Define a ratio and show the students the three ways to express ratios in written form. Write examples on the blackboard in all three forms, explaining how the numbers or quantities relate, such as 20 miles per gallon of gas (20:1), the number of girls in the class: the number of boys in the class, or the ratio of 10 red to 21 blue marbles (10:21).
Arrange some manipulatives in a place visible to all the students. Ask the students to write the ratio you describe, such as the ratio of pictures of cars to pictures of trucks, in all three forms. When the students understand the concept, allow one student at a time to set up some manipulatives for another student to express in all three forms.
Define a proportion, showing the students how to write a proportion, such as 1/2 = 3/6, that does not require solving for "x." Point out that the ratios on both sides of the equal sign are equivalent fractions, a familiar concept for sixth graders. If the ratios are equal, they are proportional. The relationship between the two sides of the proportion is multiplicative. Proportions are always related by multiplication. Continue showing examples until the students grasp the concept.
Draw four stick figures, one each of six, 12, 18 and 24 cm. Label the figures in size order from shortest to tallest, Ms. Jones, Mr. Smith, Ms. Bell and Mr. Ward. Duplicate this page and give one to each student. On the blackboard, draw a proportional reasoning table with three columns for the students to copy and fill out. Label the first column "Person," the second column "Height in Red Rods" and the last column "Height in Green Rods." Students should measure each stick figure and fill out the table accordingly.
Present the story problem: "Ms. Mills is 33 red rods tall. What is her height in green rods?" Ask the students to study the relationships in the table. Ask them to write a formula, or rule, to find the answer. Red rods are two centimeters, and green rods are three centimeters. (G = 2/3 R) Proportional relationships always involve either multiplication or division.
Tell the students to plot the data from the table on a pair of coordinate axes, and draw a line to connect each point. The graph is a straight line through the origin. Proportional relationships always result in a graph that is a straight line through the origin.
Create other problems to reinforce the concept. Begin to include problems that appear to be proportional relationships, but are not.