Math Self-Study Diagnostic: Six Worked Exercises Before Calculus

An informal algebra and trigonometry worksheet: try first, check your reasoning, and choose what to review.

Use this informal math self-study diagnostic to identify specific algebra and trigonometry gaps. Start by trying the six original exercises below on paper, without looking at the worked solutions. Use any hesitation to choose a review topic—not to calculate a placement score. MIT OCW’s 18.01SC Single Variable Calculus lists high school algebra and trigonometry as prerequisites; prior calculus experience is helpful but not essential. This worksheet is a small self-study sample, not an MIT assessment or a mastery guarantee. [1.1] [5.2]

Use the self-study mathematics roadmap to choose the broader branch after checking these prerequisites. Practice the same care with units and assumptions in the server-energy calculator.

In this article

How to use this worksheet
1. Fractions: combine before simplifying
2. Order of operations: make the grouping visible
3. A linear equation: preserve equality
4. Quadratic factoring: connect factors to solutions
5. A square-root domain: identify allowed inputs
6. Right-triangle sine: name the sides first
Use MIT OCW resources for your next step

How to use this worksheet

• Try before checking. Use paper and pencil, show your intermediate steps, and try without a calculator so you can see which operations need attention. [5.2]
• Notice the sticking point. Beside each attempt, write something specific, such as “forgot the common denominator” or “unsure which side is the hypotenuse.” Treat that note as a review prompt rather than a grade. [5.1] [5.2] [5.3]
• Explain, then revisit. After checking a solution, cover it and explain the steps in your own words. MIT’s own worked-example guidance similarly asks learners to solve problems before checking detailed solutions. [1.4]

1. Fractions: combine before simplifying

Exercise. Simplify 2/3 + 5/6 − 1/4. Give your answer as a fraction in lowest terms. This original prompt checks fraction arithmetic, one of the topics included in the University of Arizona’s calculus-preparation review. [5.3]

Worked solution. Use 12 as a common denominator: 2/3 = 8/12, 5/6 = 10/12, and 1/4 = 3/12. Therefore, 2/3 + 5/6 − 1/4 = (8 + 10 − 3)/12 = 15/12 = 5/4. The denominator describes the size of the pieces; once those pieces match, combine the numerators. If you added the denominators, review equivalent fractions and common denominators before attempting more involved expressions. [5.3] [6.1]

Turn each sticking point into a review topic

Swipe horizontally to explore the map

Read the mind map as an outline

Turn each sticking point into a review topic

Arithmetic foundations
- Fractions: equivalent forms and common denominators
- Order of operations: grouping and powers
Algebraic transformations
- Linear equations: distribute and preserve equality
- Quadratics: factor, expand to check, then solve
Allowed inputs
- Square roots: nonnegative radicand
- Inequalities: include valid endpoints
Trigonometry
- Identify sides relative to the angle
- Sine: opposite divided by hypotenuse
Continue with MIT OCW
- Read the 18.01SC syllabus
- Attempt worked examples before checking solutions
- Investigate the trigonometric identities review

Use this topic map to organize review after your attempt. It connects the worksheet’s algebra and trigonometry prompts to the prerequisite areas, without assigning scores or predicting readiness. [1.1] [6.1] [10.3]

2. Order of operations: make the grouping visible

Exercise. Evaluate 18 − 2 × (3 + 4)² ÷ 7. Write one intermediate line before giving the answer. This original prompt isolates arithmetic inside an algebraic expression, rather than asking you to learn a calculus rule. [1.1] [5.3]

Worked solution. First evaluate the parentheses: 3 + 4 = 7. Next square: 7² = 49. The expression becomes 18 − 2 × 49 ÷ 7. Perform multiplication and division from left to right: 2 × 49 ÷ 7 = 98 ÷ 7 = 14. Finally, 18 − 14 = 4. If your result differs, locate the first line where your calculation changed the grouping or performed subtraction too early. [1.1] [5.3]

3. A linear equation: preserve equality

Exercise. Solve 3(x − 2) + 5 = 2x + 9, then substitute your answer into the original equation. Solving equations and expanding expressions are both included in Columbia University’s precalculus review topics; the particular equation here is original. [6.1]

Worked solution. Distribute the 3 to obtain 3x − 6 + 5 = 2x + 9, then combine constants: 3x − 1 = 2x + 9. Subtract 2x from both sides to get x − 1 = 9. Add 1 to both sides: x = 10. Check in the original equation: the left side is 3(10 − 2) + 5 = 29, and the right side is 2(10) + 9 = 29. If you stalled, distinguish a distribution error from uncertainty about doing the same operation to both sides. [6.1] [10.1]

