- =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
**How Many Points?**- Students work with: (a) a context =E2=80= =94 distance as a function of time; (b) generating coordinates. =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
**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

- =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
**Comparing Strips of Unmeasured Lengths I**- The class is the first of a series that will focus directly up= on the algebraic representation of measurements and their multiplicative re= lations. Children are asked to compare the lengths of strips, to describe t= he relationships between them in multiple ways, and to demonstrate that the= relationships they represent are true. =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
**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 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
**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
**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
**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
**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 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

- =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
**Graphing Equations - Nonlinear Functions**- Students write equations for three graphs and examine their sl= opes by comparing and contrasting the graphs. Students also look at the sam= e functions graphed on differently scaled coordinate planes. =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
**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
**= x2 and x**- Students will generate tables and graphs from quadr= atic and cubic equations with different multipliers on the highest degree t= erm. =20