Equivalent Expressions - Grades 6-8

• Rationale
• Summary
• Goals and Standards
• Progression of Skills (outline)
• Materials List

• Lesson Introduction & Pre-Assessment
• Lesson
• Lesson Closure & Post-Assessment
• Practice Questions
• Real-World Applications & Project Ideas

What?

Students will be engaged in opportunities to process their thinking during dialogue with others. They will apply prior knowledge and skills to create and use multiple representations of equivalent algebraic expressions in various contexts. Students will use factoring, combining like terms, integer operations, and geometric applications during the lesson.

Why?

Students bring both understanding as well as misunderstandings into new learning opportunities. Surfacing all thoughts, both correct and incorrect,(without affirming or negating) provides an important component in the learning process. The opportunity to correct misunderstandings, replacing them with new learning and thinking supports success and mastery of current and future learning.

"Teachers should be sensitive and recognize that students come into their classrooms with misunderstandings and misconceptions. It is imperative that they work toward detecting the existing misconceptions that students may have and work purposefully to correct them. This is to avoid a situation whereby students move from one grade to another with these harmful misconceptions. Teachers should acknowledge that learners can overcome misconceptions by planning and consciously providing opportunities for learning through effective teaching strategies.”

(Ojose, Bobby. "Students' Misconceptions in Mathematics: Analysis of Remedies and What Research Says.” Ohio Journal of School Mathematics, Vol. 72, 2015, pp. 30–34, https://doi.org/https://kb.osu.edu/server/api/core/bitstreams/5c4f895c-64a7-5018-921a-475db38d03ca/content. )

How?

Place the responsibility for learning in the hands of the students, using the teacher as a facilitator, to empower students to formulate expressions and confirm their understanding. A good facilitator helps students to identify and correct misunderstandings, enabling confidence to apply their knowledge in future lessons.

Before the lesson:

• Students should be fluent in simplifying numerical expressions. They should also have a firm grasp on variables as numbers, including exposure to (but not mastery of) equivalent expressions with positive integers. If a student can complete the pre-assessment, they are ready for this toolkit lesson.

During this toolkit lesson:

• Focuses on identifying and generating equivalent expressions, including understanding the rules for multiplying negative numbers, distribution, factoring and combining like terms.

After the lesson:

• Students are ready to learn to solve linear equations in one variable.

Lesson

Read the teacher's guides for directions for each session.

Anticipate Misconceptions:

• Distributing a common factor to the first term in the parentheses only
• Misusing operations of numbers
• Confusing order of operations (e.g., multiplication and division are applied together)
• Confusing integer operations, especially operations with negative numbers
• Correctly identifying a common factor
• Overlooking multiple equivalent expressions
• Confusion that a number next to parentheses denotes multiplication

Optional Activities:

1. Review of area and perimeter of rectangles
2. Desmos for review of order of operations
3. Review of Fractions, Decimals and Percents
4. Tax Tip Discount practice
5. Error Analysis for distribution and combining like terms

Progress Monitoring:

• Listening for misconceptions during student dialogue
• Attending to the claims initially made during each task
• Asking questions to clarify thinking and correct misconceptions
• Using pre and post assessments to document mastering and identify needed clarification.

• Area and perimeter problems: fence around a garden, painting a wall, carpeting a floor
• Consumer applications: Amazon prime day problem, discounts, tipping on a meal.