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Write and interpret numerical expressions
Learn step-by-step with practice exercises built right in.
A numerical expression is a math problem with numbers and operations (+, −, ×, ÷) but NO equal sign.
Examples of numerical expressions:
NOT expressions (these are equations):
Key: Expressions show a calculation but don't have an answer written yet!
Numbers: 3, 15, 100, etc. Operations: +, −, ×, ÷ Grouping symbols: ( ), [ ], { }
Example: (12 + 3) × 5 - 2
Evaluate the expression: 15 + 8 - 3
Work from left to right since we only have addition and subtraction.
Step 1: Add first 15 + 8 = 23
Step 2: Subtract 23 - 3 = 20
Answer: 20
Write a numerical expression for: "five times the sum of 6 and 4"
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To evaluate means to find the value (solve it).
Example: Evaluate 7 + 3
7 + 3 = 10
The value of the expression is 10 ✓
To evaluate expressions with multiple operations, use PEMDAS!
P - Parentheses E - Exponents M - Multiplication D - Division A - Addition S - Subtraction
Example: Evaluate 6 + 3 × 2
Wrong way: 6 + 3 = 9, then 9 × 2 = 18 ✗
Right way: 3 × 2 = 6, then 6 + 6 = 12 ✓
Always multiply BEFORE adding!
Example 1: 8 + 5 Value = 13 ✓
Example 2: 20 - 7 Value = 13 ✓
Example 3: 6 × 4 Value = 24 ✓
Example 4: 15 ÷ 3 Value = 5 ✓
Easy! Just do the one operation.
Example: 12 + 5 + 3
Work left to right: 12 + 5 = 17 17 + 3 = 20 ✓
Example: 20 − 8 − 4
Work left to right: 20 − 8 = 12 12 − 4 = 8 ✓
Work left to right
Example: 15 + 7 − 3
15 + 7 = 22 22 − 3 = 19 ✓
Work left to right
Example: 20 ÷ 4 × 2
20 ÷ 4 = 5 5 × 2 = 10 ✓
Example: 5 + 3 × 4
Multiply first: 3 × 4 = 12 Then add: 5 + 12 = 17 ✓
Example: 18 ÷ 3 + 2
Divide first: 18 ÷ 3 = 6 Then add: 6 + 2 = 8 ✓
Parentheses tell you do this first!
Example: (5 + 3) × 4
Step 1: Parentheses first 5 + 3 = 8
Step 2: Multiply 8 × 4 = 32 ✓
Compare without parentheses: 5 + 3 × 4 3 × 4 = 12 5 + 12 = 17 ✓
Different answer! Parentheses matter!
Addition (+):
Subtraction (−):
Multiplication (×):
Division (÷):
Words: "The sum of 8 and 5" Expression: 8 + 5 ✓
Words: "The product of 6 and 4" Expression: 6 × 4 ✓
Words: "12 decreased by 7" Expression: 12 − 7 ✓
Words: "20 divided by 4" Expression: 20 ÷ 4 ✓
Words: "3 times the sum of 5 and 2" Expression: 3 × (5 + 2) ✓ (Parentheses show "sum of 5 and 2" happens first!)
Example 1: Evaluate (15 − 3) ÷ 4 + 5
Step 1: Parentheses 15 − 3 = 12
Step 2: Division 12 ÷ 4 = 3
Step 3: Addition 3 + 5 = 8 ✓
Example 2: Evaluate 20 − 4 × 3 + 6
Step 1: Multiply (M comes before A and S) 4 × 3 = 12
Step 2: Now we have: 20 − 12 + 6
Step 3: Work left to right 20 − 12 = 8 8 + 6 = 14 ✓
Example: 5 × [(12 − 4) ÷ 2]
Step 1: Innermost parentheses 12 − 4 = 8
Step 2: Now: 5 × [8 ÷ 2]
Step 3: Division inside brackets 8 ÷ 2 = 4
Step 4: Multiply 5 × 4 = 20 ✓
Rule: Work from the inside out!
