Keeping the Equation Balanced
This lesson is the first of a sequence of eight lessons where students learn to work with equations that have variables on each side. In this lesson, students recall a representation that they have seen in prior grades: the balanced hanger. The hanger is balanced because the total weight on each side, hanging at the same distance from the center, is equal in measure to the total weight on the other side.
In the warm-up, students encounter two real hangers, one balanced and one slanted, and notice and wonder about what could cause the hangers’ appearance. This leads into the first activity where students consider two questions about a balanced hanger: first, whether a change of the number of weights keeps the hanger in balance, and second, how to find the unknown weight of one of the shapes if the weight of the other shape is known. Students learn that adding or removing the same weight from each side is analogous to writing an equation to represent the hanger and adding or subtracting the same amount from each side of the equation. They reason similarly about how halving the weight on each side of the hanger is analogous to multiplying by \(\frac12\) or dividing by 2. In both the hanger and the equation, these kinds of moves will produce new balanced hangers and equations that ultimately reveal the value of the unknown quantity.
In the second activity, students encounter a hanger with an unknown weight that cannot be determined. This situation parallels the situation of an equation where the variable can take on any value and the equation will always be true, which is a topic explored in more depth in later lessons.
As students use concrete quantities to develop their power of abstract reasoning about equations, they engage in MP2.
- Calculate the weight of an unknown object using a hanger diagram, and explain (orally) the solution method.
- Comprehend that adding and removing equal items from each side of a hanger diagram or multiplying and dividing items on each side of the hanger by the same amount are moves that keep the hanger balanced.
Let's figure out unknown weights on balanced hangers.
- I can add or remove blocks from a hanger and keep the hanger balanced.
- I can represent balanced hangers with equations.
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