Mathematics · 7th Grade · Expressions and Linear Equations · Weeks 10-18

Writing Algebraic Expressions

Students will translate verbal phrases into algebraic expressions and identify parts of an expression.

• Common Core State Standards
• CCSS.Math.Content.7.EE.A.2

About This Topic

Equivalent expressions are the building blocks of algebraic fluency. Students learn to use properties of operations, distributive, associative, and commutative, to rewrite linear expressions in different forms. The Common Core focus is not just on 'simplifying,' but on understanding that rewriting an expression can reveal new information about the problem's context, such as seeing 1.05p as p + 0.05p to represent a 5% increase.

This topic is essential for solving equations and understanding functions in later grades. It teaches students to look at the structure of an expression rather than just seeing a string of symbols. This topic comes alive when students can physically manipulate terms or engage in 'matching' games that require them to justify why two different-looking expressions are actually the same.

Key Questions

1. Differentiate between terms, coefficients, and constants in an algebraic expression.
2. Explain how to translate complex verbal phrases into mathematical expressions.
3. Construct an algebraic expression to represent a real-world situation.

Learning Objectives

• Identify the terms, coefficients, and constants within a given algebraic expression.
• Translate verbal phrases involving addition, subtraction, multiplication, and division into algebraic expressions.
• Construct an algebraic expression to represent a real-world scenario involving a single unknown quantity.
• Explain the relationship between a verbal phrase and its corresponding algebraic expression, justifying each component.
• Compare and contrast different verbal phrases that translate to the same algebraic expression.

Before You Start

Introduction to Variables

Why: Students need to understand the concept of a variable as a placeholder for an unknown value before they can write expressions.

Order of Operations (PEMDAS/BODMAS)

Why: Understanding the order of operations is crucial for correctly interpreting and writing expressions, especially those involving multiple operations.

Key Vocabulary

Variable	A symbol, usually a letter, that represents an unknown number or quantity in an algebraic expression.
Coefficient	A numerical factor that multiplies a variable in an algebraic term. For example, in the term 3x, the coefficient is 3.
Constant	A term in an algebraic expression that does not contain a variable; its value remains fixed. For example, in the expression x + 5, the constant is 5.
Term	A single number or variable, or numbers and variables multiplied together. Terms are separated by addition or subtraction signs. For example, in 2x + 7, the terms are 2x and 7.

Watch Out for These Misconceptions

Common Misconception

Students often forget to distribute to the second term in a parenthesis.

What to Teach Instead

They might write 2(x+3) as 2x+3. Using hands-on modeling with algebra tiles or area models surfaces this error visually, as they can see the 'missing' pieces that weren't multiplied.

Common Misconception

Thinking that 'unlike terms' can be combined.

What to Teach Instead

Students might try to simplify 3x + 4 into 7x. Peer explanation tasks where students have to 'sort' physical objects (like 3 apples and 4 oranges) help them internalize why we can only combine terms with the same variable.

Active Learning Ideas

Gallery Walk: Expression Match-Up

Post various expressions around the room (e.g., 2(x+3), 2x+6, x+x+6). Students move in pairs to find all the equivalent versions and must write down which property (like Distributive) proves they are the same. They leave 'critique' sticky notes on matches they disagree with.

30 min·Pairs

Inquiry Circle: Algebra Tile Modeling

Groups use physical or digital algebra tiles to model an expression like 3(x-2). They must then rearrange the tiles to show the expanded form 3x-6. This tactile approach helps them visualize the distributive property as 'three groups of x and three groups of negative two.'

25 min·Small Groups

Think-Pair-Share: Contextual Rewriting

Give students a scenario: 'A shirt costs x dollars and is on sale for 20% off.' Students write two different expressions for the sale price (e.g., x - 0.20x and 0.80x). They pair up to explain why both are correct and which one is easier to use for a quick calculation.

20 min·Pairs

Real-World Connections

• A retail manager might write an expression like '2c + 5s' to represent the total cost of 'c' shirts at $2 each and 's' scarves at $5 each, helping to track inventory and sales.
• A city planner could use an expression such as '1500p' to estimate the total water needed for 'p' new housing units, where each unit requires 1500 gallons.
• A baker might calculate the cost of ingredients for cookies using an expression like '0.50b + 0.25c', where 'b' is the number of batches of dough and 'c' is the number of chocolate chip bags used.

Assessment Ideas

Quick Check

Provide students with a list of verbal phrases (e.g., 'five more than a number', 'twice a number decreased by three'). Ask them to write the corresponding algebraic expression for each and identify the variable, coefficient, and constant in at least two of them.

Exit Ticket

Present students with a real-world scenario, such as 'A group of friends bought 3 pizzas at $12 each and shared a $5 delivery fee.' Ask them to write an algebraic expression representing the total cost, clearly defining what their variable represents.

Discussion Prompt

Pose the question: 'Why is it important to be precise when translating verbal phrases into algebraic expressions?' Facilitate a class discussion where students share examples of how ambiguity in language can lead to incorrect mathematical representations.

Frequently Asked Questions

What makes two algebraic expressions equivalent?

Two expressions are equivalent if they yield the same value regardless of which number is substituted for the variable. They can be transformed into each other using the properties of operations, like the distributive or commutative properties.

What are the best hands-on strategies for teaching equivalent expressions?

Algebra tiles are the most effective hands-on tool for this topic. They allow students to physically see the 'area' of an expression and rearrange the pieces to find different forms. Matching games and 'sorting' activities also help students recognize patterns in expression structure.

How does the distributive property work with negative numbers?

When distributing a negative number, like -2(x - 3), you must multiply both terms inside the parentheses by -2. This changes the signs: -2 times x is -2x, and -2 times -3 is +6, resulting in -2x + 6.

Why would I want to rewrite an expression in a different way?

Rewriting helps clarify the meaning. For example, writing 'total cost' as 1.08x clearly shows a tax of 8%, whereas x + 0.08x shows the original price plus the tax amount. Different forms are useful for different types of problem-solving.