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**Wallet Review Problem**- This activity is a review of the Wallet Problem done in fourth grade. It is intended to introduce new students to some of the concepts we have covered and to refresh the memories of our old students. Students compare the amounts of money two students have. The amounts are described relationally but not through specific dollar amounts.**Arcade**- Students are told a story about two children, each of whom has a certain amount of money, but only one of whom has an amount known to us. After a series of events they happen to end up with the same amount of money.**Solving Equations with One Variable**- Students work on a story about two children who each have a certain amount of money. The amount of one of the children is known but the other is not. After a sequence of transformations they end with the same amount of money. Students will be led to solve for the starting value by relating the equation to the events in the story. After that, they will be asked to solve another similar problem.**Enacting and Solving Equations**- Students enact and discuss a situation where two children have amounts of candies. Some of the candies are visible, others are inside opaque tubes or boxes. After considering multiple possibilities they are told that the children have the same amount of candies. The situation corresponds to the equation 3x + y + 6 = x + y + 20, where x is the amount of candies per tube and y is the amount of candies per box. Students will be asked to discuss and to represent the situation, to solve the equation that corresponds to the situation, and to solve other written equations with similar structure.**Assessment I**- This assessment will focus on writing equations to solve verbal problems and on solving equations using syntactic rules. It is intended as a diagnostic tool to assist teachers in planning future activities.**Assessment I Review**- This lesson will focus on reviewing the recent in-class assessment, on writing equations for word problems, and on solving equations.**Equations in Groups**- Students first discuss equality situations and how equal changes on both sides of the equality do not change the equality or the solution to the equation. In a second activity, A pair of students begins with a solved equation (e.g. N = 4) and passes the equation to their neighbor; the neighbor operates equally on each side of the equation and passes the equations to the following neighbor. They continue this process until the series of equations return to the first two students who, then, check whether the solution still holds. They also check the logic and correctness of their colleagues operations on the initial equation.**Equations in Groups II**- A student (or a pair of students) begins with a solved equation (e.g. N = 4) and pass(es) the equation to neighbor (or pair of neighbors); the neighbor(s) operate(s) equally on each side of the equation. And so on, around the table. There should be at least three students or pair of students at each table. When the series of equations returns to the first students, each student (or pair of students) check whether the solution still holds for the solution they had proposed at the beginning. They also check the logic and correctness of the changes implemented by their classmates.**Solving Equations I**- Students will be asked to use the syntactic rules of algebra to solve equations with variables on both sides of the equals sign.**Solving Equations II**- Students will be asked to represent and solve verbal problems requiring algebra and to use the syntactic rules of algebra to solve equations with variables on both sides of the equals sign.**Assessment II**- This assessment will focus on writing equations to solve verbal problems and on solving equations using the syntactic rules of algebra.**Phone Plans**- Students will compare two linear functions in the context of evaluating phone plans. One plan has two parts: a basic charge plus a charge based upon the number of minutes used. The other plan has no basic charge; it only charges according to the minutes used. However the per-minute charge is higher than in the other plan. Students are asked to determine the circumstances in which the monthly bill from each plan would be the same. They then examine the graph of the two functions and discuss how equations and inequalities relate to the graph.**Equations and Graphs**- Students will further compare two linear functions in the context of evaluating two plans for shoveling snow. One plan has two parts: a basic charge plus a charge based on the number of square meters cleared. The other plan has no basic charge; it only charges according to the number of square meters cleared. However the per-meter charge is higher than in the other plan. Students are asked to determine the circumstances in which the bill from each plan would be the same. They then examine the graph of the two functions and discuss how equations and inequalities relate to the graph.**Trains**- Students will compare two linear functions represented in a graph. They reason about the problem using (a) the word problem and two diagrams; (b) a graph of position vs. time; (c) a table of values (d) making expressions for each position function; and (e) solving the equation algebraically.**Elapsed Time**- A variant of the train crash problem is used to address questions about elapsed time. The task is to determine where a train is, given a certain time.**Assessment III**- This assessment will focus on writing equations to solve verbal problems and on solving equations using the syntactic rules of algebra.**Varying Rates of Change**- Students will compare three functions, two of which are nonlinear, that tell the story of three cousins who all save $1,000 in one year. One saves a lot the first day and less and less each day as time goes on; one saves very little the first day and more and more each day throughout the year; the last cousin saves the same amount each day. Students will be asked to predict the shape of the graph for each function and, later, to look at and describe graphs of all three cousins' savings.**Basic Function Shapes**- In this lesson, the students will (a) discuss, represent, and solve a verbal problem involving the choice between two functions; (b) choose, among 8 basic graphs (7 distinct shapes), the one that matches specific situations; and (c) write stories to match a specific graph shape.**Review on Graphs and Equations**- In this lesson, the students will solve individually or in small groups the set of problems. For each problem, the teacher will lead a discussion based on the students' work (the teacher should identify strong and weak points in the students' work). The class is organized around four main problems. Within each problem students will answer different questions.

