeter of a Triangle
Workspace
A triangular community dog park has three sides measuring \(34\) feet, \(28\) feet, and \(42\) feet. What is the total perimeter of the dog park?
A. 94 feet
B. 104 feet
C. 114 feet
D. 120 feet
13 Area of a Square
Workspace
A square ceramic floor tile measures \(12\) inches on each side. What is the total surface area of the tile in square inches?
A. 24 sq inches
B. 48 sq inches
C. 120 sq inches
D. 144 sq inches
14 Circle Diameter and Radius
Workspace
The diameter of a circular outdoor swimming pool measures \(18\) feet across its center. What is the radius of the pool?
A. 6 feet
B. 9 feet
C. 36 feet
D. 54 feet
TSIA2 Core Skills Practice Packet (Score 900–920) Page 5 of 7
Parts III & IV: Geometry & Data
Rectangle Area, Angle Sums & Mean Averages
Questions 15 – 17
15 Area of a Rectangle
Workspace
A rectangular bedroom has a length of \(14\) feet and a width of \(11\) feet. What is the total floor area of the bedroom in square feet?
A. 50 sq ft
B. 144 sq ft
C. 154 sq ft
D. 165 sq ft
16 Interior Angles of a Triangle
Workspace
In triangle \(\triangle ABC\), the measure of \(\angle A\) is \(50^\circ\) and the measure of \(\angle B\) is \(75^\circ\). What is the measure of the third angle, \(\angle C\)?
A. \(45^\circ\)
B. \(55^\circ\)
C. \(65^\circ\)
D. \(125^\circ\)
17 Calculating the Mean (Average)
Workspace
Four students earned quiz scores of \(78,\; 85,\; 92,\) and \(85\). What is the mean (average) score of the four students?
A. 83
B. 84
C. 85
D. 86
TSIA2 Core Skills Practice Packet (Score 900–920) Page 6 of 7
Part IV: Statistics & Probability
Mode, Chance, Survey Counts & Final Bubble Sheet
Questions 18 – 20
18 Finding the Mode of a Dataset
Daily high temperatures (in \(^\circ\text{F}\)) recorded in Dallas over one week were: \(72,\; 75,\; 68,\; 75,\; 78,\; 71,\; 75\). What is the mode of these recorded temperatures?
A. \(68^\circ\text{F}\)
B. \(72^\circ\text{F}\)
C. \(75^\circ\text{F}\)
D. \(78^\circ\text{F}\)
19 Probability of a Simple Event
A standard fair six-sided die with faces numbered \(1\) through \(6\) is rolled once. What is the probability that the number rolled is an even number?
A. \(\frac{1}{6}\)
B. \(\frac{1}{3}\)
C. \(\frac{1}{2}\)
D. \(\frac{2}{3}\)
20 Survey Data Reading & Fraction of Total
Students were surveyed regarding cafeteria choices: Pizza (24), Tacos (18), Chicken (12), Salad (6). What is the total count, and what fraction chose Tacos?
Tacos = 18
A. Total = 50 students; Fraction = \(\frac{18}{50}\)
B. Total = 60 students; Fraction = \(\frac{3}{10}\)
C. Total = 60 students; Fraction = \(\frac{1}{4}\)
D. Total = 70 students; Fraction = \(\frac{2}{7}\)
TSIA2 Core Examination Sheet
Student Answer Record & Skills Diagnostic
Fill bubble:
(A)(B)(C)(D) 06. (A)(B)(C)(D)
(A)(B)(C)(D) 07. (A)(B)(C)(D)
(A)(B)(C)(D) 08. (A)(B)(C)(D)
(A)(B)(C)(D) 09. (A)(B)(C)(D)
(A)(B)(C)(D) 10. (A)(B)(C)(D)
(A)(B)(C)(D) 16. (A)(B)(C)(D)
(A)(B)(C)(D) 17. (A)(B)(C)(D)
(A)(B)(C)(D) 18. (A)(B)(C)(D)
(A)(B)(C)(D) 19. (A)(B)(C)(D)
(A)(B)(C)(D) 20. (A)(B)(C)(D)
Quantitative Q1–Q5: ___ / 5
Algebraic Q6–Q11: ___ / 6
Geometric Q12–Q16: ___ / 5
Statistical Q17–Q20: ___ / 4
TSIA2 Core Skills Practice Packet (Score 900–920) Page 7 of 7
TSIA2 Core Teacher Guide Teacher Instructional Resource
TSIA2 Core Skills Solutions & Scoring Guide
Foundational Target Score Range 900 – 920
Core Skills Quick-Scan Master Key 20 Multiple Choice Questions
Q01B
Q02B
Q03C
Q04B
Q05C
Q06C
Q07D
Q08A
Q09C
Q10B
Q11B
Q12B
Q13D
Q14B
Q15C
Q16B
Q17C
Q18C
Q19C
Q20B
Solutions: Quantitative Arithmetic (Questions 1 – 3)
1 Signed Integer Operations
Correct: B (\(-23\))
Step 1 (Double negative to positive): \(-18 - (-7) = -18 + 7 = -11\).
Step 2 (Add negative integer): \(-11 + (-12) = -11 - 12 = -23\).
Common Student Error: Treating \(-(-7)\) as subtraction (\(-18 - 7 - 12 = -37\), distractor A). Review the rule that subtracting a negative number is equivalent to adding.
2 Calculating Percent of a Number
Correct: B (\$6.75)
Step 1 (Convert percent to decimal): \(15\% = \frac{15}{100} = 0.15\).
Step 2 (Multiply by base amount): \(\$45.00 \times 0.15 = \$6.75\). (Mental math strategy: \(10\%\) of \(\$45 = \$4.50\); \(5\%\) of \(\$45 = \$2.25\); \(\$4.50 + \$2.25 = \$6.75\)).
Common Student Error: Misplacing the decimal point during multiplication or guessing without computing the fraction of 45.
3 Adding Fractions with Unlike Denominators
Correct: C (\(\frac{5}{8}\))
Step 1 (Find least common denominator): The LCD of 8 and 4 is 8.
Step 2 (Convert fraction): \(\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}\).
Step 3 (Add numerators): \(\frac{3}{8} + \frac{2}{8} = \frac{5}{8}\).
Common Student Error: Adding numerators and denominators straight across: \(\frac{3 + 1}{8 + 4} = \frac{4}{12} = \frac{1}{3}\) (distractors A and B).
TSIA2 Core Skills Solutions Guide (Score 900–920) Page 1 of 5
Teacher Solutions Guide
Decimals, Distance Rates & One-Step Equations
Questions 4 – 7
4 Multi-Step Decimal Arithmetic
Correct: B (\$12.75)
Step 1 (Calculate item costs): Notebooks = \(3 \times \$4.25 = \$12.75\). Backpack = \(\$24.50\).
Step 2 (Sum total spending): \(\$12.75 + \$24.50 = \$37.25\).
Step 3 (Calculate change from $50): \(\$50.00 - \$37.25 = \$12.75\).
Common Student Error: Forgetting to multiply \(\$4.25\) by 3, resulting in \(\$50.00 - (\$4.25 + \$24.50) = \$21.25\) (distractor D).
5 Speed, Time & Distance (\(d = rt\))
Correct: C (192.5 miles)
Step 1 (Formula): \(\text{Distance} = \text{rate} \times \text{time}\).
Step 2 (Calculate): \(55\text{ mph} \times 3.5\text{ hours} = 55 \times 3 + 55 \times 0.5 = 165 + 27.5 = 192.5\text{ miles}\).
Common Student Error: Multiplying only by 3 hours (\(55 \times 3 = 165\), distractor A) or miscomputing the half-hour fraction.
6 One-Step Equation (Inverse Addition)
Correct: C (\(y = 59\))
Step 1 (Apply inverse operation): Since 17 is subtracted, add 17 to both sides of the equation.
Step 2 (Evaluate): \(y = 42 + 17 = 59\).
Common Student Error: Subtracting 17 from 42 instead of adding: \(42 - 17 = 25\) (distractor A).
7 One-Step Equation (Inverse Multiplication)
Correct: D (\(k = 84\))
Step 1 (Apply inverse operation): Since \(k\) is divided by 6, multiply both sides by 6.
Step 2 (Evaluate): \(k = 14 \times 6 = 84\).
Common Student Error: Dividing 14 by 6 (\(k \approx 2.33\), distractor A) or adding 6 (\(k = 20\), distractor B).
