Proportional Relationships (5)

M6A2. Students will consider relationships between varying quantities.
c. Use proportions (a/b=c/d) to describe relationships and solve problems, including percent problems.
While discussing "segment/tape diagrams," I discussed how those diagrams can be used to solve problems involving percents. In the last post, while discussing models for proportional problems, I discussed how double number line may be used to represent problems involving proportional relationships.

Percents describes the relative size of quantities compared to the base quantity. It turns out that "percents" and the actual quantities are in a proportional relationship. For example, suppose the base quantity is 80. The table below summarizes the relationship between the size of quantities being compared and corresponding percentages.



You can see that they are in a proportional relationship because as the quantity becomes 2, 3, 4, ... times as much, the percentages also become 2, 3, 4, ... times as much.

So, if quantities and percentages are in a proportional relationship, then we can also use double number lines to represent problems involving percents, too. So, here are 3 examples.






The first double number line representation may be for a problem like the following:
At Jackson Elementary School, there were 80 fifth grade students this year. Next year, they anticipate that the fifth grade class to be 115% of this year's fifth grade class. How many fifth graders will there be?
The second represents a problem like this:
At Jackson Elementary School, there were 80 fifth grade students this year. Next year, they are expecting 92 fifth grade students. What percents of this year's fifth grade class will the next year's class be?
Finally, the third one represents a problem like this one:
At Jackson Elementary School, they are expecting 92 fifth graders next school year. This is 115% of this year's fifth grade class. How many fifth graders are there this year?
While discussing Process Standards 5, I shared how a segment/tape diagram to represent and solve problems involving percents. The double number line is a different representation. Double number lines representing multiplication or division problems always included a "1" on one of the number lines. In these situations, there is no "1," but by placing a "1" on either number line, a solution approach that combines division and multiplication - the approach discussed in the previous entry - may become apparent.