- =20
**Comparing Different Functions**- The stude= nts will discuss, represent, and solve a verbal problem involving the choic= e between two functions. =20
**Comparing Graphs**- Students are given an hourly rate o= f pay and infer coordinates for (h, $) over a range of hours. They produce = a table and a graph of work-pay. Then they produce another graph for anothe= r rate of pay and discuss differences in time and pay. =20
**Dinner Tables I**- Students work with a function relatin= g number of tables to the number of available seats. One table seats 4, two= tables seat 8, three tables seat 12.... =20
**Dinner Tables II**- Students work with a function relat= ing the number of tables (in a straight line) to the number of available se= ats. One table seats 4, two tables seat 6, three tables seat 8.... =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
**Formulas and Stories**- The students will be requir= ed to work with the relation between different mathematical expressions (fo= rmulas) and stories. =20
**Functioning Together**- Students work together to d= evelop multiple representations of a function. The students split up into g= roups of three with each student having a separate responsibility. When all= the input values have been used up, the students are asked to, together, m= ake up a story that describes their function. =20
**Functions - Earning Money**- The students will= create tables and equations from given stories. The functions are additive= and multiplicative. =20
**Functions II**- The students will use three functions that = are represented as a sequence of patterns and create a sequence of hops on = the number line, a data table, and an algebraic expression to express the f= unctions. =20
**Functions from Tables**- Students work with a func= tion, beginning with a table and then a formula, to generate ordered pairs = that follow the rule of the function. =20
**How Many Points?**- Students work with: (a) a context =E2=80= =94 distance as a function of time; (b) generating coordinates. =20
**Human Graph I**- Students plot themselves on a Cartesian p= lane. Each student will get a large card with a place for an ordered pair: = (x, y), where x refers to hours worked, and y refers to amount earned. The = students must name the coordinate pair for the point they themselves are st= anding on. =20
**Linear vs Quadratic Functions**- The stude= nts will use two functions (a linear and a quadratic) that are represented = as a sequence of patterns and create a sequence of hops on the number line = and an algebraic expression to express the functions. =20
**Rates vs Totals**- Students compare points on an hours/p= ay Cartesian space. The main challenge lies in recognizing that, although o= ne student earned more, the other student was paid better, that is, at a hi= gher rate of pay. They must indicate the difference in pay and the differen= ces in amount worked. =20
**Recipes that Exchange**- The lesson focuses on a f= unction that multiplies input by two but also changes the ingredient to ano= ther type of ingredient. =20
**Starting With A Rule**- Students focus on whether g= iven outputs are consistent with a given rule. =20
**Times Two**- The lesson focuses on a function that multiplies = the input by two. New notations are introduced. =20

- =20
**Cartesian Candy Bars II**- Children work on shar= ing different amounts of candy bars among different numbers of people. They= compare ratios (candy bars per person) and plot points in a Cartesian grid= . =20
**Comparing Functions**- This lesson is split into two= days. In the first class, the students will analyze eight basic graph shap= es and will represent and solve a verbal problem involving the choice betwe= en two functions. In the second one they will be asked to choose, among the= eight basic graph shapes, the ones that matches specific situations. =
=20
**Equations and Inequalities**- Students will w= ork with equations and inequalities, first with simple ones and later with = comparisons of two functions. The Wallet Problem, introduced in a previous = lesson, will provide the background context. =20
**Evaluation Problem**- Students will be given a proble= m that asks about the amount of money each person has, based on the amount = in a piggy bank. They will be given one graph and asked to draw the second = graph. =20
**Fourth Grade Assessment III**- This is a wri= tten assessment where children will be asked to interpret graphs and to int= erpret and determine the truth or falsehood of equations and statements tha= t describe comparisons between quantities. =20
**Fourth Grade Assessment IV**- This is a writt= en assessment where children will compare two students. One of the students= ' speed can be represented linearly while the other's speed is represented = by a non-linear graph. =20
**Graphing Halves and Doubles**- Children work= on a problem about distance and time and compare two rates: half a meter p= er second and two meters per second. =20
**Graphing Thirds and Triples**- Children work= on a problem about distance and time and compare two rates: one third of a= meter per second and three meters per second. =20
**Intervals**- Students reason about graphs showing growth over = time. They compare heights of children and heights of two animals at differ= ent time intervals. =20
**Running Race I**- Compare a race between two students: on= e who runs at a constant pace, the other who tires out as the race proceeds= . =20
**Running Race II**- Compare a race between two students: = one who runs at a constant pace and one who changes pace as the race procee= ds. =20
**Swimming Pools I**- Compare how two swimming pools fill= up with water over several hours. =20
**Swimming Pools II**- Students will examine the rate of= pools filling over several hours. =20
**The Better Paying Job I**- Children work on a pr= oblem about rate of pay per hour of work. They compare ratios (dollars earn= ed per hour of work) and discuss and plot points in a Cartesian plane. =
=20
**The Better Paying Job II**- Children work on a = problem about rate of pay per hour of work. They compare ratios (dollars ea= rned per hour of work) and discuss and plot points in a Cartesian plane.=20**Three Heights Review**- In this class we will explo= re: (a) How children deal with comparisons, (b) How they draw inferences fr= om comparisons, and (c) How they represent comparisons between three unknow= n amounts. =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
**Two Phone Plans I**- Students compare two phone plans,= one of which has a lower rate, but a monthly basic charge; the other has a= higher rate but no basic charge. =20
**Two Phone Plans II**- Students will work on the compa= rison between two phone plans (also used in the lesson "Two Phone Plans I")= , one of which has a lower rate, but a monthly basic charge, the other has = a higher rate but no basic charge. =20
**Wallet Problem I**- Students compare the amounts of mon= ey two students have. The amounts are described relationally but not throug= h precise dollar amounts. =20
**Wallet Problem II**- Students will be given a wallet p= roblem. They will be asked to compare the amounts of money two students hav= e. The amounts are described relationally but not through precise dollar am= ounts. =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
**Basic Function Shapes**- In this lesson, the stude= nts will (a) discuss, represent, and solve a verbal problem involving the c= hoice between two functions; (b) choose, among 8 basic graphs (7 distinct s= hapes), the one that matches specific situations; and (c) write stories to = match a specific graph shape. =20
**Elapsed Time**- A variant of the train crash problem is use= d to address questions about elapsed time. The task is to determine where a= train is, given a certain time. =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 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
**Fifth Grade Assessment III**- This assessment= will focus on writing equations to solve verbal problems and on solving eq= uations 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 I**- Students will be asked to use= the syntactic rules of algebra to solve equations with variables on both s= ides of the equals sign. =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
**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 gr= aph for each function and, later, to look at and describe graphs of all thr= ee cousins' savings. =20
**Wallet Review Problem**- This activity is a review= of the Wallet Problem done in fourth grade. It is intended to introduce ne= w students to some of the concepts we have covered and to refresh the memor= ies of our old students. Students compare the amounts of money two students= have. The amounts are described relationally but not through specific doll= ar amounts. =20

- =20
**Biggest Output**- Students will decide on what linear and= quadratic functions will result in the greatest output, starting from an a= lgebraic expression, and using tables and graphs to help them make these de= cisions. =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
**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
**Compare and Contrast**- Students will identify the = y-intercept and slope using equations and then use that data to create corr= esponding tables and graphs. =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
**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
**What Will Happen**- Students will work with equations o= f functions (both linear and non-linear) to find the y-intercept without gr= aphing. =20
**Who Shares My Function? - Linear with All Representations**- S= tudents 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. T= hey will attempt to explain and discuss why the different representations r= efer to the same function. =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
**Who Shares My Function? - Linear with Graphs, Tables, and Equations - Students will make groups by finding other students who have the s= ame linear function, as shown in representations of graphs, tables, or equa= tions. They will then generate a story to go with the function.****Who Shares My Function? - Linear with Negative and Fractional Slope - Students will find other functions that are the same as theirs, st= arting 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 tha= t describes all of the different representations of the same function.****Who Shares My Function? - Quadratics**- Students will make gro= ups by finding other students who have the same quadratic or linear functio= n in different representations. =20
**Wind-Up Car**- Students will produce an equation from a grap= h, based on an engineering-context. =20
**= 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.=20