Ratio, Proportion, and Linear Relationship
Make sure you know the difference between the three
A ratio compares two given numbers.
Example:
Country A imported 400 tons of grains last year while Country B imported 600 tons. We see a ratio of 2:3 when we compare their grain imports.
A proportion states the equality of two ratios.
Example:
A recipe calls for 3 eggs for one cup of milk. If we want to scale up the recipe to make full use of 3 cups of milk, we can use proportion to calculate how many eggs we need then. The answer is 9.
A linear relationship describes the mathematical relationship between two variables, say x and y, that can be expressed as y=mx+b where m and b are constants.
The simplest case of a linear relationship is y=mx when b=0.
Please take note of the difference between the 3 terms:
A ratio concerns two numbers; A proportion concerns two ratios; A linear relationship concerns two variables.
You cannot immediately jump from the concept of either a ratio or a proportion to a linear relationship without knowing what ‘variables’ are.
What are variables?
A variable is a quantity that can take up different numerical values as the circumstance changes, say your height h and your weight w. As a child grows, both h and w will change. Many people want to know if h and w relate to each other. In other words, can you always predict a child’s weight from his/her height or vice versa? The answer is no, scientists do not see any evidence that such a relationship exists.
Another more interesting example of variables comes from sea turtles. It is known that the temperature at which a sea turtle egg is incubated determines the gender of the baby when hatched, therefore the gender ratio r of sea turtles on a beach is related to the average temperature T of the sand.
In our universe many physical quantities of interest do have a fundamental relationship with each other, meaning knowing the values of a few of them will allow us to predict the values of the others.
Examples of Linear Relationships in Basic Science
The famous Newton’s Second Law of Motion states that the net force F on an object relates to its mass m and its acceleration a by
F=ma
One can see that if the net force on an object is doubled, the acceleration will too.
Another example is Ohm’s Law in a circuit, which states that voltage V and the current I in an electric circuit are related linearly
V=IR
where R is the resistance.
Linear Relationship as a Graph
Young students are first introduced to a linear relationship in middle school in a graphical way. They will learn that when you plot the linear equation y=mx+b on a coordinate plane, it will be a straight line will a slope m and y-intercept b. This explains why the word ‘linear’ is used to describe such a mathematical relationship.
Recommendations
Ratio, proportion, and linear relationship are closely related but very different mathematical concepts, if your children find them confusing, the information above can be useful.
Common middle school math curriculum tightly binds the concept of a linear relationship with the coordinate plane. On my desk, there is a middle school math workbook bought at a local bookstore. There is nothing I can see in the book that relates y=mx+b with any real-world applications. There are no word problems on linear equations in the book, only graphical problems like finding the slope or the intercept of a given linear equation. Since linear relationship plays an important role in basic science and statistics, I recommend educators and parents help children see its significance through real-life examples of practical applications.
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