Ratios and Proportions / Double Number Lines

Understanding

Students use double number lines to solve questions that involve finding a value for x when a:b is equivalent to c:x.

What to look for

At this level, students are investigating informal but useful strategies for solving problems involving proportions. Moving to the formal equation can lead to misunderstanding and support longstanding student misconceptions.

Sample Assessment

8 km is approximately 5 miles. How many km is 32 miles?

Answer: 32 miles is about 51 $\frac{1}{5}$ km. This is because 5:8 is equivalent to 1:( $\frac{8}{5}$ ) ; multiplying by 32 gives 32:( $\frac{\mathrm{256}}{5}$ ) .

The Big Idea

The graph of a collection of equivalent ratios lies on a line through the origin.

What are the students doing?

Students notice patterns that occur - in tables and on a graph - when a collection of equivalent ratios is graphed in a coordinate plane.

What is the teacher doing?

Emphasize the connection of the ordered pairs along a line as “for every 2 over to the right, go up 3” when moving from point to point, using an additive strategy. Moving horizontally first then vertically supports thinking about the horizontal axis as representing the independent variable and the vertical axis as the dependent variable.