Course Guide - USAID Teacher Education Project

Unit 2 AlgebraWeek 4, Session 3: Solving for x, the Unknown1. What are the important concepts?a) "x as an unknown" is a different concept from "x as a variable."b) The properties of arithmetic that students have already learned (commutative,associative, distributive, and identity) also hold true for (and can be applied to)algebraic equations.c) An expression without a given value for x relates to functions, where x is anunspecified variable.d) On the function's table of values, any given value for x has a corresponding "yvalue."e) Evaluating expressions with a given value for x by using substitution is a startingpoint for finding x as the unknown in an equation.f) Solving an equation for x relates to the "balance model" of equivalency.g) In order to create this balance students need to use 1) symbol manipulation, 2)arithmetic properties, and 3) order of operations.2. How do children think about these concepts?a) In arithmetic, equations are usually written so that the "answer" comes on the righthand side of the equation with the typical syntax of a number sentence being "4 plus 3equals 7."In algebra's use of symbolic notation, that syntax is often reversed so that the"answer" is on the left of the equals sign, which would result in the arithmeticequation "7 = 4 + 3."After many years of working with equations in the format of a + b = c, it can bedifficult for youngsters to shift to a syntax where the answer is on the left hand side ofthe equation and where the expressions and operation signs are on the right hand side,such as in 7 = 4 + 3 (or its corresponding algebraic equation, 7 = 2 x + 3).b) When working with linear equations in the format y = mx + b, youngsters may notunderstand that the dependent variable "y" can have a coefficient. For example in theequation 4y = 2x, youngsters may not realize they need to divide both sides of theequation by 4 in order to maintain equivalence. Instead, they may interpret theequation as either y = 2x + 4, or 4y = 4(2x).c) Even if youngsters know that variables can have a coefficient, they may not realizethat a coefficient does not need to be a whole number. They need to understand thatcoefficients can be any type of the numbers they have studied: integers, fractions, ordecimals.