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In programs that adopt the recommendations in Principles and Standards, middle-grades students will have explored and discovered relationships among geometric shapes, often using dynamic geometry software. Using features of polygons and polyhedra, they will have had experience in comparing and classifying shapes. High school students should conduct increasingly independent explorations, which will allow them to develop a deeper understanding of important geometric ideas such as transformation and symmetry. These understandings will help students address questions that have always been central to the study of Euclidean geometry: Are two geometric figures congruent, and if so, why? Are they similar, and if so, why? Given that a geometric object has certain properties, what other properties can be inferred?
Geometry offers a means of describing, analyzing, and understanding the world and seeing beauty in its structures. Geometric ideas can be useful both in other areas of mathematics and in applied settings. For example, symmetry can be useful in looking at functions; it also figures heavily in the arts, in design, and in the sciences. Properties of geometric objects, trigonometric relationships, and other geometric theorems give students additional resources to solve mathematical problems.
High school students should develop facility with a broad range of ways of representing
geometric ideas—including coordinates, networks, transformations,
vectors, and matrices—that allow multiple approaches to geometric
problems and that connect geometric interpretations to other contexts.
Students should recognize connections among different representations,
thus enabling them to use these representations flexibly. For example,
in one set of circumstances it might be most useful to think about an
object's properties from the perspective of Euclidean geometry, whereas
in other circumstances, a coordinate or transformational approach might
be more useful. This ability to use different representations advantageously
is part of students' developing geometric sophistication.
Geometry has always been a rich arena in which students can discover patterns
and formulate conjectures. The use of dynamic geometry software enables
students to examine many cases, thus extending »
their ability to formulate and explore conjectures. Judging, constructing,
and communicating mathematically appropriate arguments, however, remain
central to the study of geometry. Students should see the power of deductive
proof in establishing the validity of general results from given conditions.
The focus should be on producing logical arguments and presenting them
effectively with careful explanation of the reasoning, rather than on
the form of proof used (e.g., paragraph proof or two-column proof). A
particular challenge for high school teachers is to integrate technology
in their teaching as a way of encouraging students to explore ideas and
develop conjectures while continuing to help them understand the need
for proofs or counterexamples of conjectures.
Students should enter high school understanding the properties of, and relationships among, basic geometric objects. In high school, this knowledge can be extended and applied in various ways. Students should become increasingly able to use deductive reasoning to establish or refute conjectures and should be able to use established knowledge to deduce information about other situations. For example, a teacher might ask students to solve problems like that in figure 7.12.
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First, I noticed that since
and
are parallel, angles B and E must be congruent. Also, angles ACB and DCE are congruent, since they are vertical. So now I know that the two triangles (ABC corresponds to DEC) are similar by angle-angle similarity. But that tells me that their corresponding sides are proportional. Since DE = 4(AB), I know that all the sides of triangle DEC are 4 times as large as the corresponding sides of triangle ABC, so CD = 4(15) = 60.
Now I just need to find the other side of triangle DEC to find its perimeter. But
makes it into 2 right triangles, so I can use the » Pythagorean theorem on each of those. FE2 + 482 = 522, so FE is 20. (Actually, I just noticed that this is just 4 times a 5-12-13 triangle, but I saw that too late.) Then looking at CDF, this is 12 times a 3-4-5 triangle, so CF must be 36. (I checked using the Pythagorean theorem and got the same answer.) So the perimeter is 52 + 60 + 56 = 168.
Once I find the perimeter of ABC, I'm done. But that's easy, since the scale factor from DEC to ABC is 25%. I can just divide 168 by 4 and get 42. The reason that works is that each of the sides of ABC is 25% of its corresponding side in DEC, so the whole perimeter of ABC will be 25% of DEC. We already proved that in class anyway.
