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Teacher Guide: Variables and Algebraic Expressions

Learn what variables are and how to write, read, and evaluate algebraic expressions.

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10 questions on Algebraic Expressions. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Define variables and explain their purpose in algebra
  • Identify the parts of an algebraic expression (variables, coefficients, constants, terms)
  • Translate verbal phrases into algebraic expressions
  • Evaluate algebraic expressions by substituting given values
  • Apply algebraic expressions to real-world situations
Prerequisites
  • • Understanding of basic arithmetic operations
  • • Familiarity with order of operations (PEMDAS/BODMAS)
  • • Ability to work with whole numbers and basic fractions
Discussion Starters
  • 1. Why do you think mathematicians started using letters instead of just words to describe unknown numbers?
  • 2. Can you think of a situation in your life where a number changes or is unknown?
  • 3. If a video game costs gg dollars, what would an expression for buying 3 copies look like?
  • 4. Why might it be useful to have a formula like d=60td = 60t instead of calculating distance separately each time?
Common Misconceptions

Thinking xx always equals a specific number (like 24)

Remediation: Emphasize that xx can be ANY number. Show the same expression with different values of xx to demonstrate it changes.

Confusing 2x2x (two times xx) with x2x^2 (xx times xx)

Remediation: Use concrete examples: if x=3x = 3, then 2x=62x = 6 but x2=9x^2 = 9. Show that these give different results.

Thinking expressions and equations are the same thing

Remediation: Point out that expressions are like phrases (no equals sign), while equations are like complete sentences with an equals sign.

Differentiation Ideas

For Struggling Students:

  • • Start with single-variable expressions only
  • • Use physical objects (like blocks) to represent variables
  • • Provide a word-to-symbol reference chart
  • • Focus on single-operation expressions before combining operations

For On-Level Students:

  • • Evaluate expressions with two variables
  • • Translate multi-step word problems into expressions
  • • Write expressions from real-world scenarios
  • • Compare different expressions for the same situation

For Advanced Students:

  • • Work with expressions containing fractions and decimals
  • • Create their own word problems that match given expressions
  • • Explore when two different expressions give the same result
  • • Introduce the concept of like terms and combining them
Standards Alignment
  • 6.EE.A.2 (CCSS.MATH.CONTENT.6.EE.A.2)

    Write, read, and evaluate expressions in which letters stand for numbers

  • 6.EE.A.2a (CCSS.MATH.CONTENT.6.EE.A.2.A)

    Write expressions that record operations with numbers and letters

  • 6.EE.A.2c (CCSS.MATH.CONTENT.6.EE.A.2.C)

    Evaluate expressions at specific values of their variables

  • 6.EE.B.6 (CCSS.MATH.CONTENT.6.EE.B.6)

    Use variables to represent numbers and write expressions for real-world problems

Lesson Resources
  • visualAlgebra Tiles and Value Table

    Model 3x+73x + 7 with tiles and see how its value changes as xx changes

  • activityExpression Builder

    Translate word problems into algebraic expressions

  • worksheetEvaluate Expressions Practice

    Practice substituting values and calculating results

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A variable is a letter (like xx, yy, or nn) that represents an unknown number or a number that can change.
An algebraic expression is a mathematical phrase that combines:
  • Numbers (constants like 55, 1212, 12\frac{1}{2})
  • Variables (letters like xx, yy, aa)
  • Operations (like ++, −-, ×\times, ÷\div)
Examples of algebraic expressions:
3x+53x + 5
2y−72y - 7
n4+10\frac{n}{4} + 10
Unlike equations, expressions do not have an equals sign.

Worked Examples

Write an algebraic expression for: "five more than a number"
1

Choose a variable for the unknown number

Let nn represent the unknown number → Variable: nn
2

Identify the operation

"More than" means addition → Operation: ++
3

Write the expression

Five more than nn means n+5n + 5

Common Mistakes

Writing 3x3x as 3+x3 + x instead of 3×x3 \times x

Why it's wrong: In algebra, when a number is written next to a variable without a sign, it means multiplication. 3x3x means "3 times xx", not "3 plus xx".

