Unit 2: Proportional Relationships
In this unit, students develop the idea of a proportional relationship out of the grade 6 idea of equivalent ratios. Proportional relationships prepare the way for the study of linear functions in grade 8.
In grade 6, students learned two ways of looking at equivalent ratios. First, if you multiply both values in a ratio a:b by the same positive number s(called the scale factor) you get an equivalent ratio sa:sb. Second, two ratios are equivalent if they have the same unit rate. A unit rate is the “amount per 1” in a ratio; the ratio a:b is equivalent to ab:1, and ab is a unit rate giving the amount of the first quantity per unit of the second quantity. You could also talk about the amount of the second quantity per unit of the first quantity, which is the unit rate ba, coming from the equivalent ratio 1:ba.
In a table of equivalent ratios, a multiplicative relationship between the pair of rows is given by a scale factor. By contrast, the multiplicative relationship between the columns is given by a unit rate. Every number in the second column is obtained by multiplying the corresponding number in the first column by one of the unit rates, and every number in the first column is obtained by multiplying the number in the second column by the other unit rate. The relationship between pairs of values in the two columns is called a proportional relationship, the unit rate that describes this relationship is called a constant of proportionality, and the quantity represented by the right column is said to be proportional to the quantity represented by the left. (Although a proportional relationship between two quantities represented by a and b is associated with two constants of proportionality, ab and ba, throughout the unit, the convention is if a and b are, respectively, in the left and right columns of a table, then ba is the constant of proportionality for the relationship represented by the table.)
For example, if a person runs at a constant speed and travels 12 miles in 2 hours, then the distance traveled is proportional to the time elapsed, with constant of proportionality 6, because
The time elapsed is proportional to distance traveled with constant of proportionality 16, because
Students learn that any proportional relationship can be represented by an equation of the form y=kx where k is the constant of proportionality, that its graph lies on a line through the origin that passes through Quadrant I, and that the constant of proportionality indicates the steepness of the line. By the end of the unit, students should be able to easily work with common contexts associated with proportional relationships (such as constant speed, unit pricing, and measurement conversions) and be able to determine whether a relationship is proportional or not.
Because this unit focuses on understanding what a proportional relationship is, how it is represented, and what types of contexts give rise to proportional relationships, the contexts have been carefully chosen. The first tasks in the unit employ contexts such as servings of food, recipes, constant speed, and measurement conversion, that should be familiar to students from the grade 6 course. These contexts are revisited throughout the unit as new aspects of proportional relationships are introduced.
Associated with the contexts from the grade 6 course are derived units: miles per hour; meters per second; dollars per pound; or cents per minute. In this unit, students build on their grade 6 experiences in working with a wider variety of derived units, such as cups of flour per tablespoon of honey, hot dogs eaten per minute, and centimeters per millimeter. The tasks in this unit avoid discussion of measurement error and statistical variability, which will be addressed in later units.
On using the terms quantity, ratio, proportional relationship, unit rate, and fraction. In these materials, a quantity is a measurement that is or can be specified by a number and a unit, e.g., 4 oranges, 4 centimeters, “my height in feet,” or “my height” (with the understanding that a unit of measurement will need to be chosen, MP6). The term ratio is used to mean a type of association between two or more quantities. A proportional relationship is a collection of equivalent ratios.
A unit rate is the numerical part of a rate per 1 unit, e.g., the 6 in 6 miles per hour. The fractions ab and ba are never called ratios. The fractions ab and ba are identified as “unit rates” for the ratio a:b. In high school—after their study of ratios, rates, and proportional relationships—students discard the term “unit rate,” referring to a to b, a:b, and ab as “ratios.”
In grades 6–8, students write rates without abbreviated units, for example as “3 miles per hour” or “3 miles in every 1 hour.” Use of notation for derived units such as mihr waits for high school—except for the special cases of area and volume. Students have worked with area since grade 3 and volume since grade 5. Before grade 6, they have learned the meanings of such things as sq cm and cu cm. After students learn exponent notation in grade 6, they also use cm2 and cm3.
A fraction is a point on the number line that can be located by partitioning the segment between 0 and 1 into equal parts, then finding a point that is a whole number of those parts away from 0. A fraction can be written in the form ab or as a decimal.