Grade 6 • Unit 6 HyperDoc

Inside the Expression Engine: Powers, Variables & the Distributive Property

You have just been hired as a coder at Pixel Forge Games. Game designers do not know the final numbers yet — how many coins, how many lives, how many players — so they write the rules with letters that stand for those unknown numbers. Your job: read, build, and find the value of the expressions that power the game. Hoy eres programador en Pixel Forge Games: vas a leer, construir y evaluar expresiones.

Standard: 6.EE.A.1–4 Topic: Expressions & Exponents MCAP-aligned Work time: ~45 min

1 Engage

Hook & essential question

In your new game, every player starts with the same number of gold coins. The designers have not decided the starting amount yet, so they call it g. Right now there are 4 players in the match.

g g g g 4 players · each starts with g coins

Each chest holds g coins. There are 4 of them.

Quick think (no calculator): Write one short expression for the total number of coins in the whole match. Then decide: if g = 50, is the total closer to 54, 200, or 2500? Jot your guess and one reason. You will test it in the Apply section.

Essential question: How can a few letters and symbols describe a rule that still works no matter what numbers we plug in later?

2 Explore

Investigate before the lesson

Explore each tool below for a few minutes. As you go, hunt for one big idea: an expression is a recipe — it tells you what to do, and you only get a number once you decide what the letter equals.

🎮 Expression Engine (3D Game) Feed number-blocks and letter-blocks into the machine and watch it build & evaluate expressions. 📐 Unit 6 Math Hub Worked examples for exponents, evaluating, and the distributive property. 📖 Unit 6 Math Novel See variables and powers solve a story problem in pictures.

Power-up experiment: why exponents save space

A "shield power-up" doubles its strength every second: 2 → 2×2 → 2×2×2 … Writing all those 2's gets long fast. An exponent is just shorthand for "multiply this number by itself again and again."

2¹ = 2 2² = 4 2³ = 8 2⁴ = 16

The base (2) stays the same; the exponent counts the factors.

Notice: 2⁴ = 2 × 2 × 2 × 2 = 16 — four 2's, not "2 times 4." The exponent counts how many times you multiply, it is not a regular multiplier.

3 Explain

The math, in plain language

An expression is a math phrase made of numbers, variables, and operations — and it has no equals sign. Below are the key words for this unit, each with a quick example.

variable
a letter that stands for an unknown number · la variable
coefficient
the number multiplied by a variable — in 3b it is 3 · el coeficiente
term
a single number or variable part, separated by + or − · el término
base & exponent
in 5³, 5 is the base, 3 is the exponent · la base y el exponente
evaluate
substitute a value for the letter and compute · evaluar
equivalent
two expressions that give the same value for every number · equivalente

Read the parts of an expression

3n + 7 coefficient variable constant 2 terms: 3n and 7

3n + 7 has 2 terms.

Write with exponents

5 × 5 × 5 = 5³

Count the equal factors. Three 5's becomes 5³ ("5 to the third power"). Then it equals 125.

Evaluate by substitution

4x → 4(3) = 12

Replace the letter with its value, then follow the order of operations. If x = 3, then 4x = 12.

Order of operations

2 + 3² = 2 + 9 = 11

Powers come before adding. Do 3² = 9 first, then add 2 — the answer is 11, not 25.

Distributive property

3(x + 5) = 3x + 15

Multiply the outside number by each term inside. This builds an equivalent expression.

Worked example — total coins in the match

Back to Engage: 4 players, each with g coins.

  1. Write the expression: 4g (means 4 × g).
  2. Evaluate when g = 50: 4g = 4 × 50 = 200 coins.
  3. So the answer was closest to 200. (How did your Engage guess do?)

