# Midpoint Shapes

## Article excerpt

This department recognizes the importance of children's exploring hands-on and minds-on mathematics and presents teachers with open-ended explorations to enhance mathematics instruction. These explorations invoke problem solving and reasoning, require communication skills, and connect various mathematical concepts and principles. The ideas presented here have been tested in various classroom settings.

A mathematical investigation--

* has multidimensional content;

* is open ended, with several acceptable solutions;

* is an exploration requiring a full period or longer to complete;

* is centered on a theme or event; and

* is often embedded in a focus question.

In addition, a mathematical investigation involves processes that include--

* researching outside sources,

* collecting data,

* collaborating with peers, and

* using multiple strategies to reach conclusions.

This investigation invites students to formulate and test conjectures about the shapes formed by consecutively connecting the midpoints of the sides of various types of quadrilaterals, for example, squares, rectangles, rhombuses, parallelograms, and general quadrilaterals. The midpoint shapes of all these quadrilaterals have two properties: (1) they are all parallelograms and (2) their areas are half the areas of the original quadrilaterals. Extension problems ask students to explore midpoint shapes of triangles and of concave versus convex quadrilaterals and to look for patterns of consecutive midpoint shapes.

Throughout the activities, students will work with cutout pieces of paper in the shape of squares. rhombuses, rectangles, and parallellograms. These paper cutouts are actually three-dimensional objects: very thin prisms. But students will use the cutouts to represent polygons, and the language in this article will reflect students' actual usage, for example, we will refer to the cutout paper squares as "squares."

The investigation involves the midpoint problems discussed by Thomas Banchoff in his article "Mathematicians as Children, Children as Mathematicians" in the February 2000 focus issue of Teaching Children Mathematics. There Banchoff describes versions of the midpoint problem that can be used with elementary, secondary, and even college students, each working as mathematicians at their own levels. This investigation is designed for students in grades 4-6, but students in grades 2-3 could explore the midpoint shapes of familiar quadrilaterals, such as squares and rectangles.

Midpoint Shapes

Objectives

The students will--

* describe squares, rectangles, and other quadrilaterals in terms of their properties;

* find and compare ways to locate the midpoints of the sides of a quadrilateral;

* compare the areas of shapes;

* formulate and justify conjectures about the midpoint shapes of various types of quadrilaterals;

* compare and test conjectures about the midpoint shapes of convex versus concave quadrilaterals;

* describe types of triangles--for instance, isosceles, equilateral, scalene--in terms of their properties and explore their midpoint shapes and their properties; and

* look for patterns in successive midpoint shapes of shapes.

Materials

Each student will need scissors and a ruler.

For the first part of the investigation, each student will need a cutout paper square. Prepare four sizes of squares, for instance, with sides of 8 cm, 10 cm, 12 cm, and 15 cm. For the second part of the investigation, students will work in groups, each examining one of the following types of quadrilaterals: rectangles, rhombuses, parallelograms. Prepare one cutout paper shape for each student in a group--using rectangles in three sizes, for example, 8 cm by 10 cm, 8 cm by 12 cm, 10 cm by 14 cm; and rhombuses and parallelograms in three sizes. …