Algebra & Data Skills for a 15-year-old: Units, Expressions, Equations, Inequalities, and Interpreting Data (Common Core HS)

Lesson and practice for a 15-year-old that teaches units and quantities, seeing structure in expressions, creating and solving linear equations and inequalities, and interpreting categorical and quantitative data. Aligned to Common Core HS standards N-Q, A-SSE, A-CED, A-REI, and S-ID.

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Overview and learning goals

This lesson teaches important high-school algebra and data skills for a 15-year-old. By the end you should be able to:

  • Use units to understand problems and choose appropriate quantities (N-Q.1, N-Q.2).
  • Read and rewrite algebraic expressions by identifying their parts and structure (A-SSE.1, A-SSE.2, A-SSE.3).
  • Create equations from word problems and known formulas, then rearrange to solve for a chosen variable (A-CED.1, A-CED.4).
  • Solve linear equations and inequalities, and explain steps using inverse operations and properties (A-REI.1, A-REI.3).
  • Create, display, and interpret categorical and quantitative data representations and compare distributions using center and spread (S-ID.1, S-ID.2, S-ID.3).

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

Key idea: Names for quantities and units guide how you set up and solve a problem. Always define a variable with a unit.

Steps:

  1. Read the problem and identify what is being asked. Choose a variable and include its unit. Example: "Let d be distance in kilometers."
  2. Write relationships using units. For example, speed = distance/time, so d = v * t where d is km, v is km/h, t is hours.
  3. Check units at the end to make sure your answer makes sense.

Quick example: A bike travels at 12 km/h for 2.5 hours. What distance did it cover?

Let d = distance in km. d = v * t = 12 km/h * 2.5 h = 30 km. Units cancel correctly (km/h * h = km).

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

Key idea: Expressions are built from parts; recognizing those parts helps simplify or rewrite them.

Examples of "seeing structure":

  • Interpret parts: In 3(x+4), 3 is a scale factor multiplying the quantity (x+4).
  • Rewrite using structure: 4x^2 + 8x = 4x(x+2). You factored out the common factor 4x.
  • Use structure to substitute: If x^2 + 2x + 1 = (x+1)^2, you can replace one form with the other when convenient.

Step-by-step factoring example:

  1. Expression: 6x^2 + 9x
  2. Find common factor: both terms divisible by 3x
  3. Factor: 6x^2 + 9x = 3x(2x + 3)

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

Key idea: Translate words and real contexts into algebra, then solve or isolate variables.

Steps to create an equation:

  1. Define variables with units.
  2. Write relationships using known formulas or logic.
  3. Solve for the variable asked or rearrange the formula as required.

Example 1 (word problem): A concert hall has rows with 18 seats each and 7 rows. If x additional seats are added in a new row, write an expression for total seats and solve if total must be at least 160.

Let S = total seats. Existing seats = 18*7 = 126. New total S = 126 + 18 (if a full row added) or more generally 126 + x where x is number of extra seats. If each row must have 18 seats and adding one new full row gives 126 + 18 = 144, which is not yet 160. To reach 160 you need at least 34 more seats; if only full rows of 18 are allowed, two full rows add 36, giving 162 >= 160.

Example 2 (rearrange formula): Solve for t in distance formula d = vt + 3.

  1. Subtract 3: d - 3 = vt.
  2. Divide by v: t = (d - 3)/v.

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

Key idea: Use inverse operations and properties of equality/inequality to isolate variables and explain each step.

Rules to remember:

  • You can add/subtract the same number to both sides of an equation or inequality.
  • You can multiply/divide both sides by the same positive number. If you multiply/divide both sides of an inequality by a negative number, reverse the inequality sign.

Example equation: Solve 4x - 7 = 13

  1. Add 7 to both sides: 4x = 20
  2. Divide by 4: x = 5

Example inequality: Solve -2x + 5 > 1

  1. Subtract 5: -2x > -4
  2. Divide by -2 (reverse inequality): x < 2

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

Key idea: Choose the right display for the data, describe center and spread, and compare two distributions by shape, center, and spread.

