6.1.2 Ratios & Rates

Teacher Notes

• Students that struggle to identify equivalent ratios written in the form a:b will benefit from representing the ratios with fractions before determining equivalency. For example, by writing 8:4 as $\frac{8}{4}$ and 2:1 as $\frac{2}{1}$, it quickly becomes apparent that the two ratios are equivalent. Similarly, by writing 3:1 as $\frac{3}{1}$ and 1:3 as $\frac{1}{3}$, it becomes apparent that the ratios are not equivalent and order matters;
• Visual models can help students understand that ratios represent a comparison of two quantities by division, rather than two unique quantities. The following pictures show that if the ratio of number of pieces of pizza eaten to the total number of pieces of pizza is $\frac{1}{2}$, that does not mean that the pizza was necessarily cut into 2 pieces;

• As students begin to use what they know about fractions to make comparisons, they will learn to make other comparisons. For example, if the ratio of children to adults in a family is 4 to 2, they can compare the number of adults to the total number of people in the family with the ratio 2 to 6, and so on. In general, students will learn that they can write part-to-part, part-to-whole, and whole-to-part ratios.
• Students must pay close attention to language to determine whether they are being asked to represent a part-to-part, part-to-whole, or whole to part ratio;
• Students also need to build a connection from division to ratio and rate. This can be accomplished when they learn that a ratio is a comparison of two numbers by division; for example, the ratio 4:5 can be written as $\frac{4}{5}$ and is the quotient of 4÷5. Students will use this relationship when they write ratios as decimals.
• Before connecting multiplication and division to the concepts of ratio and rate, students first need to build a connection from additive reasoning to multiplicative reasoning. That is, they need to understand that 2 times has a different meaning than 2 more than, and so on.
• Teachers must help students make the transition from only thinking about a fraction as representing the comparison of a part to a whole to thinking about a fraction as a representation of many types of ratios. Expanding this thinking is particularly important in dealing with the difference between the representations of equivalent fractions and equivalent ratios. In common representations of equivalent fractions, the whole stays the same. To create the equivalence, the whole is divided into smaller and smaller parts.
• Teachers need to demonstrate that any fraction can be expressed in an infinite number of ways.

By contrast, in common representations of equivalent ratios, the whole is replicated over and over, or multiplied. In the equivalent ratios 1:2 = 2:4 = 3:6 = 5:10, the 1-for-every-2 relationship is replicated over and over, or multiplied, and the relationship of the new parts to the new whole does not change.  This concept of scaling lays the foundation of proportionality in later grades.

• Visuals can be used to confront the misconception that adding or subtracting the same number in the numerator and denominator does not produce equivalent ratios, rather multiplying or dividing does.

 Example:  $\frac{4}{8}$ is not equivalent to $\frac{4-2}{8-2}$, or $\frac{2}{6}$.  However, $\frac{4}{8}$ is equivalent to $\frac{4÷2}{8÷2}$, or $\frac{2}{4}$.

• The multiplication table is a powerful model to show equivalent fractions, and therefore, equivalent ratios. For example, following the rows for 4 and 5 to the right on the table shown below shows the equivalent ratios $\frac{4}{5}, \frac{8}{10}, \frac{12}{15},$ and so on.

• A useful activity to help students build a connection from multiplication to ratio and rate is to explore multiplication as scaling. When a quantity is multiplied by a number greater than 1, it is scaled up. When a number is multiplied by a positive number less than 1, it is scaled down. Opportunities to examine such problems as those shown below can help students understand the concept of scaling.

●     Ask students to recall when they have heard or used expressions with the word per, and make a list the expressions.  Explain that these are all rates and have a denominator of 1.  Ask students how they would find a unit rate from a rate such as $32.50 for 10 gallons of gas.  Conversely, ask how they would use a unit rate of 25 miles per gallon to determine the number of miles traveled using 10 gallons.

●     The words used to describe unit rates (e.g., 25 students per bus), and the fact that students often see unit rates described with only one number visible, make it difficult for students to think about unit rates in fraction form. It is beneficial for students to understand that every rate situation can be written in two ways with two different unit rates, with either unit as 1.  For example, the situation in which 6 pounds of bananas cost $3 can be written as the rate $\frac{6 pounds}{3 dollars}$  and as the unit rate in terms of pounds per dollar:  $\frac{2 pounds}{1 dollar}$.  The same situation can be described by the rate $\frac{3 dollars}{6 pounds}$ and as the unit rate in terms of dollars per pound:  $\frac{0.5 dollars}{1 pound}$ or $\frac{$0.50}{1 pound}$.  When the unit rate is written in terms of amount of money for 1 quantity, then it is called the unit cost or unit price.  In either instance, to find the unit rate, one must find the equivalent ratio whose denominator is 1 unit. Two ways to find equivalent unit rates is to use equivalent ratios or divide the numerator by the denominator.

●     Students will benefit from constructing various representations of additive relationships and multiplicative relationships, as shown below.