Bogglers

By Scott Kim|Sunday, December 01, 2002

Hands-on Learning
You can't learn to play basketball from a book. You have to get out there and shoot a few hoops. Mathematics, like basketball, can benefit from hands-on experience. Classrooms around the world use props—called manipulatives—to give students a concrete understanding of abstract concepts. But manipulatives aren't just for kids: Watson and Crick used balls and sticks to analyze and develop a model of the DNA double helix.

Cuisenaire Rods are rectangular blocks devised by Belgian schoolteacher Emile-Georges Cuisenaire about 50 years ago to teach arithmetic basics. As seen above, each of the 10 rods has a different color and length: The white rod is one centimeter long, the red two centimeters, the light green three, and so on, up to a 10-centimeter-long orange rod.

 A. [Easy] B. [Challenging] C. [Challenging] D. [Difficult]

1. [Easy] Cuisenaire rods give students a tangible model of addition. As shown below, the six-centimeter-long dark green rod can be expressed as the sum of two light green rods (3+3), three red rods (2+2+2), six white rods (1+1+1+1+1+1), or a combination of one white, one red, and one light green rod (1+2+3). Using any sequence of rods of any color, how many other ways are there to line up rods to equal the length of the dark green rod (6)? Count different sequences of the same rods separately. [Difficult] What about for the 10-centimeter orange rod?

2. How many ways can you fill the figures at right using each of the rods only once? Count solutions that are mirror-reverses of each other separately. Hint: Place the long rods first.

Pattern Blocks consist of six color-coded shapes: green triangles, orange squares, blue rhombuses (with a 60-degree angle), tan rhombuses (with a sharper 30-degree angle), red trapezoids, and yellow hexagons. The blocks have similar angles and lengths so they can fit together to make larger solid shapes. All edges are one inch long, except for the long side of the red trapezoid, which is two inches long. All angles are multiples of 30 degrees.

1. [Easy] The blocks fit together to make many different shapes, but they can also combine to make both larger and same-size versions of the basic pattern block shapes. The illustration at right shows ways to assemble
green triangles into a rhombus and a trapezoid, and orange squares into one larger orange square. How many other ways are there to duplicate the shapes of the pattern blocks using two, three, or four copies of a single pattern block? The individual pattern blocks used do not have to have the same shape as the overall form, but they must be the same as each other.

2. [From Easy to Difficult] For each figure below, find a different way to make the same overall shape using the same pieces but with no piece in the same position and orientation. For figure D, find a solution in which the orange squares don't touch each other at the corners.

 A. B. C. D.

Geoboards are wood or plastic squares with 25 pegs arranged in five rows of five pegs each. By snapping rubber bands around the pegs, students can make geometric shapes and learn concepts like distance and area. The construction at right shows a concave hexagon with six interior pegs. (A concave polygon has at least one inward-pointing angle; all other shapes are convex.)

[Easy to Challenging] Can all 10 geometric constructions below be made on a geoboard? Imagine that the rubber bands have no thickness and perfectly connect one point to another.

1. A triangle with five interior pegs
2. A triangle with six interior pegs
3. A triangle with seven interior pegs
4. A square with an area of four
5. A square with an area of five
6. A square with an area of six
7. A hexagon with no parallel sides
8. A hexagon with three parallel sides
9. A hexagon with no sides congruent
10. A hexagon with all sides congruent

Solution

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Got new solutions for the puzzle? Want to see other people's solutions? Talk to the puzzle master in his discussion forum at www.scottkim.com.

This month's puzzles are based on activities found in the Super Source teacher-resource books, from educational publisher ETA/Cuisenaire: etacuisenaire.com.

For books and other teaching resources, check out mathsolutions.com, run by math educator and manipulatives advocate Marilyn Burns.