Numerical Expressions

🔢 Numerical Expressions

Part 1 of 5 — What Is an Expression?

Topics in This Part

🔑 Key Concept: A numerical expression is a math phrase built from numbers and operation signs — like $3 + 4 \times 5$. It has a single value, but it is not a sentence: there is no equals sign telling you the answer yet.

Expression or Equation?

A numerical expression is a phrase: just numbers and operations.

An equation is a sentence: it has an equals sign ($=$) and says two things are equal.

Think of it like words. "Three apples plus two apples" is a phrase. "Three apples plus two apples equals five apples" is a sentence.

💡 Quick test: No equals sign? It's an expression. We evaluate an expression to find its value — but the expression itself is just the recipe, not the finished meal.

Concept Check 🎯

The Four Operations and Their Words

Expressions are built from the four basic operations. In word problems, each operation hides behind several "signal words":

For example, "the sum of $6$ and $4$" means $6 + 4$, and "the product of $3$ and $5$" means $3 \times 5$.

⚠️ Watch "less than" and "more than." "$5$ less than $12$" means $12 - 5$, not $5 - 12$. The number you start with is the bigger one mentioned, and you take away the other.

Signal Words 🎯

Reading an Expression Out Loud

You can also go the other way — read an expression and say it in words.

Being able to translate both directions — words to symbols and symbols to words — is the whole goal of this lesson's first half.

🔑 Coming up: In Part 2 you'll learn the order in which to do the operations, because $3 + 4 \times 5$ does not mean "do it left to right." The order matters, and there are rules.

Match Words to Symbols 🔽

Pick the expression that matches each phrase.

🔢 Numerical Expressions

Part 2 of 5 — Order of Operations

🔑 The Big Rule: When an expression has more than one operation, everyone must follow the same order so we all get the same answer. That order is the order of operations.

The Order: PEMDAS

Do operations in this order, from top to bottom:

A common way to remember it: Please Excuse My Dear Aunt Sally.

⚠️ Two tricky parts:

• Multiply and divide are a team — do them together, left to right, whichever comes first. $\div$ is not always after $\times$.
• Add and subtract are a team too — left to right.

Why the Order Matters

Take the expression $3 + 4 \times 5$.

Wrong way — straight left to right: $3 + 4 = 7,\quad 7 \times 5 = 35 \quad ❌$

Right way — multiply first: $4 \times 5 = 20,\quad 3 + 20 = 23 \quad ✓$

The correct value is $23$. If everyone just went left to right, some people would get $35$ and some would get $23$ — and math would stop working! The rules keep every answer the same.

💡 Multiplication is a shortcut for repeated addition, so it "happens" before plain addition. That's the idea behind doing $\times$ before $+$.

Concept Check 🎯

Worked Example: $15 - 2 \times 3 + 1$

There are no parentheses or exponents, so go straight to multiply/divide, then add/subtract.

$15 - 2 \times 3 + 1$

Step 1 — multiply: $2 \times 3 = 6$ $15 - 6 + 1$

Step 2 — add/subtract, left to right: $15 - 6 = 9$, then $9 + 1 = 10$ $= 10$

✅ Check the order: the $\times$ was done before either the $-$ or the $+$. Then $-$ and $+$ were done left to right. The value is $10$.

Which Operation Goes First? 🔽

For each expression, choose the step you must do first.

Your PEMDAS Checklist

Before you evaluate, scan the whole expression and ask, in order:

1. Any grouping symbols? Do what's inside first.
2. Any exponents? Do those next.
3. Any $\times$ or $\div$? Do them left to right.
4. Now just $+$ and $-$? Finish left to right.

⚠️ Don't auto-pilot left to right. The single most common mistake on these problems is adding or subtracting before you've cleared all the $\times$ and $\div$. Scan first, then compute.

Evaluate Each Expression 🧮

Use the order of operations. Enter just the final value.

1) $8 + 12 \div 4$ 2) $36 \div 6 + 2 \times 5$ 3) $40 - 5 \times 4 + 2$

🔢 Numerical Expressions

Part 3 of 5 — Grouping Symbols

🔑 The Power of Grouping: Parentheses $(\;)$, brackets $[\;]$, and braces $\{\;\}$ let you say "do this part first," even overriding the normal order. They are the P in PEMDAS.

