With the teacher's guidance, students can thus begin to develop a more formal definition of similarity in terms of relationships among sides and angles. Linking Length, Perimeter, Area, and Volume Investigations into the properties of, and relationships among, similar shapes can afford students many opportunities to develop and evaluate conjectures inductively and deductively. Teachers might encourage students to formulate conjectures about the ratios of the side lengths, of the perimeters, and of the areas of the four similar triangles. They might conjecture that the ratio of the perimeters is the same as the scale factor relating the side lengths and that the ratio of the areas is the square of that scale factor. Then students could use dynamic geometry software to test the conjectures with other examples.
Students can formulate deductive arguments about their conjectures.
Communicating such reasoning accurately and clearly prepares students for creating and understanding more-formal proofs in subsequent grades. Specify locations and describe spatial relationships using coordinate geometry and other representational systems Geometric and algebraic representations of problems can be linked using coordinate geometry. Students could draw on the coordinate plane examples of the parallelograms discussed previously, examine their characteristic features using coordinates, and then interpret their properties algebraically. Such an investigation might include finding the slopes of the lines containing the segments that compose the shapes. From many examples of these shapes, students could make important observations about the slopes of parallel lines and perpendicular lines.