# Representations

## How are representations used in the Inquiry Project?

Within the Inquiry Project, students use many types of representations to communicate their ideas as well as to interpret the ideas of others. These representations include verbal discussions, written explanations, drawings, data tables, charts, and graphs. Students use multiple representations to express their ideas in alternate ways and to make connections between multiple expressions of the same idea. Students learn to choose the best representation for a particular concept or to answer a specific question.

Students’ representations enable teachers to see the development of and gaps in children’s ideas beyond spoken or written words. Teachers may use students’ representations as visual structures to scaffold conceptual development.

### The Weight Line

The weight line is an important representation in the third grade curriculum. Students use the weight line, to organize cubes by weight, similar to using a number line. When students first order the cubes from lightest to heaviest, students often place them adjacent to one another without considering spacing or representing how much more one weighs than another. Students need a unit of measurement to determine if one block is 2x or 3x heavier than another. The use of ratio and proportionality does not come easily for many third graders. Using the scaffold of the weight line, students can place the cubes at the measured location and interpret the spacing as 2x heavier or 3x heavier.

Relative spacing on the weight line also brings forth issues of scale and origin, which can be a challenge for grade 3 students. Using zero as a starting point is not always immediately apparent to children. The difference in how students begin their weight line fosters debate and discussion as children develop a measurement model including the concept of weight as a quantity.

As students investigate smaller pieces of material, interesting weight line issues are raised. For instance, many students do not immediately understand that there can be values on the weight line between 3 and 4, or more interestingly, between 0 and 1. Research has shown that students who doubt the existence of numbers between 0 and 1 also doubt that very small pieces of matter have weight. Conversely, those who understand that even invisible pieces of matter have weight also understand that there are an infinite number of values between 0 and 1.

When students repeatedly cut material into smaller pieces, they begin to understand that each piece is not only made of the same material, but also continues to have weight and volume. When placing smaller pieces of material on the weight line, students see the need for values between the integers and begin to make the connection between fractions, the number line, and their increasingly smaller measurements of weight.

The Inquiry Project uses representations, such as the weight line, to assess and scaffold students understanding of numeracy, mathematics, and also scientific principles. By examining and evaluating the students’ representations, teachers are able to judge students’ conceptual development, make decisions about what needs to be further emphasized, and determine when students are ready to move on to more advanced investigations.