- =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
**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
**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 Different Functions**- The stude= nts will discuss, represent, and solve a verbal problem involving the choic= e between two functions. =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 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
**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 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
**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
**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
**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 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
**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
**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

- =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
**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 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
**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
**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
**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
**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
**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
**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 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
**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
**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
**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
**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
**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 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