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**All Things Being Equal II**- The equals sign signifies that amounts on each side are the same. The students will use Unifix blocks and the corresponding equations to represent equalities between additive amounts.**All Things Being Equal III**- The students will write equations to represent verbal statements and successive transformations that maintain or do not maintain the equality.**Box Extremum**- Students will start by finding average rates of change for a non-linear function over increments of the independent variable. The size of the increments will decrease to introduce the idea of using tangent lines to find instantaneous rates of change of linear and non-linear functions. Students will see what a tangent looks like at the extrema 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 volume.**Candy Boxes**- This class centers on the possible amounts of candies two children, John and Maria, have. They each have the same, unspecified 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 total candies they might have?**Comparing Different Functions**- The students will discuss, represent, and solve a verbal problem involving the choice between two functions.**Comparing Discrete Quantities**- Students compare amounts of tokens and unknown amounts of discrete quantities. In both cases they are guided to adopt line segments to represent discrete amounts and the differences between them. They are also asked to discuss composition of measures: "the difference plus the smaller amount is equal to the larger amount" and, "the larger amount minus the difference is equal to the smaller amount".**Comparing Heights I**- Students compare the heights of two children, measure, compare, and represent one's own height in relation to a peer's height, and focus on the differences between heights.**Comparison Problems & Tables**- This class will be used to review concepts and representations as applied to the solution of verbal comparison problems and to work on function tables.**Comparisons**- Comparisons and comparison operators: =, ≠, <, >.**Comparisons and Attributes**- Work with comparisons and comparison operators (=, ≠, <, >).**Dots Problem**- We present to the students a problem dealing with a growing pattern over time. To begin, there is one dot. With each passing minute four more dots are drawn around the previous dot(s).**Formulas and Stories**- The students will be required to work with the relation between different mathematical expressions (formulas) and stories.**Functioning Together**- Students work together to develop multiple representations of a function. The students split up into groups of three with each student having a separate responsibility. When all the input values have been used up, the students are asked to, together, make up a story that describes their function.**Functions - Earning Money**- The students will create tables and equations from given stories. The functions are additive and multiplicative.**Functions from Tables**- Students work with a function, beginning with a table and then a formula, to generate ordered pairs that follow the rule of the function.**Heights as Functions**- In this class children will work on the functional representation of two unknown heights and on the composition of the shorter height plus the difference between the heights as equal to the second height.**How Many Points?**- Students work with: (a) a context — distance as a function of time; (b) generating coordinates.**Interpreting Graphs**- Students will be given two linear 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 line.**Linear vs Quadratic Functions**- The students 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.**Maps to Graphs**- Students will be given two linear 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 line.**Multiple Number Lines**- Students continue to learn that two partial changes that start at different points on the number line are equivalent. At the end, they will work with notation for variables (N + 5 - 3 or N + 2).**N-Number Line I**- Students work with the table they built in the previous class for multiple number lines, focusing on the notation for variables (N + 5 - 3 or N + 2).**N-Number Line II**- Students use the N-Number line to make generalizations about an unknown amount of money in a piggy bank.**Number Line Shortcuts**- The students will use a number line to see how two addends or subtrahends are equivalent to one single change once combined.**Part-Whole Relations**- This class follows the discussion from the Candy Boxes I class. The challenge is to work with a visual representation of the relationships among the various quantities in the candy box problem and to relate the visual and numerical information contained in visual diagram(s) to verbal descriptions and to algorithms for finding unknown values.**Piggy Banks**- The whole lesson revolves around a multipart story problem involving changes in two quantities over several days of a week. The initial quantities are equal yet unknown. Then transformations are applied to the quantities. Students are asked to compare the quantities throughout the week even though only their relative relationship can be determined.**Rates vs Totals**- Students compare points on an hours/pay Cartesian space. The main challenge lies in recognizing that, although one student earned more, the other student was paid better, that is, at a higher rate of pay. They must indicate the difference in pay and the differences in amount worked.**Starting With A Rule**- Students focus on whether given outputs are consistent with a given rule.**Symbols**- Discussion about what symbols are; writing messages or "stories" with symbols; interpreting symbols.**Three Heights**- In this class we will explore: (a) How the children deal with comparisons, (b) How they draw inferences from comparisons, and (c) How they represent comparisons between three unknown amounts.**Time and Time Lines**- Students will discuss and learn about points and intervals on time lines of various sorts.

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