Proportional Thinking Put Right!

Three of the hardest concepts to grasp in mathematics for students of any age are proportional thinking, ratio and rate. However, they are quite simple once we know a little bit more about them.

Small explains proportional thinking or proportional reasoning as the way two amounts relate to each other multiplicatively (pp. 312). That means, the relationship between two amounts, when the value of one amount is always a constant multiple of the corresponding value of the other. It is the main topic with the subgroups ratio and rate.

Ratio describes the relationship between quantities of two values with the same unit. This relationship indicates how many times the first number contains the second. There are two different types of ratios:

            a) Part-to-Part ratios, which compare two parts of something

            b) Part-to-Whole ratios, which compare a part of something to its whole.

Zapanta, D. (2015, September 22). Determining Ratios
[Online Slide]. Retrieved from http://bit.ly/2yLXsOR 

Rate, on the other hand, is a comparison of two values with different units. Check out Math Gains for more information. One of the most useful abilities in life is being able to determine unit rate. Small describes a unit rate as an equivalent rate where the second term is 1 of that unit (pp. 320).

Ms. Smith. Unit Rates R' Us [Online Slide]. 

Retrieved from http://bit.ly/2y3EeAl

Now let's compare. Both ratio and rate are often expressed as fractions, contain two terms, and can be used to create equivalencies, such as equivalent ratios or equivalent rates.

And they both are both constantly used in everyday life. For me, I use ratios when I am baking muffins. I need to know the ratio of eggs to flour, so that when I make more or less than the recipe calls for, the muffins still come out fluffy and delicious. Rates are great when determining how many calories I am eating per serving of pizza, or my hourly wage, or determining my running pace in meters per hour. I am also constantly finding myself comparing products in the grocery store about the better; which item can I buy more of for less cost.

Ratio of an igloo's circumference to its diameter = Eskimo Pi

References:

Small, M. (2017). Making Math Meaningful to Canadian Students K-8, (3rd ed.). Nelson Education.