Introduction: Activities - Length
Determining the length of a horizontal or vertical line segment on the geoboard is easy; even primary students can do it - by counting. A diagonal line segment is more difficult to measure, however. Many students are content to estimate the length of diagonal line segments, which should be encouraged. More mathematically sophisticated students will want to compute such lengths directly.
One calculates the length of an arbitrary line segment using the Pythagorean theorem, a concept whose understanding is a primary objective of this unit. (We are not suggesting that first graders be taught the Pythagorean theorem. The lessons provided will almost certainly be spread out over many years, starting with the elementary grades on up to middle school or even high school.) As a teacher, you must be able to apply the Pythagorean theorem on sight, since students are surely going to ask you questions regarding length early in the unit.
Here's how it's done. For any given line segment, construct a right triangle having that line segment as hypotenuse. For example, to calculate the length of line segment AB in Figure 3:
Figure 3. Computing the length
of a diagonal line segment.
construct the right triangle ABC. The base of this triangle (line segment AC) is 4 units long, whereas the height (line segment BC) is 3 units. Using the Pythagorean theorem, the length of line segment AB is found by calculating the square root of the sum of the squares of the lengths of the base and height of the triangle, that is,
Thus the length of line segment AB is 5 units. Of course, the sum of the squares is not always a perfect square (in fact, this will seldom be the case), so the length will be an irrational number, in general. (An irrational number is not a rational number (or fraction.) Not all students will be ready for square roots, but the teacher is advised to know how to do such calculations nonetheless.
Additional examples of the Pythagorean theorem are given in Figure 4:
Figure 4. More examples of
length calculations on the geoboard.
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