Evaluating Expressions

🔢 Evaluating Expressions

Part 1 of 5 — Variables, Substitution & the Big Idea

Topics in This Part

🔑 Key Concept: To evaluate an expression means to find its value. You do this by substituting a number in for each variable, then doing the arithmetic.

What Is an Expression?

An algebraic expression is a combination of numbers, variables, and operations — but it has no equals sign.

A variable is a letter that stands for a number we don't know yet — like $x$, $n$, or $a$.

💡 $3n$ means $3 \cdot n$ (three times $n$). A number written right next to a variable always means multiply. There is no hidden plus.

Concept Check 🎯

Substitution: Swapping In a Number

To evaluate an expression, we replace the variable with its value. This swap is called substitution.

Worked Example: Evaluate $x + 8$ when $x = 5$

1. Write the expression: $x + 8$
2. Substitute $5$ for $x$: $5 + 8$
3. Compute: $13$

$x + 8 = 5 + 8 = 13$

Worked Example: Evaluate $4n$ when $n = 6$

Remember $4n$ means $4 \cdot n$:

$4n = 4 \cdot 6 = 24$

⚠️ Watch out: When you substitute, put the number exactly where the letter was. $4n$ with $n = 6$ is $4 \cdot 6 = 24$ — not $46$.

Substitute and Evaluate 🧮

Find the value of each expression for the given number.

1) $x + 9$ when $x = 7$ 2) $6n$ when $n = 5$ 3) $20 - a$ when $a = 12$

Read the Expression 🔽

Pick the correct meaning or value for each.

Recap

You now have the core move:

🔑 Evaluate = Substitute, then Compute. Replace each variable with its number, then do the arithmetic.

So far each expression had just one operation. In Part 2 we handle expressions with several operations at once — which means we need a set of rules for what to do first. That set of rules is the order of operations.

🔢 Evaluating Expressions

Part 2 of 5 — Order of Operations (PEMDAS)

🔑 The Idea: When an expression has more than one operation, the answer depends on the order you work in. Everyone follows the same order so everyone gets the same answer.

The Order of Operations

Work through an expression in this order:

⚠️ Multiply and divide are a team — do them together, left to right. Same for add and subtract. You do not do all multiplication before all division.

Worked Example: $3 + 4 \cdot 2$

Multiplication comes before addition:

$3 + 4 \cdot 2 = 3 + 8 = 11$

If you (wrongly) added first you'd get $7 \cdot 2 = 14$ — that's the classic mistake.

Which Comes First? 🔽

For each expression, pick the operation you do first.

A Longer Walkthrough

Worked Example: $2 + 3 \cdot (6 - 1)$

1. Parentheses first: $6 - 1 = 5$, giving $2 + 3 \cdot 5$
2. Multiply: $3 \cdot 5 = 15$, giving $2 + 15$
3. Add: $2 + 15 = 17$

$2 + 3 \cdot (6 - 1) = 2 + 3 \cdot 5 = 2 + 15 = 17$

Worked Example: $24 \div 6 \cdot 2$

Divide and multiply are a team — go left to right:

$24 \div 6 \cdot 2 = 4 \cdot 2 = 8$

💡 If you had multiplied first ($6 \cdot 2 = 12$, then $24 \div 12 = 2$) you'd get the wrong answer. Left-to-right matters!

Concept Check 🎯

Order of Operations Drill 🧮

Evaluate each expression. Follow PEMDAS carefully.

1) $5 + 2 \cdot 4$ 2) $3 \cdot (7 - 2)$ 3) $18 \div 3 + 1$

🔢 Evaluating Expressions

Part 3 of 5 — Substitution Meets the Order of Operations

🔑 The Combo: Most Grade 6 problems ask you to substitute a number and follow the order of operations. The trick: substitute first, wrap your number in parentheses, then evaluate.

The Safe 3-Step Method

1. Substitute — replace each variable with its number, using parentheses around it.
2. Order — apply PEMDAS.
3. Answer — finish the arithmetic.

