Mathematics · 6th Grade · The Number System, Rational Numbers, and Expressions · Weeks 10-18

Writing Algebraic Expressions

Students will use variables to represent numbers and write expressions for real-world problems.

• Common Core State Standards
• CCSS.Math.Content.6.EE.A.2a

About This Topic

Writing algebraic expressions is where students begin the formal language of algebra. CCSS 6.EE.A.2a asks students to use variables to represent unknown quantities and to translate verbal descriptions into mathematical notation. This involves understanding that a variable stands for a number that may or may not be known, and that operations can be expressed symbolically using +, −, ×, and ÷ alongside variables and constants.

Translating between verbal and symbolic representations is a foundational algebraic skill. Students learn to read phrases like 'five more than a number' (n + 5) or 'a number decreased by eight' (n − 8) and write expressions that capture the relationship precisely. A frequent challenge is with subtraction and division, which are order-sensitive -- 'a number minus five' and 'five minus a number' produce different expressions.

Active learning is well-suited to this topic because expression writing is about communication and precision, not just computation. When students write expressions for real-world scenarios and share them for peer interpretation, they quickly discover whether their expression means what they intended. This feedback loop -- write an expression, have a partner try to decode it, compare -- builds both fluency and precision more efficiently than independent practice alone.

Key Questions

1. Differentiate between a variable and a constant in a mathematical sentence.
2. Explain how to translate verbal phrases into accurate mathematical symbols.
3. Construct an algebraic expression to represent a given real-world scenario.

Learning Objectives

• Identify the variable and constant in a given algebraic expression.
• Translate verbal phrases representing addition, subtraction, multiplication, and division into algebraic expressions.
• Construct an algebraic expression to represent a real-world scenario involving a single unknown quantity.
• Compare two different verbal phrases to determine if they result in the same algebraic expression.
• Explain the meaning of a variable in the context of a real-world problem.

Before You Start

Order of Operations (PEMDAS/BODMAS)

Why: Students need to understand the order of operations to correctly interpret and write expressions, especially when multiple operations are involved.

Introduction to Variables

Why: Students should have a basic understanding of what a variable is and how it represents an unknown quantity before writing expressions.

Key Vocabulary

Variable	A symbol, usually a letter, that represents a number that can change or is unknown.
Constant	A fixed value that does not change, represented by a number in an expression.
Algebraic Expression	A mathematical phrase that contains variables, constants, and operation symbols.
Translate	To convert a verbal phrase into a mathematical expression using symbols and variables.

Watch Out for These Misconceptions

Common Misconception

Students write 'five minus a number' as n − 5 instead of 5 − n.

What to Teach Instead

Subtraction is order-sensitive: the first number named is subtracted from. 'Five minus a number' means 5 is reduced by n, so the expression is 5 − n. When the number named after 'minus' is the variable, students must resist the urge to write the variable first. Role-playing the phrase as a verbal action ('start with 5, take away n') helps anchor the order.

Common Misconception

Students use multiplication signs (×) inappropriately next to variables and confuse '3x' with '3 times x written incorrectly.'

What to Teach Instead

In algebra, juxtaposition means multiplication: 3x means 3 times x. The × symbol is avoided when variables are present because it resembles x. Explicitly introduce the conventions: 3n, 3·n, and (3)(n) all mean the same thing. Provide practice writing and reading both forms until the juxtaposition convention feels natural.

Active Learning Ideas

Think-Pair-Share: Verbal to Symbolic Translation

Give pairs a list of 8 verbal phrases and ask each student to independently write the corresponding algebraic expression. Partners then compare and discuss any differences, focusing on order-sensitive operations like subtraction and division. The class debrief covers the most common disagreements.

20 min·Pairs

Collaborative Task: Expression from a Story

Provide each small group with a short scenario (e.g., 'A store sells items at price p. A customer buys 4 items and uses a $3 coupon'). Groups must identify all variables and constants, write an expression for the total cost, and explain each term in words. Groups share their expressions and compare for equivalence.

30 min·Small Groups

Gallery Walk: Decode the Expression

Post algebraic expressions around the room (e.g., 2x + 7, n/3, 5 − k). Students write a verbal phrase that matches each expression, then a real-world scenario it could represent. Pairs compare their scenarios and discuss whether multiple verbal interpretations of the same expression are all valid.

25 min·Pairs

Real-World Connections

• Retail workers use expressions to calculate total costs, such as 'the price of one shirt plus a 7% sales tax on that price.' This helps them quickly determine the final amount a customer owes.
• Coaches might use expressions to track player statistics, for example, 'the total number of points scored by a player minus the number of fouls committed' to understand player performance.
• Event planners use expressions to estimate expenses, like 'the cost of renting a venue plus the number of guests multiplied by the catering fee per person.'

Assessment Ideas

Exit Ticket

Provide students with three verbal phrases: 'a number increased by 10', 'twice a number', and 'a number divided by 3'. Ask them to write the corresponding algebraic expression for each and identify the variable and constant in each expression.

Quick Check

Write the expression '3n + 5' on the board. Ask students to write two different verbal phrases that could be represented by this expression. Discuss the variety of correct answers and why they are equivalent.

Discussion Prompt

Present the scenario: 'Sarah is saving money. She already has $20 and saves $5 each week.' Ask students to write an algebraic expression to represent the total amount of money Sarah will have after 'w' weeks. Facilitate a discussion on why '5w + 20' is the correct expression and why '20w + 5' would be incorrect.

Frequently Asked Questions

How do you write an algebraic expression from a word problem?

Identify the unknown quantity and assign it a variable (usually n or x). Then identify the operation described: 'more than' signals addition, 'less than' or 'decreased by' signals subtraction, 'times' or 'product of' signals multiplication, and 'divided by' or 'quotient of' signals division. Write the variable and constant with the correct operation, paying careful attention to order for subtraction and division.

What is the difference between a variable and a constant in algebra?

A variable is a letter that represents an unknown or changing quantity -- its value is not fixed. A constant is a specific number whose value does not change. In the expression 4x + 7, x is the variable (its value depends on the situation) and 7 is the constant (it is always 7). This distinction matters when evaluating expressions, because only the variable's value changes.

Why does order matter when writing subtraction and division expressions?

Subtraction and division are not commutative -- changing the order changes the meaning. 'A number decreased by 5' is n − 5, while 'five decreased by a number' is 5 − n. These are different expressions with different values for any given n. Similarly, 'a number divided by 3' is n/3, while 'three divided by a number' is 3/n. Students need to practice identifying what is being subtracted from what or divided by what.

How does active learning help students write algebraic expressions?

Expression writing is a communication skill as much as a mathematical one. When students write an expression and a partner tries to decode it or reconstruct the original phrase, they get immediate feedback on whether their notation is unambiguous. Gallery walk tasks where students write scenarios for posted expressions help students develop flexibility -- seeing that the same expression can describe multiple real-world situations deepens understanding of what the variable actually represents.