Writing and Interpreting Numerical Expressions

Lesson 1 of 1: From Words to Symbols and Back

In this lesson:

• Translate verbal descriptions into expressions
• Interpret expressions without evaluating
• Write nested expressions for multi-step calculations

Goals for Writing and Interpreting Expressions

By the end of this lesson, you will:

1. Translate verbal descriptions into grouped expressions
2. Map operation words to correct symbols
3. Place grouping symbols to preserve order
4. Interpret an expression without evaluating
5. Compare two expressions by their structure
6. Recognize different structures yield different values

Review: Grouping Symbols You Already Know

You already know from 5.OA.A.1:

• Parentheses ( ) tell you what to compute first
• Brackets [ ] wrap around the next layer
• Braces { } wrap around everything else

Today you'll use these symbols to write expressions.

From Words to Symbols: The Translation Process

Operation Words and Their Symbols

• "the sum of A and B" → A + B
• "the product of A and B" → A × B
• "subtract A from B" → B - A (order reverses!)
• "divide A by B" → A ÷ B

Example: Add Then Multiply Step by Step

"Add 8 and 7, then multiply by 2"

• Step 1: Add 8 and 7 → (8 + 7)
• Step 2: Multiply by 2 → 2 × (8 + 7)

The word "then" signals the order — before "then" goes inside the group.

Watch the Reversal: Subtract From

"Subtract 3 from 10, then multiply by 4"

• Step 1: Subtract 3 from 10 → (10 - 3), not (3 - 10)
• Step 2: Multiply by 4 → (10 - 3) × 4

"From" tells you the starting number — 10 is where you begin.

Check Yourself: Write This Expression Now

Write an expression for this description:

"Divide 20 by 4, then add 9"

Think about it before advancing...

Answer: Division Goes Inside the Group

"Divide 20 by 4, then add 9"

• Step 1: Divide 20 by 4 → (20 ÷ 4)
• Step 2: Add 9 → (20 ÷ 4) + 9

Division goes inside the parentheses because it happens first.

Your Turn: Translate These Descriptions

Write an expression for each:

1. "Add 12 and 8, then divide by 5"
2. "Multiply 6 by 3, then subtract 7"
3. "Subtract 5 from 15, then multiply by 6"
4. "The product of 4 and the sum of 9 and 3"

Pause and write all four before advancing.

Answers: Four Verbal Descriptions to Expressions

1. (12 + 8) ÷ 5
2. (6 × 3) - 7
3. (15 - 5) × 6 — remember "from" reverses
4. 4 × (9 + 3)

Reading Structure, Not Computing Values

Without computing: "Three times as large as the sum of 18,932 and 921"

Paired Comparison: Times as Large

Expression A: 5 × (200 + 47)
Expression B: (200 + 47)

• Both contain the sub-expression (200 + 47)
• Expression A is five times as large as Expression B
• The "5 ×" tells you the relationship — no computation needed

Additive Versus Multiplicative: Know the Difference

Pair 1: (30 - 7) + 12 and (30 - 7)

• The first is 12 more than the second

Pair 2: 2 × (6 × 8) and (6 × 8)

• The first is twice as large as the second

"More than" = addition. "Times as large" = multiplication.

Describe This Expression Without Computing Anything

What does this expression mean? Do not evaluate.

Describe the relationship in words before advancing...

Answer: Seven Times the Sum Describes Everything

"Seven times as large as the sum of 43,291 and 8,706"

• The sub-expression is (43,291 + 8,706)
• The outer operation is 7 × — seven times as large
• Structure tells you everything without computing.

Your Turn: Describe These Relationships

For each pair, describe the relationship without evaluating:

1. 8 × (459 + 37) and (459 + 37)
2. (100 - 28) + 15 and (100 - 28)
3. (50 + 25) ÷ 3 and (50 + 25)

Write your descriptions, then advance.

Answers: Outer Operation Reveals the Relationship

1. 8 × (459 + 37) is eight times as large as (459 + 37)
2. (100 - 28) + 15 is 15 more than (100 - 28)
3. (50 + 25) ÷ 3 is one-third of (50 + 25)

Multi-Step Descriptions: The Step-Numbering Strategy

For descriptions with three or more steps, number the steps first:

• Step 1 → innermost group (parentheses)
• Step 2 → next layer (brackets)
• Step 3 → outermost operation

Break the description apart before writing any symbols.

Building Layers: Three Steps to Symbols

"Add 6 and 14, multiply the sum by 3, then subtract 10"

• Step 1: (6 + 14) — innermost group
• Step 2: [(6 + 14) × 3] — brackets wrap around Step 1
• Step 3: [(6 + 14) × 3] - 10 — final operation

Reverse Direction: Expression to Words

Read this expression and describe it in words:

• Innermost: (20 - 8) → "Subtract 8 from 20"
• Next layer: [(20 - 8) × 5] → "Multiply the difference by 5"
• Outermost: + 2 → "Then add 2"

Quick Check: Write a Nested Expression

Write an expression for:

"Multiply 5 by 3, add 7 to the product, then divide by 2"

Number your steps first, then write the expression.

Answer: Three Steps Build One Nested Expression

"Multiply 5 by 3, add 7 to the product, then divide by 2"

• Step 1: (5 × 3)
• Step 2: [(5 × 3) + 7]
• Step 3: [(5 × 3) + 7] ÷ 2

Re-read to verify each step matches the description.

Comparing Nested Expressions Without Computing Values

• Both share the sub-expression (20 - 8) × 5
• The first is 2 more than the second
• The "+ 2" on the outside tells you the relationship

Practice: Write, Read, and Compare These Expressions

1. Write: "Add 9 and 11, multiply by 4, then subtract 6"
2. Read in words: [(15 - 3) × 2] + 8
3. How do [(8 + 2) × 6] and (8 + 2) × 6 relate?

Answers: Write, Read, and Compare Results

1. [(9 + 11) × 4] - 6
2. "Subtract 3 from 15, multiply the difference by 2, then add 8"
3. They are the same expression — the brackets are optional when there is only one level of grouping

Match Each Description to Its Correct Expression

Mixed Practice: Write, Interpret, and Compare

1. Write: "The sum of 14 and 6, divided by 4"
2. Interpret: What does 5 × (827 + 3946) mean?
3. Compare: (40 + 10) × 2 and (40 + 10)
4. Write: "Subtract 8 from 20, multiply by 3, add 5"

Try all four, then advance for answers.

Answers: Mixed Practice Four Key Skills

1. (14 + 6) ÷ 4
2. "Five times as large as the sum of 827 and 3,946"
3. The first is twice as large as the second (× 2)
4. [(20 - 8) × 3] + 5

Five Key Ideas for Writing and Interpreting

• Read first — identify operations and order before writing
• "Subtract A from B" means B - A, not A - B
• Grouping symbols show what happens first
• Structure tells the relationship — no computing needed
• "More than" = addition; "times as large" = multiplication

Preview: Patterns and Relationships Come Next

Next lesson: Patterns and Relationships (5.OA.B.3)

• Generate two numerical patterns using given rules
• Identify relationships between corresponding terms

Exit ticket: Write an expression for "multiply the sum of 9 and 6 by 4." Then describe what 7 × (5382 + 1209) tells you without evaluating.