Course Guide - USAID Teacher Education Project

Unit 2 AlgebraWeek 2, Session 3: Equivalence, Multiple Representations1. What are the important concepts?a) The idea of the equals sign, discussed in Week 1 Session 3 of the Number andOperations unit takes on new and extended meaning in this Algebra unit wheresymbols (not just numbers) are added to students' thinking about equivalenceb) Not only can numerical equations such as 3 + 5 = 7 + 1 be proved to showequivalence, so can algebraic equations such as 4s + 4 = 4 (s + 1).This idea of determining proof of equivalence, justified by symbol manipulation, isone of the cornerstones of algebra, moving young children's informal algebraicthinking to more formal ways to think about equivalence.c) Multiple representations of a problem show how the a table of values and itscorresponding graph can give rise to different symbolic expressions that areequivalent to each other.d) The distributive property of multiplication over addition becomes an importantsolution strategy when comparing expressions to investigate their equivalence.e) Symbolic representation allows students to move from the calculation of specificdata to a generalized formula that can be expressed in variables.2. How do children think about these concepts?a) When youngsters are given the "tiling" problem for a pool with a side length of 5,their first instinct usually is to assume that the solution must be either:1) The perimeter surrounding the border tiles (28) or2) The perimeter of the pool (20) (not considering that there are 4 cornertiles).Because of what they have learned about perimeter in earlier grades, they think thatone of these two lengths can be translated into the number of tiles surrounding thepool.Teachers need to be aware of and anticipate such common student responses so thatthey can sensitively redirect a youngster's thinking to the question posed.b) In the middle grades, students will begin to have a formal sense of the distributiveproperty of multiplication over addition. They will need to be reminded of theinformal way they have used the distributive property in earlier grades and then bedirected to the procedural way in which it is applied when using symbols.c) In addition to the use of multiple representations, the Tiling the Pool probleminvolves the informal use of symbolic notation and symbol manipulation. Researchshows that students need opportunities to solve problems involving symbols and