TSIA2 Core Skills Solutions Guide (Score 900–920) Page 2 of 5
Teacher Solutions Guide
Like Terms, Distribution & Coordinate Planes
Questions 8 – 11
8 Combining Like Terms
Correct: A (\(3x + 11\))
Step 1 (Group variable terms): \(5x - 2x = 3x\).
Step 2 (Group constant terms): \(+8 + 3 = +11\).
Step 3 (Combine): \(3x + 11\).
Common Student Error: Adding all numerical values regardless of whether they have variables (\(5 + 8 - 2 + 3 = 14x\), distractor D).
9 Distributive Property Expansion
Correct: C (\(8a - 20\))
Step 1 (Distribute 4 to the first term): \(4 \times 2a = 8a\).
Step 2 (Distribute 4 to the second term): \(4 \times (-5) = -20\).
Step 3 (Combine): \(8a - 20\).
Common Student Error: Multiplying only the first term and not distributing to the constant: \(8a - 5\) (distractor B).
10 Evaluating Expressions with Two Variables
Correct: B (21)
Step 1 (Substitute \(x = 5\) and \(y = 3\)): \(4(5) - 2(3) + 7\).
Step 2 (Perform multiplications first): \(20 - 6 + 7\).
Step 3 (Add and subtract left to right): \(20 - 6 = 14\); \(14 + 7 = 21\).
Common Student Error: Adding \(6 + 7 = 13\) first, then evaluating \(20 - 13 = 7\). Remind students that addition and subtraction have equal precedence left-to-right.
11 Identifying Coordinate Quadrants
Correct: B (Quadrant II)
Step 1 (Sign conventions): Point \((-4, 6)\) has negative \(x\) and positive \(y\): \((-, +)\).
Step 2 (Locate quadrant): Quadrant I: \((+, +)\); Quadrant II: \((-, +)\); Quadrant III: \((-, -)\); Quadrant IV: \((+, -)\). Therefore, \((-4, 6)\) lies in Quadrant II.
Common Student Error: Counting quadrants clockwise instead of counterclockwise (confusing Quadrant II with Quadrant IV).
TSIA2 Core Skills Solutions Guide (Score 900–920) Page 3 of 5
Teacher Solutions Guide
Triangle Perimeter, Tile Area & Circle Geometry
Questions 12 – 15
12 Perimeter of a Triangle
Correct: B (104 feet)
Step 1 (Definition): Perimeter is the total distance around the outside boundary (sum of all three side lengths).
Step 2 (Addition): \(34 + 28 + 42 = 104\text{ feet}\).
Common Student Error: Regrouping errors in multi-digit vertical addition (e.g., getting 94 or 114).
13 Area of a Square
Correct: D (144 sq inches)
Step 1 (Formula): \(\text{Area} = \text{side} \times \text{side} = s^2\).
Step 2 (Evaluate): \(12 \times 12 = 144\text{ square inches}\).
Common Student Error: Calculating perimeter (\(4 \times 12 = 48\), distractor B) or multiplying by 2 (\(2 \times 12 = 24\), distractor A) instead of squaring the side length.
14 Radius from Diameter
Correct: B (9 feet)
Step 1 (Relationship): The radius \(r\) is half the diameter \(d\): \(r = \frac{d}{2}\).
Step 2 (Calculate): \(r = \frac{18}{2} = 9\text{ feet}\).
Common Student Error: Doubling the diameter instead of halving it (\(18 \times 2 = 36\), distractor C).
15 Area of a Rectangle
Correct: C (154 sq ft)
Step 1 (Formula): \(\text{Area} = \text{length} \times \text{width}\).
Step 2 (Evaluate): \(14 \times 11 = 14 \times 10 + 14 \times 1 = 140 + 14 = 154\text{ square feet}\).
Common Student Error: Calculating perimeter (\(2(14 + 11) = 50\), distractor A) instead of multiplying the two dimensions.
TSIA2 Core Skills Solutions Guide (Score 900–920) Page 4 of 5
Teacher Solutions Guide
Angle Sums, Mean, Mode, Probability & Progression
Questions 16 – 20
Q16: Interior Angles of a Triangle Correct: B (\(55^\circ\))
Triangle angles sum to \(180^\circ\). Sum of known angles: \(50^\circ + 75^\circ = 125^\circ\). Unknown angle \(\angle C = 180^\circ - 125^\circ = 55^\circ\). Error: Subtracting from \(180^\circ\) without summing first.
Q17: Mean (Average) Calculation Correct: C (85)
Sum of scores = \(78 + 85 + 92 + 85 = 340\). Divide by count (4): \(\frac{340}{4} = 85\).
Q18: Identifying the Mode Correct: C (\(75^\circ\text{F}\))
The mode is the most frequently occurring value. In the set \(\{72, 75, 68, 75, 78, 71, 75\}\), \(75\) appears 3 times, while all other numbers appear once.
Q19: Basic Dice Probability Correct: C (\(\frac{1}{2}\))
Even outcomes on a six-sided die are \(\{2, 4, 6\}\) (3 favorable outcomes). Total possible outcomes = \(6\). Probability = \(\frac{3}{6} = \frac{1}{2}\).
Q20: Survey Frequency & Fraction of Total Correct: B (Total = 60; \(\frac{3}{10}\))
Total count = \(24 + 18 + 12 + 6 = 60\). Tacos fraction = \(\frac{18}{60} = \frac{18 \div 6}{60 \div 6} = \frac{3}{10}\).
TSIA2 Core Skills Placement & Progression Rubric
Core Mastery: Score \(\ge 14/20\)
Emerging Core (< 10)
Provide targeted manipulatives and integer number line practice. Review place values, basic fact fluency, and perimeter vs area distinction.
Proficient Core (10–13)
Demonstrates command of basic integer arithmetic and one-step equations. Begin introducing two-step equations and ratio proportions.
Ready for 920–945 (14–20)
Has mastered foundational pre-algebra. Advance immediately to the TSIA2 Foundations Practice Packet (score band 920–945).
TSIA2 Core Skills Solutions Guide (Score 900–920) Page 5 of 5
TSIA2 Form B Practice Packet Texas Success Initiative Assessment 2.0 Form B
Mathematics Benchmark Practice Packet
College Benchmark CRC 950+ or Diagnostic 6
Student Name
Test Date
Class / Period
Raw Score / 20
TSIA2 Mathematics Formula Reference Calculators strictly permitted on designated items only
Slope & Line: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
Geometry & Right \(\triangle\): \(a^2 + b^2 = c^2\)
Solids & Circles: Cylinder: \(V = \pi r^2 h\)
Section 1: Quantitative Reasoning Questions 1 & 2
1 Unit Rates & Dimensional Analysis
Workspace
An agricultural irrigation system injects micronutrient solution at a constant rate of 1.75 milligrams per liter of treated irrigation water. If the system pumps \(4.0 \times 10^5\) liters of water per day, how many kilograms of the micronutrient are needed for a 20-day crop watering cycle? (Note: \(1\text{ kg} = 10^6\text{ mg}\))
A. 1.4 kg
B. 14 kg
C. 28 kg
D. 140 kg
2 Percentages & Successive Price Adjustments
Workspace
A retail store purchased a commercial display unit at a wholesale cost of \(\$500\). The store marked up the wholesale price by \(30\%\). During a seasonal clearance event, this marked-up price was reduced by a \(20\%\) discount. At checkout, a buyer paid a \(7\%\) sales tax on the discounted purchase price. What was the total amount the buyer paid?
A. \(\$535.00\)
B. \(\$546.80\)
C. \(\$556.40\)
D. \(\$585.00\)
TSIA2 Mathematics Practice Packet • Form B Page 1 of 7
Sections 1 & 2: Quantitative & Algebraic Foundations Form B
Mixtures, Modeling & Linear Inequalities
Questions 3 – 5
3 Mixture Problems & Concentrations
Workspace
A biomedical laboratory technician blends a \(12\%\) saline solution with a \(30\%\) saline solution to prepare exactly 60 liters of a \(24\%\) saline solution. How many liters of the \(30\%\) saline solution must the technician use?
A. 20 liters
B. 35 liters
C. 40 liters
D. 45 liters
4 Linear Functions from Tabular Data
Workspace
The table displays the total production charge \(C(n)\), in dollars, for manufacturing \(n\) embroidered team jackets. Assuming the cost increases at a constant rate, which equation models \(C(n)\)?
