An example of how multiple representations might be facilitated in the mathematics (or other classrooms) is given below. This is sometimes called using the “Rule of Four” or “Rule of Five” (graphs, symbols, tables, words, pictures). The concept for this particular example is that of a relationship between two variables in which one variable is four times the other. Algebraically, it is represented by the equation. The mathematical equation for thisrelationship can be related to the table of values generated for ordered pairs, which can also berepresented graphically. In words, there are many examples from real life that might be represented bythis relationship.

Types of Representation / Example
Symbols /
Tables / x / y
1 / 4
1.1 / 4.4
1.2 / 4.8
1.3 / 5.2
1.4 / 5.6
1.5 / 6.0
Graphs
Words / “for every x-value, the y-value is
three times as much”
Or
“I spend four times as much as my brother. If he spends €1:00 spend €4:00 and if he spends €1:10 I spend€4:40. In general if he spends € x then I spend€ 3x.

Forcing students to move among these various representations can be a valuable learningexperience. A simple lesson is to give students one of the representations and have them construct the others. Or, preferably, allow students to make up one of the forms of representations and give to classmates to construct the others.

Working in groups of four, the students are presented with an appropriate topic to analyse for example, an English teacher might select a graph form an advertising campaign, or a physical education teacher might present his/her students with data relating to their performance in a given physical exercise.

Participants then divide a sheet of paper into four sections by folding the paper in half, first horizontally and then vertically. Each participant then chooses one representation and completes one of the four sections. The papers are then rotated to the left and the next person completes a new section of his/her choosing, using a representation that has not been used. Papers are rotated twice more so that all quadrants are complete and then returned to the original author. The group then discusses each and how the use of multiple representations led to a greater understanding of the concept.

A template for the activity is given below

Rule of Four: Multiple Representations
Words / Symbols
Table / Graph

Adapted from NCTM 2012