- =20
**Dots Problem**- We present to the students a problem dealin= g with a growing pattern over time. To begin, there is one dot. With each p= assing minute four more dots are drawn around the previous dot(s). =20
**Functions - Earning Money**- The students will= create tables and equations from given stories. The functions are additive= and multiplicative. =20

- =20
**Comparing Strips of Unmeasured Lengths II<= /a>**- The class is the second of the "Strips of Unmeasured Lengths= " series that will focus directly upon the algebraic representation of meas= urements and their multiplicative relations. Children are asked to compare = the lengths of strips, to use algebraic notation to describe the relationsh= ips between them, and to demonstrate that the relationships they represent = are true. =20
**Comparing Strips of Unmeasured Lengths II= I**- This is the third lesson in the "Strips of Unmeasured Leng= ths" series that focuses directly upon the algebraic representation of meas= urements and their multiplicative relations. We will work with the relation= ship B =3D 3S, focusing on equations and their verbal descriptions and on t= rue and false equations and statements. =20
**Fourth Grade Assessment II**- This is a writt= en assessment where children will interpret and determine the truth or fals= ehood of equations and of statements that describe comparisons between quan= tities. =20
**Piggy Banks**- The whole lesson revolves around a multipart = story problem involving changes in two quantities over several days of a we= ek. The initial quantities are equal yet unknown. Then transformations are = applied to the quantities. Students are asked to compare the quantities thr= oughout the week even though only their relative relationship can be determ= ined. =20
**Three to One**- Children discuss and produce verbal and mat= hematical statements on the proportion, S:L :: 1:3, that is, on the functio= n f( x ) =3D 3x and on its inverse f^{-1}( x ) =3D 1/3 x =20
**Wallet Problem III**- Students will continue working = with the wallet problem. They will be shown a graph for Mike's amounts and = asked to (a) determine whether it represents Robin's or Mike's money and (b= ) to predict where the line for Mike would fall. Later they will plot Mike'= s amounts and will discuss why the lines cross. =20

- =20
**Ar= cade**- Students are told a story about two children, each of w= hom 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. =20
**Enacting and Solving Equations**- Student= s 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 ha= ve the same amount of candies. The situation corresponds to the equation 3x= + y + 6 =3D 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 re= present the situation, to solve the equation that corresponds to the situat= ion, and to solve other written equations with similar structure. =20
**Equations and Graphs**- Students will further compa= re two linear functions in the context of evaluating two plans for shovelin= g snow. One plan has two parts: a basic charge plus a charge based on the n= umber 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 determ= ine the circumstances in which the bill from each plan would be the same. T= hey then examine the graph of the two functions and discuss how equations a= nd inequalities relate to the graph. =20
**Equations in Groups**- Students first discuss equali= ty situations and how equal changes on both sides of the equality do not ch= ange the equality or the solution to the equation. In a second activity, A = pair of students begins with a solved equation (e.g. N =3D 4) and passes th= e equation to their neighbor; the neighbor operates equally on each side of= the equation and passes the equations to the following neighbor. They cont= inue this process until the series of equations return to the first two stu= dents who, then, check whether the solution still holds. They also check th= e logic and correctness of their colleagues operations on the initial equat= ion. =20
**Equations in Groups II**- A student (or a pair of= students) begins with a solved equation (e.g. N =3D 4) and pass(es) the eq= uation to neighbor (or pair of neighbors); the neighbor(s) operate(s) equal= ly 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 seri= es of equations returns to the first students, each student (or pair of stu= dents) check whether the solution still holds for the solution they had pro= posed at the beginning. They also check the logic and correctness of the ch= anges implemented by their classmates. =20
**Fifth Grade Assessment I**- This assessment wil= l focus on writing equations to solve verbal problems and on solving equati= ons using syntactic rules. It is intended as a diagnostic tool to assist te= achers in planning future activities. =20
**Fifth Grade Assessment I Review**- This = lesson will focus on reviewing the recent in-class assessment, on writing e= quations for word problems, and on solving equations. =20
**Fifth Grade Assessment II**- This assessment w= ill focus on writing equations to solve verbal problems and on solving equa= tions using the syntactic rules of algebra. =20
**Phone Plans**- Students will compare two linear functions in= the context of evaluating phone plans. One plan has two parts: a basic cha= rge 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 th= e 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. =20
**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 th= e 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. =20
**Solving Equations II**- Students will be asked to r= epresent and solve verbal problems requiring algebra and to use the syntact= ic rules of algebra to solve equations with variables on both sides of the = equals sign. =20
**Solving Equations with One Variable**= - Students work on a story about two children who each have a certain amou= nt of money. The amount of one of the children is known but the other is no= t. After a sequence of transformations they end with the same amount of mon= ey. Students will be led to solve for the starting value by relating the eq= uation to the events in the story. After that, they will be asked to solve = another similar problem. =20
**Train Crash**- Students will compare two linear functions re= presented in a graph. They reason about the problem using (a) the word prob= lem and two diagrams; (b) a graph of position vs. time; (c) a table of valu= es (d) making expressions for each position function; and (e) solving the e= quation algebraically. =20

