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Visualizing with colored markers allows students to investigate a wide variety of shapes and patterns.

Perceiving and visualizing sets of same-color shapes, students can "map" them. They become kaleidoscopic designs. Sectors of the circle made into repeating units can be rotated around the center of the circle and/or reflected symmetrically across it.When we do this, we fold the circle so that only one sector at a time is visible. Children visualize a shape and color it; then they may either

rotateorreflectthis shape by copying it, its color, and its locus to the next sector.Now they add another and then another shape to the sector, doing the same to successive sectors. Having done this consecutively in clockwise or counterclockwise order, when many shapes and colors have been added, unfolding the paper is a revelation. We see symmetry and rotation.

Rotation yields a pattern like that of a pinwheel, symmetry like that of a kaleidoscope.A simple rule for coloring is to:

Give allThe skills involved in mapping with color relate art to the famous 'four color theorem' in mathematics: any map can be made using no more than four colors.congruentshapes the same color, except when they areadjacentto one another, i.e., the sides of the shapes are constructed with the same line.

This is a

heuristicmethod of learning: literally, a 'learning by finding out' experience. The learner gains insight into mathematical logic by perceiving how different grids develop, beginning with a finite set of vertices on the circumference of the circle, and moving to new vertices made when line segments cross.Allow your students plenty of time to play with the variables. They'll need opportunities to look at their classmates' work, share and compare their discoveries.

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