Different expressions can have the same value.
Example:
All these expressions equal 15, so they're equivalent! ✓
The distributive property lets you rewrite expressions.
Example: 3 × (4 + 2)
Method 1: Parentheses first 3 × (4 + 2) = 3 × 6 = 18 ✓
Method 2: Distribute (multiply each term inside) 3 × (4 + 2) = (3 × 4) + (3 × 2) = 12 + 6 = 18 ✓
Both give the same answer!
General form: a × (b + c) = (a × b) + (a × c)
Shopping: You buy 3 shirts at 8 each. Total cost? Expression: 3 × 12 + 2 × 8 Value: 36 + 16 = $52 ✓
Lunch: You have 7 on lunch and 8 ✓
Sharing: 24 cookies divided equally among 4 friends, then each friend gets 2 more. Expression: 24 ÷ 4 + 2 Value: 6 + 2 = 8 cookies each ✓
❌ Mistake 1: Ignoring order of operations Wrong: 5 + 3 × 2 = 8 × 2 = 16 ✗ Right: 5 + 3 × 2 = 5 + 6 = 11 ✓
❌ Mistake 2: Forgetting parentheses change everything (5 + 3) × 2 ≠ 5 + 3 × 2 16 ≠ 11
❌ Mistake 3: Working right to left instead of left to right For same-level operations, go LEFT to RIGHT!
Challenge: Write three different expressions that equal 24.
Possible answers:
Many possibilities! ✓
Step-by-step process:
Problem 1: Evaluate 12 ÷ 4 + 5 12 ÷ 4 = 3 3 + 5 = 8 ✓
Problem 2: Evaluate 7 × (3 + 2) 3 + 2 = 5 7 × 5 = 35 ✓
Problem 3: Evaluate 18 − 6 ÷ 3 6 ÷ 3 = 2 18 − 2 = 16 ✓
Problem 4: Write an expression for "8 more than the product of 5 and 3" Expression: 5 × 3 + 8 Or: (5 × 3) + 8 ✓
Expression: 5 + 3 × 2 (no equal sign)
Equation: 5 + 3 × 2 = 11 (has equal sign)
Remember: Expressions don't have = signs!
Numerical Expression:
To Evaluate:
Key Points:
Remember: Expressions are like recipes - they tell you what to do, step by step! ✓
Break down the phrase:
Expression: 5 × (6 + 4)
We can also evaluate it: 5 × (6 + 4) = 5 × 10 = 50
Answer: 5 × (6 + 4) or 5(6 + 4)
Simplify: 3(4 + 2) using the distributive property.
The distributive property says: a(b + c) = ab + ac
Apply it to 3(4 + 2):
Step 1: Multiply 3 by each number inside the parentheses 3 × 4 = 12 3 × 2 = 6
Step 2: Add the products 12 + 6 = 18
Check: 3(4 + 2) = 3(6) = 18 ✓
Answer: 18
Write a numerical expression for: "20 decreased by the product of 3 and 4"
Break down the phrase:
Expression: 20 - (3 × 4)
Evaluate: 20 - (3 × 4) = 20 - 12 = 8
Answer: 20 - (3 × 4) or 20 - 3 × 4
Are these expressions equivalent? Explain. Expression A: 2(5 + 3) Expression B: 2 × 5 + 3
Let's evaluate both expressions:
Expression A: 2(5 + 3) Step 1: Parentheses first 5 + 3 = 8 Step 2: Multiply 2 × 8 = 16
Expression B: 2 × 5 + 3 Step 1: Multiply first (PEMDAS) 2 × 5 = 10 Step 2: Add 10 + 3 = 13
Expression A = 16 Expression B = 13
Answer: NO, they are NOT equivalent. The parentheses in Expression A make a big difference! However, 2(5 + 3) IS equivalent to 2 × 5 + 2 × 3 (distributive property).