High school students should begin to organize their knowledge about classes of objects more formally. Finding precise descriptions of conditions that characterize a class of objects is an important first step. For example, students might define a trapezoid as a quadrilateral with at least one pair of parallel sides. They should realize that such a definition includes parallelograms, rectangles, and squares as special classes of trapezoids. Students might also ask, "How much information do I need to be sure a quadrilateral is a trapezoid? Do I need also to know something about its diagonals and angles? Can I get by with just some of this information?" As their ability to make logical deductions grows, students should be able to develop characterizations that follow directly from the properties of parallel lines and similar triangles. Alternatively, the class of trapezoids could be characterized in terms of its diagonals: If the diagonals of a quadrilateral cut each other so that the ratios of the corresponding segments of the diagonal are equal, then the quadrilateral is a trapezoid.
A teacher might ask a class to consider, on the basis of this characterization, how trapezoids are related to other classes of quadrilaterals. In considering parallelograms, students may note that the diagonals bisect each other, so each is cut in a 1:1 ratio and therefore the parts are proportional. The obvious conclusion that parallelograms (and many other classes of quadrilaterals) may be considered a special kind of trapezoid may seem unusual to those who think of a trapezoid as a quadrilateral with exactly one pair of parallel sides. However, it is important for students to see that the definition chosen will determine the conclusions that can be drawn.
One of the most important challenges in mathematics teaching has to do with the
roles of evidence and justification, especially in increasingly technological
environments. Using dynamic geometry software, students can quickly generate
and explore a range of geometric examples. If they have not learned the
appropriate uses of proof and mathematical argumentation, they might argue
that a conjecture must be valid simply because it worked in all the examples
they tried. Despite the possibility of students' developing such a misconception,
in a classroom in which students understand the roles of experimentation,
conjecture, and proof, being able to generate and explore many examples
can result in deeper and more-extended mathematical investigations than
might otherwise be possible. The following hypothetical example illustrates
how students might investigate relationships in a dynamic geometry environment
and justify or refute conclusions.
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The students are asked to draw a triangle, construct a new triangle by joining the midpoints of its three sides, and calculate the ratio of » the area of the midpoint triangle to the area of the original triangle (see fig. 7.13a). As they drag one vertex to create many different triangles, the students notice that the ratio of the two areas appears to remain constant at 0.25.
Berta agrees with Jake's answer and thinks she can show that it must be true. She explains how she has extended the midlines and the sides of the triangles to form three pairs of parallel lines and is now able to determine many pairs of congruent angles. She reasons, using parallelism, that the corresponding sides are congruent and determines that the three small triangles formed at the vertices of the original triangle are congruent by angle-side-angle. She is confident that the midpoint triangle should be congruent to the other three, but when the teacher asks her how she can be sure, she is unable to give an explanation. The teacher asks one question: "Do you know anything about the sides of that triangle?" Her friend Dawn quickly notes that all its sides are shared with the sides of the other three, which indicates that it would have to be the same size.
Hope has a somewhat different way of looking at the situation. She notices that
the lengths of the corresponding sides of the midpoint triangle
and the original triangle are in a ratio of 1:2, so they must be
similar. Thus, the area of the midpoint triangle must be one-fourth
the area of the original on the basis of the class's earlier observation
that the areas of similar triangles are related by the square of
their scale factor. The teacher asks the class to think about the
relationship between Hope's method and Jake's method. |
p. 312
Applied problems can furnish both rich contexts for using geometric ideas and practice in modeling and problem solving. For example, right-triangle trigonometry is useful in solving a range of practical problems. Teachers can introduce students to problems such as the following, which is adapted from Hamilton and Hamilton (1993):
People working in the building trades sometimes need to divert their construction around obstacles. Although they often use simple offsets of 90°, 45°, or 30°, other angles are sometimes needed in areas where space is limited.
A construction worker needs to reroute an underground pipe in order to avoid the root systems of two trees. She needs to raise the path of her pipe 23 inches over a distance of over 86 inches [see fig. 7.14], and then continue on a course parallel to that of the original pipe. What angles should she cut in order to accomplish this?
