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On your way home from school you travel 5 miles on the highway at a speed of 60 mph. Once you exit the highway, you travel 7 miles at 35 mph. Once you arrive at your subdivision, you travel 1/2 a mile at a speed of 10 mph.
Your speed changes at different points throughout your trip so it is not invariant.

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The ratio of boys to girls in a class is 1:3.
This is a part-to-part ratio.

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This is not a comparison of two values.

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The boy ran 4 meters less than the girl.

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The boy ran 1/3 as far as the girl.

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If the image below represents 1 and 2/3 (One and two-thirds) What is the unit?
Incorrect Response:
The whole circle is 1 unit.

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This represents absolute reasoning:
Both trees grew the same amount because 4=4.

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This is relative reasoning.

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These are similar figures. If you change the measure of one side, the measure of the other sides change also. The side measurements co-vary.

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Definition: To determine the measure of 1 unit.
Example:
If the image below represents 1 and 2/3 (One and two-thirds) What is the unit?
Correct Response:
3 sections would represent 1 whole or 1 unit.

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Definition: Seeing the multiplicative relationship between two quantites.
Real World Example:
Tree A is 4 feet tall. Tree B is 8 feet tall. Both trees grow 4 feet in the next month.
A relative thinker might claim that the Tree A grew more that Tree B relative to its original height because tree A grew 2 twice its original height.

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Definition: Seeing the additive relationship between two quantities.
Real World Example:
A tree is 4 feet tall. The tree next to it is 8 feet tall.
Both trees grow 4 feet in the next month.
An absolute thinker might claim that both trees grew the same amount.

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These are NOT similar figures. If you change the measurement of the sides on one figure, the measures of the sides on the other figures do not change, therefore they do not co-vary.

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Definition:Multiplying any x value by the constant to arrive at the corresponding y value.
Example:
There are two-thirds as many girls as there are boys.

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Definition: Covariation means that two quantities (a ratio) vary together.
Real World Example: five mangos cost $2.00 (two quantities in a multiplicative relationship); as the number of mangos varies (for example, to 10 mangos), so does the cost. And as the cost changes, so does the number of mangos you will get.

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Definition:
When two quantities covary, they are invariant meaning the ratio between the two quantities stays the same even when they are changing.
Example: If one Triple Cheese Burger costs $5, the 5 Triple Cheese Burgers will cost $25. The ratio between Cheeseburgers and Cost does not change, it stays constant (1:5).

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Definition: a comparison of two measurements in which
one of the terms has a value of 1.
Inches per foot, milliliters per liter, and centimeters per inch, miles per gallon, square yards of wall coverage per gallon of paint, passengers per busload, and roses per bouquet are all unit rates.

Example:
50 words per minute.
Non Example:
100 words in two minutes.

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Definition: •one quantity is directly proportional to another when the ratio of the two quantities is constant (the same)

• one variable "varies directly" in line with the other.
Always crosses the origin (0, 0).
y=kx where k is constant.
Example:

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Definition: Adding a constant about to an x value to get to the corresponding y value.
Example:
For every 2 girls there are 3 boys.

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Definition: the relationship between two quantities where as one quantity (x) increases the other quantity (y) decreases in proportion.
•The graph of an inverse variation (indirect proportion) is a curve with no x-intercept or y-intercept.
xy = k where k is the constant of variation.
Example:

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Definition: a part-whole ratio compares a number of parts of a whole
Example: The ratio or girls to  students in a class.

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Definition: a part-part ratio compares a number of parts of a whole
to other parts in the whole.
Example: Ratio of males to females in a team.
Non Example: Ratio of males to all players on the team.

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Definition:
Commonly a ratio is the comparison of two values of the same kind,
which may be written as a to b, a:b or as a fraction a/b.
The values can be compared as a part-part ratio or a part-whole ratio
Example: