- =20
**All Things Being Equal II**- The equals sign s= ignifies that amounts on each side are the same. The students will use Unif= ix blocks and the corresponding equations to represent equalities between a= dditive amounts. =20
**All Things Being Equal III**- The students wi= ll write equations to represent verbal statements and successive transforma= tions that maintain or do not maintain the equality. =20
**Candy Boxes**- This class centers on the possible amounts of= candies two children, John and Maria, have. They each have the same, unspe= cified number of candies inside their own candy box. John has, in addition,= one extra candy and Maria has three extra candies. What are the possible t= otal candies they might have? =20
**Comparing Discrete Quantities**- Students = compare amounts of tokens and unknown amounts of discrete quantities. In bo= th cases they are guided to adopt line segments to represent discrete amoun= ts and the differences between them. They are also asked to discuss composi= tion of measures: "the difference plus the smaller amount is equal to the l= arger amount" and, "the larger amount minus the difference is equal to the = smaller amount". =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
**Comparing Heights I**- Students compare the heights = of two children, measure, compare, and represent one's own height in relati= on to a peer's height, and focus on the differences between heights. =
=20
**Comparison Problems & Tables**- This class will be used to= review concepts and representations as applied to the solution of verbal c= omparison problems and to work on function tables. =20
**Comparisons**- Comparisons and comparison operators: =3D, = =E2=89=A0, <, >. =20
**Comparisons and Attributes**- Work with compa= risons and comparison operators (=3D, =E2=89=A0, <, >). =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
**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
**Guess my Rule - Multiplicative Tables - Two children create secret rules for transforming input numbers. The= teacher uses a doubling or tripling rule.****Guess my Rule - Tables**- Two children create sec= ret rules for transforming input numbers. The teacher uses a doubling rule.= =20
**Heights as Functions**- In this class children will= work on the functional representation of two unknown heights and on the co= mposition of the shorter height plus the difference between the heights as = equal to the second height. =20
**How Many Points?**- Students work with: (a) a context =E2=80= =94 distance as a function of time; (b) generating coordinates. =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
**Human Graph II**- Students graph the functions k x 2 $/h = and k x 3 $/h. The idea is to show that for each linear function the points= fall onto a straight line. =20
**Interpreting Graphs**- Students will be given two li= near distance-time graphs and asked to tell a story about each graph and to= compare them. They will later explore comparisons between points in each l= ine. =20
**Interpreting Maps**- Students construct a narrative of= a trip, given a simplified map and a table of arrival and departure times.= They also determine how much time was spent along each segment of the trip= (and how much time was spent at each place along the way.) If time permits= , they construct a table ordered by time, showing the duration of each segm= ent and the accumulated times. =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
**Maps to Graphs**- Students will be given two linear dista= nce-time graphs and asked to tell a story about each graph and to compare t= hem. They will later explore comparisons between points in each line. =
=20
**Multiple Number Lines**- Students continue to lear= n that two partial changes that start at different points on the number lin= e are equivalent. At the end, they will work with notation for variables (N= + 5 - 3 or N + 2). =20
**N-Number Line I**- Students work with the table they bui= lt in the previous class for multiple number lines, focusing on the notatio= n for variables (N + 5 - 3 or N + 2). =20
**N-Number Line II**- Students use the N-Number line to m= ake generalizations about an unknown amount of money in a piggy bank. =
=20
**Number Line - Locations**- Students place themse= lves at points on the number line. Main contexts: stairs, age, money, tempe= rature, and pure number. =20
**Number Line Shortcuts**- The students will use a n= umber line to see how two addends or subtrahends are equivalent to one sing= le change once combined. =20
**Part-Whole Relations**- This class follows the disc= ussion from the Candy Boxes I class. The challenge is to work with a visual= representation of the relationships among the various quantities in the ca= ndy box problem and to relate the visual and numerical information containe= d in visual diagram(s) to verbal descriptions and to algorithms for finding= unknown values. =20
**Partial and Total Changes**- Students learn th= at two partial changes are equivalent to a single total change. On the numb= er line, this corresponds to the idea of a shortcut. Three notations are em= phasized: words, number lines with hopping arrows, and numerical expression= s. =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
**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
**Rules and Formulas**- Students are given a rule and a= data table supposedly generated according to the rule. Students evaluate w= hether: (1) the proper rule has been applied and (2) the result is correct.= =20
**Starting With A Rule**- Students focus on whether g= iven outputs are consistent with a given rule. =20
**S= ymbols**- Discussion about what symbols are; writing messages o= r "stories" with symbols; interpreting symbols. =20
**Three Heights**- In this class we will explore: (a) How th= e children deal with comparisons, (b) How they draw inferences from compari= sons, and (c) How they represent comparisons between three unknown amounts.= =20
**Time and Time Lines**- Students will discuss and lea= rn about points and intervals on time lines of various sorts. =20
**Times Two**- The lesson focuses on a function that multiplies = the input by two. New notations are introduced. =20

- =20
**Cartesian Candy Bars I**- We compare ratios of va= rious ordered pairs in a Cartesian grid. The initial discussion concerns th= e space as a whole; the task will focus on selected points and on the ratio= of the dependent variable to the independent variable. =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
**Consistency**- Children choose pairs of numbers that maintai= n the relationship of 1 to 3 that is given in a statement, and they explain= why they believe the relationship is maintained. =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
**Graphing A Story**- A trip is described in miles, hours= , and miles/hr. Students produce a graph from the description. They then pr= oduce a table from the graph and answer questions about the trip. =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
**Multiplicative Candy Boxes I**- This class = centers on the possible amounts of candies two children, Juan and Marcia, h= ave. Juan has a box of candy and Marcia has twice as much candy. What are t= he possible amounts of candies they might have? =20
**Multiplicative Candy Boxes II**- This clas= s is a continuation of the Multiplicative Candy Boxes I lesson. It centers = on the possible amounts of candies two children, Juan and Marcia, have. Jua= n has a box of candy and Marcia has twice as much candy. What are the possi= ble amounts of candies they might have? =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
**Varying Speed**- Children are asked to tell a story about = a trip depicted through a graph that has varying slopes/speeds. =20
**Varying Velocity**- Children are asked to tell a story = about a trip depicted through a graph that has varying slopes/velocities.=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**- 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 Review**- This = lesson will focus on reviewing the recent in-class assessment, on writing e= quations for word problems, and on solving equations. =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
**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
**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
**Curves in a Cubic**- Students will explore different k= inds of cubic functions through graphs and tables. =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
**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
**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
**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
**= 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