A. \(C(n) = 9.6n\)
B. \(C(n) = 8n + 20\)
C. \(C(n) = 7n + 32\)
D. \(C(n) = 10n - 4\)
5 Multi-Step Linear Inequalities
Workspace
Which inequality represents the complete solution set for \(-\frac{3}{5}(10x - 15) \ge 3x - 18\)?
A. \(x \ge 3\)
B. \(x \le 3\)
C. \(x \le -3\)
D. \(x \ge -1\)
TSIA2 Mathematics Practice Packet • Form B Page 2 of 7
Section 2: Algebraic Reasoning Form B
Linear Systems, Literal Formulas & Quadratic Roots
Questions 6 – 8
6 Systems of Two Linear Equations
Workspace
A high school theater department sold 280 total tickets for the opening night performance of a musical, taking in \(\$2,760\) in revenue. General admission tickets were \(\$12\) each, and student admission tickets were \(\$7\) each. Exactly how many general admission tickets were sold?
A. 120 tickets
B. 140 tickets
C. 160 tickets
D. 180 tickets
7 Rearranging Literal Equations
Workspace
The illumination intensity produced on a projection screen by a calibrated spotlight is given by \(I = \frac{L^2 k}{(d + h)^2}\). If \(I\), \(L\), \(k\), \(d\), and \(h\) are all positive real numbers, which equation correctly expresses \(L\) in terms of \(I\), \(k\), \(d\), and \(h\)?
A. \(L = \frac{(d + h)\sqrt{I}}{k}\)
B. \(L = (d + h)\sqrt{\frac{I}{k}}\)
C. \(L = \frac{I(d + h)}{\sqrt{k}}\)
D. \(L = \sqrt{I(d + h)^2 - k}\)
8 Solving Quadratic Equations by Factoring
Workspace
What are all the real solutions to the quadratic equation \(2x^2 - 9x - 18 = 0\)?
A. \(x = -\frac{3}{2}\) and \(x = 6\)
B. \(x = \frac{3}{2}\) and \(x = -6\)
C. \(x = -2\) and \(x = \frac{9}{2}\)
D. \(x = -\frac{2}{3}\) and \(x = 6\)
TSIA2 Mathematics Practice Packet • Form B Page 3 of 7
Section 3: Advanced Algebraic Reasoning Form B
Rational Expressions, Function Composition & Vertices
Questions 9 – 11
9 Simplifying Rational Algebraic Expressions
Workspace
Which expression is equivalent to \(\frac{3x^2 - x - 2}{9x^2 - 4} \div \frac{x^2 - x}{3x^2 + 2x}\) for all real values of \(x\) where the expression is defined?
A. \(\frac{x - 1}{x + 1}\)
B. \(\frac{3x + 2}{3x - 2}\)
C. \(\frac{3x - 2}{3x + 2}\)
D. \(\frac{x + 2}{3x - 2}\)
10 Function Notation & Composition
Workspace
Given the functions \(f(x) = 3x^2 - 2x + 4\) and \(g(x) = 2x - 1\), what is the value of the difference \(f(g(2)) - g(f(2))\)?
A. \(-2\)
B. 2
C. 6
D. 14
11 Quadratic Vertex & Maximum Profit
Workspace
A software startup models its weekly operating profit by \(P(x) = -4x^2 + 240x - 1600\), where \(x\) is the selling price per subscription tier in dollars. What subscription price \(x\) maximizes the company's weekly profit, and what is the maximum profit?
A. Price \(\$25\); Max Profit \(\$1,900\)
B. Price \(\$30\); Max Profit \(\$2,000\)
C. Price \(\$35\); Max Profit \(\$1,900\)
D. Price \(\$30\); Max Profit \(\$2,400\)
TSIA2 Mathematics Practice Packet • Form B Page 4 of 7
Section 4: Geometric & Spatial Reasoning Form B
Coordinate Geometry, Right Triangles & Composite Area
Questions 12 – 14
12 Perpendicular Lines & Coordinate Intercepts
Workspace
In the \(xy\)-plane, line \(L_1\) passes through points \((-2, 7)\) and \((2, -1)\). Line \(L_2\) is perpendicular to \(L_1\) and passes through the point \((6, 1)\). What is the \(y\)-intercept of line \(L_2\)?
A. \(-2\)
B. \(-\frac{3}{2}\)
C. \(1\)
D. \(3\)
13 Pythagorean Theorem & Right Triangles
Workspace
A telecommunications guy wire is anchored to the level ground at a point 45 feet away from the base of a cellular transmission tower. The wire fastens to the tower at an elevation of 60 feet. What is the total length of the taut guy wire from the ground anchor to the tower fastening point?
A. 68 feet
B. 72 feet
C. 75 feet
D. 80 feet
14 Composite Geometric Area
Workspace
A community park playground surface consists of a rectangle adjoining a semicircle along one of its shorter sides. The rectangle measures 24 meters long and 10 meters wide. The adjoining semicircle has a diameter equal to the 10-meter width. Which value is closest to the total surface area of the playground? (Use \(\pi \approx 3.14\))
A. \(265\text{ m}^2\)
B. \(279\text{ m}^2\)
C. \(319\text{ m}^2\)
D. \(397\text{ m}^2\)
TSIA2 Mathematics Practice Packet • Form B Page 5 of 7
Sections 4 & 5: Geometry & Data Analysis Form B
Solid Geometry, Proportionality & Distribution Measures
Questions 15 – 17
15 3D Volume Scaling & Dimensions
Workspace
Right circular Cylinder A has radius \(r\) and height \(h\). Cylinder B has a radius that is \(25\%\) greater than Cylinder A and a height that is \(20\%\) less than Cylinder A. What is the ratio of the volume of Cylinder B to the volume of Cylinder A?
A. \(\frac{21}{20}\)
B. \(\frac{6}{5}\)
C. \(\frac{5}{4}\)
D. \(\frac{3}{2}\)
16 Similar Triangles & Proportional Sides
Workspace
In \(\triangle ABC\), segment \(DE\) is parallel to base \(BC\), with point \(D\) located on side \(AB\) and point \(E\) located on side \(AC\). If \(AD = 9\), \(DB = 6\), and the base \(BC = 25\), what is the length of segment \(DE\)?
A. 12.5
B. 15.0
C. 16.5
D. 18.0
17 Central Tendency & Impact of Outliers
Workspace
An initial sample of 7 laboratory measurements is: \(62, 65, 68, 68, 71, 74, 76\). An 8th observation is recorded with a value of \(18\). Which statement correctly describes the resulting change in the dataset's summary statistics?
A. Both the mean and median decrease by \(\approx 6\) points.
B. The mean decreases by \(> 6\) points; median is unchanged.
C. The median decreases, but the mean resists the outlier.
D. The interquartile range triples while the mean stays constant.
TSIA2 Mathematics Practice Packet • Form B Page 6 of 7
Section 5: Probabilistic & Statistical Reasoning Form B
Tables, Probability, Regression & Final Scoring
Questions 18 – 20
18 Two-Way Table Conditional Probability
Given that a randomly selected student prefers evening study sessions, what is the probability that they study primarily at a coffee shop?
A. \(\frac{47}{125}\)
B. \(\frac{47}{65}\)
C. \(\frac{47}{80}\)
D. \(\frac{33}{80}\)
19 Compound Probability without Replacement
A storage crate holds 16 precision drone propellers: 12 are perfectly balanced and 4 have minor calibration flaws. An assembly robot selects 2 propellers at random one after another without replacement. What is the probability that at least one of the selected propellers has a calibration flaw?
A. \(\frac{1}{4}\)
B. \(\frac{7}{20}\)
C. \(\frac{9}{20}\)
D. \(\frac{1}{2}\)
20 Linear Regression & Residual Analysis
A least-squares linear regression line modeling hours of weekly exam review \(x\) and final assessment score \(y\) is \(\hat{y} = 5.2x + 58.0\). A student who logged 5.0 hours of review earned an actual score of 88. What is this student's residual, and how is it interpreted?