4. Quadratic factoring: connect factors to solutions

Exercise. Factor x² − x − 12, then solve x² − x − 12 = 0. This original exercise separates two related tasks: rewriting an expression as a product and using that product to solve an equation. [6.1] [10.1]

Worked solution. Look for two numbers whose product is −12 and whose sum is −1. They are −4 and 3, so x² − x − 12 = (x − 4)(x + 3). Check by expanding: x² + 3x − 4x − 12 = x² − x − 12. For the equation, (x − 4)(x + 3) = 0 means at least one factor is zero. Thus x = 4 or x = −3. If you found the factors but missed a solution, review the step from a zero product to its individual factors. [6.1] [10.1]

5. A square-root domain: identify allowed inputs

Exercise. Working over the real numbers, find the domain of f(x) = √(5 − 2x). Give the answer as an inequality and in interval notation. This original prompt combines square roots, inequalities, and domains—topics identified in the supplied precalculus-readiness materials. [6.1] [10.1] [10.5]

Worked solution. A real square root requires its radicand—the expression inside the root—to be nonnegative. Require 5 − 2x ≥ 0. Subtract 5: −2x ≥ −5. Divide by −2 and reverse the inequality: x ≤ 5/2. In interval notation, the domain is (−∞, 5/2]. The endpoint is included because it makes the radicand zero, and √0 is defined. If you obtained x ≥ 5/2, focus your review on dividing inequalities by a negative number. [6.1] [10.1] [10.5]

6. Right-triangle sine: name the sides first

Exercise. In a right triangle, θ is an acute angle. The side opposite θ has length 7, and the hypotenuse has length 25. Find sin θ exactly. This original exercise checks a triangle relationship, not an inverse-trigonometric calculation. [1.1] [10.3]

Worked solution. For an acute angle in a right triangle, sine is the opposite side divided by the hypotenuse. Therefore, sin θ = 7/25. You do not need the remaining side or the angle’s degree measure. If you wrote 25/7 or wanted to use the adjacent side, sketch the triangle and label the sides relative to θ before choosing a ratio. [1.1] [10.3]

Use MIT OCW resources for your next step

Start with the 18.01SC syllabus. It states the algebra and trigonometry prerequisites and explains how to use the course’s problem sets and solutions. It also says the OCW Scholar course is self-contained and requires no textbook. Use the linked source to understand the course expectations, rather than treating this worksheet as a substitute for them. [1.1] [1.2]

Use Single Variable Calculus, Fall 2010, as the main study course. MIT describes 18.01SC as designed for independent study. Its listed materials include lecture videos, written notes, recitation videos, worked examples, problem sets, and solutions. A practical next step is to begin the course and keep a separate list of algebra or trigonometry steps you need to revisit, checking solutions only after making an attempt. [4.1] [4.3] [1.4]

Bookmark Review of Trigonometric Identities. MIT OCW lists a resource with that title within 18.01SC. It is an appropriate topic-specific resource to investigate when reviewing identities, but its supplied listing does not establish that it teaches elementary right-triangle ratios. For broader review planning, remember that the six exercises here do not sample everything: Columbia’s preparation outline also includes exponents, logarithms, unit-circle values, inverse trigonometric functions, and trigonometric graphs. [3.1] [6.2] [6.5]

Single Variable Calculus · ocw.mit.edu

Frequently asked questions

Do I need to learn calculus before starting 18.01SC?

No prior calculus course is listed as a prerequisite. MIT’s syllabus says prior calculus experience is helpful but not essential; the stated prerequisites are high school algebra and trigonometry. Focus your preparation on those foundations rather than assuming you must already know differentiation or integration. [1.1]

What should I do if I understand a solution but could not start the problem?

Identify the first decision you could not make—such as selecting a common denominator or setting up the square-root inequality. Review that specific topic, then cover the solution and attempt the problem again. This follows the attempt-first approach encouraged by the Arizona review and MIT’s worked-example guidance; it is a study routine, not a readiness test. [5.2] [1.4]

Does getting all six right mean I have covered the prerequisites?

No. These exercises sample only six small areas. Broader precalculus preparation also includes topics such as exponents, logarithms, unit-circle values, identities, inverse trigonometric functions, and graphs. Use successful attempts as information about these particular tasks, not as proof of complete preparation. [6.2] [6.5]

Conclusion

Use this worksheet to make your next review session specific: identify the step that caused trouble, revisit that topic, and try again without the solution visible. There is no placement score to calculate. MIT OCW’s 18.01SC syllabus and independent-study materials provide the next context: attempt problems, check the supplied solutions, and keep track of prerequisite techniques that need more practice. The goal here is a useful starting point—not a guarantee of mastery. [1.1] [1.2] [1.4] [4.3]

Source notes

The six exercises and their worked solutions are original instructional examples, not questions taken from MIT or a validated diagnostic. Citations identify the prerequisite topics and study-method premises, not the source of the original calculations.