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When a section of pipe is cut at an angle, the cross section is an ellipse. If one of the two resulting pieces is rotated 180 degrees about the axis going through the center of the pipe and then repositioned, the elliptical cross sections match each other, so the two pipes can be joined smoothly [see fig. 7.15].
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A second cut will be needed to send the pipe on in the original direction, as in [figure 7.14]. The angles markedare equal in measure, as are the angles marked
. In order to proceed, the pipe fitter needs to find the two angles at which to cut the materials as well as the length C of the connecting piece.
To find and , students need to
observe that 2 + 
= 180°,
2 + (90° – ) = 270°,
and tan() = 23/86.
Working through this kind of problem can help students develop visualization
skills and see how what they have learned can be applied in meaningful
contexts.
Geometric problems can be presented and approached in various ways. For example, many problems from Euclidean geometry, such as showing that the medians of any triangle intersect at a point, can be » approached through coordinate geometry. Although it is possible to use a different variable for the coordinates of each vertex (especially if a CAS package is effectively used), a "without loss of generality" argument can be used to lower the level of symbolic complexity. This type of argument relies on conveniently placing a coordinate plane over a general triangle (or other figure), often so that the coordinate axis coincides with a side of the triangle. By making clever choices about naming the coordinates of the vertices, taking care, of course, to be sure that the choices do not introduce unintended conditions, the calculations can be quite reasonable. Once they have obtained a representation like that in figure 7.16, students could determine the equations of two of the medians, find the point at which they intersect, and show that the third median passes through that point. Although proofs of this kind can be difficult for high school students, grappling with them may stimulate growth in students' understanding of geometry, algebraic variables, and generality.
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for –3 
x 3. With polar coordinates,
these pairs are represented more simply as (3, ) for 0 2
, where is measured in
radians. Students should be able to explain why both of these forms describe
the points on a circle. The polar-coordinate representation is simpler
in this example and may be more useful for solving certain problems.
Middle-grades students should have had experience with such basic geometric transformations as translations, reflections, rotations, and dilations (including contractions). In high school they will learn to represent these transformations with matrices, exploring the properties of the transformations using both graph paper and dynamic geometry » tools. For example, students who are familiar with matrix multiplication could be introduced to matrix representations of transformations through tasks such as the one in figure 7.17.
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3 matrix for convenience:
They might solve the resulting set of equations, showing that the transformation matrix is
Alternatively, they might observe that they need to find an M so that
and explore various possibilities until observing that
Other transformations with easily accessible matrix representations include reflections
over either axis and about the line y = –x,
rotations about the origin that are multiples of 90 degrees, and
dilations from the origin. Students should understand that multiplying
transformation matrices corresponds to composing the transformations represented.
They should also understand that transformations have many practical applications.
Creating and analyzing perspective drawings, thinking about how lines or angles are formed on a spherical surface, and working to understand orientation and drawings in a three-dimensional rectangular coordinate » system all afford opportunities for students to think and reason spatially. With the expanding role of computer graphics in the workplace, students will have increased needs and opportunities to use visualization as a problem-solving tool. Schooling should provide rich mathematical settings in which they can hone their visualization skills. Visualizing a building represented in architectural plans, the shape of a cross section formed when a plane slices through a cone (a conic section) or another solid object, or the shape of the solid swept out when a plane figure is rotated about an axis become easier when students work with physical models, drawings, and software capable of manipulating three-dimensional representations.
Geometric relationships explain procedures used by artists for drawing in perspective (see Smith [1995]), as demonstrated by the following perspective problem adapted from Consortium for Mathematics and Its Applications (1999, pp. 65–67):
An artist wants to draw a set of evenly spaced telephone poles along the side of a straight road, starting with two telephone poles as shown in figure 7.18a below. Where should the third telephone pole be placed so that it appears as far from the second as the second is from the first?
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The midpoint of the second telephone pole is also the center of the rectangle whose sides are the first and third telephone poles (fig. 7.18c). Thus the line from the top of the first telephone pole through the midpoint of the second pole intersects the ground at the bottom of the third telephone pole (fig. 7.18c); the top of the third pole can be located similarly. Finally, removing the lines used in the construction yields the » desired perspective drawing (fig. 7.18d). The process can be continued to locate other telephone poles along the same line.
