Information for course "Unit 2 - Ratio And Proportion"

Description: Work with ratios in grade 6 draws on earlier work with numbers and operations. In elementary school, students worked to understand, represent, and solve arithmetic problems involving quantities with the same units. In grade 4, students began to use two-column tables, e.g., to record conversions between measurements in inches and yards. In grade 5, they began to plot points on the coordinate plane, building on their work with length and area. These early experiences were a brief introduction to two key representations used to study relationships between quantities, a major focus of work that begins in grade 6 with the study of ratios.

Starting in grade 3, students worked with relationships that can be expressed in terms of ratios and rates (e.g., conversions between measurements in inches and in yards), however, they did not use these terms. In grade 4, students studied multiplicative comparison. In grade 5, they began to interpret multiplication as scaling, preparing them to think about simultaneously scaling two quantities by the same factor. They learned what it means to divide one whole number by another, so they are well equipped to consider the quotients ab and ba associated with a ratio a:b for non-zero whole numbers a and b.

In this unit, students learn that a ratio is an association between two quantities, e.g., “1 teaspoon of drink mix to 2 cups of water.” Students analyze contexts that are often expressed in terms of ratios, such as recipes, mixtures of different paint colors, constant speed (an association of time measurements with distance measurements), and uniform pricing (an association of item amounts with prices).

One of the principles that guided the development of these materials is that students should encounter examples of a mathematical concept in various contexts before the concept is named and studied as an object in its own right. The development of ratios, equivalent ratios, and unit rates in this unit and the next unit is in accordance with that principle. In this unit, equivalent ratios are first encountered in terms of multiple batches of a recipe and “equivalent” is first used to describe a perceivable sameness of two ratios, for example, two mixtures of drink mix and water taste the same or two mixtures of red and blue paint are the same shade of purple. Building on these experiences, students analyze situations involving both discrete and continuous quantities, and involving ratios of quantities with units that are the same and that are different. Several lessons later, equivalent acquires a more precise meaning (MP6): All ratios that are equivalent to a:b can be made by multiplying both a and b by the same non-zero number (note that students are not yet considering negative numbers).

This unit introduces discrete diagrams and double number line diagrams, representations that students use to support thinking about equivalent ratios before their work with tables of equivalent ratios.

Initially, discrete diagrams are used because they are similar to the kinds of diagrams students might have used to represent multiplication in earlier grades. Next come double number line diagrams. These can be drawn more quickly than discrete diagrams, but are more similar to tables while allowing reasoning based on the lengths of intervals on the number lines. After some work with double number line diagrams, students use tables to represent equivalent ratios. Because equivalent pairs of ratios can be written in any order in a table and there is no need to attend to the distance between values, tables are the most flexible and concise of the three representations for equivalent ratios, but they are also the most abstract. Use of tables to represent equivalent ratios is an important stepping stone toward use of tables to represent linear and other functional relationships in grade 8 and beyond. Because of this, students should learn to use tables to solve all kinds of ratio problems, but they should always have the option of using discrete diagrams and double number line diagrams to support their thinking.

When a ratio involves two quantities with the same units, we can ask and answer questions about ratios of each quantity and the total of the two. Such ratios are sometimes called “part-part-whole” ratios and are often used to introduce ratio work. However, students often struggle with them so, in this unit, the study of part-part-whole ratios occurs at the end. (Note that tape diagrams are reserved for ratios in which all quantities have the same units.) The major use of part-part-whole ratios occurs with certain kinds of percentage problems, which comes in the next unit.

On using the terms ratio, rate, and proportion. 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). The term ratio is used to mean an association between two or more quantities and the fractions ab and ba are never called ratios. Ratios of the form 1:ba or ab:1 (which are equivalent to a:b) are highlighted as useful but ab and ba are not identified as unit rates for the ratio a:b until the next unit. However, the meanings of these fractions in contexts is very carefully developed. The word “per” is used with students in interpreting a unit rate in context, as in “$3 per ounce,” and “at the same rate” is used to signify a situation characterized by equivalent ratios.

In the next unit, students learn the term “unit rate” and that if two ratios a:b and c:d are equivalent, then the unit rates ab and cd are equal.

The terms proportion and proportional relationship are not used anywhere in the grade 6 materials. A proportional relationship is a collection of equivalent ratios, and such collections are objects of study in grade 7. 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.”