⚠ Watch out for these traps

  • 3² is 9 (3 × 3), not 6 (3 × 2).
  • 3(x + 5) is 3x + 15, not 3x + 5 — distribute to both terms.
  • An expression has no equals sign. With an equals sign it becomes an equation.
Level 1 · support Step-by-step helper / Ayuda paso a paso
  1. Exponents: say it out loud — "5 to the third means three 5's multiplied." Write 5 × 5 × 5 first, then find the value.
  2. Evaluate: wherever you see the letter, draw a box and write the number inside, e.g. 4x → 4(▢). Then compute.
  3. Distribute: draw two arrows from the outside number to each term inside the parentheses, then multiply along each arrow.
  4. Lee la expresión en voz alta. Dibuja una caja para la letra y escribe el número adentro.
Level 2 · enrichment Push your thinking

Are 2(x + 3) and 2x + 6 the same expression, or just equal for one value of x? Test both with x = 0, 1, 5. If they match for every value, explain why the distributive property guarantees it — without testing more numbers.

4 Apply

Show what you can do — auto-checked

1. Write an expression for the total coins: 4 players, each with g coins. (Use the coefficient and the letter, like 4g.)

2. Evaluate 4g when g = 50. (number only)

3. The shield power-up multiplies 5 × 5 × 5. Write this using an exponent. (Type with the ^ key, like 5^3.)

4. Order of operations. Evaluate 2 + 3². (Do the power first. Number only.)

5. In the score rule 7m + 4, what is the coefficient of the variable?

6. Apply the distributive property to expand 5(x + 3). (Write the full expression, like 5x+15.)

5(x + 3) 5 × x 5 × 3

7. Which expression is equivalent to 6(n + 2)?

Level 1 · support Sentence starters for explaining

"I wrote ____ as an expression because the coefficient is ____ and the variable is ____."

"To evaluate, I replaced the letter ____ with the number ____, then I ____. My answer is ____."

"Escribí ____ como expresión porque el coeficiente es ____ y la variable es ____. Para evaluar, reemplacé la letra ____ con el número ____."

Level 2 · enrichment Reverse it & factor

A teammate wrote 8x + 12. Work the distributive property backwards: pull out the greatest common factor to write it as __(__ + __). Then explain how you could check your factored form is equivalent.

Teacher Notes & Answer Key (not printed)

Stage-by-Stage Notes (Engage → Explore → Explain → Apply → Reflect)

  • Engage: a game cost rule motivates writing an expression.
  • Explore: evaluate expressions and powers for different inputs.
  • Explain: exponents = repeated multiplication; distributive property generates equivalent forms.
  • Apply: answer key below.
  • Reflect: students explain why 5² ≠ 5 × 2.

Apply — Answer Key

  • Q1 — Write expression: 4 players each with g → 4g.
  • Q2 — Evaluate 4g (g = 50): 200.
  • Q3 — Power: a 5×5×5 stack → 5³ (= 125).
  • Q4 — Order of operations: 11.
  • Q5 — Coefficient of 7m (MC): answer c.
  • Q6 — Distribute 5(x + 3): 5x + 15.
  • Q7 — Equivalent (MC): answer b.

Standard

CCSS 6.EE.A.1–3.

5 Reflect

Think about your thinking

A. In your own words, what is the difference between an expression and an equation? Give one example of each.

B. Someone says "3² means 3 × 2 = 6." Explain why that is wrong, and what 3² really means.

C. One thing that is now clear to me / Una cosa que ahora entiendo bien:

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6 Extend

Optional challenge project

Design a scoring rule for Pixel Forge Games

Invent a small game. Write a scoring expression that uses at least:

  1. one variable with a coefficient (like 10c for coins),
  2. one exponent (like 2ˡ or l² for a level bonus),
  3. and a constant.

Then pick values for your variables, evaluate the score, and use the distributive property to rewrite one part of your rule in an equivalent form. Be ready to defend that the two forms are truly equivalent.

Inventa un juego, escribe una expresión de puntaje con una variable, un exponente y una constante; evalúala y reescribe una parte usando la propiedad distributiva.