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

Simplifying Expressions: Combining Like Terms

Students will simplify algebraic expressions by combining like terms.

• Common Core State Standards
• CCSS.Math.Content.7.EE.A.1

About This Topic

Modeling with inequalities introduces students to the idea that some problems have a range of solutions rather than a single answer. Students learn to write, solve, and graph inequalities of the form px + q > r or px + q < r. A major focus in the Common Core is the unique rule: when multiplying or dividing by a negative number, the inequality sign must flip to maintain the truth of the statement.

Inequalities are essential for understanding constraints in engineering, economics, and daily life (like budget limits or speed limits). This topic bridges algebra and real-world decision making. Students grasp this concept faster through structured discussion and peer explanation, especially when they have to test points on a number line to 'prove' which direction the arrow should point.

Key Questions

1. Explain why only like terms can be combined in an algebraic expression.
2. Analyze the impact of combining like terms on the structure and value of an expression.
3. Construct a simplified expression from a given complex expression.

Learning Objectives

• Identify like terms within an algebraic expression based on variable and exponent matching.
• Calculate the simplified form of an algebraic expression by combining like terms.
• Explain the distributive property's role in combining like terms.
• Construct a simplified algebraic expression from a given complex expression containing multiple terms.

Before You Start

Introduction to Algebraic Expressions

Why: Students need to be familiar with variables, constants, and the basic structure of algebraic expressions before they can combine terms within them.

Order of Operations (PEMDAS/BODMAS)

Why: Understanding the order of operations is foundational for manipulating algebraic expressions, although combining like terms is often done before applying multiplication or division.

Key Vocabulary

Term	A term is a single number, a variable, or a product of numbers and variables. For example, in the expression 3x + 5, '3x' and '5' are terms.
Like Terms	Like terms are terms that have the same variable(s) raised to the same power(s). For example, 4x and -2x are like terms, but 4x and 4x² are not.
Coefficient	The coefficient is the numerical factor of a term that contains a variable. In the term 7y, the coefficient is 7.
Constant	A constant is a term that does not contain a variable. In the expression 2x + 9, the constant is 9.

Watch Out for These Misconceptions

Common Misconception

Forgetting to flip the inequality sign when dividing by a negative.

What to Teach Instead

This is the most common error. Using a 'testing points' strategy, where students pick a number from their solution set and plug it back in, helps them see immediately if their sign is facing the wrong way.

Common Misconception

Confusing 'open' and 'closed' circles on the number line.

What to Teach Instead

Students often treat 'greater than' and 'greater than or equal to' the same. Peer teaching using real-world examples (like 'you must be at least 48 inches tall') helps clarify whether the boundary number itself is included.

Active Learning Ideas

Formal Debate: The Sign Flip Mystery

Students are given a simple inequality like -2x < 6. Half the class solves it without flipping the sign, the other half flips it. They then test numbers (like 0 or -10) in the original inequality to see which group's solution set actually works, debating the results.

25 min·Whole Class

Inquiry Circle: Real World Constraints

Groups are given scenarios like 'You have $50 to spend at a fair; admission is $10 and rides are $3 each.' They must write an inequality, solve it, and then create a 'solution poster' showing the maximum number of rides they can afford and graphing it on a number line.

40 min·Small Groups

Gallery Walk: Inequality Graphs

Students create 'mystery' inequalities and their corresponding number line graphs on separate cards. The cards are posted around the room, and students must walk around to match the correct inequality to its graph, explaining their reasoning to a partner.

30 min·Pairs

Real-World Connections

• Retail inventory management uses simplified expressions to track stock. For example, a store might start with '50 shirts + 30 pants', sell '10 shirts' and '5 pants', and then receive '20 shirts'. Combining like terms simplifies the calculation to find the total number of shirts remaining (50 - 10 + 20 = 60 shirts) and pants (30 - 5 = 25 pants).
• In budgeting and personal finance, simplifying expressions helps track expenses. If a person plans to spend '$100 on groceries + $50 on gas' each week for 4 weeks, they can combine like terms to calculate the total cost for groceries (4 * $100 = $400) and gas (4 * $50 = $200) separately before summing them for a total budget.

Assessment Ideas

Exit Ticket

Provide students with the expression 5x + 3y - 2x + 7. Ask them to: 1. Identify all the 'x' terms. 2. Identify all the 'y' terms. 3. Write the simplified expression.

Quick Check

Write several expressions on the board, such as '8a + 2b - 3a', '4m² + 5m - m²', and '12 + 3c - 5'. Ask students to hold up fingers corresponding to the number of like terms they can combine in each expression. Then, ask a few students to verbally explain their reasoning for one expression.

Discussion Prompt

Pose the question: 'Imagine you have the expression 3 apples + 2 oranges + 4 apples. Why can you combine the apples to get 7 apples, but you cannot combine apples and oranges into a single category like 'fruit units' without more information?' Guide students to explain the concept of 'like terms' in their own words.

Frequently Asked Questions

Why does the inequality sign flip when you multiply by a negative?

Multiplying by a negative changes the side of zero the number is on. For example, 5 is greater than 2, but -5 is less than -2. To keep the mathematical statement true after the signs change, the direction of the inequality must also change.

What are the best hands-on strategies for teaching inequalities?

Using 'test points' on a physical number line is highly effective. Have students stand on the boundary number and then 'test' a number to their left and right by plugging it into the inequality. This physical movement helps them internalize which direction the solution set extends.

What is the difference between an open and closed circle?

An open circle means the boundary number is not part of the solution (used for < or >). A closed circle means the boundary number is included in the solution (used for ≤ or ≥).

How do I translate 'at most' and 'at least' into math symbols?

'At most' means that value or less (≤). 'At least' means that value or more (≥). These are often confusing for students, so connecting them to real-world limits like 'at least 18 to vote' is helpful.