Although problems formulated for mathematics classes are typically stated with great precision, underspecified problems that students need to formulate clearly for themselves also play an important role. The following problem (Keynes 1998, p. 109) draws on students' knowledge of geometric and trigonometric relationships and on their spatial-visualization skills. The way it is posed compels students to determine what additional information is needed, an important aspect of working problems in real-world contexts.
You are installing track lighting in an old warehouse that is being remodeled into a restaurant. The lights can adequately illuminate up to 15 feet from the bulbs and, at that distance, illuminate a circle with a 6-foot diameter. Figure out where to place the tracks and the bulbs to provide for maximum illumination of the customer area.
Students might compile a list of questions such as these: Is the ceiling flat or vaulted? What is the height of the ceiling? What is the square footage of the customer area? A discussion of this type furthers students' abilities to solve real problems, which are typically much more open-ended than those found in textbooks.
The following problem comes from discrete mathematics. Vertex-edge graphs can be used to find optimal solutions to problems involving paths, networks, or relationships among a finite number of objects. This example, adapted from Coxford et al. (1998, p. 326), illustrates these ideas.
Seven small towns in Smith County are connected to one another by dirt roads, as shown in the diagram in figure 7.19. (The diagram depicts only the beginnings, ends, and lengths of the roads. The roads may be straight or curved.) The distances are given in kilometers. The county, which has a limited budget, wants to pave some roads so that people can get from every town to every other town on paved roads, either directly or indirectly, but they want to minimize the total number of kilometers paved. Find a network of paved roads that will fulfill these requirements.
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The diagram in figure 7.19 is a vertex-edge graph modeling the road network. The goal is to find a subnetwork of the given network that is connected, contains no circuits, includes all the cities (vertices), and minimizes the sum of the distances represented by each edge in the network. Such a network is called a minimal spanning tree. There are » many ways to approach this problem, and students should be encouraged to share as many different approaches as possible with the class. Students can and will approach this problem in many ways, many of which will match the standard, formal solution methods. Students should first be given the opportunity to think about the characteristics of a paved road network that will satisfy the given requirements. This will often lead them to formulate the definition of a minimal spanning tree for themselves. Then they can begin to find a solution systematically. For example, students often proceed as follows: First, choose the shortest edge (AB). Then choose the shortest remaining edge (BD). Continue in this way, but never choose an edge that closes a circuit. One solution found by using this method is AB, BD, EF, BF, FG, CD (total length = 88).
Another commonly tried method is to start at A, then go to the "nearest neighbor," then the nearest neighbor from there, and so on. This leads to AB, BD, DC, and then a dead end, since moving on from C creates a circuit. Students might try to resolve this dead-end problem by starting at a different vertex, or they might try to modify their method by finding the nearest neighbor to any vertex that has already been reached (and, as always, not creating a circuit). This latter approach leads to AB, BD, DC, BF, EF, FG (total length = 88). As students try these different methods, they should be encouraged to fully specify their methods as algorithms, compare algorithms with other students, and consider which algorithms produce the desired results and which seem to be easier or more efficient. As it turns out, these two solution methods commonly used by students are the two most commonly used standard algorithms, namely, Kruskal's algorithm and Prim's algorithm.
Students should learn how to formulate and apply other vertex-edge-graph models to solve problems. For example, they can use critical paths to optimally schedule large projects like a school dance or a construction project. Using each edge exactly once (an Euler path), students can plan an optimal snow-plowing route or an optimal layout for moving people efficiently through a museum. Using each vertex exactly once (a Hamiltonian path), students can find an optimal route for collecting money from ATM machines or an optimal path for a robot to follow in a manufacturing plant. Vertex-coloring methods can be used to solve problems that involve conflict management, such as optimally assigning frequencies to radio stations or scheduling committee meetings.
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