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4. Digital Pan Balance:
Applying proportional reasoning using an online manipulative
Look in the Classroom
Ms. Daken knows that her students have a good conceptual understanding of equivalent fractions that express part-to-whole relationships but that they need more work with ratios that express part-to-part relationships. She decides to use a digital pan balance investigation to help her students develop a conceptual understanding of these types of comparisons. Her goals for this investigation are for her students to engage in solving problems that require proportional reasoning skills and to identify a ratio as representing a multiplicative relationship. She also wants her students to understand the inverse relationship between the number of shapes on each side of the pan balance and the relative weights of the shapes.
Ms. Daken introduces her students to the NCTM's Shape Pan Balance and gives them time to work independently and experiment with the tool. Many students simply add shapes to the pans and then observe the changes in the position of the balance. After the initial exploration period, Ms. Daken has her students focus on specific tasks: determining the heaviest shape and lightest shape; determining the order of the shapes according to weight; and balancing the scale (Figure 2). Students record the strategies they use in their journals.
As the students investigate and share their strategies, Ms. Daken finds that some groups are ordering the weights of the shapes and then verifying by finding equivalencies; others are first balancing, then determining the order.
S1: To find out how heavy a square is, I start with one red square on one pan and test out the other shapes compared to the square.
Ms. D: What did you find out?
S2: We found out that the square was heavier than the circle and lighter than the triangle or diamond.
S1: So, we know the circle is the lightest, but we still don't know which is heaviest.
Ms. D: What would you try next?
S2: I think we could try the triangle on one side and the diamond on the other.
The investigation continues, with students working in pairs, trying to find equivalencies and ordering the shapes. Ms. Daken then challenges them to identify more sophisticated relationships among the shapes by asking questions that lead to writing expressions involving ratios.
Ms. D: I see you found it takes three circles to balance two squares. Can you make it balance with more circles and squares?
S3: If we only put on one square, then we know we can't get it to balance.
Ms. D: Why not?
S4: Because when we started out with one square, we found that one circle was too light and two circles were too heavy.
S3: So, we could put on two more squares, that's twice as many. Then we would need twice as many circles, or six, to balance.
Ms. D: What are some other equivalencies for squares and circles?
S3: We could use three times as many, or six squares with nine circles.
S4: If we had room on the scale, we could put as many as we want, like if we used 200 squares, we would need 300 circles-that's 100 times as many of each.
Satisfied that the students have a good understanding of the multiplicative relationship, Ms. Daken challenges them to write expressions comparing the weights of the shapes.
Ms. D: Is there any shape that is twice as heavy as another?
S5: It takes two triangles to balance one square, so the square is twice as heavy as the triangle.
Ms. D: How would you express the ratio that compares the weight of the square to the weight of the triangle?
S5: That would be 1 to 2 because one square balances two triangles, so I'll write 1:2.
S6: No, 1 to 2 stands for the number of squares compared to the number of triangles it takes to balance.
S5: But wouldn't it be the same?
S6: No, it's just the opposite-that's 2 to 1.
S5: I'm confused; I have to think about this some more.
Before the end of the class, Ms. Daken initiates a class discussion and invites students to share their strategies and observations. She finds that although her students are able to identify multiplicative relationships expressed by ratios, they are having difficulty understanding the inverse relationship between the number of shapes on each side of the pan balance and the relative weights of the shapes. She plans to continue with this investigation, with the goal of helping her students understand this important idea.
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