Common displays:

  • Categorical: bar chart, pie chart
  • Quantitative: dot plot, histogram, box plot

Measures:

  • Center: mean and median.
  • Spread: range, interquartile range (IQR), standard deviation (idea).

Example: Two classes took the same test. Class A scores: 65, 70, 72, 75, 80. Class B scores: 72, 72, 73, 78, 87.

  • Mean A = (65+70+72+75+80)/5 = 362/5 = 72.4. Median A = 72.
  • Mean B = (72+72+73+78+87)/5 = 382/5 = 76.4. Median B = 73.
  • Interpretation: Class B has a higher mean and median and one higher outlier (87). Class A is slightly lower but more centered around low 70s.

Worked practice problems and step-by-step solutions

  1. Problem: A car uses gasoline at a rate of 6 liters per 100 km. How many liters does it use for a 350 km trip? (Use units and define variables.)

    Solution: Let L = liters, d = distance in km. Rate r = 6 L per 100 km. L = r * (d / 100) = 6 * (350/100) = 6 * 3.5 = 21 liters.

  2. Problem: Factor 12x^2 - 8x.

    Solution: Common factor = 4x. 12x^2 - 8x = 4x(3x - 2).

  3. Problem: Translate to an equation and solve: "Three more than twice a number is 17."

    Solution: Let n = the number. Equation: 2n + 3 = 17. Subtract 3: 2n = 14. Divide by 2: n = 7.

  4. Problem: Rearrange the formula A = (1/2)bh to solve for b.

    Solution: A = (1/2) b h. Multiply both sides by 2: 2A = b h. Divide by h: b = 2A/h.

  5. Problem: Solve the inequality 3( x - 2 ) < 9.

    Solution: Distribute: 3x - 6 < 9. Add 6: 3x < 15. Divide by 3: x < 5.

  6. Problem: Given data for number of books read by students last month: categorical counts by genre: Fiction 12, Nonfiction 5, Fantasy 8. Draw a bar chart (conceptually) and identify which genre was most read.

    Solution: A bar chart with each genre on x-axis and counts on y-axis shows Fiction highest (12), so Fiction most read.

Practice set (try these)

  1. Define variable and solve: A runner's average speed is 9 km/h. How long to run 15 km? (Give units.)
  2. Factor: 9y^2 - 6y.
  3. Write and solve: Five less than three times a number equals 16.
  4. Rearrange formula: Solve for v in s = ut + (1/2)vt^2 (think of v as a coefficient). Note: treat u,t,s as known.
  5. Solve inequality and explain each step: -4x + 2 >= 10.
  6. Given two small data sets, compute mean and median and compare: Set1: 2, 4, 6, 8. Set2: 1, 5, 5, 9.

Solutions (brief):

  1. Time = distance/speed = 15 km / 9 km/h = 15/9 h = 5/3 h = 1 hour 40 minutes.
  2. 9y^2 - 6y = 3y(3y - 2).
  3. Let n be number. 3n - 5 = 16 => 3n = 21 => n = 7.
  4. Equation: s = ut + (1/2) v t^2. Subtract ut: s - ut = (1/2) v t^2. Multiply by 2: 2(s - ut) = v t^2. Divide by t^2: v = 2(s - ut)/t^2.
  5. -4x + 2 >= 10. Subtract 2: -4x >= 8. Divide by -4 (flip inequality): x <= -2.
  6. Set1 mean = (2+4+6+8)/4 = 20/4 = 5, median = (4+6)/2 = 5. Set2 mean = (1+5+5+9)/4 = 20/4 = 5, median = (5+5)/2 = 5. Both have same mean and median, but Set2 has more spread (values 1 and 9). Use range or IQR to compare spread.

Study tips and next steps

  • Always start by defining variables and units; write them on your paper.
  • When you see an expression, ask: can I factor something out? Can I recognize a square or difference of squares?
  • When solving inequalities, remember to flip the inequality sign when multiplying or dividing by a negative.
  • For data, make quick plots (dot plots or box plots) to visualize center and spread before calculating numbers.

If you want, tell me which practice problems you tried and I will check your work step by step, or I can create a short quiz with instant feedback aligned to any of the specific standards listed.

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