A. \(-4.0\); the student scored 4.0 points lower than model.
B. \(+4.0\); the student scored 4.0 points higher than model.
C. \(+5.2\); the student scored 5.2 points higher than model.
D. \(-5.2\); the student scored 5.2 points lower than model.
Official Examination Document • Form B
Student Answer Record & Diagnostic Summary
Fill bubble completely:
(A)(B)(C)(D) 06. (A)(B)(C)(D)
(A)(B)(C)(D) 07. (A)(B)(C)(D)
(A)(B)(C)(D) 08. (A)(B)(C)(D)
(A)(B)(C)(D) 09. (A)(B)(C)(D)
(A)(B)(C)(D) 10. (A)(B)(C)(D)
(A)(B)(C)(D) 16. (A)(B)(C)(D)
(A)(B)(C)(D) 17. (A)(B)(C)(D)
(A)(B)(C)(D) 18. (A)(B)(C)(D)
(A)(B)(C)(D) 19. (A)(B)(C)(D)
(A)(B)(C)(D) 20. (A)(B)(C)(D)
Quantitative Q1–Q4: ___ / 4
Algebraic Q5–Q11: ___ / 7
Geometric Q12–Q16: ___ / 5
Statistical Q17–Q20: ___ / 4
TSIA2 Mathematics Practice Packet • Form B Page 7 of 7
TSIA2 Form B Teacher Guide Instructor Assessment Resource Form B Key
TSIA2 Solutions & Diagnostic Guide (Form B)
College Readiness Target 14+ / 20 (70% Benchmark)
Form B Quick-Scan Master Key 20 Multiple Choice Questions
Q01B
Q02C
Q03C
Q04B
Q05B
Q06C
Q07B
Q08A
Q09B
Q10B
Q11B
Q12A
Q13C
Q14B
Q15C
Q16B
Q17B
Q18C
Q19C
Q20B
Detailed Solutions: Quantitative Reasoning (Questions 1 – 3)
1 Unit Rates & Scientific Notation
Correct: B (14 kg)
Step 1 (Daily chemical mass): Multiply daily volume by injection rate: \(4.0 \times 10^5\text{ L/day} \times 1.75\text{ mg/L} = 7.0 \times 10^5\text{ mg/day}\).
Step 2 (20-day cycle requirement): Multiply daily mass by 20 days: \(20 \times 7.0 \times 10^5\text{ mg} = 1.4 \times 10^7\text{ mg}\).
Step 3 (Convert to kilograms): Divide by \(10^6\text{ mg/kg}\): \(\frac{1.4 \times 10^7}{10^6} = 14\text{ kg}\).
Common Student Trap: Miscounting decimal places when converting scientific notation gives distractor A (1.4 kg). Forgetting the 20-day factor leads to options under 1 kg.
2 Percentages & Successive Price Changes
Correct: C ($556.40)
Step 1 (Marked-up retail price): Wholesale \(\$500\) increased by \(30\%\): \(\$500 \times 1.30 = \$650.00\).
Step 2 (Sale discount): \(20\%\) discount on \(\$650\): \(\$650 \times (1 - 0.20) = \$650 \times 0.80 = \$520.00\).
Step 3 (Sales tax addition): Add \(7\%\) tax to the discounted price: \(\$520.00 \times 1.07 = \$556.40\).
Common Student Trap: Additive combination of percentages (\(30\% - 20\% + 7\% = 17\%\) net increase \(\rightarrow \$500 \times 1.17 = \$585.00\), distractor D). Emphasize compounding successive factors.
3 Mixtures & Concentration Equations
Correct: C (40 liters)
Step 1 (Set up equation): Let \(x\) be liters of \(30\%\) solution; then \((60 - x)\) is liters of \(12\%\) solution.
Step 2 (Balance pure solute): \(0.12(60 - x) + 0.30x = 0.24(60)\).
Step 3 (Solve for \(x\)): \(7.2 - 0.12x + 0.30x = 14.4 \implies 0.18x = 7.2 \implies x = \frac{7.2}{0.18} = 40\text{ liters}\).
Common Student Trap: Solving for the \(12\%\) portion \((60 - 40 = 20\text{ liters})\) results in choosing distractor A.
TSIA2 Mathematics Solutions Guide • Form B Page 1 of 5
Teacher Solutions Guide Form B
Linear Models, Inequalities & Literal Formulas
Questions 4 – 7
4 Linear Modeling from Data Tables
Correct: B (\(C(n) = 8n + 20\))
Step 1 (Find constant rate of change): Slope \(m = \frac{260 - 116}{30 - 12} = \frac{144}{18} = 8\text{ dollars per jacket}\).
Step 2 (Determine fixed setup charge): Using point \((12, 116)\): \(116 = 8(12) + b \implies b = 116 - 96 = 20\).
Common Student Trap: Choosing A (\(9.6n\)) assumes a purely proportional relationship through \((0,0)\) by dividing \(116 / 12 \approx 9.6\).
5 Multi-Step Linear Inequalities
Correct: B (\(x \le 3\))
Step 1 (Distribute fraction): \(-\frac{3}{5}(10x) - (-\frac{3}{5})(15) = -6x + 9\). Inequality: \(-6x + 9 \ge 3x - 18\).
Step 2 (Collect variables and constants): Subtract \(3x\) and subtract 9: \(-9x \ge -27\).
Step 3 (Divide by negative coefficient): Divide by \(-9\) and flip the inequality sign: \(x \le \frac{-27}{-9} \implies x \le 3\).
Common Student Trap: Forgetting to reverse the inequality direction when dividing by \(-9\) leads directly to distractor A (\(x \ge 3\)).
6 Linear Systems in Context
Correct: C (160 adult tickets)
Step 1 (System formulation): Total tickets: \(G + S = 280\); Total revenue: \(12G + 7S = 2760\).
Step 2 (Substitution method): Substitute \(S = 280 - G\) into revenue equation: \(12G + 7(280 - G) = 2760\).
Step 3 (Simplify): \(12G + 1960 - 7G = 2760 \implies 5G = 800 \implies G = 160\).
Common Student Trap: Solving for student tickets (\(S = 280 - 160 = 120\)) and selecting distractor A.
7 Rearranging Literal Formulas
Correct: B (\(L = (d + h)\sqrt{\frac{I}{k}}\))
Step 1 (Isolate \(L^2\)): Multiply both sides by \((d + h)^2\) and divide by \(k\): \(L^2 = \frac{I(d + h)^2}{k}\).
Step 2 (Extract square root): \(L = \sqrt{\frac{I(d + h)^2}{k}} = \sqrt{(d + h)^2} \cdot \sqrt{\frac{I}{k}} = (d + h)\sqrt{\frac{I}{k}}\).
Common Student Trap: Forgetting to keep \(k\) under the radical (distractor A) or mishandling the binomial factor.
TSIA2 Mathematics Solutions Guide • Form B Page 2 of 5
Teacher Solutions Guide Form B
Factoring, Rational Expressions & Functions
Questions 8 – 11
8 Solving Quadratics by Factoring
Correct: A (\(x = -\frac{3}{2}, 6\))
Step 1 (Factor trinomial): For \(2x^2 - 9x - 18 = 0\), find factors of \(2(-18) = -36\) that sum to \(-9\): \(-12\) and \(+3\).
Step 2 (Group and factor): \(2x(x - 6) + 3(x - 6) = (2x + 3)(x - 6) = 0\).
Step 3 (Solve roots): \(2x + 3 = 0 \implies x = -\frac{3}{2}\); \(x - 6 = 0 \implies x = 6\).
Common Student Trap: Inverting signs when setting factors to zero yields distractor B (\(x = \frac{3}{2}, -6\)).
9 Dividing Rational Algebraic Expressions
Correct: B (\(\frac{3x + 2}{3x - 2}\))
Step 1 (Factor all polynomials): \(3x^2 - x - 2 = (3x + 2)(x - 1)\); \(9x^2 - 4 = (3x - 2)(3x + 2)\); \(x^2 - x = x(x - 1)\); \(3x^2 + 2x = x(3x + 2)\).
Step 2 (Multiply by reciprocal): \(\frac{(3x + 2)(x - 1)}{(3x - 2)(3x + 2)} \times \frac{x(3x + 2)}{x(x - 1)}\).
Step 3 (Cancel common factors): Canceling \((3x + 2)\), \((x - 1)\), and \(x\) leaves \(\frac{3x + 2}{3x - 2}\).
Common Student Trap: Inverting the fraction incorrectly or canceling individual squared terms across subtraction signs.
10 Function Notation & Composition
Correct: B (2)
Step 1 (Evaluate \(f(g(2))\)): \(g(2) = 2(2) - 1 = 3\). Then \(f(3) = 3(3)^2 - 2(3) + 4 = 27 - 6 + 4 = 25\).
