중학교 수학, 논리 및 문제 해결: 방정식 쓰기 및 풀기

This is the third course in year two of a series of middle school mathematics courses. These courses are taught in small-groups to provide individual instruction and social-learning opportunities aligned with a social constructionist or situated cognition view of learning. This series is based on an accelerated math curriculum that covers three years of content, aligned to Common Core Math Standards, over the course of two school years. It is perfect for students preparing to begin Algebra ahead of time, or those who need review and remedial support. The curriculum is problem-based this means instead of lectures or videos students work together as a small group to solve problems to discover principles and strategies with teacher guidance, as necessary. Therefore, we will spend approximately 90% of each class period working on problems and discussing them as a group. The use of discussion and problem-solving leads to generalized mathematics or proofs. This process prepares students well who may seek advanced mathematics in high-school or beyond. Students will complete a math notebook In the style of a main lesson book to help with recall and long-term retention.
In this unit, students solve equations in the form of px+q=r and p(x+q)=r using only rational numbers. They will draw and write equations with a single variable to match diagrams and real world scenarios. They will use tape and hanger diagrams to represent real world problems and equations. They will practice algebraic equations for solving equations. 
Week 1: 
Day 1: We will begin with a readiness check to understand students existing conception of geometry and algebra ideas that may need to be addressed and inform instruction throughout the unit. 
Day 2: We will explore engaging contexts that are characterized by non-proportional relationships. The goal of this lesson is to demonstrate that strategies from year 1 dealing with proportional relationships need adjusting in order to be efficient.
Day 3: We will work on representing and reasoning with the real world context of an algebraic equation and representing our understanding using tape diagrams. We will use these tape diagrams to solve more complex word problems. 
Day 4: We will shift our focus to equations as we connect them to our tape diagrams. We will match tape diagrams with equations, sort equations, and draw tape diagrams to match provided equations. We will then use the tape diagram to support us in solving equations using student's existing strategies and reasoning. 
Day 5: We will work to standardized our equations and understand the importance of standard ways of communicating information in math. We will learn what sorts of situations we can represent with the px+q=r equations and reason about solving them to prepare for generalizing solution methods. 
Week 2:
Day 1: This lesson is very similar to week 1 day 5 except our focus is on equations of the form p(x+q)=r. We will focus on understanding the meaning of equations in context and choosing an appropriate representation for real world problems. 
Day 2: We will learn the differences between the two types of equations and work on distinguishing them both in context and mathematically using corresponding tape diagrams and the situations. We will sort equations  and write our own equations to match stories. We will then solve the equations to answer questions about complex word problems. 
Day 3: We will begin the work of generalizing problem solving strategies using hanger diagrams. Students will learn to approach problems using inverse operations and concrete numbers before building the skill. We are also going to practice generating equivalent expressions to broaden our concept of how equations can be used. 
Day 4:  We will continue our work of developing equation solving strategies using hanger diagrams with the goal of developing two strategies to solve an equation of the form p(x+q)=r.  We will also focus on using the distributive property to accomplish that. We will work on matching equations to diagrams and solving them based on complex word problems. 
Day 5: We will practice what we have learned up to this point and our generalized problem solving strategies by working with negative numbers in equations through a variety of problem-solving activities including generating equations and chaining equations. 
Week 3
Day 1: We will review what we have learned so far through a focus on solving equations and understanding clues which can help us decide which of our two strategies will be more efficient. 
Day 2: The will bring together all the skills and concepts of the unit and expand on them so that students can solve complex word problems which can be represented with equations of the two forms without diagrams or additional scaffolding. They will choose appropriate representations to communicate their work to their peers.
Day 3: We will further our understanding of equations and review percentages by using our skills to solve word problems related to percent change. Students will communicate their understanding using visuals and written explanations. 
Day 4: We will practice using equations to represent the relationships between angles and practice writing and solving equations to find unknown angle measures. This reviews content from unit 2 and builds fluency with both concepts. 
Day 5: Learners will independently take a final assessment to assess mastery and inform their next steps. They will demonstrate this through problem-solving and written/drawn explanations of their work and key concepts.