### Connections Standard for Grades 9–12

 Instructional programs from prekindergarten through grade 12 should enable all students to— recognize and use connections among mathematical ideas; understand how mathematical ideas interconnect and build on one another to produce a coherent whole; recognize and apply mathematics in contexts outside of mathematics.

When students can see the connections across different mathematical content areas, they develop a view of mathematics as an integrated whole. As they build on their previous mathematical understandings while learning new concepts, students become increasingly aware of the connections among various mathematical topics. As students' knowledge of mathematics, their ability to use a wide range of mathematical representations, and their access to sophisticated technology and software increase, the connections they make with other academic disciplines, especially the sciences and social sciences, give them greater mathematical power.

#### What should connections look like in grades 9 through 12?

Students in grades 9–12 should develop an increased capacity to link mathematical ideas and a deeper understanding of how more than one approach to the same problem can lead to equivalent results, even though the approaches might look quite different. (See, e.g., the "counting rectangles" problem in the "Problem Solving" section in this chapter.) Students can use insights gained in one context to prove or disprove conjectures generated in another, and by linking mathematical ideas, they can develop robust understandings of problems.

The following hypothetical example highlights the connections among what would appear to be very different representations of, and approaches to, a mathematical problem.

 The students in Mr. Robinson's tenth-grade mathematics class suspect they are in for some interesting problem solving when he starts class with this story: "I have a dilemma. As you may know, I have a faithful dog and a yard shaped like a right triangle. When I go away for short periods of time, I want Fido to guard the yard. Because I don't want him to get loose, I want to put him on a leash and secure the leash somewhere on the lot. I want to use the shortest leash possible, but wherever I secure the leash, I need to make sure the dog can reach every corner of the lot. Where should I secure the leash?"
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After Mr. Robinson responds to the usual array of questions and comments (such as "Do you really have a dog?" "Only a math teacher would have a triangle-shaped lot—or notice that the lot was » triangular!" "What type of dog is it?"), he asks the students to work in groups of three. All their usual tools, including compass, straightedge, calculator, and computer with geometry software, are available. They are to come up with a plan to solve the problem.

Jennifer dives into the problem right away, saying, "Let's make a sketch using the computer." With her group's agreement, she produces the sketch in figure 7.36.

 Fig. 7.36. Jennifer's computer-drawn sketch of the "dog in the yard" problem

As Mr. Robinson circulates around the room, he observes each group long enough to monitor its progress. On his first pass, Jennifer's group seems to be experimenting somewhat randomly with dragging the point D to various places, but on his second pass, their work seems more systematic. To assess what members of the group understand, he asks how they are doing:

 Mr. R: Joe, can you bring me up-to-date on the progress of your group? Joe: We're trying to find out where to put the point. Jeff: We don't want the point too close to the corners of the triangle. Jennifer: I get it! We want all the lengths to be equal! They all work against each other.

Before moving on to work with other groups, Mr. Robinson works with the members of Jennifer's group on clarifying their ideas, using more-standard mathematical language, and checking with one another for shared understanding. Jennifer clarifies her idea, and the group decides that it seems reasonable. They set a goal of finding the position for D that results in the line segments DA, DB, and DC all being the same length. When Mr. Robinson returns, the group has concluded that point D has to be the midpoint of the hypotenuse, otherwise, they say, it could not be equidistant from B and C. (Mr. Robinson notes to himself that the group's conclusion is not adequately justified, but he decides not to intervene at this point; the work they will do later in creating a proof will ensure that they examine this reasoning.)

 Mr. R: What else would you need to know?

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 Jeff: We're not sure yet whether D is the same distance from all three vertices. » Jennifer: It has to be! At least I think it is. It looks like it's the center of a circle.

Small-group conversations continue until several groups have made observations and conjectures similar to those made in Jennifer's group. Mr. Robinson pulls the class back together to discuss the problem. When the students converge on a conjecture, he writes it on the board as follows:

Conjecture: The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices of the triangle.

He then asks the students to return to their groups and work toward providing either a proof or a counterexample. The groups continue to work on the problem, settling on proofs and selecting group members to present them on the overhead projector. As always, Mr. Robinson emphasizes the fact that there might be a number of different ways to prove the conjecture.

Remembering Mr. Robinson's mantra about placing the coordinate system to "make things eeeasy," one group places the coordinates as shown in figure 7.37a, yielding a common distance of . Alfonse, who is explaining this solution, proudly remarks that it reminds him of the Pythagorean theorem. Mr. Robinson builds on that observation, noting to the class that if the students drop a perpendicular from M to AC, each of the two right triangles that result has legs of length a and b; thus the length of the hypotenuses, MC and MA, are indeed .

