중학교 수학, 논리 및 문제 해결: 선형 관계 및 플롯

This is the fifth 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. 
In this unit, we expand our math vocabulary to include terms like rate of change, linear relationship, and vertical intercept. We deepen our understanding of slope and connect it to concepts like the constant of proportionality and proportional relationships. We will represent linear relationships in a variety of ways including tables, equations, and graphs with positive/negative slops, vertical intercepts, and horizontal or vertical lines. We define solution of an equation and learn to solve systems with many, one, or no solutions. We also work with bivariate data sets and work with scatter plots, outliers, and associations. We will work on understanding "goodness of fit" for lines on a scatter plot. 
Week 1, Day 1: We will begin with completing a readiness check that covers prerequisite skills as well as preview skills used in the unit. This will allow modification of pacing and instruction to support students throughout the unit with additional review or acceleration. 
Week 1, Day 2: We will work on understanding what makes a good graph by considering the components of a graph and then adding scale to graphs. We also graph lines based on verbal description and comparison. 
Week 1, Day 3: We will work to understand that there are many successful ways to graph a proportional relationship but that we may make choices to see specific information. We will practice identifying features of a graph relating to proportional relationships and equations. 
Week 1, Day 4: We will expand our work by comparing two situations represented in different ways and how we can find needed information from each type with context-related questions and problem solving. 
Week 1, Day 5: We will shift from proportional relationships to linear relationships with positive rates of change. We use the same tools we've learned from proportional relationships (graphs, tables, and equations). We will investigate similarities and differences. We will make connections between the rate of change and the slope of a line. 

Week 2, Day 1: We expand our understanding of slope as the rate of change in a linear relationship and explore the vertical intercept or y intercept as part of a problem. 
Week 2, Day 2: We will develop an equation for a linear relationship by expressing regularity in repeated calculations. We will graph the relationships and interpret the vertical intercept and slope. We will work on a procedure to compute the slope of a line from any two points that lie on the line. 
Week 2, Day 3: We will introduce y= mx+b  and generalize a third way of understanding the equation of a line using vertical translations (unit 1). We will also introduce the idea of a negative y intercept. 
Week 2, Day 4: We will explore lines with non positive slopes including flat lines through graphs and context. We will consider these in real world situations. 
Week 2, Day 5: We will extend our work with slope traingles to develop a method for the line using any two lines and then extend our thinking of formulas for equations to include horizontal and vertical lines. 
Week 3, Day 1 We will work on exploring solutions where variables are not dependent but where both variables have to satisfy a constraint. We will learn that the solutions to the equations are pairs of numbers that make it true and can be represented as points on the graph. 
Week 3, Day 2: We will continue to focus on the relationship between variables, solutions, and graphs for linear equations. We will prepare for work on systems of equations by exploring graphs and relationships of two simultaneously satisfied equations. 
Week 3, Day 3: We will complete a mid-unit assessment. This checkpoint allows us to identify skills that need additional support, misconceptions that need to be addressed, and review previous lessons. 
Week 3, Day 4: We will study systems of linear equations in the context of distance versus time and using the form of y=mx+b. We practice graphical interpretation of these systems. 
Week 3, Day 5: We will formally introduce the concept of systems of equations with different contexts. We recognize that we have found solutions using graphing by examining the intersection. We explore systems with no solutions. 

Week 4, Day 1:  We will continue to explore systems where equations are both of the form y=mx+b and connect algebraic to graphical representations. We will introduce how tow equations with the same rate of change can have 0 or infinitely many solutions. 
Week 4, Day 2: We  graduate to other types of systems with different structures. We learn that examining structure is a good first step since it is sometimes possible to recognize an efficient method for solving the system through observation.
Week  4, Day 3: We learn to write their own systems representing different contexts, and to interpret the solutions for those systems. Different contexts can lead to systems in different forms, so we also continue to practice looking at different systems and thinking ahead about how to solve them. 
Week 4, Day 4: We begin a mini-unit on finding associations using tables and scatter plots to organize data. 
Week 4, Day 5: We continue our investigation of scatter plots. We interpret points in a scatter plot in terms of a context, and add points to a scatter plot given information about an individual in the population. We compare individuals represented by different points and informally discuss trends in the data.

Week 5, Day 1: We will gain a more holistic focus and see a set of data as a single things to be analyzed. We will analyze connections between the scatter plot and line by comparing individual points and reason abstractly and quantitatively. 
Week 5, Day 2: We will introduce the terms  positive association and negative association. We use  fitted lines to help them understand this language and tie it back to their work in an earlier unit on linear relationships. We start to use language to describe trends like, "Cars made in a later year tend to have a higher price."
Week 5, Day 3: We focus on the language “As the independent variable increases, the dependent variable tends to decrease.”. We interpret the slopes of fitted lines in context and identify positive and negative associations without a fitted line. 
Week 5, Day 4: We visually identify clusters in data and do a card sort to distinguish linear and non linear associates in data and rely on visual patterns. We bring everything we've studied to the unit so far to analyze and interpret bivariate data in context using scatter plots, identifying outliers, fitting a line, determining and interpreting the slope of the like.
Week 5, Day 5: We study categorical data displayed in two-way tables, bar graphs, and segmented bar graphs. The different graphical representations help students visualize the frequencies and relative frequencies, aiding them in making judgement about whether there is evidence of an association or not in the next lesson.

Week 6, Day 1: In this project based lesson, students use two-way tables, bar graphs, and segmented bar graphs to decide whether there is evidence of an association in categorical data or not.
Week 6, Day 2: In this culminating lesson for the unit, students put what they have learned to work in solving real-world problems, using all the different forms of equations they have studied. 
Week 6, Day 3: In this final lesson on systems of equations, students work in groups as they apply what they have learned to solve three problems with different structures and then create a new problem similar in structure to one of the ones they solved. Groups trade problems, prepare well-explained solutions, and take turns sharing their solutions with the class. 
Week 6, Day 4: In this project based lesson, students collect, represent, and analyze quantitative bivariate data. Then they group the quantitative data to create two categories, and look for evidence of an association in the resulting categorical data. This lesson allows students the opportunity to carryout the entire process of data analysis beginning with data collections and ending with an analysis of graphical representations to determine whether there is an association between two variables in the data.
Week 6, Day 5: We will finish the unit with an independent assessment.