Algebra Foundations

Overview

In this first lesson of a 20-lesson Algebra I unit, students connect familiar arithmetic properties to algebraic expressions. They use algebra tiles, area models, and symbolic notation to identify parts of expressions, simplify equivalent forms, and explain why their methods work.

Learning intentions

Students will be able to:

• Evaluate numerical and algebraic expressions using substitution and order of operations.
• Identify variables, terms, coefficients, constants, and factors.
• Use the commutative, associative, and distributive properties to rewrite expressions.
• Explain why two expressions are equivalent and distinguish an expression from an equation.

Success criteria

• I can label the parts of an algebraic expression accurately.
• I can simplify expressions by combining like terms and using the distributive property.
• I can show two equivalent forms of an expression.
• I can explain each step using a property or mathematical representation.

Curriculum links

• Seeing Structure in Expressions: interpret expressions in context and choose equivalent forms that reveal structure and properties.
• Reasoning with Equations and Inequalities: explain algebraic steps using equality and construct a viable justification.
• Creating Equations: represent relationships among quantities with equations or expressions in one or more variables.
• New Jersey Algebra I expectations for symbolic reasoning, mathematical argument, precision, and multiple representations.

Lesson structure (50 minutes)

1. 0–6 min · Fluency warm-up. Display integer and fraction problems in the warm-up slides; students solve independently on whiteboards, then compare strategies with a partner. Check sign rules, fraction operations, and order of operations, addressing errors without turning the warm-up into a lengthy review.

2. 6–14 min · Launch and vocabulary. Use the vocabulary and essential-question slides to introduce the question, “How can symbols represent quantities and relationships?” Teacher models (3x+5-2x), labeling variable, coefficient, terms, constant, and factors; students annotate the guided notes and expression-analysis worksheet and identify the difference between an expression and an equation.

3. 14–24 min · Concrete and pictorial modeling. In pairs, students build (2x+3) and (x+1) with algebra tiles, then combine and rearrange tiles to represent (3x+4). Teacher connects the tiles to an area model and symbolic notation using the algebra-tile and area-model slides. Ask the hinge question: “Which expression is equivalent to (2(x+3)): (2x+3), (2x+6), or (x+6)? Explain using the model.”

4. 24–35 min · Partner practice. Distribute the guided notes and expression-analysis worksheet and provide algebra tiles to pairs. Students simplify expressions such as (4x+2x+3), (3(a+4)), and (2(3y-1)+y), recording a symbolic method and, where useful, a tile or area-model representation. Teacher uses an observation checklist for correct identification of like terms, complete distribution, and accurate notation.

5. 35–44 min · Discourse routine: compare methods. Display selected solutions with the comparison and discussion slides. Pairs first rehearse an explanation, then discuss: “Which method makes equivalence easiest to see?” Students compare rearranging terms, grouping terms, and distributing, using sentence frames such as “These expressions are equivalent because…” and “The property used in this step is….” Emphasize that unlike terms cannot be combined.

6. 44–50 min · Independent check and exit ticket. Students complete the final problems on the guided notes and expression-analysis worksheet and submit an exit response: simplify (2(3x+4)+x), name the properties used, and explain why (3x+8) is not equivalent. They also correct one misconception: “To distribute (4(x+2)), multiply 4 only by (x).” Review responses to plan the next lesson.

Resources

• the Algebra Foundations slide deck
• the guided notes and expression-analysis worksheet
• Algebra tiles
• Whiteboards and markers
• Expression cards
• Teacher observation checklist
• Student notebooks or pencils
• Calculators for students whose plans permit them

Assessment

• During the warm-up and modeling, use whiteboards and hinge questions to check prerequisite fluency and conceptual understanding.
• During partner work, record observations about vocabulary, combining like terms, distribution, and the quality of justifications.
• Use the exit ticket to identify students who confuse expressions and equations, combine unlike terms, or distribute to only one term.

Differentiation

• For multilingual learners, provide labeled visuals of algebra tiles, cognate connections where appropriate, partner rehearsal, and sentence frames: “The coefficient is ___ because ___” and “The expressions are equivalent because ___.”
• For students with IEPs, provide chunked examples, color coding for like terms and factors, guided notes, manipulatives, and calculator access when appropriate. Read directions aloud and check understanding after each step.
• Provide a reference box defining variable, coefficient, constant, term, factor, equivalent, and distributive property. Pair students strategically and assign roles such as builder, recorder, and explainer.
• For enrichment, students must prove equivalence using two different methods and create a misleading-looking expression that is nevertheless equivalent, then explain why it works.