Step 2 (Evaluate \(g(f(2))\)): \(f(2) = 3(2)^2 - 2(2) + 4 = 12 - 4 + 4 = 12\). Then \(g(12) = 2(12) - 1 = 23\).
Step 3 (Calculate difference): \(25 - 23 = 2\).
Common Student Trap: Subtracting in reverse order gives \(-2\) (distractor A).
11 Parabola Vertex & Maximum Optimization
Correct: B ($30; $2,000)
Step 1 (Find maximizing price): Vertex \(x = -\frac{b}{2a} = -\frac{240}{2(-4)} = \frac{240}{8} = \$30\).
Step 2 (Calculate maximum profit): \(P(30) = -4(30)^2 + 240(30) - 1600 = -4(900) + 7200 - 1600 = -3600 + 7200 - 1600 = \$2,000\).
Common Student Trap: Omitting the negative constant \(-1600\) when computing the vertex output yields distractor D (\$2,400).
TSIA2 Mathematics Solutions Guide • Form B Page 3 of 5
Teacher Solutions Guide Form B
Coordinate Geometry, Pythagorean & 3D Solids
Questions 12 – 15
12 Perpendicular Lines & Coordinate Intercepts
Correct: A (\(-2\))
Step 1 (Find slope of \(L_1\)): \(m_1 = \frac{-1 - 7}{2 - (-2)} = \frac{-8}{4} = -2\).
Step 2 (Determine perpendicular slope): \(m_2 = -\frac{1}{m_1} = -\frac{1}{-2} = \frac{1}{2}\).
Step 3 (Find equation of \(L_2\)): Through \((6, 1)\): \(y - 1 = \frac{1}{2}(x - 6) \implies y - 1 = \frac{1}{2}x - 3 \implies y = \frac{1}{2}x - 2\). The \(y\)-intercept is \(-2\).
Common Student Trap: Forgetting the negative reciprocal operation and applying \(m = -2\) directly.
13 Pythagorean Theorem & Right Triangles
Correct: C (75 feet)
Step 1 (Set up right triangle legs): Ground distance \(a = 45\text{ ft}\), vertical height \(b = 60\text{ ft}\).
Step 2 (Pythagorean calculation): \(c = \sqrt{45^2 + 60^2} = \sqrt{2025 + 3600} = \sqrt{5625} = 75\text{ feet}\).
Note: Scaled 3-4-5 right triangle: \(15 \times (3, 4, 5) = (45, 60, 75)\).
Common Student Trap: Adding the two dimensions linearly (\(45 + 60 = 105\)) or arithmetic errors squaring 45.
14 Composite Two-Dimensional Area
Correct: B (\(279\text{ m}^2\))
Step 1 (Area of rectangle): \(A_{\text{rect}} = 24 \times 10 = 240\text{ m}^2\).
Step 2 (Area of semicircle): Semicircle diameter = 10 m \(\implies\) radius \(r = 5\text{ m}\). \(A_{\text{semi}} = \frac{1}{2}\pi r^2 = 0.5 \times 3.14 \times 25 = 39.25\text{ m}^2\).
Step 3 (Total surface sum): \(240 + 39.25 = 279.25 \approx 279\text{ m}^2\).
Common Student Trap: Computing the full circle area (\(78.5\)) without taking half, yielding distractor C (\(319\text{ m}^2\)).
15 3D Volume Scaling & Dimensions
Correct: C (\(\frac{5}{4}\))
Step 1 (Express new dimensions): \(r_B = 1.25r_A\) and \(h_B = 0.80h_A\).
Step 2 (Evaluate volume ratio): \(\frac{V_B}{V_A} = \frac{\pi (1.25r)^2 (0.80h)}{\pi r^2 h} = (1.25)^2 \times 0.80 = 1.5625 \times 0.80 = 1.25\).
Step 3 (Convert to fraction): \(1.25 = \frac{125}{100} = \frac{5}{4}\).
Common Student Trap: Neglecting to square the radius factor (\(1.25 \times 0.80 = 1.0\)), assuming volume remains unchanged.
TSIA2 Mathematics Solutions Guide • Form B Page 4 of 5
Teacher Solutions Guide Form B
Similarity, Probability, Regression & Diagnostic Action
Questions 16 – 20
Q16: Similar Triangles & Side Ratios Correct: B (15.0)
Total side \(AB = 9 + 6 = 15\). By triangle similarity \(\triangle ADE \sim \triangle ABC\): \(\frac{DE}{BC} = \frac{AD}{AB} \implies \frac{DE}{25} = \frac{9}{15} = \frac{3}{5} \implies DE = 25 \times \frac{3}{5} = 15.0\). Trap: Using \(\frac{AD}{DB} = \frac{9}{6} = 1.5\) yields distractor D (18.0).
Q17: Central Tendency & Outlier Resistance Correct: B (Mean drops > 6 pts; median unchanged)
Original: Mean \(= 484 / 7 \approx 69.14\), Median = 68. Adding 18: new sum = 502; new Mean \(= 502 / 8 = 62.75\) (drops \(6.39\) pts); ordered data: \(18, 62, 65, \mathbf{68, 68}, 71, 74, 76 \implies\) new Median = 68 (unchanged). Trap: Assuming median always shifts with sample size expansion.
Q18: Two-Way Conditional Probability Correct: C (\(\frac{47}{80}\))
Given evening preference, condition is restricted to the evening column total (160). Numerator is coffee shop evening students (94): \(P(\text{Coffee Shop} \mid \text{Evening}) = \frac{94}{160} = \frac{47}{80}\). Trap: Dividing by overall survey total 250 yields distractor A (\(\frac{47}{125}\)).
Q19: Compound Probability without Replacement Correct: C (\(\frac{9}{20}\))
Apply complement rule: \(P(\ge 1\text{ flawed}) = 1 - P(\text{both balanced}) = 1 - \left(\frac{12}{16} \times \frac{11}{15}\right) = 1 - \left(\frac{3}{4} \times \frac{11}{15}\right) = 1 - \frac{11}{20} = \frac{9}{20}\). Trap: Using replacement gives \(\frac{7}{16}\); omitting the two-flaw outcome gives distractor B.
Q20: Linear Regression & Residuals Correct: B (+4.0; scored 4.0 pts higher than model)
Model predicted \(\hat{y} = 5.2(5.0) + 58.0 = 26.0 + 58.0 = 84.0\). Residual = \(\text{Actual} - \text{Predicted} = 88 - 84 = +4.0\). Positive residual signifies performance above model expectation. Trap: \(\text{Predicted} - \text{Actual} = -4.0\) flips sign (distractor A).
TSIA2 Readiness Placement & Action Rubric (Form B)
Passing Threshold: Score \(\ge 14\)
Needs Remediation (0–9)
Intensive practice needed on sign reversal in multi-step inequalities, percentage compounding, and two-variable systems formulation.
Approaching Benchmark (10–13)
Competent in direct calculations; focus targeted intervention on composite geometry partition, conditional probability denominators, and quadratic vertex calculations.
College Ready (14–20)
Meets Texas TSI 2.0 College Readiness standard. Ready for collegiate algebra, elementary analysis, or college-level business statistics.
TSIA2 Mathematics Solutions Guide • Form B Page 5 of 5
TSIA2 Form C Practice Packet Texas Success Initiative Assessment 2.0 Form C
Mathematics Benchmark Practice Packet
College Benchmark CRC 950+ or Diagnostic 6
Student Name
Test Date
Class / Period
Raw Score / 20
TSIA2 Mathematics Formula Reference Calculators strictly permitted on designated items only
Slope & Line: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
Geometry & Right \(\triangle\): \(a^2 + b^2 = c^2\)
Solids & Circles: Cylinder: \(V = \pi r^2 h\)
Section 1: Quantitative Reasoning Questions 1 & 2
1 Unit Rates & Scientific Notation
Workspace
A commercial beverage bottling facility injects a mineral electrolyte solution at a constant rate of 1.50 milligrams per liter of purified water. If the facility processes \(6.0 \times 10^5\) liters of water each day, how many kilograms of the mineral solution are required for a 25-day production cycle? (Note: \(1\text{ kg} = 10^6\text{ mg}\))
A. 2.25 kg
B. 22.5 kg
C. 45.0 kg
D. 225 kg
2 Percentages & Successive Price Adjustments
Workspace
A supply distributor purchased a digital projector at a wholesale cost of \(\$450\). The supplier marked up the wholesale cost by \(50\%\). During a seasonal promotional sale, this marked-up price was discounted by \(20\%\). A buyer purchased the projector and paid a \(6\%\) sales tax on the discounted price. What was the total amount the buyer paid?