The available evidence for MIT’s Review of Trigonometric Identities establishes its title and location, not its detailed contents.

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Generated with Ithy Pro. Editorial arithmetic and primary-source checks: 2026-10-23. AI-assisted; this is not a named human expert review. References open the original sources.

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GPT-6 Luna

MIT OCW’s 18.01SC Single Variable Calculus lists high school algebra and trigonometry as prerequisites. The six short exercises below are an informal way to notice topics to review—not a placement score or a guarantee of readiness. MIT OCW offers an appropriate self-study course with problem sets and solutions, plus a review of trigonometric identities.

Use these as pencil-and-paper prompts: try each before reading its solution, then note which steps felt uncertain. This is a self-study check, not a placement test, score, or mastery guarantee.
Fractions — Simplify 3/4 + 5/6. Solution: use denominator 12: 9/12 + 10/12 = 19/12 = 1 7/12.
Order of operations — Evaluate 18 − 2(3 + 4)² ÷ 7. Solution: parentheses give 7; the power is 49; multiplication and division from left to right give 2 × 49 ÷ 7 = 14; then 18 − 14 = 4.
Linear equation — Solve 3(x − 2) + 5 = 2x + 9. Solution: expand to 3x − 1 = 2x + 9; subtract 2x and add 1 to get x = 10.
Quadratic factoring — Solve x² − x − 12 = 0. Solution: factor as (x − 4)(x + 3) = 0, so x = 4 or x = −3.
Square-root domain — For real numbers, find the domain of f(x) = √(5 − 2x). Solution: the radicand must be nonnegative, so 5 − 2x ≥ 0, hence x ≤ 5/2. The domain is (−∞, 5/2].
Right-triangle sine — In a right triangle, the side opposite angle θ is 3 and the hypotenuse is 5. Find sin θ. Solution: sine is opposite over hypotenuse, so sin θ = 3/5.
For primary preparation, MIT OCW’s 18.01SC syllabus identifies high school algebra and trigonometry as prerequisites. Its independent-study course provides problem sets with solutions to check after attempting them. MIT OCW also has a resource titled “Review of Trigonometric Identities.” [1.1] [1.2] [3.1]

The supplied MIT trigonometry-review excerpt gives its title but not its exercises or coverage; it does not establish that the resource reviews right-triangle sine ratios specifically.

Claude Haiku 5.5

The supplied MIT OCW pages state that Single Variable Calculus (18.01SC) is designed for independent study with prerequisites of high school algebra and trigonometry, and that its problem sets come from MIT homework and can be checked against provided solutions. Other supplied sources (Arizona, Columbia, Lamar) offer self-check quizzes or review sets covering the same algebra and trigonometry skills. The worksheet below is an original informal self-study exercise set; the supplied evidence does not establish any placement score, mastery threshold or guaranteed outcome.

MIT OpenCourseWare describes Single Variable Calculus as a first-year, first-semester MIT course whose prerequisites are high school algebra and trigonometry, and says prior calculus experience is helpful but not essential. [1.1]
The MIT OCW 18.01SC course is designed for independent study. Its problem sets were taken from MIT homework, and you can check your answers against provided solutions after completing a problem set. [1.1] [1.2]
The MIT OCW 18.01SC materials listed include lecture videos with written notes, recitation videos on problem-solving tips, worked examples with detailed solutions, problem sets with solutions, exams with solutions, and interactive Mathlets. [4.3]
The University of Arizona review states that a student needs a good command of college algebra and trigonometry for Calculus I, offers 10-question web quizzes with online help, and states that no results are recorded or collected. [5.1]
The Arizona review advises working with paper and pencil, starting from the question rather than the answers, and not using a calculator. [5.2]
Columbia's precalculus bootcamp covers algebra (factoring, simplifying, solving equations, inequalities), rational expressions and radicals (including square roots), exponents and logarithms, and trigonometry (unit circle values and identities), and offers six focused worksheets. [6.1] [6.2] [6.6]
The Lamar algebra and trig review was written for Calculus I students and presents problem sets with detailed solutions, noting that much of calculus work involves algebra. [7.3] [7.5]
Suggested worksheet (original exercises, informal, no placement or mastery guarantee): 1) Fractions: simplify (3/4 + 5/6) × 2/5. Worked solution: 3/4 + 5/6 = 9/12 + 10/12 = 19/12; times 2/5 gives 38/60 = 19/30. 2) Order of operations: evaluate 12 − 2(3 + 4²) ÷ 2. Worked solution: 4² = 16; 3 + 16 = 19; 2 × 19 = 38; 38 ÷ 2 = 19; 12 − 19 = −7. 3) Linear equation: solve 5x − 3 = 2x + 9. Worked solution: 3x = 12, so x = 4. 4) Quadratic factoring: factor x² − 5x + 6. Worked solution: (x − 2)(x − 3), roots 2 and 3. 5) Square-root domain: find the domain of √(x − 4). Worked solution: need x − 4 ≥ 0, so x ≥ 4, i.e. [4, ∞). 6) Right-triangle sine: in a right triangle with opposite side 3 and hypotenuse 5, sin θ = 3/5. Worked solution: sine is opposite over hypotenuse, 3/5.

