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Turning Manipulatives Inside Out
By Gary S. Stager

Dust Off Your Old Math Manipulatives
Tactile math manipulatives are a great tool for counting, measuring, representing numerical relationships and exploring geometric patterns. Logo provides an opportunity for manipulatives to be used in new ways. While the following classroom activity appears simple, it can yield quite complex mathematical thinking and problem solving. Any manipulatives may be used including pattern blocks, Cuisenaire Rods, Infix Cubes, tangrams and pentominoes. Manipulatives which tessellate provide some of the richest opportunities for problem solving. I am rather fond of the “Puzzelations,” foam-rubber tessellating manipulatives, available from: These manipulatives have a finite number of shape pieces which may be combined in infinite ways.

The activity described below is appropriate for 4th grade through adult learners. Past experience and mathematical knowledge may be brought to the problem solving process, regardless of age.

Off-Computer Preparatory Activity

  • Each group of 3-4 students should be given a representative set of manipulatives with a few copies of each piece.
  • The same type of manipulatives may be given to each student team or different sets may be used, depending on the educational goals of the project.
  • Students should spend some time moving the pieces around and exploring their physical relationships.
  • Decide upon names for each shape (invented names are OK for strange pieces or with young learners)

Logo-based Activity
Teams of students will design “manipulative” software to be used by others. The user should have a satisfying experience manipulating the shapes on the screen.

Each team of students should pursue the following three goals:

  1. Create a Logo procedure to draw each shape individually (polygons should all have equal sides)
  2. Write procedures to combine/connect two or more of the pieces.
  3. Design an interface for a user to manipulate the shapes on the screen and create interesting patterns. Commands, buttons, sliders and text instructions may be used to assist the user.

Extension Activities

  • Use variables to control the size of each piece.
  • Use sliders to control the variable size of each piece or even to zoom in/out of a tessellation.
  • Use Find/Replace to replace constant fd and bk distances with variables, such as: Replace FD with FD Scale * . Be sure to do the same for BK and use :SCALE if global variables are used instead of sliders.
  • Write procedures to randomly tessellate the pieces!
By using language to describe the attributes and measurements of each piece students enhance their ability to understand and communicate geometric relationships to the computer. Issues of variable, measurement, angle and orientation are made concrete. Adults who did well in high school math classes are often hamstrung by their recollections of half-memorized algorithms while their colleagues (or students) use their bodies, common sense and turtle geometry to solve the same problems.

At the end of the activity, kids can trade software and explore the work of their peers.

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