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Jars and Jars - Interpret relationships to compare weights

In each problem, two pan balances are shown with jars of different types. All jars of the same type have the same weight. Students have to figure out the relative weights of objects from the relationships displayed in the pan balances. To begin, be sure that students recognize differences in the types of jars. Draw students’ attention to the position of the pans. If the two pans are not at the same level, then one pan–the lower one is heavier. If the pans are level, then they hold equal weight. One of the difficulties students often experience is recognizing that if one block (call it A) has the same weight as, for example, two others, (B + B), then the one block (A) is heavier. Point out to students that lb stands for pounds, and that all jars with the same letter are the same number of pounds.

Solutions

  1. m is more than 5 pounds.
  2. t is more than 6 pounds.
  3. j is more than 3 pounds
  4. z is less than 4 pounds
  5. v is more than 3 pounds
  6. t is less than 8 pounds

& \mathbf{Describe} && \text{I see two pan balances. I see two jars} \ x \ \text{and} \ z.\\ &&& \text{Pan Balance} \ A: \ \text{The pans are not balanced}.\\ &&& \text{Pan Balance} \ B: \ \text{The pans are balanced. One pan has a} \ 12 \ \text{pound jar}.\\& \mathbf{My \ Job} && \text{Figure out a weight for} \ x. \ \text{Are other weights possible?}\\& \mathbf{Plan} && \text{Pan Balance} \ A: \ \text{The pan with} \ z \ \text{is lower so,} \ z \ \text{is heavier than} \ x.\\&&& \text{Pan Balance} \ B: \ 3 \ z \ \text{weigh} \ 12 \ \text{pounds. Start with the fact.}\\& \mathbf{Solve} && \text{Pan Balance} \ B: \ z + z + z = 12 \ \text{pounds, so} \ z = 4 \ \text{pounds}.\\&&& \text{Pan Balance} \ A: \ x \ \text{has to weigh less than} \ 4 \ \text{pounds}.\\&&& \qquad \qquad \qquad \ \ \ \ \ x \ \text{could be} \ 1, 2 \ \text{or} \ 3 \ \text{pounds}.\\&&& \text{Pan Balance} \ A: \ \text{The picture shows that} \ z \ \text{is heavier than} \ x.\\& \mathbf{Check} && \text{Pan Balance} \ B: \ 4 + 4 + 4 = 12 \ \text{pounds}.\\&&& \qquad \qquad \qquad \qquad  \ z=4 \ \text{pounds}.\\&&& \qquad \qquad \qquad \qquad  \ 1, 2, \ \text{and} \ 3 \ \text{are all less than} \ 4. \ \text{So} \ x \ \text{can be} \ 1, 2, \ \text{or} \ 3 \ \text{pounds}.

What could be the weight? Tell how you figured it out.

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CK.MAT.ENG.SE.1.Algebra-Explorations-K-7.5.2

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