Writing and Evaluating Expressions

🧩 Writing and Evaluating Expressions

Part 1 of 5 — The Language of Algebra

Topics in This Part

🔑 Key Concept: An expression is a math phrase built from numbers, variables, and operations — but it has no equals sign. Learning to read and build expressions is the first real step into algebra.

What Is an Expression?

An algebraic expression mixes numbers and variables (letters that stand for unknown values) using operations like $+$, $-$, $\times$, and $\div$.

💡 Expression vs. Equation: An expression is a phrase (like "three more than a number"). An equation is a full sentence that says two things are equal (it has an $=$). This lesson is all about expressions.

Concept Check 🎯

The Parts of an Expression

Look at the expression $\;4x + 7\;$. Every piece has a name:

So $4x + 7$ has two terms: the term $4x$ and the term $7$.

🔑 Spotting the coefficient: In $4x$, the coefficient is $4$. A variable alone, like $x$, secretly has a coefficient of $1$, because $x = 1x$.

Name the Parts 🔽

Use the expression $\;6m + 9\;$ to answer each one.

Count the Terms 🧮

Terms are separated by $+$ and $-$ signs. How many terms does each expression have?

1) $3x + 8$ 2) $5a - 2b + 7$ 3) $10y$

🧩 Writing and Evaluating Expressions

Part 2 of 5 — Turning Words into Expressions

🔑 The Goal: Real problems come as words. Your job is to translate those words into a math expression. The secret is learning which words mean which operations.

Operation Keywords

Each operation has a set of "signal words." Memorize these — they unlock almost every word problem.

Watch the Order!

Most phrases translate left to right, but "more than" and "less than" flip the order:

• "$5$ more than a number" → $\;n + 5\;$ ✅ (not $5 + n$ wrong order, though it equals the same)
• "$5$ less than a number" → $\;n - 5\;$ ✅ (not $5 - n$ — that's backwards!)

⚠️ The #1 trap: "$3$ less than $x$" means $x - 3$, not $3 - x$. The "less than" amount gets subtracted from the number, so it goes second.

Concept Check 🎯

Worked Examples: Words → Expressions

Example 1: "the sum of a number $x$ and $12$" $x + 12$

Example 2: "$8$ less than twice a number $y$"

Break it into pieces:

• "twice a number $y$" $\Rightarrow 2y$
• "$8$ less than" that $\Rightarrow$ subtract $8$ from $2y$

$2y - 8$

Example 3: "a number $n$ divided by $5$, then increased by $3$" $\frac{n}{5} + 3$

💡 Strategy: Translate one chunk at a time, then assemble. Underline the signal words first so you don't miss a flip.

Translate Each Phrase 🔽

Build the Expression 🧮

Write each as an expression using the variable shown. Type it with no spaces (for example: 2x+5).

1) "$10$ more than a number $x$" 2) "$3$ less than a number $y$" 3) "the product of $6$ and a number $z$"

🧩 Writing and Evaluating Expressions

Part 3 of 5 — Evaluating Expressions

🔑 The Big Idea: To evaluate an expression means to plug in a number for each variable and calculate the result. "$x = 4$" means "wherever you see $x$, write $4$."

Substitution: Swap the Variable for a Number

To evaluate $\;3x + 5\;$ when $x = 4$:

1. Replace every $x$ with $4$: $\;3(4) + 5$
2. Multiply first: $\;12 + 5$
3. Add: $\;17$

$3x + 5 \;\text{ at }\; x = 4 \;=\; 3(4) + 5 = 12 + 5 = 17$

⚠️ Hidden multiplication: $3x$ means $3 \times x$. After you substitute, write $3(4)$ and multiply — don't accidentally write $34$. The number next to a variable is always a times.

Concept Check 🎯

Use the Order of Operations

After you substitute, follow PEMDAS to finish:

Worked Example: $\;4a^2 - 7\;$ when $a = 3$

$4a^2 - 7 = 4(3)^2 - 7$

Do the exponent first: $3^2 = 9$.

$= 4(9) - 7 = 36 - 7 = 29$

💡 Order matters: $4(3)^2$ is not $(4 \cdot 3)^2$. The exponent only touches the $3$, so square first, then multiply by $4$.

Evaluate Carefully 🧮

Follow the order of operations. Enter each numeric answer.

1) $5x - 2$ when $x = 6$ 2) $\dfrac{m}{4} + 9$ when $m = 12$ 3) $2n^2$ when $n = 5$

Two Variables 🎯

Some expressions have more than one variable. Substitute each one.

🧩 Writing and Evaluating Expressions

Part 4 of 5 — Real-World Expressions

🔑 Why This Matters: Expressions describe real situations — money saved, distance traveled, items bought. When the situation can change (the "how many" is unknown), we use a variable to hold its place.

Modeling a Situation

Pick a variable for the unknown, then build the expression around it.

Example: Buying Notebooks

Notebooks cost $3 each, and you also pay a flat $2 shipping fee. Write an expression for the total cost of $n$ notebooks.

• Cost of notebooks: $3n$ ($3 times how many)
• Plus shipping: $+\,2$

$\text{Total cost} = 3n + 2$

How much for 5 notebooks? Evaluate at $n = 5$:

$3(5) + 2 = 15 + 2 = 17 \;\text{dollars}$

💡 Two pieces, two parts: A "cost per item" becomes a coefficient ($3n$), and a one-time flat fee becomes a constant ($+2$).

Match the Situation 🔽

A gym charges a $20 sign-up fee plus $15 each month. Let $m$ be the number of months.

Use Your Expression 🧮

A taxi charges a $4 base fare plus $2 per mile. The cost for $d$ miles is $\;2d + 4\;$ dollars.

1) What is the cost for $d = 6$ miles? (dollars) 2) What is the cost for $d = 10$ miles? (dollars) 3) What is the base fare (the cost for $d = 0$ miles)? (dollars)

Concept Check 🎯

🧩 Writing and Evaluating Expressions

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) name the parts of an expression, (2) translate words into expressions, (3) evaluate by substituting, and (4) model real situations. Let's put it all together.

Quick Reference

⚠️ Top traps: "$5$ less than $x$" is $x - 5$ (not $5 - x$); $3x$ means $3 \times x$ (not $34$ after subbing $4$); and exponents come before multiplication in PEMDAS.

Mixed Practice 🎯

Evaluation Drill 🧮

Substitute and follow the order of operations. Enter each numeric answer.

1) $7x + 1$ when $x = 3$ 2) $\dfrac{n}{5} - 2$ when $n = 30$ 3) $a^2 + 4$ when $a = 6$

Exit Quiz ✅

Answer all three to finish the lesson.