Worked Example: Evaluate $3x + 5$ when $x = 4$

$3x + 5 = 3(4) + 5 = 12 + 5 = 17$

Multiplication ($3 \cdot 4$) happens before the addition.

Worked Example: Evaluate $2n - 1$ when $n = 9$

$2n - 1 = 2(9) - 1 = 18 - 1 = 17$

⚠️ Use parentheses when you substitute. Writing $3(4)$ instead of $34$ keeps the multiplication visible and stops careless errors.

Concept Check 🎯

Two Variables at Once

Some expressions use two different letters. Substitute both numbers, then evaluate.

Worked Example: Evaluate $ab + 4$ when $a = 3$ and $b = 5$

$ab + 4 = (3)(5) + 4 = 15 + 4 = 19$

Worked Example: Evaluate $\dfrac{x + y}{2}$ when $x = 8$ and $y = 6$

The fraction bar acts like a grouping symbol — do the top first:

$\frac{x + y}{2} = \frac{8 + 6}{2} = \frac{14}{2} = 7$

💡 A fraction bar groups the whole top and the whole bottom — finish each before you divide.

Step Through It 🔽

Evaluate $4x - 3$ when $x = 5$. Choose what happens at each stage.

Substitute & Evaluate Drill 🧮

1) $6m + 7$ when $m = 2$ 2) $30 - 4k$ when $k = 5$ 3) $\dfrac{p + q}{3}$ when $p = 7$ and $q = 5$

🔢 Evaluating Expressions

Part 4 of 5 — Exponents & Real-World Formulas

🔑 Why this matters: Expressions with exponents and real formulas (like area and perimeter) are everywhere. The same method still works — just remember exponents come before multiplication.

Exponents in Expressions

An exponent tells you how many times to multiply the base by itself:

$x^2 = x \cdot x \qquad x^3 = x \cdot x \cdot x$

Worked Example: Evaluate $x^2$ when $x = 6$

$x^2 = 6^2 = 6 \cdot 6 = 36$

Worked Example: Evaluate $2x^2$ when $x = 3$

The exponent applies to $x$ only. Do the power before multiplying by $2$:

$2x^2 = 2(3)^2 = 2 \cdot 9 = 18$

⚠️ Huge mistake to avoid: $2x^2$ is not $(2x)^2$. With $x = 3$, $2x^2 = 18$, but $(2 \cdot 3)^2 = 36$. The exponent grips only the symbol it sits on.

Concept Check 🎯

Using Real Formulas

A formula is just an expression with a meaning. You evaluate it by substituting the measurements.

Worked Example: Area of a Rectangle

Find $A = l \cdot w$ when $l = 7$ and $w = 4$:

$A = (7)(4) = 28 \text{ square units}$

Worked Example: Perimeter of a Rectangle

Find $P = 2l + 2w$ when $l = 7$ and $w = 4$:

$P = 2(7) + 2(4) = 14 + 8 = 22 \text{ units}$

💡 Substitute every measurement, then let the order of operations finish the job.

Pick the Right Value 🔽

Exponents & Formulas Drill 🧮

1) $x^2 + 1$ when $x = 7$ 2) $4a^2$ when $a = 3$ 3) Perimeter $P = 2l + 2w$ when $l = 6$ and $w = 5$

🔢 Evaluating Expressions

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) substitute values for variables, (2) follow the order of operations, (3) combine both moves, and (4) handle exponents and real formulas. Time to put it all together.

Quick Reference

⚠️ Top mistakes: adding before multiplying, treating $2x^2$ as $(2x)^2$, and reading $4n$ as a two-digit number instead of $4 \cdot n$.

Mixed Practice 🎯

Mixed Drill 🧮

1) $x^2 - 4$ when $x = 5$ 2) $2(a + b)$ when $a = 3$ and $b = 6$ 3) $50 - 3n$ when $n = 7$

Exit Quiz ✅

Answer all three to finish the lesson.