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In earlier grades, students write and evaluate numerical expressions and equations to represent operations with numbers. Their experience with the meaning of addition, subtraction, multiplication, and division as well as the properties of operations prepares them to understand algebra as “generalized arithmetic.” Since numerical expressions can only have one value, an equation with only numbers, such as $2^3 = 9-1$, is either true or false, depending on whether the expression on the left has the same value as the expression on the right. On the other hand, there is an important distinction between being equal and being equivalent for algebraic expressions. For example, the expressions $3x + 5$ and $5x + 3$ are equal when $x = 1$, but they are not equivalent. The expressions $x(2 + 3)$ and $3x + 2x$ are equivalent because they are equal no matter the value of $x$. So an algebriac equation is sometimes viewed as an implict question: “For what values of the variable is the equation true?”
This is a rich discussion question that spotlights the different possibilities for an equation when a variable is involved. It’s important to encourage students to understand what a solution to an equation is and what it means to solve an equation before they learn general and efficient procedures. Students should think about what an equation is saying and what solution(s) are reasonable before solving it; for example when trying to solve $10=2x$, students think about what number times $2$ would equal $10$. Throughout grade 6, students should see a variety of equations, some of which have only one solution, some which are identities, and some with no solution; these states correspond with being true for one value, all values, and no values of a variable. Sometimes when algebraic equations are introduced in grade 6, students are just asked to write an equation for a problem that they could easily solve numerically. This task represents a way to have a meaningful conversation about equations as algebraic objects with potentially different characteristics.
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Considering all three equations at once or in the same class period might be a too much for one day; the discussion can be spread out over three different class periods. If a class has a “number talk” or “algebra talk” routine, these items might fit nicely in that structure.
These equations are somewhat complicated relative to the other work of sixth grade, so if appropriate, a teacher may decide to replace or preface this task with a task that asks the same question (“all, some, or none”) but uses simpler equations. For example, you could use $20 cdot x div 4 = 5x$ for the identity, $x=x+2$ which is true for no values of $x$, and $12=3x$ which is only true for one value of $x$.
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Category: WHICH
