Modeling with Expressions, Equations, Inequalities, and Data Interpretation — High School Math (Age 16)

Overview — What you'll learn

This lesson ties together: (1) interpreting quantities and units, (2) seeing and rewriting structure in expressions, (3) creating equations from word problems, (4) solving linear equations and inequalities and reasoning about solutions, and (5) summarizing and interpreting quantitative data (center and spread). Each step has clear examples and practice problems with solutions.

1. Quantities and units — think in ratios and unit analysis

Key idea: When a problem gives two related quantities (like cost and time, or ingredient amount and number of servings), write the rate as a ratio and use unit cancellation to scale.

Example: A recipe uses 2 cups of sugar to make 12 cookies. How much sugar per cookie?

1. Rate = 2 cups / 12 cookies = 1/6 cup per cookie.
2. For 30 cookies: sugar = (1/6) * 30 = 5 cups.

Always state units: cups per cookie, dollars per hour, kilometers per hour, etc.

2. Seeing structure in expressions

Key ideas: Expand and factor expressions to reveal structure. Recognize forms like a(b + c), ax + b, or difference of squares when appropriate. This helps when you translate a situation into an equation or when solving.

Example: Rewrite and factor.

Start with 4(2x - 3) + 5.

1. Distribute: 4(2x) - 4*3 + 5 = 8x - 12 + 5 = 8x - 7.
2. If you instead had 12x - 18, factor common factor 6: 12x - 18 = 6(2x - 3).

Seeing the factor 2x - 3 in both expressions tells you these expressions are related; factoring and expanding are inverse operations.

3. Creating equations from words

Key idea: Define variables, translate phrases to algebra, then build an equation or inequality from the relationships.

Translation tips:

• Let a variable represent the unknown (e.g., h = hours used).
• Identify fixed parts (flat fees) and variable parts (per-unit rates).
• Build an equation like total = fixed + rate * quantity or an inequality if a limit is involved.

Example: A bike rental charges a $8 registration fee plus $2.50 per hour. If you have at most $20, how many hours can you rent?

1. Let h = hours. Cost C = 8 + 2.5h.
2. Inequality for budget: 8 + 2.5h ≤ 20.
3. Solve: 2.5h ≤ 12 ⟹ h ≤ 12/2.5 = 4.8 hours.
4. Interpretation: You can rent up to 4.8 hours (if time is continuous). If hours must be whole, at most 4 hours.

4. Solving equations and inequalities — step by step

Linear equation example:

Solve 3(2x - 1) = 5x + 4.

1. Distribute: 6x - 3 = 5x + 4.
2. Collect x terms: 6x - 5x = 4 + 3 ⟹ x = 7.
3. Check: Left 3(2*7 -1)=3(13)=39; Right 5*7+4=35+4=39. Works.

Inequality example and important rule:

Solve -2x + 3 > 7.

1. Subtract 3: -2x > 4.
2. Divide by -2. Because we divided by a negative, flip the inequality: x < -2.
3. Interpretation: All x less than -2 satisfy the inequality.

5. Interpreting quantitative data (center and spread)

Key idea: To compare two data sets, compute measures of center (mean, median) and spread (range, interquartile range IQR). Use appropriate measures depending on skew/outliers: median and IQR are robust to outliers; mean and standard deviation are sensitive to outliers.

Example: Two classes took the same test. Scores (out of 100):

Class A: 72, 75, 78, 81, 85, 88, 92

Class B: 60, 70, 80, 82, 84, 86, 98

Compute center and spread:

• Mean A = (72+75+78+81+85+88+92)/7 = 571/7 ≈ 81.6
• Median A = middle value = 81
• Q1 (A) = median of lower half (72,75,78) = 75; Q3 (A) = median of upper half (85,88,92) = 88; IQR = 88 - 75 = 13
• Range A = 92 - 72 = 20
• Mean B = (60+70+80+82+84+86+98)/7 = 560/7 = 80
• Median B = 82
• Q1 (B) = 70; Q3 (B) = 86; IQR = 86 - 70 = 16
• Range B = 98 - 60 = 38

Interpretation:

• Centers: The class means are similar (81.6 vs 80). Medians are also close (81 vs 82).
• Spread: Class B is more variable (larger IQR and much larger range). Class B has an extreme high score (98) and a low score (60), increasing spread.
• Conclusion in context: Although average performance is similar, Class B has more varied performance with both lower low scores and a very high top score; Class A is more consistent.

Practice problems (with brief answers)

1. Translate and solve: A phone plan costs $12 per month plus $0.10 per text message. If you want the monthly bill to be no more than $25, how many texts can you send?
   Answer: 12 + 0.10t ≤ 25 ⟹ 0.10t ≤ 13 ⟹ t ≤ 130 texts.
2. Rewrite/explain: Show two algebraic forms are equal: 9x - 6 = 3(3x - 2).
   Answer: Expand right side 3(3x - 2) = 9x - 6. They are identical; factoring 9x - 6 = 3(3x - 2) shows common factor 3.
3. Data comparison: Two groups have test scores. Group 1: 55, 60, 65, 70, 75 Group 2: 50, 60, 70, 80, 90 Which group has greater spread? Which has higher median?
   Answer: Group1 median 65, Group2 median 70 (Group2 higher median). Range1 = 75-55 = 20, Range2 = 90-50 = 40 (Group2 greater spread).

How this maps to the Common Core standards you gave

• Number & Quantity (Quantity): interpreting and using units, rates, and scaling (writing rates and unit analysis).
• Algebra (Seeing Structure in Expressions): expanding, factoring, and recognizing structure in linear expressions and using that to manipulate equations.
• Algebra (Creating Equations; Reasoning with Equations & Inequalities): translating real situations to equations/inequalities, solving and interpreting solutions in context, and handling sign rules when solving inequalities.
• Statistics & Probability (Interpreting Data): summarizing single-variable data with center and spread, using median/IQR for skewed data and interpreting differences in context.

Next steps and practice suggestions

Work 6–8 problems that mix skills: (a) translate a short word problem into an equation, (b) solve an inequality with a negative coefficient, (c) expand and factor similar expressions, and (d) compute mean/median/IQR for small datasets and write a one-sentence interpretation comparing two groups.

If you want, give me one word problem or one dataset and I will walk through it step-by-step with you.