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Representing Variables

Third Grade Lessons
  1. 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?
  2. 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.
  3. Functions - Earning Money - The students will create tables and equations from given stories. The functions are additive and multiplicative.
  4. 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).
  5. 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).
  6. N-Number Line II - Students use the N-Number line to make generalizations about an unknown amount of money in a piggy bank.
  7. 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.
  8. Symbols - Discussion about what symbols are; writing messages or "stories" with symbols; interpreting symbols.
  9. 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.
Fourth Grade Lessons
  1. Cartesian Candy Bars I - We compare ratios of various ordered pairs in a Cartesian grid. The initial discussion concerns the space as a whole; the task will focus on selected points and on the ratio of the dependent variable to the independent variable.
  2. Three Heights Review - In this class we will explore: (a) How children deal with comparisons, (b) How they draw inferences from comparisons, and (c) How they represent comparisons between three unknown amounts.
  3. Wallet Problem I - Students compare the amounts of money two students have. The amounts are described relationally but not through precise dollar amounts.
  4. Wallet Problem II - Students will be given a wallet problem. They will be asked to compare the amounts of money two students have. The amounts are described relationally but not through precise dollar amounts.
Fifth Grade Lessons
  1. Arcade - Students are told a story about two children, each of whom has a certain amount of money, but only one of whom has an amount known to us. After a series of events they happen to end up with the same amount of money.
  2. Enacting and Solving Equations - Students enact and discuss a situation where two children have amounts of candies. Some of the candies are visible, others are inside opaque tubes or boxes. After considering multiple possibilities they are told that the children have the same amount of candies. The situation corresponds to the equation 3x + y + 6 = x + y + 20, where x is the amount of candies per tube and y is the amount of candies per box. Students will be asked to discuss and to represent the situation, to solve the equation that corresponds to the situation, and to solve other written equations with similar structure.
  3. Fifth Grade Assessment I - This assessment will focus on writing equations to solve verbal problems and on solving equations using syntactic rules. It is intended as a diagnostic tool to assist teachers in planning future activities.
  4. Fifth Grade Assessment I Review - This lesson will focus on reviewing the recent in-class assessment, on writing equations for word problems, and on solving equations.
  5. Fifth Grade Assessment II - This assessment will focus on writing equations to solve verbal problems and on solving equations using the syntactic rules of algebra.
  6. Fifth Grade Assessment III - This assessment will focus on writing equations to solve verbal problems and on solving equations using the syntactic rules of algebra.
  7. Solving Equations with One Variable - Students work on a story about two children who each have a certain amount of money. The amount of one of the children is known but the other is not. After a sequence of transformations they end with the same amount of money. Students will be led to solve for the starting value by relating the equation to the events in the story. After that, they will be asked to solve another similar problem.
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