Each of our activities is designed to be flexible and self-conta= ined. Please feel free to use any of the activities as the basis of your te= aching or as supplementary materials.

=20You do not need to do more than one of the activities in order for them = to be useful. However, we have listed them here in the approximate order th= at we have used them with students. This may be helpful for teachers lookin= g for a series of activities.

=20Please explore our categorizations by activity type, process, and math c= oncept if you are looking for something specific!

=20- =20
**S= ymbols**- Discussion about what symbols are; writing messages o= r "stories" with symbols; interpreting symbols. =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
**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
**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
**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
**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
**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
**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
**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
**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
**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
**Guess my Rule - Tables**- Two children create sec= ret rules for transforming input numbers. The teacher uses a doubling rule.= =20
**Guess my Rule - Multiplicative Tables - Two children create secret rules for transforming input numbers. The= teacher uses a doubling or tripling rule.****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
**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
**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
**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
**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
**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
**Comparing Different Functions**- The stude= nts will discuss, represent, and solve a verbal problem involving the choic= e between two functions. =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
**Starting With A Rule**- Students focus on whether g= iven outputs are consistent with a given rule. =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
**Formulas and Stories**- The students will be requir= ed to work with the relation between different mathematical expressions (fo= rmulas) and stories. =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
**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
**Times Two**- The lesson focuses on a function that multiplies = the input by two. New notations are introduced. =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
**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
**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
**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
**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
**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
**Time and Time Lines**- Students will discuss and lea= rn about points and intervals on time lines of various sorts. =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