M6A2b - Proportional Relationships (4)

M6A2. Students will consider relationships between varying quantities.
b. Use manipulatives or draw pictures to solve problems involving proportional relationships.
In the last three posts, we considered two proportional situations. They are,
c) the distance traveled and the time of travel (at a constant speed), and
f) the amount of meat and the price of meatThe tables below show the values of these quantities:





"Problems involving proportional relationships" from these contexts might be something like the following:
Jim can walk 9 miles in 3 hours. If he maintained the same speed, how far can he walk in 6 hours?

4 pounds of meat cost $18. How much will 10 pounds of the same meat cost?So, what kinds of pictures might we draw to solve these problems? Actually, you may find it rather difficult to draw pictures for these problems. We can draw pictures that might represent the contexts of the problems, but those pictures may not be too helpful in actually solving the problems. What about manipulatives? What manipulatives would you use to solve these problems? I am not sure what I would use.

If it is difficult to use a picture or manipulative to solve these problems, what is this standard talking about? Perhaps "pictures" here are really referring to diagrams. One particular form of diagrams is double number lines. I used double number lines extensively to talk about multiplication and division of decimal numbers (November 2008). But they can be useful to represent problems involving proportional relationships. Here are the double number line representations of the two problems above.





When students are familiar with double number lines with multiplication and division, they will notice the difference between these double number line representations and those of typical multiplication and division problems. Here are examples of multiplication and division double number line representations:

Do you notice the difference? In the two representations of the problems involving proportional relationship, there is no "1" in the representation. If we put a "1" in the representation, then we can see a solution path. For example, let's use the second problem. If we put a "1" on the top number line, it will look like this:

Now, the left side part of the representation,

is really a partitive division situation. Thus, by dividing 18 by 4, we can find that # = 4.5. Now, double number line representation looks like this:

Now, if we can ignore the middle part of this representation, it will look like this:

and this is a multiplication situation, isn't it. So, multiplying 4.5 by 10, we can obtain the missing quantity.

So, double number line representations can not only represent problems involving proportional relationships, they can also suggest ways to solve the problems, too. Of course, if we want students to be able to use double number lines as their thinking tool at this stage, they do need to be familiar with double number lines. Thus, it is important for teachers of different grade levels to discuss what representations they want to emphasize. It is important for students to be able to use multiple representations. But, if there is any representation, like double number line, that may be used across grades, then that representation should be consistently introduced/developed/used across grades.