Math II (Несколько текстов для зачёта), страница 10

The instructions indicate that the toy works because the monkey is constructed around a plane mechanical linkage and because the products and sums are arranged on the plate in a "special" order. The linkage, which ensures that moving the feet to different factors forces the hands to move, consists of two upper arm-leg pieces; two arm-hand pieces; and a "tail" with an answer window that moves up and down with different products or sums. The two arm-hand pieces are attached below the answer window to the tail. The two upper arm-leg pieces are attached to the two arm-hand pieces at the "elbow" and to the upper part of the tail. The monkey's head is attached on top of this piece. Moreover, the feet on the upper arm-leg pieces slide along the straight-line opening at the bottom of the toy's plate. The instructions do not give any mathematical explanation indicating why the linkage and the "special" placement of products or sums work the way they do.

This article explains why the Educated Monkey works by looking at the geometry of the linkage, as well as at the special placement of the products and sums on the plate of the toy. We believe that this problem is an interesting one to present to plane geometry students, since it reviews many important geometric, algebraic, and arithmetic concepts. Moreover, students may want to recreate the linkage, or create their own linkages, from strips of cardboard and paper fasteners.

In explaining why the toy works, we eventually show that the entire multiplication table forms a 45 degrees-45 degrees-90 degrees triangle (see fig. 5). Moreover, we show that for any particular choice of factors, the product is found at the right-angle vertex of the 45 degrees-45 degrees-90 degrees triangle defined by the factors.

EXPLANATION

We first refer to the photograph and figure 6. Points A and B are at the tips of the monkey's feet; they can move only along the straight-line opening, line AB, at the bottom of the toy's plate. Point C is directly below the window where the monkey's hands point to the product or sum. Points D and F are at the monkey's elbows. Point E is hidden behind the monkey's nose. At all these points, the linkage can rotate. The toy is constructed so that angle ADE and angle BFE are constant, congruent right angles. However, most--but, surprisingly, not all--of the other angles vary as the monkey's hands and feet move. Segments AD, DE, EF, FB, DC, and FC are congruent in the toy; we mark these segments as congruent in figures 6 and 7. No physical links AD, BF, AC, or BC exist; these segments are auxiliary ones for the explanation.

We assume that A is stationary; that is, we have chosen one factor using the left foot and are about to slide the right foot at B to the other factor to obtain a product; the product will appear in a window directly above C, as shown in the photograph. We wish to show that for a fixed choice of A--that is, the origin, or left factor--as B points to different second factors beyond--that is, to the right of A--triangle ACB is always a 45 degrees-45 degrees-90 degrees triangle and the product appears directly above C in the window on the tail.

Without loss of generality, we assume that A is at the origin of a Cartesian coordinate system. We are given that

(1) m angle ADC + m angle CDE = m angle BFC + m angle CFE = 90 degrees.

CDEF is a parallelogram. More specifically, it is a rhombus. Therefore, its opposite angles, which are labeled 1 and 3, are respectively congruent.

m angle ADE - m angle 1 = m angle BFE - m angle 1;

therefore, the angles labeled 2 are congruent. Triangles ACD and BCF are congruent by SAS. Moreover, they are both isosceles. Therefore, all the angles labeled 4 are congruent. Since triangles ACD and BCF are congruent, segments AC and BC are congruent by corresponding parts. Therefore, triangle ABC is isosceles, and the angles labeled 5 are congruent by the isosceles triangle theorem.

We have established that triangle ABC is isosceles. We next establish that it is a 45 degrees-45 degrees-90 degrees triangle. Since the sum of the measures of the angles of a triangle equals 180 degrees,

(2) m angle 2 + m angle 4 + m angle 4 = 180 degrees.

Since consecutive angles of a parallelogram are supplementary,

(3) m angle 1 + m angle 3 = 180 degrees.

Because ABFED is a pentagon, its interior angles sum to (n - 2)180 degrees = (5 - 2)180 degrees = 540 degrees, where n is the number of sides. Thus,

(4) {m angle 3 + m angle 1 + m angle 1 + m angle 2 + m angle 2

{+ m angle 4 + m angle 4 + m angle 5 + m angle 5

= 540 degrees.