A. \(\$540.00\)
B. \(\$562.50\)
C. \(\$572.40\)
D. \(\$612.00\)
TSIA2 Mathematics Practice Packet • Form C Page 1 of 7
Sections 1 & 2: Quantitative & Algebraic Foundations Form C
Mixtures, Modeling & Linear Inequalities
Questions 3 – 5
3 Mixture Problems & Concentrations
Workspace
A laboratory chemist blends an \(18\%\) glucose solution with a \(45\%\) glucose solution to prepare exactly 90 milliliters of a \(30\%\) glucose solution. How many milliliters of the \(45\%\) glucose solution must the chemist use?
A. 30 milliliters
B. 35 milliliters
C. 40 milliliters
D. 50 milliliters
4 Linear Functions from Tabular Data
Workspace
The table displays the total print charge \(C(n)\), in dollars, for manufacturing \(n\) customized conference workbooks. Assuming the cost increases at a constant rate, which equation models \(C(n)\)?
A. \(C(n) = 8.67n\)
B. \(C(n) = 7n + 25\)
C. \(C(n) = 6n + 40\)
D. \(C(n) = 8n + 10\)
5 Multi-Step Linear Inequalities
Workspace
Which inequality represents the complete solution set for \(-\frac{3}{4}(8x - 16) \ge 2x - 20\)?
A. \(x \ge 4\)
B. \(x \le 4\)
C. \(x \le -4\)
D. \(x \ge -1\)
TSIA2 Mathematics Practice Packet • Form C Page 2 of 7
Section 2: Algebraic Reasoning Form C
Linear Systems, Literal Formulas & Quadratic Roots
Questions 6 – 8
6 Systems of Two Linear Equations
Workspace
A university symphony orchestra sold 350 total tickets for an evening gala, collecting \(\$4,450\) in total gate receipts. Reserved seating tickets were \(\$16\) each, and general admission tickets were \(\$11\) each. Exactly how many reserved seating tickets were sold?
A. 100 tickets
B. 120 tickets
C. 140 tickets
D. 230 tickets
7 Rearranging Literal Equations
Workspace
The discharge fluid pressure delivered by a hydraulic pump nozzle is modeled by \(P = \frac{v^2 \rho}{2(g + k)^2}\). If \(P\), \(v\), \(\rho\), \(g\), and \(k\) are all positive real numbers, which equation correctly expresses velocity \(v\) in terms of \(P\), \(\rho\), \(g\), and \(k\)?
A. \(v = \frac{(g + k)\sqrt{2P}}{\rho}\)
B. \(v = (g + k)\sqrt{\frac{2P}{\rho}}\)
C. \(v = \frac{2P(g + k)}{\sqrt{\rho}}\)
D. \(v = \sqrt{2P(g + k)^2 - \rho}\)
8 Solving Quadratic Equations by Factoring
Workspace
What are all the real solutions to the quadratic equation \(3x^2 + 10x - 25 = 0\)?
A. \(x = \frac{5}{3}\) and \(x = -5\)
B. \(x = -\frac{5}{3}\) and \(x = 5\)
C. \(x = -3\) and \(x = \frac{25}{3}\)
D. \(x = 5\) and \(x = -\frac{3}{5}\)
TSIA2 Mathematics Practice Packet • Form C Page 3 of 7
Section 3: Advanced Algebraic Reasoning Form C
Rational Expressions, Function Composition & Vertices
Questions 9 – 11
9 Simplifying Rational Algebraic Expressions
Workspace
Which expression is equivalent to \(\frac{2x^2 + 7x + 3}{4x^2 - 1} \div \frac{x^2 + 3x}{2x^2 + 3x}\) for all real values of \(x\) where the expression is defined?
A. \(\frac{x + 3}{x - 1}\)
B. \(\frac{2x + 3}{2x - 1}\)
C. \(\frac{2x - 1}{2x + 3}\)
D. \(\frac{x + 1}{2x - 1}\)
10 Function Notation & Composition
Workspace
Given the functions \(f(x) = 2x^2 - 5x + 3\) and \(g(x) = 3x - 4\), what is the value of the difference \(f(g(3)) - g(f(3))\)?
A. \(-14\)
B. 14
C. 18
D. 24
11 Quadratic Vertex & Maximum Profit
Workspace
An online learning portal models its monthly operating profit by \(P(x) = -5x^2 + 300x - 2500\), where \(x\) is the monthly course subscription fee in dollars. What subscription fee \(x\) maximizes monthly profit, and what is the maximum monthly profit?
A. Fee \(\$25\); Max Profit \(\$1,875\)
B. Fee \(\$30\); Max Profit \(\$2,000\)
C. Fee \(\$35\); Max Profit \(\$1,875\)
D. Fee \(\$30\); Max Profit \(\$2,500\)
TSIA2 Mathematics Practice Packet • Form C Page 4 of 7
Section 4: Geometric & Spatial Reasoning Form C
Coordinate Geometry, Right Triangles & Composite Area
Questions 12 – 14
12 Perpendicular Lines & Coordinate Intercepts
Workspace
In the \(xy\)-plane, line \(L_1\) passes through points \((-4, 9)\) and \((2, -3)\). Line \(L_2\) is perpendicular to \(L_1\) and passes through the point \((4, 1)\). What is the \(y\)-intercept of line \(L_2\)?
A. \(-1\)
B. \(-\frac{3}{2}\)
C. \(0\)
D. \(2\)
13 Pythagorean Theorem & Right Triangles
Workspace
A construction surveyor inspects a heavy-duty diagonal steel strut for an athletic stadium lighting tower. The strut anchors to a concrete footing 32 feet horizontally from the tower base and fastens to the vertical tower at an elevation of 24 feet. What is the length of the diagonal strut?
A. 36 feet
B. 38 feet
C. 40 feet
D. 44 feet
14 Composite Geometric Area
Workspace
A botanical conservatory walkway area consists of a rectangle adjoining a semicircle along one of its shorter sides. The rectangle measures 30 meters long and 12 meters wide. The adjoining semicircle has a diameter equal to the 12-meter width. Which value is closest to the total surface area of the walkway? (Use \(\pi \approx 3.14\))
A. \(394\text{ m}^2\)
B. \(417\text{ m}^2\)
C. \(473\text{ m}^2\)
D. \(528\text{ m}^2\)
TSIA2 Mathematics Practice Packet • Form C Page 5 of 7
Sections 4 & 5: Geometry & Data Analysis Form C
Solid Geometry, Proportionality & Distribution Measures
Questions 15 – 17
15 3D Volume Scaling & Dimensions
Workspace
Right circular Cylinder A has radius \(r\) and height \(h\). Cylinder B has a radius that is \(50\%\) greater than Cylinder A and a height that is \(50\%\) less than Cylinder A. What is the ratio of the volume of Cylinder B to the volume of Cylinder A?
A. 1
B. \(\frac{7}{6}\)
C. \(\frac{9}{8}\)
D. \(\frac{5}{4}\)
16 Similar Triangles & Proportional Sides
Workspace
In \(\triangle ABC\), segment \(DE\) is parallel to base \(BC\), with point \(D\) located on side \(AB\) and point \(E\) located on side \(AC\). If \(AD = 10\), \(DB = 5\), and the base \(BC = 27\), what is the length of segment \(DE\)?
A. 15.0
B. 18.0
C. 20.0
D. 21.5
17 Central Tendency & Impact of Outliers
Workspace
An initial sample of 7 diagnostic quiz scores is: \(70, 72, 75, 75, 78, 81, 84\). An 8th participant completes the quiz and scores a \(23\). Which statement correctly describes the resulting change in the dataset's summary statistics?
A. Both the mean and median decrease by \(\approx 6\) points.
B. The mean decreases by \(> 6\) points; median is unchanged.
C. The median decreases, but the mean resists the outlier.
D. The interquartile range triples while the mean stays constant.
TSIA2 Mathematics Practice Packet • Form C Page 6 of 7
Section 5: Probabilistic & Statistical Reasoning Form C
Tables, Probability, Regression & Final Scoring
Questions 18 – 20
18 Two-Way Table Conditional Probability
Given that a randomly selected student is a Humanities major, what is the probability that they are enrolled primarily in online coursework?
