Use Variables to Represent Numbers

Grade 6 Expressions and Equations — 6.EE.B.6

In this lesson:

• Identify whether a variable is an unknown or a ranging value
• Write the "Let…" statement before any expression
• Translate verbal descriptions into algebraic expressions

What You Will Learn Today

By the end of this lesson, you should be able to:

1. Identify whether a variable is an unknown or a ranging value
2. State the variable's meaning before writing any expression
3. Translate verbal descriptions into algebraic expressions
4. Translate expressions back into verbal descriptions

Maria Earned $60 — How Many Hours?

Maria earns $12 per hour.

Last Tuesday she earned exactly $60.

Let = the number of hours she worked.

• Her earnings:
• We want to find

Can you figure out ? What does substitution tell you?

Same Expression — What Does h Mean?

Both cards use . What does represent in each?

A Variable Can Play Two Roles

• Unknown — one specific value to find

  • satisfies ; substitution works

• Ranging value — any number in a set

  • could be 10, 20, 30 — no single answer

Same expression . Different role. Different next step.

Always Define the Variable First

Before writing any expression, write a declaration:

"Let = the number of hours Maria works."

• If is unknown: the declaration sets up your solve
• If is ranging: the declaration describes the relationship

Without it, is ambiguous — it could mean solve or describe.

Quick Check: Unknown or Ranging?

Classify the variable in each scenario.

1. A bag holds marbles — 14 total. Let = marbles in bag.
2. A painter charges $25/hr. Let = hours this week.
3. A number squared equals 49. Let = the number.

One specific value, or could be any value?

The Role Determines Your Next Step

Ask first: "Specific value, or all possibilities?"

Four Patterns for Translating Phrases

English word order does not always match symbol order.

Additive Translation Goes Both Ways

Words → Expression:

• "6 more than " →
• "A number decreased by 4" →

Expression → Words:

• : a number increased by 6
• : 4 less than a number

Any valid description counts.

Predict: Which Expression Is Correct?

"4 less than " — which matches?

• A.
• B.

Pick A or B before turning the page.

Hint: "less than" tells you where to start.

Subtraction Order: Reference Goes First

"4 less than " → start at , take 4 away:

Check with :

• ✓ Four less than ten is six.
• ✗ Negative — wrong direction.

Multiplicative and Division Translation Patterns

"3 times a number" →

"A number divided by 2" →

Numerator rule:

• "Number divided by 2" — divided → on top
• "2 divided by a number" — 2 divided → on top

The divided quantity goes in the numerator.

Your Turn: Write Four Expressions

Write an expression. Define the variable first with a "Let…" statement.

1. "Seven more than a number"
2. "A number decreased by 9"
3. "The product of 5 and a number"
4. "A number divided by 4"

Try all four, then advance for the answers.

Two-Operation Expressions: Building Inside Out

"5 less than twice a number" →

Step 1: "Twice " →

Step 2: "5 less than that" →

Test : ✓

Wrong: at — fails the check.

Error Analysis: Spot the Mistake

A student wrote this for "4 less than a number":

Your tasks:

1. Apply the reasonableness check — use
2. Explain in one sentence why it is wrong
3. Write the correct expression

Work it out, then advance.

One Expression Describes Many Situations

Given:

Task: Write a word problem this could model.

Two student examples:

• "I bought packs of stickers (3 each) plus 2 extra"
• "A plumber charges $3/foot of pipe plus a $2 fee"

Both correct — one expression, many real-world stories.

Your Turn: Translate and Create

No scaffolding. Write your "Let…" statement first for each.

Part 1 — Write expressions:

1. "8 more than twice a number"
2. "A number divided by 3, decreased by 5"

Part 2 — Write a word problem for .

Try all three independently, then advance.

Two Common Mistakes Worth Remembering

Mistake 1: "6 less than " → ✗ → ✓

Reasonableness: gives , not 4.

Mistake 2: "Write an expression" → ✓ not ✗

No value asked? Stop at the expression.

What You Can Now Do

✓ Classify a variable: unknown or ranging

✓ Define first — "Let = …" — always

✓ Subtraction: "4 less than " →

✓ Two-operation: inside-out →

Reasonableness check catches subtraction reversal

Don't set equal unless a value is asked for

Next Lesson: Expressions Become Equations

In 6.EE.B.7 you will set expressions equal to a given value and solve.

The "define the variable" step from today is the setup.

Every equation you write in 6.EE.B.7 starts with a correct expression.