Number & Quantity, Expressions, Equations, Inequalities, and Data — Age 15 (AoPS Prealgebra 1 & 2)

Clear, step-by-step lessons for a 15-year-old on using units and defining quantities, spotting structure in expressions, creating and rearranging equations, reasoning with equations and inequalities, and interpreting categorical and quantitative data. Includes worked examples and practice problems with answers.

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Overview

These five ideas are tools you use all the time in algebra and problem solving. I'll explain each idea, give a step-by-step example, and then give short practice problems you can try.

1. Number and Quantity: Using units and defining quantities (N-Q.1, N-Q.2)

Key idea: Always keep track of units (meters, seconds, dollars) and name derived quantities clearly (speed = distance per time). Units help check if an answer makes sense.

Example 1 — Converting units:

  1. Problem: A car travels 90 kilometers in 2 hours. What is its speed in meters per second?
  2. Step 1: Find speed in km/h: 90 km / 2 h = 45 km/h.
  3. Step 2: Convert km/h to m/s. 1 km = 1000 m and 1 hour = 3600 s, so multiply by 1000/3600 = 5/18.
  4. Step 3: 45 * (5/18) = 45/18 * 5 = (2.5) * 5 = 12.5 m/s.
  5. Answer: 12.5 m/s.

Example 2 — Defining a quantity:

  1. Problem: You have 3 liters of paint and each liter covers 8 square meters. Define total area A covered.
  2. Step 1: Let A be the area in square meters. Then A = (liters) * (coverage per liter) = 3 * 8.
  3. Step 2: A = 24 square meters. Always write the unit: 24 m2.

Practice (try them):

  1. Convert 5 km/h to m/s. (Answer: 5 * 5/18 = 25/18 ≈ 1.39 m/s.)
  2. A recipe uses 200 grams of flour for 4 muffins. Define the flour per muffin and compute it. (Answer: 200/4 = 50 g per muffin.)

2. Seeing structure in expressions (A-SSE.1, A-SSE.2, A-SSE.3)

Key idea: Recognize patterns in expressions so you can simplify, factor, or rewrite them. Look for common factors, binomial products, and ways to group terms.

Example — Factoring by grouping:

  1. Problem: Factor the expression 6x + 9 - 2x^2 - 3x.
  2. Step 1: Rearrange terms by degree: -2x^2 + (6x - 3x) + 9 = -2x^2 + 3x + 9.
  3. Step 2: Factor out common factor -1 if helpful: -(2x^2 - 3x - 9).
  4. Step 3: Try factoring 2x^2 - 3x - 9. Look for numbers a and b with a*b = 2 * (-9) = -18 and a + b = -3. Numbers -6 and 3 work. Rewrite: 2x^2 - 6x + 3x - 9 = 2x(x - 3) + 3(x - 3) = (x - 3)(2x + 3).
  5. Step 4: Include the minus sign: Original = -(x - 3)(2x + 3) = (3 - x)(2x + 3).

Tip: Rewriting an expression can make solving equations easier or reveal geometry relationships.

Practice:

  1. Factor 4x^2 - 12. (Answer: 4(x^2 - 3) = 4(x - sqrt3)(x + sqrt3) if over reals you leave as 4(x^2 - 3) or factor as difference of squares when appropriate.)
  2. Expand (x + 5)(x - 2) and then factor it back. (Answer: x^2 + 3x - 10, factors to (x + 5)(x - 2).)

3. Creating equations and rearranging formulas (A-CED.1, A-CED.4)

Key idea: Translate word problems into equations. Then solve or rearrange formulas to get the variable you need.

Example — Translate a word problem:

  1. Problem: Three consecutive integers add up to 48. Find them.
  2. Step 1: Let the first integer be n. Then the next two are n+1 and n+2.
  3. Step 2: Equation: n + (n+1) + (n+2) = 48.
  4. Step 3: Combine: 3n + 3 = 48 → 3n = 45 → n = 15.
  5. Answer: 15, 16, 17.

Example — Rearranging a formula:

  1. Problem: The area of a rectangle is A = lw. Solve for w.
  2. Step 1: Start with A = l w.
  3. Step 2: Divide both sides by l (assuming l ≠ 0): w = A / l.
  4. Answer: w = A / l.