- =20
**Box Extremum**- Students will start by finding average rate= s of change for a non-linear function over increments of the independent va= riable. The size of the increments will decrease to introduce the idea of u= sing tangent lines to find instantaneous rates of change of linear and non-= linear functions. Students will see what a tangent looks like at the extrem= a of a graph. Students will then create a box that maximizes the volume and= see how determining the extrema of a graph can help to find the maximum vo= lume. =20
**Area of a Square as a Function**- Student= s will develop a quadratic equation to represent the area of a square. =
=20
**Box of Clay Activity**- Students will compare two c= ubic functions based on the context of the volumes of a box of clay. =
=20
**Can We Predict Differences?**- Students will predict, produce,= and compare linear and non-linear function graphs used to represent the nu= mber of punches on a balloon. =20
**Candy Experiment**- Students will create their own data= to construct a graph and equation of negative and fractional slope. =
=20
**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 sca= led coordinate planes. =20
**Coupon Activity**- Students will create graphs, tables a= nd equations to explain their stories and look at how a graph changes depen= ding on the y-intercept. =20
**Function Challenges - 20 Questions**-= Students will compete in a game to generate equations for functions that m= eet certain criteria, as given by the instructor. =20
**Graphing Equations**- Students will practice moving b= etween graphs and equations of functions, as well as identifying the y-inte= rcept and slope. =20
**Guess My Rule - Linear**- Students will try to de= termine the equation to match their partner's created graph and work togeth= er to correct their own mistakes. =20
**Guess My Rule - Non-Linear**- Students will p= roduce algebraic expressions starting from non-linear graphs produced by ot= her students in the class. =20
**It Depends**- Students will think about how we can show a dep= endent relationship between two quantities, using a variety of representati= ons. =20
**Jason's Tree House**- Students will extract data fr= om a story and use tables and graphs to answers questions about proposed sc= enarios. =20
**Lotto Winnings**- Students will generate a graph for a no= nlinear function, point by point, in order to realize that there are differ= ent types of functions that they might not know about yet. =20
**Playground Construction**- Students will create = a quadratic equation based on the context of building a playground referrin= g to surface, fencing, and equipment needed, to create an equation of y =3D= ax^{2}+ bx + c form. =20
**Race Car Activity**- Students will look at four differ= ent graphs to determine which two describe the scenario proposed by the tea= cher displaying parallel lines and the correct y-intercepts. =20
**Relating Graphs and E= quations - Linear and Quadratic Functions**- Students will gene= rate graphs from given equations and equations from given graphs. =20
**Same and Different**- Students will compare graphs of= linear functions, looking for similarities and differences, and will produ= ce algebraic expressions, again identifying what is the same and what is di= fferent about each one. =20
**Sound Loudness**- Students will examine a non-linear func= tion depicted in a graph and generate the corresponding function table and = equation. =20
**Who Shares My Function? - Linear with Graphs and Stories**- St= udents will make groups by finding other students who have the same quadrat= ic or linear function in different representations. =20
**Wind-Up Car**- Students will produce an equation from a grap= h, based on an engineering-context. =20