Looking at equation (4) and substituting equation (3) for the first two terms, equation (1) for the next two terms, and equation (2) for the next three terms gives us

180 degrees + 90 degrees + 180 degrees + 2m angle 5 = 540 degrees,

or

m angle 5 = 45 degrees.

Therefore, triangle ABC is always a 45 degrees-45 degrees-90 degrees triangle. Although angles 1, 2, 3, and 4 vary as the monkey's feet and hands move, angle CAB remains a 45 degree angle no matter where we move B, the second factor. Since we assumed that A is at the origin, then C, the product or sum, must move on the line y = x--on a line through the origin, A, at 45 degrees from the x-axis--as we move B to different factors beyond A. The multiplication table on the toy, shown in figure 1 is arranged so that ---,

• the 1's tables from 1 x 2 to 1 x 12 are on a 45 degree line whose origin is the location of point A when the left foot is directly above 1.

• the 2's tables from 2 x 3 to 2 x 12 are on a 45 degree line whose origin is the location of point A when the left foot is directly above 2.

• the 3's tables from 3 x 4 to 3 x 12 are on a 45 degree line whose origin is the location of point A when the left foot is directly above 3.

• the 10's tables from 10 x 11 to 10 x 12 are on a 45 degree line whose origin is the location of point A when the left foot is directly above 10.

• the 11's table for 11 x 12 is on a 45 degree line whose origin is the location of point A when the left foot is directly above 11.

• the 12's tables are obtained by the commutative property.

Thus, the multiplication tables on the plate are a family of 45 degree lines, each going from N x (N + 1) to N x (12), where N is the number to which the left foot, A, points, and 1 less than or equal to N less than or equal to 11. In fact, the entire multiplication table is itself arranged as a 45 degrees-45 degrees-90 degrees right triangle, as shown in figure 5. Any missing products are accomplished by the commutative property. For example, although 3 x 2 is not possible to compute with the monkey, 2 x 3 is possible.

In reality, the toy is arranged so that the product or sum lies directly above C, in the window between the monkey's hands. Therefore, the entire table is raised up one unit.

CHALLENGES

We pose the following challenges to students and teachers:

1. Prove why the toy works if you keep B stationary and vary A. (The commutative property gives the explanation.)

2. Prove why squaring a number works using the square symbol. (Note the position of the perfect squares on the toy's plate.)

3. Work with and construct such other plane linkages as the pantograph, which draws figures similar to those traced, and explore their geometric properties.

4. How many different 45 degrees-45 degrees-90 degrees triangles can you find in the entire multiplication or addition table?

5. Prove that line segment DF is always parallel to line segment AB.

6. Prove that the monkey's tail, line segment CE, is perpendicular to line segment AB.

CONCLUSION

"Consul," the Educated Monkey, is an outstanding, practical example of a plane linkage. In learning why the monkey works the way it does, students are required to review many important concepts from plane geometry, algebra, and arithmetic. Making their own "monkey" linkage similar to Consul, which one of the authors has done with construction paper and paper fasteners, would give students additional, hands-on experience with many important mathematical concepts. An outstanding primary resource for mathematical models, including linkages, is Cundy and Rollett (1961). When examining why this toy works, your students will not just be monkeying around; they will be learning some very interesting, highly motivating hands-on mathematics.

The authors would like to thank Susan Cisco for preparing all the figures used in this article.

ZOOMING IN ON SLOPE IN CURVED GRAPHS

Source: Mathematics Teaching in the Middle School, Jan2000, Vol. 5 Issue 5, p330, 5p, 1 chart, 5 graphs

Author(s): Beigie, Darin

STUDENTS IN A MIDDLE SCHOOL MATHematics club used the zooming technology of Green Globs and Graphing Equations (Dugdale and Kibbey 1996) to study slope in curved graphs. Seventh and eighth graders investigated some elementary curved graphs by zooming in on evenly spaced points along a graph until the graph appeared linear and slope could be calculated. The slopes at the various points were then plotted on a separate grid and joined to make a graph of the slope itself and to discover the algebraic equation describing the new graph. The zooming technology gave the students a concrete, visual context in which to learn about the idea of slope in a curved graph and to study how slope varies along a curved graph.