Practice:

  1. A total of $t is split: 40% to A and the rest to B. Write B in terms of t. (Answer: B = 0.6t.)
  2. Given the formula for distance d = rt (rate times time), solve for t. (Answer: t = d / r.)

4. Reasoning with equations and inequalities (A-REI.1, A-REI.3)

Key idea: Solve equations step-by-step and use inverse operations. For inequalities, treat like equations but reverse the inequality when multiplying or dividing by a negative number.

Example — Solving an equation:

  1. Problem: Solve 4(x - 2) = 3x + 8.
  2. Step 1: Expand left: 4x - 8 = 3x + 8.
  3. Step 2: Subtract 3x from both sides: x - 8 = 8.
  4. Step 3: Add 8 to both sides: x = 16.
  5. Check: Left 4(16 - 2) = 4*14 = 56, right 3*16 + 8 = 48 + 8 = 56.

Example — Solving an inequality:

  1. Problem: Solve -2(3x - 5) < 4.
  2. Step 1: Expand: -6x + 10 < 4.
  3. Step 2: Subtract 10: -6x < -6.
  4. Step 3: Divide by -6. Remember: dividing by a negative flips the inequality: x > 1.
  5. Answer: x > 1. (Graphically, all numbers greater than 1.)

Practice:

  1. Solve 5x + 2 = 22. (Answer: x = 4.)
  2. Solve 3 - 4x ≥ 11. (Work: 3 - 4x ≥ 11 → -4x ≥ 8 → x ≤ -2.)

5. Interpreting categorical and quantitative data (S-ID.1, S-ID.2, S-ID.3)

Key idea: Understand what different data displays tell you. For categorical data use counts or percentages; for quantitative data use center (mean, median), spread (range, IQR), and look for relationships (scatter plots, correlation).

Example — Categorical data:

  1. Problem: Survey of favorite fruit among 50 students: 20 like apples, 15 like bananas, 10 like grapes, 5 like oranges. What percent like apples?
  2. Step: Percent = 20/50 = 0.4 = 40%.

Example — Quantitative data (mean and median):

  1. Problem: Test scores: 70, 80, 85, 90, 95.
  2. Mean: (70 + 80 + 85 + 90 + 95) / 5 = 420 / 5 = 84.
  3. Median: the middle value when ordered = 85.

Example — Scatter plot idea:

If a scatter plot of hours studied vs test score shows points rising from left to right, that suggests a positive association: more hours tends to mean higher scores. Correlation does not prove causation.

Practice:

  1. Given data 10, 12, 12, 14, 20: find mean and median. (Answer: mean = 68/5 = 13.6, median = 12.)
  2. If 30% of 200 people prefer chocolate, how many is that? (Answer: 0.3 * 200 = 60 people.)

Putting it together: Sample multi-step problem

Problem: A bike rental charges a $5 base fee plus $8 per hour. You have $37. How many full hours can you rent the bike?

  1. Translate to an inequality. Let h = number of hours. Cost C = 5 + 8h. You need 5 + 8h ≤ 37.
  2. Solve: 8h ≤ 32 → h ≤ 4. So you can rent for up to 4 full hours.
  3. Interpret units: hours. Check by plugging back: cost = 5 + 8*4 = 37, fits exactly.

Quick checklist to use when solving problems

  • Write down units and label variables.
  • Look for structure in expressions to simplify or factor.
  • Translate words into equations carefully (watch keywords: total, difference, product, per).
  • Use inverse operations step-by-step when solving equations; flip inequality when multiplying/dividing by a negative.
  • When reading data, ask: Is the data categorical or quantitative? What does the center and spread tell me? Is there an association or causation?

Extra practice (answers below)

  1. Convert 72 km/h to m/s. (Answer: 72 * 5/18 = 20 m/s.)
  2. Factor 9x^2 - 6x. (Answer: 3x(3x - 2).)
  3. Write an equation: Five more than twice a number is 27. Find the number. (Equation: 2n + 5 = 27 → n = 11.)
  4. Solve inequality: 2(x + 3) > 10. (Work: 2x + 6 > 10 → 2x > 4 → x > 2.)
  5. From scores: 78, 82, 90, 90, 95, find mean, median, and mode. (Mean = 85.833..., median = (90+90)/2 = 90, mode = 90.)

If you want, tell me which of the five areas you struggle with most and I will give more focused practice problems and step-by-step solutions at your level.

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