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November 21, 2017
Integral vs. Non-integral Multiplicative Relationships

Proportional situations involve multiplicative relationships. Simple cases involve unit rates (4 erasers/$1). This unit rate can be used to determine the cost of more or fewer erasers by scaling up or down using this multiplicative relationship (e.g., 8 erasers for $2, 2 erasers for $0.50). The multiplicative relationship in unit rate problems is given. On the other hand, solving missing values problems demands recognizing and applying the multiplicative relationships either within the ratio or between the equal ratios. Notice the multiplicative relationships in the problem below. In this case the multiplicative relationships between boxes and bushels is non-integral (x1.5) while the multiplicative relationship between bushels is integral (x4).
Problems that involve non-integral relationship can be harder to solve particularly if students do not flexibly analyze the problem situation to determine which multiplicative relationship is easier to apply. To help students, researchers suggest varying the multiplicative relationship 'within' and 'between' ratios as it influences a student's ability to solve problems involving proportionality.  Ultimately we want our students to be flexible in solving proportion problems. We want them to be able to recognize and identify both multiplicative relationships found in a proportion problem.  
Action Item

Analyze the proportion problems you are having your students solve. What are the multiplicative relationships in the problem?  Are they both integral (whole number), both non-integral, or some non-integral (like the 'Carrie packing problem' above)?

Give the 'Carrrie packing problem' above (PD1) to your students to solve. Sort the evidence of student understanding using your OGAP Proportional Reasoning learning progression.  If many students struggle to find the integral relationship, try giving a problem where the multiplicative relationships are both integral (whole number) and analyze their strategies.
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