Teacher Guide: Variables and Algebraic Expressions
Learn what variables are and how to write, read, and evaluate algebraic expressions.
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Class quiz
10 questions on Algebraic Expressions. Students join with a name, you see everyone's score.
For Teachers
- 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
- • Understanding of basic arithmetic operations
- • Familiarity with order of operations (PEMDAS/BODMAS)
- • Ability to work with whole numbers and basic fractions
- 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 euros, what would an expression for buying 3 copies look like?
- 4. Why might it be useful to have a formula like instead of calculating distance separately each time?
Thinking always equals a specific number (like 24)
Confusing (two times ) with ( times )
Thinking expressions and equations are the same thing
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
- 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
- visualInteractive Balance Scale
Explore how variables represent unknown quantities
- 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.
Lesson Content
Everything students see: definition, examples, common mistakes, applications. Tap to open.
Definition
- Numbers (constants like , , )
- Variables (letters like , , )
- Operations (like , , , )
Worked Examples
Write an algebraic expression for: "five more than a number"
Choose a variable for the unknown number
Let represent the unknown number → Variable:
Identify the operation
"More than" means addition → Operation:
Write the expression
Five more than means →
Answer: (or equivalently )
Common Mistakes
Writing as instead of
Why it's wrong: In algebra, when a number is written next to a variable without a sign, it means multiplication. means "3 times ", not "3 plus ".
Correct: . So if , then (not ).
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 with : First , then .
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 for age and for height, not for both.
Why It Matters
- Formulas: The area of a rectangle is (length times width)
- Real life: If a movie ticket costs 12 dollars, the cost for people is dollars
- Science: Distance equals speed times time:
- Computer programming: Variables store changing values in every program
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: dollars total.
A pizza shop charges 12 dollars for a pizza plus 2 dollars per topping.
Write an expression for the cost with 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: km.
A train travels at 80 kilometers per hour.
Write an expression for the distance after 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 euros and save $w$ euros each week, after one week you have $50 + w$ euros.
Example:
Saving 15€ per week: After 1 week: .
You have 100€ and spend 8€ each day on lunch.
Write an expression for the money left after 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 ), it means multiplication:
- 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?
Does it matter which letter I use for a variable?
What is the difference between an expression and an equation?
Glossary
- Variable
- A letter that represents an unknown or changing number (e.g., , , )
- Algebraic expression
- A combination of numbers, variables, and operations without an equals sign
- Constant
- A number that does not change (e.g., , , )
- Coefficient
- The number multiplied by a variable (in , the coefficient is )
- Term
- A single part of an expression separated by or (in , the terms are and )
- Evaluate
- To find the value of an expression by substituting numbers for variables