Slope of a Curved Graph

WHEN STUDYING TWO-POINT CALCULATIONS OF the slope of a line, a seventh grader once asked in class, "What do you do if the graph is curved?" Another student responded without hesitation, "If you make the two points really close together, the graph will look straight and you can still find the slope." Such a penetrating insight led me to wonder if an age-appropriate way was available to introduce middle schoolers to the idea of slope in curved graphs. Indeed, after playing such a central role in middle school study of linear graphs, the idea of slope is effectively abandoned with nonlinear graphs until the introduction of calculus later in high school. An appropriate middle school exposure to slope in its more general context would be helpful in conveying the utility and flexibility of the concept.

The zooming technology of graphing calculators and certain graphing software offers an ideal environment for middle schoolers to extend their understanding of slope to curved graphs. Such technology allows a student to explicitly see a curved graph becoming effectively straight as one zooms in closer and closer on any point on the graph, much as the curved surface of the earth can appear flat from close range. The idea of slope of a curved graph can be illustrated by a skateboard on a curved ramp (see fig. 1). Even though the ramp is curved, the flat skateboard has a well-defined slope anywhere on the ramp, one that changes with location along the ramp. In a similar manner, one can always make a first definition of slope in a curved graph by simply selecting any two points on the graph and calculating the slope of the line segment joining the two points. By zooming in on any point on a curved graph, the student sees the graph becoming increasingly straight and the two-point slope calculation making sense.

For example, consider the graph of the quadratic equation y = x[sup 2] and the result of successive zooms on the point (2, 4), shown in figure 2. The graph appears less and less curved as the number of zooms increases, resulting in an effectively straight graph by the eighth zoom. Within the window of the eighth zoom, the student selects two points and registers the coordinates with the trace feature of the software: A = (1.98513, 3.93898) and B = (2.01511, 4.05906). The slope m of the graph at the point (2, 4) is then calculated as follows, rounded to the nearest thousandth:

change in y 4.05906 - 3.93898

m = ----------- = ----------------- = 4.005

change in x 2.01511 - 1.98513

By using this method, the mathematics-club members calculated the slope along various points of a curved graph.

Investigating Slopes and Finding Patterns

BY USING THE ZOOM AND TRACE FEATURES OF the graphing software, students investigated some elementary curved graphs in detail. The students had gained some familiarity with graphs of different powers of x through previous open-ended explorations using graphing calculators and software. They could recognize, for example, the graphs of the quadratic equation y = x[sup 2] and the cubic equation y = x[sup 3]. The study began with these two graphs, and the students zoomed in and performed two-point calculations to determine the slopes of these graphs at integral values of x ranging from -4 to 4. As in the previous example, the slopes were quite close to integral values, and the students were asked to round their answers to the nearest integer; for example, in the previous calculation, 4.005 would be rounded to 4. Once the slopes were determined, both the original equation and the corresponding slopes were graphed by hand, as shown in figure 3. Hand-drawn graphs helped students absorb the meaning of their calculations and the relationship between the original graphs and the slope graphs.

The students discovered that the quadratic graph y = x[sup 2] had a slope graph that was a line with a slope of 2 passing through the origin. The equation for the slope graph was thus determined to be m = 2x. The students then discovered that the cubic graph y = x[sup 3] had a slope graph with the familiar shape of a parabola. Determining the equation of this slope graph was like solving a puzzle. The students suspected that the parabolic shape meant that the equation involved an x[sup 2] somehow, and they tried to find an equation that would match their calculated table of values for slope. Soon they caught on that the slope m was triple the familiar x[sup 2] pattern, and they were able to deduce the slope equation m = 3x[sub 2].

After investigating more graphs, for example, y = x and y = x[sup 4], the students picked up on some general patterns between equations of the form y = x[sup n] and the corresponding slope equation:

• The exponent in the slope equation is one less than the exponent in the original equation.

• The coefficient in the slope equation is equal to the exponent in the original equation.

Their principal finding, summarized in table 1, was that the graph of the equation y = x[sup n] has a corresponding slope graph described by the equation m = nx[sup n-1]. Although such a finding is certainly advanced for a middle schooler, it was the result of a concrete activity involving two-point slope calculations along curved graphs that had been magnified by the zooming technology to look effectively linear. Discovering the equation patterns for these slope graphs was quite manageable for the students, who enjoyed pattern problems and were comfortable working with algebraic expressions. Some had even guessed the general pattern after studying the cubic equation.