A. \(\frac{8}{25}\)
B. \(\frac{16}{50}\)
C. \(\frac{16}{25}\)
D. \(\frac{3}{5}\)
19 Compound Probability without Replacement
A parts organizer contains 12 precision aerospace fasteners: 9 meet exact tolerance specifications and 3 have minor thread imperfections. A robotic arm selects 2 fasteners at random one after another without replacement. What is the probability that at least one of the selected fasteners has a thread imperfection?
A. \(\frac{1}{4}\)
B. \(\frac{4}{11}\)
C. \(\frac{5}{11}\)
D. \(\frac{1}{2}\)
20 Linear Regression & Residual Analysis
A least-squares linear regression line modeling hours of weekly lab practice \(x\) and procedural exam score \(y\) is \(\hat{y} = 4.8x + 62.0\). A student who completed 5.0 hours of lab practice earned an actual score of 91. What is this student's residual, and how is it interpreted?
A. \(-5.0\); the student scored 5.0 points lower than model.
B. \(+5.0\); the student scored 5.0 points higher than model.
C. \(+4.8\); the student scored 4.8 points higher than model.
D. \(-4.8\); the student scored 4.8 points lower than model.
Official Examination Document • Form C
Student Answer Record & Diagnostic Summary
Fill bubble completely:
(A)(B)(C)(D) 06. (A)(B)(C)(D)
(A)(B)(C)(D) 07. (A)(B)(C)(D)
(A)(B)(C)(D) 08. (A)(B)(C)(D)
(A)(B)(C)(D) 09. (A)(B)(C)(D)
(A)(B)(C)(D) 10. (A)(B)(C)(D)
(A)(B)(C)(D) 16. (A)(B)(C)(D)
(A)(B)(C)(D) 17. (A)(B)(C)(D)
(A)(B)(C)(D) 18. (A)(B)(C)(D)
(A)(B)(C)(D) 19. (A)(B)(C)(D)
(A)(B)(C)(D) 20. (A)(B)(C)(D)
Quantitative Q1–Q4: ___ / 4
Algebraic Q5–Q11: ___ / 7
Geometric Q12–Q16: ___ / 5
Statistical Q17–Q20: ___ / 4
TSIA2 Mathematics Practice Packet • Form C Page 7 of 7
TSIA2 Form C Teacher Guide Instructor Assessment Resource Form C Key
TSIA2 Solutions & Diagnostic Guide (Form C)
College Readiness Target 14+ / 20 (70% Benchmark)
Form C Quick-Scan Master Key 20 Multiple Choice Questions
Q01B
Q02C
Q03C
Q04B
Q05B
Q06B
Q07B
Q08A
Q09B
Q10B
Q11B
Q12A
Q13C
Q14B
Q15C
Q16B
Q17B
Q18C
Q19C
Q20B
Detailed Solutions: Quantitative Reasoning (Questions 1 – 3)
1 Unit Rates & Scientific Notation
Correct: B (22.5 kg)
Step 1 (Daily chemical mass): Multiply daily volume by injection rate: \(6.0 \times 10^5\text{ L/day} \times 1.50\text{ mg/L} = 9.0 \times 10^5\text{ mg/day}\).
Step 2 (25-day cycle requirement): Multiply daily mass by 25 days: \(25 \times 9.0 \times 10^5\text{ mg} = 2.25 \times 10^7\text{ mg}\).
Step 3 (Convert to kilograms): Divide by \(10^6\text{ mg/kg}\): \(\frac{2.25 \times 10^7}{10^6} = 22.5\text{ kg}\).
Common Student Trap: Decimal placement error during unit conversion leads to distractor A (2.25 kg).
2 Percentages & Successive Price Changes
Correct: C ($572.40)
Step 1 (Marked-up retail price): Wholesale \(\$450\) increased by \(50\%\): \(\$450 \times 1.50 = \$675.00\).
Step 2 (Sale discount): \(20\%\) discount on \(\$675\): \(\$675 \times (1 - 0.20) = \$675 \times 0.80 = \$540.00\).
Step 3 (Sales tax addition): Add \(6\%\) tax to discounted price: \(\$540.00 \times 1.06 = \$572.40\).
Common Student Trap: Omitting sales tax (\(\$540.00\), distractor A) or calculating additive net percentage (\(50\% - 20\% + 6\% = 36\% \implies \$450 \times 1.36 = \$612.00\), distractor D).
3 Mixtures & Concentration Equations
Correct: C (40 mL)
Step 1 (Set up equation): Let \(x\) be mL of \(45\%\) solution; then \((90 - x)\) is mL of \(18\%\) solution.
Step 2 (Balance pure solute): \(0.18(90 - x) + 0.45x = 0.30(90)\).
Step 3 (Solve for \(x\)): \(16.2 - 0.18x + 0.45x = 27 \implies 0.27x = 10.8 \implies x = \frac{10.8}{0.27} = 40\text{ mL}\).
Common Student Trap: Solving for the \(18\%\) volume \((90 - 40 = 50\text{ mL})\) and selecting distractor D.
TSIA2 Mathematics Solutions Guide • Form C Page 1 of 5
Teacher Solutions Guide Form C
Linear Models, Inequalities & Literal Formulas
Questions 4 – 7
4 Linear Modeling from Data Tables
Correct: B (\(C(n) = 7n + 25\))
Step 1 (Find constant rate of change): Slope \(m = \frac{270 - 130}{35 - 15} = \frac{140}{20} = 7\text{ dollars per workbook}\).
Step 2 (Determine fixed setup charge): Using point \((15, 130)\): \(130 = 7(15) + b \implies b = 130 - 105 = 25\).
Common Student Trap: Dividing total cost by books (\(130 / 15 \approx 8.67\)) overlooks the non-zero \(y\)-intercept (distractor A).
5 Multi-Step Linear Inequalities
Correct: B (\(x \le 4\))
Step 1 (Distribute coefficient): \(-\frac{3}{4}(8x) - (-\frac{3}{4})(16) = -6x + 12\). Inequality: \(-6x + 12 \ge 2x - 20\).
Step 2 (Collect variables and constants): Subtract \(2x\) and subtract 12: \(-8x \ge -32\).
Step 3 (Divide by negative number): Divide by \(-8\) and reverse inequality: \(x \le \frac{-32}{-8} \implies x \le 4\).
Common Student Trap: Forgetting to reverse inequality symbol leads to distractor A (\(x \ge 4\)).
6 Linear Systems in Context
Correct: B (120 tickets)
Step 1 (System formulation): Total tickets: \(R + G = 350\); Total revenue: \(16R + 11G = 4450\).
Step 2 (Substitution method): Substitute \(G = 350 - R\) into revenue equation: \(16R + 11(350 - R) = 4450\).
Step 3 (Simplify): \(16R + 3850 - 11R = 4450 \implies 5R = 600 \implies R = 120\).
Common Student Trap: Solving for general admission tickets (\(G = 350 - 120 = 230\)) and selecting distractor D.
7 Rearranging Literal Formulas
Correct: B (\(v = (g + k)\sqrt{\frac{2P}{\rho}}\))
Step 1 (Isolate \(v^2\)): Multiply by \(2(g + k)^2\) and divide by \(\rho\): \(v^2 = \frac{2P(g + k)^2}{\rho}\).
Step 2 (Extract square root): \(v = \sqrt{\frac{2P(g + k)^2}{\rho}} = \sqrt{(g + k)^2} \cdot \sqrt{\frac{2P}{\rho}} = (g + k)\sqrt{\frac{2P}{\rho}}\).
Common Student Trap: Failing to keep the density parameter \(\rho\) under the radical (distractor A).
TSIA2 Mathematics Solutions Guide • Form C Page 2 of 5
Teacher Solutions Guide Form C
Factoring, Rational Expressions & Functions
Questions 8 – 11
8 Solving Quadratics by Factoring
Correct: A (\(x = \frac{5}{3}, -5\))
Step 1 (Factor trinomial): For \(3x^2 + 10x - 25 = 0\), find factors of \(3(-25) = -75\) that sum to \(+10\): \(+15\) and \(-5\).
Step 2 (Group and factor): \(3x(x + 5) - 5(x + 5) = (3x - 5)(x + 5) = 0\).
Step 3 (Solve roots): \(3x - 5 = 0 \implies x = \frac{5}{3}\); \(x + 5 = 0 \implies x = -5\).
Common Student Trap: Setting factors equal to opposites of roots produces distractor B (\(x = -\frac{5}{3}, 5\)).
