Variables, Expressions, and Algebra as a Superpower

Warm-Up: Mental Math Sprint

Solve each of these in your head using the distributive property. Time yourself — can you finish all six in under 90 seconds?

1. 6 × 14 (think: 6 × 10 + 6 × 4)
2. 8 × 25
3. 7 × 98 (hint: 7 × 100 - 7 × 2)
4. 12 × 15
5. 99 × 6
6. 25 × 32

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Core Lesson: Algebra — The Language of Problem Solving

What Is a Variable?

In math, a variable is a letter that stands for an unknown number. You've already been using this idea — when you saw ___ in a sequence last week, that blank was acting as a variable.

But variables aren't just blanks. They're the foundation of algebra, which is arguably the most powerful tool in all of mathematics. Algebra lets you take a messy real-world problem and turn it into a clean equation you can solve.

Translating Words into Math

The hardest part of algebra isn't solving the equation — it's setting it up. Here's a cheat sheet for translating English into math:

| English Phrase | Math Operation | Example | |---|---|---| | "more than," "increased by," "total" | Addition (+) | "5 more than x" → x + 5 | | "less than," "decreased by," "fewer" | Subtraction (-) | "7 less than x" → x - 7 | | "times," "of," "product" | Multiplication (×) | "triple x" → 3x | | "per," "divided by," "split among" | Division (÷) | "x split among 4" → x / 4 | | "is," "equals," "gives," "results in" | Equals (=) | "x plus 3 is 10" → x + 3 = 10 |

Practice: Translate these sentences into equations:

1. "A number doubled, plus five, equals seventeen."
2. "If you subtract eight from three times a number, you get thirteen."
3. "The total cost of n shirts at $12 each, plus $5 shipping, is $65."

Solving Single-Variable Equations

Solving an equation means isolating the variable — getting x by itself on one side. The golden rule:

Whatever you do to one side, you must do to the other.

Let's solve 3x - 8 = 13 step by step:

| Step | Action | Result | |------|--------|--------| | Start | — | 3x - 8 = 13 | | Add 8 to both sides | Undo the subtraction | 3x = 21 | | Divide both sides by 3 | Undo the multiplication | x = 7 | | Check: 3(7) - 8 = 21 - 8 = 13 | Verified! | |

The Distance = Rate × Time Formula

One of the most useful equations in all of science:

d = r × t (Distance = Rate × Time)

This single formula connects three quantities. If you know any two, you can find the third:

• d = r × t — How far did it go?
• r = d / t — How fast was it going?
• t = d / r — How long did it take?

Example: A car travels at 60 mph for 2.5 hours. How far does it go?

d = 60 × 2.5 = 150 miles

Example: A runner covers 10 km in 50 minutes. What is their speed?

r = 10 / 50 = 0.2 km/min = 12 km/hr

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Hands-On Lab: Build a Python Equation Solver

We're going to write a program that solves equations of the form ax + b = c, where the user provides a, b, and c.

Program 1: Basic Equation Solver

# Solve ax + b = c for x
print("=== Equation Solver ===")
print("I'll solve equations of the form: ax + b = c\n")

a = float(input("Enter a (the coefficient of x): "))
b = float(input("Enter b (the constant added to x): "))
c = float(input("Enter c (what it equals): "))

if a == 0:
    if b == c:
        print("Every number is a solution! (The equation is always true.)")
    else:
        print("No solution! (The equation is never true.)")
else:
    x = (c - b) / a
    print(f"\nSolving: {a}x + {b} = {c}")
    print(f"Step 1: {a}x = {c} - {b} = {c - b}")
    print(f"Step 2: x = {c - b} / {a} = {x}")
    print(f"\nAnswer: x = {x}")

    # Verify the answer
    check = a * x + b
    print(f"\nCheck: {a} × {x} + {b} = {check} ✓" if check == c else f"\nCheck failed!")

Test it with: a = 3, b = -8, c = 13 — you should get x = 7.

Program 2: Distance-Rate-Time Calculator

# Distance-Rate-Time solver
print("=== Distance-Rate-Time Calculator ===")
print("d = r × t")
print()
print("What do you want to find?")
print("1. Distance (I know rate and time)")
print("2. Rate (I know distance and time)")
print("3. Time (I know distance and rate)")

choice = input("\nYour choice (1, 2, or 3): ")

if choice == "1":
    r = float(input("Enter rate (speed): "))
    t = float(input("Enter time: "))
    d = r * t
    print(f"\nDistance = {r} × {t} = {d}")

elif choice == "2":
    d = float(input("Enter distance: "))
    t = float(input("Enter time: "))
    if t == 0:
        print("Error: time cannot be zero!")
    else:
        r = d / t
        print(f"\nRate = {d} / {t} = {r}")

elif choice == "3":
    d = float(input("Enter distance: "))
    r = float(input("Enter rate (speed): "))
    if r == 0:
        print("Error: rate cannot be zero!")
    else:
        t = d / r
        print(f"\nTime = {d} / {r} = {t}")

else:
    print("Invalid choice!")

Program 3: Word Problem Translator

This program walks through a word problem step by step:

# Shirt problem: 12n + 5 = budget
print("=== T-Shirt Budget Problem ===")
print("Each shirt costs $12. Shipping is $5.\n")

budget = float(input("What is your total budget? $"))

# Solve: 12n + 5 = budget → n = (budget - 5) / 12
if budget < 5:
    print("You can't even afford shipping!")
else:
    n_exact = (budget - 5) / 12
    n_whole = int(n_exact)  # Can only buy whole shirts!
    leftover = budget - (12 * n_whole + 5)

    print(f"\nEquation: 12n + 5 = {budget}")
    print(f"12n = {budget - 5}")
    print(f"n = {n_exact:.2f}")
    print(f"\nYou can buy {n_whole} shirt(s).")
    print(f"Total cost: ${12 * n_whole + 5:.2f}")
    print(f"Money left over: ${leftover:.2f}")

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Challenge: The Classic Train Problem

Two trains leave different cities heading toward each other. City A and City B are 360 miles apart.

• Train A leaves City A heading east at 80 mph.
• Train B leaves City B heading west at 100 mph.
• They leave at the same time.

Part 1: How long until the trains meet? Solve by hand using algebra.

Hint: If Train A travels a distance d₁ and Train B travels d₂, then d₁ + d₂ = 360. Use d = r × t, and remember they travel for the same amount of time.

Part 2: Write a Python program to verify your answer. Bonus: Make the program animate the trains approaching each other.

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Resources

• Khan Academy: Intro to Variables — clear video lessons on variables and expressions
• Python Basics — W3Schools — reference for Python syntax
• Replit — online code editor
• Math is Fun: Algebra — great reference for equation-solving techniques
• Distance, Rate, Time Problems — Purplemath — more practice with d = r × t problems