## Middle School Lessons

**Who Shares My Function? - Linear with All Representations**- Students will work in groups after finding other students who have the same linear function represented by a story, a table, a graph, or an equation. They will attempt to explain and discuss why the different representations refer to the same function.**It Depends**- Students will think about how we can show a dependent relationship between two quantities, using a variety of representations.**Who Shares My Function? - Linear with Graphs and Stories**- Students will make groups by finding other students who have the same quadratic or linear function in different representations.**Contrasting Equations**- Students write equations for three graphs and examine their slopes by comparing and contrasting the graphs. Students also look at the same functions graphed on differently scaled coordinate planes.**Who Shares My Function? - Linear with Graphs, Tables, and Equations**- Students will make groups by finding other students who have the same linear function, as shown in representations of graphs, tables, or equations. They will then generate a story to go with the function.**Jason's Tree House**- Students will extract data from a story and use tables and graphs to answers questions about proposed scenarios.**Candy Experiment**- Students will create their own data to construct a graph and equation of negative and fractional slope.**Graphing Equations**- Students will practice moving between graphs and equations of functions, as well as identifying the y-intercept and slope.**Wind-Up Car**- Students will produce an equation from a graph, based on an engineering-context.**Who Shares My Function? - Linear with Negative and Fractional Slope**- Students will find other functions that are the same as theirs, starting from a table, a graph, or an equation. Once they have identified the same function represented in a different way, they will create a story that describes all of the different representations of the same function.**Guess My Rule - Linear**- Students will try to determine the equation to match their partner's created graph and work together to correct their own mistakes.**Compare and Contrast**- Students will identify the y-intercept and slope using equations and then use that data to create corresponding tables and graphs.**Coupon Activity**- Students will create graphs, tables and equations to explain their stories and look at how a graph changes depending on the y-intercept.**Race Car Activity**- Students will look at four different graphs to determine which two describe the scenario proposed by the teacher displaying parallel lines and the correct y-intercepts.**Same and Different**- Students will compare graphs of linear functions, looking for similarities and differences, and will produce algebraic expressions, again identifying what is the same and what is different about each one.**Can We Predict Differences?**- Students will predict, produce, and compare linear and non-linear function graphs used to represent the number of punches on a balloon.**Lotto Winnings**- Students will generate a graph for a nonlinear function, point by point, in order to realize that there are different types of functions that they might not know about yet.**Area of a Square as a Function**- Students will develop a quadratic equation to represent the area of a square.**Relating Graphs and Equations - Linear and Quadratic Functions**- Students will generate graphs from given equations and equations from given graphs.**Playground Construction**- Students will create a quadratic equation based on the context of building a playground referring to surface, fencing, and equipment needed, to create an equation of y = ax2 + bx + c form.**Who Shares My Function? - Quadratics**- Students will make groups by finding other students who have the same quadratic or linear function in different representations.**What Will Happen**- Students will work with equations of functions (both linear and non-linear) to find the y-intercept without graphing.**x2 and x**- Students will look at x squared and x as functions, and for which values of x one function value is greater than the other.**Biggest Output**- Students will decide on what linear and quadratic functions will result in the greatest output, starting from an algebraic expression, and using tables and graphs to help them make these decisions.**Sound Loudness**- Students will examine a non-linear function depicted in a graph and generate the corresponding function table and equation.**Guess My Rule - Non-Linear**- Students will produce algebraic expressions starting from non-linear graphs produced by other students in the class.**Box of Clay Activity**- Students will compare two cubic functions based on the context of the volumes of a box of clay.**Graphing Equations - Nonlinear Functions**- Students write equations for three graphs and examine their slopes by comparing and contrasting the graphs. Students also look at the same functions graphed on differently scaled coordinate planes.**Curves in a Cubic**- Students will explore different kinds of cubic functions through graphs and tables.**Function Challenges - 20 Questions**- Students will compete in a game to generate equations for functions that meet certain criteria, as given by the instructor.