Ratios and Proportions / Adding Ratios

Understanding

Students continue their investigation of ratios by exploring graphical representations of the sum of two ratios. They should recognize that the sum of two equivalent ratios results in ratio equivalent to the original; and that the coordinates representing the ratio lie on the same line.

What to look for

At the end of the lesson, have students change the ratios for the blue and pink segments and explain what each segment represents. Discuss as a class the effects the rates associated with the two ratios have on the steepness of the segments.

Sample Assessment

The ratio 50:100 is equivalent to the ratio 1:2 and the ratio is 75:100 is equivalent to the ratio 3:4. Is the sum of the ratios 50:100 and 75:100 equivalent to the sum of the ratios 1:2 and 3:4? Why or why not.

Answer: One sum is the ratio 4:6 and the second is 125:200, which is equivalent to 5:8. The ratios are not equivalent so the sums are not equivalent.

The Big Idea

The sum of two equivalent ratios is an equivalent ratio and lies on the same line as the original ratios. The sum of two ratios that are not equivalent lies on a line that is between the lines formed by the two ratios.

What are the students doing?

Students investigate different pairs of ratios within the same context. They are asked to think about the steepness of the lines generated by adding two ratios.

What is the teacher doing?

Have students consider that in some cases, the trend indicated by the ratios for a certain outcome can be greater than another in two different groups in the same context, but when the groups are combined, the trend is reversed. (This is an example of what is known as Simpson’s Paradox).