9 Dividing Rational Algebraic Expressions
Correct: B (\(\frac{2x + 3}{2x - 1}\))
Step 1 (Factor all polynomials): \(2x^2 + 7x + 3 = (2x + 1)(x + 3)\); \(4x^2 - 1 = (2x - 1)(2x + 1)\); \(x^2 + 3x = x(x + 3)\); \(2x^2 + 3x = x(2x + 3)\).
Step 2 (Multiply by reciprocal): \(\frac{(2x + 1)(x + 3)}{(2x - 1)(2x + 1)} \times \frac{x(2x + 3)}{x(x + 3)}\).
Step 3 (Cancel common factors): Canceling \((2x + 1)\), \((x + 3)\), and \(x\) leaves \(\frac{2x + 3}{2x - 1}\).
Common Student Trap: Inverting numerator and denominator inverted yields \(\frac{2x - 1}{2x + 3}\) (distractor C).
10 Function Notation & Composition
Correct: B (14)
Step 1 (Evaluate \(f(g(3))\)): \(g(3) = 3(3) - 4 = 5\). Then \(f(5) = 2(5)^2 - 5(5) + 3 = 50 - 25 + 3 = 28\).
Step 2 (Evaluate \(g(f(3))\)): \(f(3) = 2(3)^2 - 5(3) + 3 = 18 - 15 + 3 = 6\). Then \(g(6) = 3(6) - 4 = 14\).
Step 3 (Calculate difference): \(28 - 14 = 14\).
Common Student Trap: Subtracting terms in reversed sequence yields \(-14\) (distractor A).
11 Parabola Vertex & Maximum Optimization
Correct: B ($30; $2,000)
Step 1 (Find maximizing price): Vertex \(x = -\frac{b}{2a} = -\frac{300}{2(-5)} = \frac{300}{10} = \$30\).
Step 2 (Calculate maximum profit): \(P(30) = -5(30)^2 + 300(30) - 2500 = -5(900) + 9000 - 2500 = -4500 + 9000 - 2500 = \$2,000\).
Common Student Trap: Neglecting the vertical shift constant \(-2500\) gives distractor D (\$2,500).
TSIA2 Mathematics Solutions Guide • Form C Page 3 of 5
Teacher Solutions Guide Form C
Coordinate Geometry, Pythagorean & 3D Solids
Questions 12 – 15
12 Perpendicular Lines & Coordinate Intercepts
Correct: A (\(-1\))
Step 1 (Find slope of \(L_1\)): \(m_1 = \frac{-3 - 9}{2 - (-4)} = \frac{-12}{6} = -2\).
Step 2 (Determine perpendicular slope): \(m_2 = -\frac{1}{m_1} = -\frac{1}{-2} = \frac{1}{2}\).
Step 3 (Find equation of \(L_2\)): Through \((4, 1)\): \(y - 1 = \frac{1}{2}(x - 4) \implies y - 1 = \frac{1}{2}x - 2 \implies y = \frac{1}{2}x - 1\). The \(y\)-intercept is \(-1\).
Common Student Trap: Forgetting the negative reciprocal operation and applying parallel slope \(m = -2\).
13 Pythagorean Theorem & Right Triangles
Correct: C (40 feet)
Step 1 (Set up right triangle legs): Base \(a = 32\text{ ft}\), vertical mast \(b = 24\text{ ft}\).
Step 2 (Pythagorean calculation): \(c = \sqrt{32^2 + 24^2} = \sqrt{1024 + 576} = \sqrt{1600} = 40\text{ feet}\).
Note: Scaled 3-4-5 right triangle: \(8 \times (3, 4, 5) = (24, 32, 40)\).
Common Student Trap: Linear leg addition (\(32 + 24 = 56\)) or subtracting squares under the radical.
14 Composite Two-Dimensional Area
Correct: B (\(417\text{ m}^2\))
Step 1 (Area of rectangle): \(A_{\text{rect}} = 30 \times 12 = 360\text{ m}^2\).
Step 2 (Area of semicircle): Semicircle diameter = 12 m \(\implies\) radius \(r = 6\text{ m}\). \(A_{\text{semi}} = \frac{1}{2}\pi r^2 = 0.5 \times 3.14 \times 36 = 56.52\text{ m}^2\).
Step 3 (Total surface sum): \(360 + 56.52 = 416.52 \approx 417\text{ m}^2\).
Common Student Trap: Using the whole circle area (\(113.04\)) yields distractor C (\(473\text{ m}^2\)).
15 3D Volume Scaling & Dimensions
Correct: C (\(\frac{9}{8}\))
Step 1 (Express new dimensions): \(r_B = 1.50r_A\) and \(h_B = 0.50h_A\).
Step 2 (Evaluate volume ratio): \(\frac{V_B}{V_A} = \frac{\pi (1.50r)^2 (0.50h)}{\pi r^2 h} = (1.50)^2 \times 0.50 = 2.25 \times 0.50 = 1.125\).
Step 3 (Convert to fraction): \(1.125 = \frac{1125}{1000} = \frac{9}{8}\).
Common Student Trap: Forgetting to square the radius factor (\(1.50 \times 0.50 = 0.75\)) or assuming the volume remains 1.
TSIA2 Mathematics Solutions Guide • Form C Page 4 of 5
Teacher Solutions Guide Form C
Similarity, Probability, Regression & Diagnostic Action
Questions 16 – 20
Q16: Similar Triangles & Side Ratios Correct: B (18.0)
Total side \(AB = 10 + 5 = 15\). By triangle similarity \(\triangle ADE \sim \triangle ABC\): \(\frac{DE}{BC} = \frac{AD}{AB} \implies \frac{DE}{27} = \frac{10}{15} = \frac{2}{3} \implies DE = 27 \times \frac{2}{3} = 18.0\). Trap: Using \(\frac{10}{5} = 2\) yields distractor D.
Q17: Central Tendency & Outlier Resistance Correct: B (Mean drops > 6 pts; median unchanged)
Original: Mean \(= 535 / 7 \approx 76.43\), Median = 75. Adding 23: new sum = 558; new Mean \(= 558 / 8 = 69.75\) (drops \(6.68\) pts); ordered data: \(23, 70, 72, \mathbf{75, 75}, 78, 81, 84 \implies\) new Median = 75 (unchanged). Trap: Believing median always decreases with low outliers.
Q18: Two-Way Conditional Probability Correct: C (\(\frac{16}{25}\))
Given Humanities major, denominator is restricted to the Humanities column total (150). Numerator is online Humanities students (96): \(P(\text{Online} \mid \text{Humanities}) = \frac{96}{150} = \frac{16}{25}\). Trap: Dividing by survey grand total 300 gives distractor A (\(\frac{8}{25}\)).
Q19: Compound Probability without Replacement Correct: C (\(\frac{5}{11}\))
Apply complement rule: \(P(\ge 1\text{ flawed}) = 1 - P(\text{both meet}) = 1 - \left(\frac{9}{12} \times \frac{8}{11}\right) = 1 - \left(\frac{3}{4} \times \frac{8}{11}\right) = 1 - \frac{6}{11} = \frac{5}{11}\). Trap: Sampling with replacement yields \(\frac{7}{16}\); computing only single flawed pick gives distractor B (\(\frac{4}{11}\)).
Q20: Linear Regression & Residuals Correct: B (+5.0; scored 5.0 pts higher than model)
Model predicted \(\hat{y} = 4.8(5.0) + 62.0 = 24.0 + 62.0 = 86.0\). Residual = \(\text{Actual} - \text{Predicted} = 91 - 86 = +5.0\). Positive residual indicates exceeding model expectation. Trap: \(\text{Predicted} - \text{Actual} = -5.0\) reverses sign (distractor A).
TSIA2 Readiness Placement & Action Rubric (Form C)
Passing Threshold: Score \(\ge 14\)
Needs Remediation (0–9)
Intensive support required on proportional dimensional analysis, inequality sign rules with division by negatives, and systems substitution setup.
Approaching Benchmark (10–13)
Solid linear foundation; prioritize targeted remediation in rational polynomial cancellation, conditional probability table restrictions, and 3D geometric scaling.
College Ready (14–20)
Meets Texas TSI 2.0 College Readiness standard. Fully prepared for credit-bearing college algebra, finite math, or introductory statistics.
TSIA2 Mathematics Solutions Guide • Form C Page 5 of 5
