Using Variables to Represent Numbers

Grade 6 — Expressions and Equations

In this lesson:

• Use variables to represent unknown quantities and ranging values
• Translate verbal descriptions into algebraic expressions
• Interpret expressions as real-world situations

Learning Objectives for This Lesson

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

1. Use a variable to represent an unknown quantity in a real-world or mathematical problem, and write an expression for the situation
2. Distinguish between a variable representing a specific unknown value versus a variable that ranges over any number in a set
3. Translate real-world descriptions into algebraic expressions, and interpret algebraic expressions in real-world terms

Reviewing What You Already Know

You have written expressions with variables:

• — add 7 to some number
• — multiply by 3
• — twice , then add 5

Today's question: What does a variable mean in a real problem?

A Variable Can Be a Specific Unknown

Role 1: Unknown — represents one specific number not yet known.

Let = books Sarah started with.

is exactly one value — we solve to find it.

A Variable Can Range Over Many Values

Role 2: Ranging — the variable takes any value from a set.

Width = , length = .

• : length = 8
• : length = 13

describes a relationship, not one specific value.

Side-by-Side: Two Roles of One Variable

Quick Check: Ranging or Specific Unknown?

For each situation, decide: is a specific unknown or a ranging variable?

1. "A plumber charges $50 per hour. Write an expression for the cost of hours of work."

2. "A plumber charged $200 total. How many hours did she work?"

Think before the next slide.

Quick Check: Role Answers Revealed

1. "Cost of hours" → Ranging variable →

2. "$200 total — how many hours?" → Specific unknown → ,

Key insight: Same expression . The context tells you: evaluate or solve.

Translation Vocabulary for Writing Expressions

These words map verbal phrases to operations:

Example 1: Writing an Additive Expression

"A number increased by 7"

Let = the number.

• "Increased by 7" → add 7

"The sum of and 8" →

"12 less than " →

Example 2: Writing a Multiplicative Expression

"The cost of tickets at $9 each"

• tickets × $9 each =

Convention: write the coefficient first → , not

"Half of " → or

"The product of 7 and " →

Example 3: Writing a Two-Operation Expression

"5 more than twice the number of apples"

Let = number of apples.

• Step 1: "twice the number" →
• Step 2: "5 more than" → add 5

"4 more than the product of 3 and " →

Example 4: Writing a Division Expression

"The distance split equally among 3 cars"

Let = total distance.

• "Split equally among 3" → divide by 3

"A number divided by 6, then increased by 1" →

Quick Check: Translate Three Verbal Phrases

Translate each phrase into an algebraic expression:

1. "8 fewer than "
2. "The product of 6 and , decreased by 4"
3. "A recipe uses 2 cups per serving. Expression for servings."

Write your answers, then check the next slide.

Quick Check: Three Expression Answers

1. "8 fewer than " → (y is the starting quantity)

2. "Product of 6 and , decreased by 4" →

3. "Flour for servings" →

Watch Out: Order Matters in Subtraction

Ask: "What is the starting quantity?" — it goes first.

Correcting the Subtraction Order Error

Error: "5 less than " →

Correct: Start with , remove 5 →

Check : 5 less than 10 = 5.
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Reading Expressions as Real-World Situations

Expression:

Interpretation: "A box has 4 rows of donuts, plus 2 extras."

Also works: "A plumber charges $4 per pipe plus a $2 fee."

Any story with the structure multiply n by 4, then add 2 fits.

One Independent Unknown Needs One Variable

"A plank cut in two: first piece is 4 ft. Expression for second piece?"

Error: Using and for both pieces.

Correct: Let = total length. Second piece =

Rule: One independent unknown → one variable.

Independent Practice: Write the Expressions

Write an expression for each situation:

1. Taxi: $3 flat plus $2 per mile, for miles.
2. "9 more than half of "
3. You have coins; friend has 3 times as many.

Bonus: Write a word problem for .

Try all, then check next slide.

Practice Answers: Check Your Work Here

1. Taxi:

2. Half of , plus 9:

3. Friend's coins:

Bonus: "3 rows of cars, plus 4 motorcycles" →

Key Takeaways for This Lesson

• Unknown: one specific value, found by solving an equation
• Ranging: any value in a set, described by an expression
• "5 less than " = , not
• Coefficient first: by convention (not )
• One unknown → one variable

What Comes Next in the Unit

Next: 6.EE.B.7 — Writing and solving equations from word problems

• Use today's expressions — set them equal to a value
• Solve: → find the one value of

Today's ranging expressions become equations when you know the total.