Algebra & Quantities — A friendly step-by-step guide (age 13)

This lesson covers four ideas you will use a lot in high school math and real life:

• Working with quantities and units (using units to solve problems)
• Seeing structure in expressions and rewriting them (factoring and rearranging)
• Creating and reasoning with equations (writing equations from words and explaining each solving step)
• Interpreting categorical and quantitative data (comparing distributions using shape, center, and spread)

1) Quantities and units — use units to solve problems

Key idea: Always keep track of units (meters, seconds, dollars, etc.). You can often solve problems by treating units like labels that must cancel or match.

Example 1 — Speed and units:

A car travels 150 kilometers in 3 hours. What is its speed in kilometers per hour (km/h)?

Step 1: Write quantity as fraction with units: 150 km ÷ 3 h
Step 2: Compute the number: 150 ÷ 3 = 50
Step 3: Attach the unit: 50 km/h
Answer: 50 kilometers per hour.

Example 2 — Converting units (unit cancellation):

Convert 90 meters per minute to meters per second.

90 meters / minute × (1 minute / 60 seconds) = 90 / 60 meters / second = 1.5 m/s

We multiplied by a form of 1 (1 minute / 60 seconds) to cancel the minute unit.

Tip on accuracy (rounding): Choose how many decimal places based on the situation. If measuring people’s heights, you usually round to the nearest centimeter. If measuring galaxy distances, many digits are not meaningful. That’s why we think about the precision of measurements.

2) Seeing structure in expressions — rewrite and factor

Key idea: Look for patterns in expressions so you can rewrite them into simpler or more useful forms.

Common patterns:

• Common factor: 6x + 9 = 3(2x + 3)
• Grouping: ax + ay + bx + by = (a + b)(x + y)
• Perfect square, difference of squares — you will meet these later.

Example — factoring out the greatest common factor (GCF):

Rewrite 12x + 8.
GCF of 12 and 8 is 4, so 12x + 8 = 4(3x + 2).

Why useful? If you want to divide, solve, or simplify, the factored form is often easier.

Example — rewriting to see structure:

Rewrite 2x + x^2 in order that shows a structure: x^2 + 2x = x(x + 2).

This shows the expression is x times (x + 2), which can help if solving equations or simplifying.

3) Creating equations and reasoning with each solving step

Key idea: Translate a written situation into an equation. Then solve it by writing each algebra step and why it is allowed (like add the same number to both sides).

Example — creating an equation from words:

"Twice a number plus 3 equals 11."  
Let the number be n.  
Equation: 2n + 3 = 11

Solve step-by-step with reasons:
1) 2n + 3 = 11          (start)
2) 2n = 11 - 3          (subtract 3 from both sides; allowed because equality stays true)
3) 2n = 8
4) n = 8/2              (divide both sides by 2)
5) n = 4                (final answer)

Example — a word problem that needs modeling:

Sam has $5 more than twice what Lea has. Together they have $41. How much does Lea have?

Let L = amount Lea has.
Sam = 2L + 5
Equation for total: L + (2L + 5) = 41
Solve:
L + 2L + 5 = 41           (combine terms)
3L + 5 = 41               (subtract 5)
3L = 36
L = 12
Lea has $12.

Reasoning with inequalities works the same but watch signs when multiplying/dividing by a negative number (it flips the inequality sign).

4) Interpreting categorical and quantitative data — shape, center, spread

Key idea: When you look at a set of numbers (quantitative data), compare distributions by:

• Shape — is it symmetric, skewed left, skewed right, or has outliers?
• Center — typical value: mean (average) or median (middle value)
• Spread — how much values vary: range, interquartile range (IQR), or standard deviation

Example — two small datasets:

Dataset A: 2, 3, 4, 5, 6   (symmetric-ish)
Mean = (2+3+4+5+6)/5 = 20/5 = 4
Median = middle value = 4
Range = 6 - 2 = 4

Dataset B: 1, 2, 3, 4, 10  (has an outlier: 10)
Mean = (1+2+3+4+10)/5 = 20/5 = 4
Median = 3
Range = 10 - 1 = 9

Interpretation: Both means are 4, but medians differ (4 vs 3) and Dataset B is more spread out and skewed by the outlier 10. If you want a measure not affected much by outliers, use the median.

When comparing groups (like test scores for two classes), describe differences in shape, center, and spread and say which measure (mean or median) is better to use depending on outliers.

Practice Problems (Try these!)

1. Units: A runner covers 5 kilometers in 20 minutes. What is the speed in meters per second? (Remember 1 km = 1000 m.)
2. Rewrite / factor: Factor 9x + 6.
3. Create an equation: "Three more than four times a number is 23." Write and solve the equation.
4. Reason each step: Solve 3(x - 2) = 9. Write each algebraic step and say why it is allowed.
5. Data: Two sets of ages in a small club: Group 1 = {12, 13, 13, 14, 14}. Group 2 = {11, 12, 13, 15, 17}. Compute mean and median of each and say which group has more spread and whether either has an outlier.

Answers and explanations

1) 5 km in 20 min = 5000 m / 1200 s (because 20 min × 60 = 1200 s). 5000/1200 = 25/6 ≈ 4.1667 m/s.
   So about 4.17 m/s.

2) 9x + 6 = 3(3x + 2)  (factor out GCF = 3)

3) Let n = the number. Equation: 4n + 3 = 23.  
   4n = 20  (subtract 3), n = 5  (divide by 4)

4) 3(x - 2) = 9
   Step 1: 3(x - 2) = 9                          (start)
   Step 2: x - 2 = 9 ÷ 3 = 3                     (divide both sides by 3; allowed because dividing by nonzero keeps equality)
   Step 3: x = 3 + 2 = 5                         (add 2 to both sides)

5) Group 1: {12,13,13,14,14}
   Mean = (12+13+13+14+14)/5 = 66/5 = 13.2
   Median = middle value = 13
   Range = 14 - 12 = 2 (small spread)

   Group 2: {11,12,13,15,17}
   Mean = (11+12+13+15+17)/5 = 68/5 = 13.6
   Median = 13
   Range = 17 - 11 = 6 (larger spread)

   Interpretation: Group 2 has more spread and possibly higher values (17 might be considered a high value compared to others). No extreme outlier, but Group 2 is more spread out.

How this matches the Common Core standards you listed

• Number & Quantity — Quantities (HS): Using units and reasoning about precision (we used unit cancellation and discussed rounding/accuracy).
• Algebra — Seeing Structure in Expressions: We practiced recognizing structure and factoring expressions like pulling out the GCF and rewriting expressions.
• Algebra — Creating Equations & Reasoning with Equations and Inequalities: We wrote equations from word problems and showed step-by-step solutions with reasons for each algebraic move.
• Statistics & Probability — Interpreting Categorical & Quantitative Data: We compared data distributions by shape, center (mean/median), and spread and discussed when each measure is appropriate.

If you want, I can give you more practice problems at the right level, make a short quiz with answers, or explain any single step in more detail. Which part would you like to practice next?