Date: Sat, 24 Oct 2020 23:13:52 -0400 (EDT) Message-ID: <965635414.31698.1603595632600@wikis-prod-01.uit.tufts.edu> Subject: Exported From Confluence MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_Part_31697_419516040.1603595632600" ------=_Part_31697_419516040.1603595632600 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable Content-Location: file:///C:/exported.html Small Group Work

# Small Group Work

## Small Group Work

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1. 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.
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3. 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.
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5. 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.
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7. 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?
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9. Comparing Different Functions - The stude= nts will discuss, represent, and solve a verbal problem involving the choic= e between two functions.
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11. 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".
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13. 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.
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15. 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.
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17. Comparisons - Comparisons and comparison operators: =3D, = =E2=89=A0, <, >.
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19. Comparisons and Attributes - Work with compa= risons and comparison operators (=3D, =E2=89=A0, <, >).
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21. 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).
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23. Formulas and Stories - The students will be requir= ed to work with the relation between different mathematical expressions (fo= rmulas) and stories.
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25. 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.
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27. Functions - Earning Money - The students will= create tables and equations from given stories. The functions are additive= and multiplicative.
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29. 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.
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31. 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.
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33. How Many Points? - Students work with: (a) a context =E2=80= =94 distance as a function of time; (b) generating coordinates.
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35. 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.
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37. 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.
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39. 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.
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41. 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).
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43. 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).
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45. N-Number Line II - Students use the N-Number line to m= ake generalizations about an unknown amount of money in a piggy bank.
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47. 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.
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49. 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.
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51. 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.
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53. 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.
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55. Starting With A Rule - Students focus on whether g= iven outputs are consistent with a given rule.
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57. S= ymbols - Discussion about what symbols are; writing messages o= r "stories" with symbols; interpreting symbols.
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59. 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.=
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61. Time and Time Lines - Students will discuss and lea= rn about points and intervals on time lines of various sorts.
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1. 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= .
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3. 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.
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5. 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.
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7. 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.
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9. 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.
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11. 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.
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13. 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.
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15. Intervals - Students reason about graphs showing growth over = time. They compare heights of children and heights of two animals at differ= ent time intervals.
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17. 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?
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19. 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.
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21. 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
22. 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.
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24. 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
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26. 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.
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28. 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.
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30. Varying Speed - Children are asked to tell a story about = a trip depicted through a graph that has varying slopes/speeds.
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32. Varying Velocity - Children are asked to tell a story = about a trip depicted through a graph that has varying slopes/velocities.=20
33. 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.
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35. 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.
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37. 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.
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1. 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.
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3. 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.
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5. 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.
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7. 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.
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9. 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.
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11. 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.
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13. 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.
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15. 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.
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###### Middle School Lessons
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1. 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.
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3. Box of Clay Activity - Students will compare two c= ubic functions based on the context of the volumes of a box of clay.
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5. 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.
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7. Candy Experiment - Students will create their own data= to construct a graph and equation of negative and fractional slope.
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9. 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.
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11. 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.
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13. 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.
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15. 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.
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17. Guess My Rule - Non-Linear - Students will p= roduce algebraic expressions starting from non-linear graphs produced by ot= her students in the class.
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19. Jason's Tree House - Students will extract data fr= om a story and use tables and graphs to answers questions about proposed sc= enarios.
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21. 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.
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23. Sound Loudness - Students will examine a non-linear func= tion depicted in a graph and generate the corresponding function table and = equation.
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25. 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.
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27. 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.
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29. 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.
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31. 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.
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33. 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.
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