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Jennifer's group returns to her earlier comment about the three points A, B, and C being on a circle. After lengthy conversations with, and questions from, Mr. Robinson, that group produces a second proof based on the properties of inscribed angles (fig. 7.37b). » Pedro presents his group's solution showing how they constructed a rectangle that includes the three vertices of the right triangle (fig. 7.37c) and reasoned about the properties of the diagonals of a rectangle. Anna presents a solution using transformational geometry (figure 7.37d). Since M and M' are the midpoints of and , respectively, the triangle MAM' is similar to the triangle BAB', with each of the sides of the smaller triangle half the length of the corresponding side of the larger triangle. The same relationship holds for triangles BMC and BAB'. Using this fact and the fact that BAB' is isosceles (since reflects onto ), Anna shows that triangle MAM' is congruent to triangle CMB, from which it follows that CM and MA are the same length.

 Fig. 7.37. Diagrams corresponding to four proofs of the midpoint-of-hypotenuse theorem

Mr. Robinson congratulates the students on the quality of their work and on the variety of approaches they used. He points out that some basic mathematical ideas such as congruence were actually part of the mathematics in a number of their solutions and that some of their thinking, such as Alfonse's comment about the Pythagorean theorem, highlighted connections to other mathematical ideas. Taking a step backward to reflect, the students begin to see how different approaches—using coordinate geometry, Euclidean geometry, and transformational geometry—are all connected. Mr. Robinson notes that it is good to have all these ways of thinking in their mathematical "tool kit." Any one of them might be the key to solving the next problem they encounter.

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Although the students learned a great deal from working on the problem, the class was not yet finished with it. Mr. Robinson had selected this problem for the class to work on because it supports a number of interesting explorations and because the students would be exploring the properties of triangles and circles as they worked on it. And, indeed, as the students worked on the problem, they remarked that they were "seeing circles everywhere." (The following discussion is inspired by Goldenberg, Lewis, and O'Keefe [1992].) »

One group decides to look at the set of all the right triangles they can find, given a fixed hypotenuse. A group member starts by constructing a right triangle with the given hypotenuse and then dragging the right angle (fig. 7.38a). Another group decides to fix the position of the right angle and look at the set of right triangles whose hypotenuses are the given length (fig. 7.38b). They observe that the plot of the midpoints of the hypotenuses of the right triangles appears to trace out the arc of a circle. At first the students are ready to dismiss the circular pattern as a coincidence. But Mr. Robinson, seeing the potential for making a connection, asks questions such as, "Why do you think you get that pattern?" and "Does the circle in your pattern have anything to do with the circle in Jennifer's group's solution?" As the groups begin to understand Mr. Robinson's questions, they begin to see the connections among the circles in their new drawings, the definition of a circle, and the fact that their problem deals with points that are equally distant from a third point.

 Fig. 7.38. Dynamic representations of right triangles (from Goldenberg, Lewis, and O'Keefe [1992, p. 257])

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 Mr. Robinson adds a final challenge for homework: can the students connect this problem (or problems related to it) to real-world situations or to other mathematics? The students create posters illustrating the mathematical connections they see. Most of the posters depict situations similar to the original problem in which something, for some reason, needs to be positioned the same distance from the vertices of a right triangle. One group, however, creates an experiment that they demonstrate for the class in one of the dark, windowless rooms in the building. They put on the floor a large sheet of white chart paper with a right triangle drawn on it, place candles (all of the same height) at each vertex, and stand an object shorter than the candles inside the triangle. The class watches the shadows of the object change as one of the group members moves it around inside the triangle. The three shadows are of equal length only when the object is placed at the midpoint of the hypotenuse—a phenomenon that delights both Mr. Robinson and his students. This activity concludes the discussion of right triangles, but it is far from the end of the class's work. Mr. Robinson reminds the students of the problem that started their discussion and asks them how the problem might be extended. "After all," he says, "not all backyards have right angles or are triangular in shape." This comment sets the stage for abstracting and generalizing some of their work—and for making more connections. »

#### What should be the teacher's role in developing connections in grades 9 through 12?

The story of Mr. Robinson's classroom indicates many of the ways in which teachers can help students seek and make use of mathematical connections. Problem selection is especially important because students are unlikely to learn to make connections unless they are working on problems or situations that have the potential for suggesting such linkages. Teachers need to take special initiatives to find such integrative problems when instructional materials focus largely on content areas and when curricular arrangements separate the study of content areas such as geometry, algebra, and statistics. Even when curricula offer problems that cut across traditional content boundaries, teachers will need to develop expertise in making mathematical connections and in helping students develop their own capacity for doing so.

One essential aspect of helping students make connections is establishing a classroom climate that encourages students to pursue mathematical ideas in addition to solving the problem at hand. Mr. Robinson started with a problem that allowed for multiple approaches and solutions. While the students worked the problem, they were encouraged to pursue various leads. Incorrect statements weren't simply judged wrong and dismissed; Mr. Robinson helped the students find the kernels of correct ideas in what they had said, and those ideas sometimes led to new solutions and connections. The students were encouraged to reflect on and compare their solutions as a means of making connections. When they had done just about everything they were able to do with the given problem, they were encouraged to generalize what they had done. Rich problems, a climate that supports mathematical thinking, and access to a wide variety of mathematical tools all contribute to students' ability to see mathematics as a connected whole.