Correct: 3x=3×x3x = 3 \times x. So if x=5x = 5, then 3x=153x = 15 (not 88).

Forgetting order of operations when evaluating

Why it's wrong: When substituting values, you must still follow PEMDAS/BODMAS. Multiplication comes before addition.

Correct: For 2+3x2 + 3x with x=4x = 4: First 3×4=123 \times 4 = 12, then 2+12=142 + 12 = 14.

Using the same variable for different unknowns

Why it's wrong: If you have two different unknown quantities, they need different variables.

Correct: If age and height are both unknown, use aa for age and hh for height, not xx for both.

Why It Matters

Variables and expressions are the foundation of algebra and appear everywhere:
  • Formulas: The area of a rectangle is A=l×wA = l \times w (length times width)
  • Real life: If a movie ticket costs 12 dollars, the cost for nn people is 12n12n dollars
  • Science: Distance equals speed times time: d=std = st
  • Computer programming: Variables store changing values in every program
Once you understand variables, you can describe patterns, solve problems, and create formulas that work for ANY numbers!

Real World Applications

Shopping and Discounts

Stores use expressions to calculate prices. If jeans cost 40 dollars and shirts cost $s$ dollars each, buying 2 shirts with jeans costs $40 + 2s$ dollars.

Example:

If shirts are 25 dollars each: 40+2(25)=40+50=9040 + 2(25) = 40 + 50 = 90 dollars total.

1Try It Yourself

A pizza shop charges 12 dollars for a pizza plus 2 dollars per topping.

Write an expression for the cost with tt toppings, then find the cost of a pizza with 5 toppings.

Step 1: Write the mathematical expression

Cost = base price + (price per topping × number of toppings):

Distance and Travel

If you drive at $s$ kilometers per hour for $t$ hours, you travel $s \times t$ kilometers.

Example:

Driving at 60 km/h for 3 hours: 60×3=18060 \times 3 = 180 km.

2Try It Yourself

A train travels at 80 kilometers per hour.

Write an expression for the distance after hh hours, then find how far it goes in 4 hours.

Step 1: Write the mathematical expression

Distance = speed × time:

Savings and Money

Expressions help track money over time. If you have 50 dollars and save $w$ dollars each week, after one week you have $50 + w$ dollars.

Example:

Saving 15 dollars per week: after 1 week you have 50+15=6550 + 15 = 65 dollars.

3Try It Yourself

You have 100 dollars and spend 8 dollars each day on lunch.

Write an expression for the money left after dd days, then find how much is left after 5 days.

Step 1: Write the mathematical expression

Money left = starting amount - (daily spending × days):

Key Takeaways

  • 1A variable is a letter representing an unknown or changing number
  • 2An algebraic expression combines numbers, variables, and operations (no equals sign)
  • 3When a number is next to a variable (like 3x3x), it means multiplication: 3×x3 \times x
  • 4To evaluate an expression, substitute values for variables and calculate
  • 5Always follow order of operations (PEMDAS) when evaluating

Frequently Asked Questions

Why do we use letters instead of just leaving blanks?

Letters are clearer and allow us to write rules. For example, A=lwA = lw tells us exactly how to find area, while "Area = ___ times ___" is ambiguous.

Does it matter which letter I use for a variable?

Usually no, but some letters have common meanings: xx for unknowns, tt for time, dd for distance, nn for number of items. Choose letters that make sense for your problem.

What is the difference between an expression and an equation?

An expression is a mathematical phrase without an equals sign (like 3x+53x + 5). An equation has an equals sign and shows two things are equal (like 3x+5=143x + 5 = 14).

Glossary

Variable
A letter that represents an unknown or changing number (e.g., xx, nn, tt)
Algebraic expression
A combination of numbers, variables, and operations without an equals sign
Constant
A number that does not change (e.g., 55, −3-3, 12\frac{1}{2})
Coefficient
The number multiplied by a variable (in 7x7x, the coefficient is 77)
Term
A single part of an expression separated by ++ or −- (in 3x+53x + 5, the terms are 3x3x and 55)
Evaluate
To find the value of an expression by substituting numbers for variables

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