The supplied excerpts are partial page views and search snippets, so the full content of the MIT OCW preparation resources and review sites was not verified.

The evidence does not establish any placement score, pass threshold or guaranteed outcome from completing these exercises, and the worksheet is informal self-study only.

The worksheet exercises were written for this answer rather than taken from a supplied source.

Claude Sonnet 5.5

MIT OCW 18.01SC lists high school algebra and trigonometry as prerequisites and is built for independent study. A sensible self-check is to work a few short problems on paper, note where you stall, then review that topic. The six exercises below are original and unscored, and the worked solutions are in the findings. Companion review materials from other institutions (Arizona, Columbia, Lamar) can help, but they are not MIT resources.

MIT OCW 18.01SC states its prerequisites as high school algebra and trigonometry, and says prior calculus experience is helpful but not essential. Use that as the benchmark for this informal check. Links: https://ocw.mit.edu/courses/18-01sc-single-variable-calculus-fall-2010/pages/syllabus/ and the course home https://ocw.mit.edu/courses/18-01sc-single-variable-calculus-fall-2010/ [1.1] [4.1]
The course is designed for independent study and includes lecture videos, worked examples with detailed solutions, problem sets with solutions, and exams. A useful habit is to attempt a problem first and then check the detailed solution. The same habit applies to the exercises here. The MIT page also lists a trigonometric identities review clip: https://ocw.mit.edu/courses/18-01sc-single-variable-calculus-fall-2010/resources/clip-1-review-of-trigonometric-identities/ [1.4] [4.3] [3.1]
Exercises 1-3 (use paper, no calculator). (1) Compute 2/3 + 5/6 - 1/4. Solution: the common denominator is 12, so 8/12 + 10/12 - 3/12 = 15/12 = 5/4. (2) Compute 12 - 3(4 - 1)^2 / 9 + 2. Solution: the parentheses give 3, the exponent gives 9, then 3*9/9 = 3 (multiplication and division go left to right), so 12 - 3 + 2 = 11. (3) Solve 3(x - 2) + 5 = 2x + 7. Solution: 3x - 6 + 5 = 2x + 7, so 3x - 1 = 2x + 7, giving x = 8. Check: 3(6) + 5 = 23 and 2(8) + 7 = 23. [5.2] [10.1]
Exercises 4-6. (4) Factor and solve x^2 - 5x - 14 = 0. Solution: find two numbers with product -14 and sum -5, which are -7 and 2. So (x - 7)(x + 2) = 0, giving x = 7 or x = -2. (5) Find the domain of f(x) = sqrt(3x - 6). Solution: require 3x - 6 >= 0, so x >= 2, which is [2, infinity). (6) In a right triangle, the leg opposite angle A is 5 and the hypotenuse is 13. Find sin A. Solution: sine is opposite over hypotenuse, so sin A = 5/13. Also, the adjacent leg is 12. [10.1] [10.5] [10.3]
Interpret results as diagnostic, not as a score. If you stall on an exercise, note the topic, review it, and try a similar problem. The Arizona review says its quizzes record no results and advises working with pencil and paper, starting from the question rather than the answers. This is a different institution's tool, with topics such as fractions, factoring and elementary trig: https://math.arizona.edu/~dsl/rur/rurci.htm [5.1] [5.2] [5.4]
Other non-MIT review options are Columbia's Precalculus Bootcamp, which covers algebra, radicals, exponents and logarithms, and trigonometry with worksheets, and Paul's Online Notes. The Lamar review is in problem-set form with detailed solutions, and it notes that algebra often makes up much of a calculus problem. Links: https://www.math.columbia.edu/programs-math/undergraduate-program/calculus-classes/calculus-readiness/ and https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/AlgebraTrig.aspx [6.1] [6.6] [7.3] [7.5]
Practical plan: do the six exercises, then start 18.01SC. As you meet a gap, such as a function domain or trigonometry, pause and review it. This workflow is reasoned guidance, and neither the exercises nor the review resources guarantee success. [1.1] [1.2]

No MIT OCW page in the supplied evidence is a dedicated algebra or trigonometry diagnostic, so these exercises are original and uncalibrated.

The trigonometry clip evidence is only a search snippet, so its contents are unverified.