Three Kinds of Grouping Symbols

All three do the same job — they say "work me out first" — but we nest them so the layers are easy to see:

When they are nested, work from the inside out — start with the innermost parentheses and move outward.

$\{\;[\;(\;\text{innermost}\;)\;]\;\}$

💡 Think of nested grouping like Russian nesting dolls: you open the smallest one first.

Parentheses First 🎯

Worked Example: Nested Grouping

Evaluate $2 \times [\,15 - (3 + 4)\,]$.

Step 1 — innermost parentheses: $3 + 4 = 7$ $2 \times [\,15 - 7\,]$

Step 2 — the brackets: $15 - 7 = 8$ $2 \times 8$

Step 3 — multiply: $2 \times 8 = 16$ $= 16$

⚠️ Inside out, always. Don't reach for the $15 - 3$; the $(3+4)$ is wrapped tighter, so it goes first. Finish each pair of grouping symbols before touching the layer outside it.

Order the Steps 🔽

You're evaluating $3 \times [\,8 + (10 - 6)\,]$. Choose what happens at each stage.

A Tip for Long Expressions

When an expression has lots of symbols, rewrite the whole thing on a new line after each step. Replace the part you finished with its value and copy everything else down.

$30 \div [\,2 \times (1 + 4)\,] \;\to\; 30 \div [\,2 \times 5\,] \;\to\; 30 \div 10 \;\to\; 3$

Doing one operation per line keeps you from losing track and makes mistakes easy to spot.

💡 Slow is smooth. It is faster to do one step per line carefully than to rush, lose a number, and start over.

Grouping Practice 🧮

Work from the inside out. Enter just the final value.

1) $48 \div (4 + 4)$ 2) $4 \times [\,9 - (2 + 3)\,]$ 3) $30 \div [\,2 \times (1 + 4)\,]$

🔢 Numerical Expressions

Part 4 of 5 — Writing & Interpreting Expressions

🔑 The Grade-5 Skill: Write an expression that records calculations with numbers — and interpret what one means — without actually calculating it. The structure tells the whole story.

Words → Expression (using grouping)

The trick is deciding what to group. If words say "do something to a whole result," that result needs parentheses.

Notice the difference between rows 1 and 2. In row 1 we add first, so $6 + 4$ gets parentheses. In row 2 we multiply first, so no parentheses are needed — order of operations already does the $\times$ before the $+$.

💡 "The sum of … then …" is your cue to wrap the sum in parentheses, because you want it done before the next step.

Words to Symbols 🎯

Compare WITHOUT Calculating

Here's the cool part: you can compare two expressions just by reading their structure — no arithmetic needed.

Look at $3 \times (214 + 87)$ versus $(214 + 87)$.

Both contain the same sum, $(214 + 87)$. But the first one multiplies that sum by $3$. So without adding anything, you know:

$3 \times (214 + 87) \text{ is } 3 \text{ times as large as } (214 + 87)$

💡 Reading structure: $5 \times (\text{something})$ is always $5$ times bigger than that "something," no matter what the something works out to be. The expression records the relationship.

Interpret the Structure 🔽

Compare each expression to the value of $(46 + 19)$ without computing.

A Plan for Word Problems

When you turn a sentence into an expression, work in this order:

1. Find the operations using the signal words (sum, product, divide, …).
2. Decide what happens first. If a phrase says "the sum/difference … then …," that part gets parentheses.
3. Write it, then evaluate using the order of operations.

💡 Parentheses are a promise. They promise "do me first." Use them whenever the words want a result computed before the next step.

Write, Then Evaluate 🧮

Translate each phrase into an expression, then find its value. Enter just the final value.

1) Add $12$ and $8$, then triple the result. 2) Subtract $6$ from $20$, then divide by $2$. 3) Half of the sum of $24$ and $16$.

🔢 Numerical Expressions

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) tell an expression from an equation, (2) use the order of operations, (3) work through grouping symbols inside-out, and (4) write and interpret expressions. Let's put it all together.

Quick Reference

⚠️ Top traps: never just go left to right when there's a $\times$ or $\div$ in the mix, and remember multiply/divide are a team (and so are add/subtract) — handle each team left to right.

Mixed Practice 🎯

One More Set 🔽

Choose the correct value or interpretation for each.

Exit Quiz ✅

Answer all three to finish the lesson.