Classroom Case Studies, 3-5 Part C: Problems That Illustrate Geometric Reasoning (55 minutes)

As you look at the next set of problems, answer these questions:

1. What is the geometry content in this problem?
2. What skills do students need to work through this problem? What skills will this problem help them develop for later work?
3. What level of geometric thinking is expected of students in the problem? Does it ask students to bridge levels?
4. What other questions might extend students’ thinking about the problem?
5. Describe a lesson that you could develop based on the content of this problem.

Problem C4

Use any materials that allow you to construct the following shapes. Some may not be possible. If you think one of the constructions is impossible, say what makes you think so.

1. Make a four-sided shape with two opposite sides the same length but not parallel.
2. Make several six-sided shapes, some with one pair of sides parallel, some with two pairs of sides parallel, some with three pairs of sides parallel, and some with no pairs of sides parallel.
3. Make some shapes with all 90° angles. Make shapes with this property and three, four, five, six, and seven sides.

Problem C5

List all the properties of a rectangle that you can think of. From your list, choose sets of two or three statements that you think would make a good definition for a rectangle. Is there more than one possible definition? Note 8

Problem C6

Begin with a convex polygon with a given number of sides. Connect two points on the figure to form two new polygons. What is the total number of sides in the two resulting polygons?

Note 6
It’s difficult to identify the important content and how students might approach an activity without actually doing the mathematics yourself. These are, for the most part, short problems and activities. Allow yourself time to work through the mathematics, even briefly, before going on to answering the other questions.

Note 7

Additional questions you might explore (or ask students to do): What is the relationship between the right and left views? Front and back views? Will that always happen, or is there something special about this building? If you were to provide the minimum information for someone to copy the building, what would you choose? (Think about the similarity between this and questions about the minimum information necessary to determine if two triangles are congruent.) Can you build two different buildings with the same set of five views? (And how do you decide if two buildings are different?)

Note 8

You may want to review our work with creating and understanding definitions in Session 3.

Problem C1

1. This problem is mostly about visualization and relating three-dimensional objects to two-dimensional representations.
2. By building three-dimensional objects, students strengthen their spatial sense and their intuitive knowledge of shapes and their properties. Two-dimensional representations help their abstract thinking about geometric properties. It also helps students with visualization and reasoning about 3D figures, which are an important part of high school geometry. Visualization and going from 2D to 3D (and vice versa) were important parts of this course.There are no real prerequisites to working through this problem. Students should have some familiarity with the blocks as building materials and with the idea of two-dimensional drawings representing three-dimensional figures. This problem helps prepare students for later work, when they must reason about drawings, drawing conclusions about figures based on information in a sketch.
3. This problem represents level 0 (informal exploration of shapes and visualization) and level 1 thinking (the drawings of squares represent cubes in which relative position, not size, matters; this requires analyzing skills on the students’ part). There could also be possible bridges to level 2 (using “if-then” thinking to help build the tower).
4. Ways to extend students’ thinking include the following: Give a set of views that can create more than one possible structure (think of a “hole” somewhere in the middle that is undetectable by any of the views provided). Ask students to create different structures with the same front view. Ask students to eliminate extra information, realizing that the left and right views are mirror images, as are front and back, so really just three views provide the required information.
5. One possible lesson: Give groups of students sets of five views to build from. Have “answer cards” (with the structure pictured) ready for them to check their work. If they are successful, they can move to a more challenging piece. Move towards a game where students try to build the structures with the least number of views needed.

Problem C2

1. This problem deals with congruence and similarity, as well as (potentially) leading to thinking about measures like area and perimeter. Students are required to reason through different properties in order to examine which ones will ensure congruence of the smaller shapes. Students might also notice that the area and the perimeter of the smaller shapes are, in each case, equal. This may be helpful to students when finding the solution, but this is not enough to be the test for congruence. One possible solution follows:
   Note that with the square, it could be cut into four congruent rectangles by making equally spaced cuts parallel to one pair of sides. This is certainly a solution to the problem posed, though the shapes in this case are not similar to the original. Students will also be looking at the properties of the smaller shapes and comparing them to those of the original, larger shapes, and deciding what makes the two figures similar. They may notice, for example, that the four smaller triangles all have the same angles as the larger one, as well as the sides that are in proportion with the sides of the larger one.
2. The openness of the problem prepares students for thinking more creatively about geometric problems. The problem is similar to some of the cutting and dissection work from Session 5 of this course. This kind of problem prepares students for working more formally with ideas of congruence and similarity later. (If they have a solid grasp of the concept through informal activities, it is much easier to formalize these notions and build up to mathematical proof.) It also helps extend students’ visualization and problem solving. With the first shape in particular, they will probably have to think creatively and try several different ideas before getting it right.
3. This represents level 1 thinking (seeing figures as congruent or similar, even if they are differently oriented).
4. To extend students’ thinking, you might make a table comparing perimeters and areas of the smaller and larger figures. Several ideas should come up:
   • If shapes are congruent, they must have the same perimeters and the same areas, so it’s only necessary to check one.
   • If four congruent shapes fit inside the larger, each must contribute 1/4 to the area. There’s no need to measure here.
   • If the four triangles really are congruent, for example, the sides must each be half as long as the original. (Two small sides make up a large side on each side of the triangle.) Again, no need to measure; if each side is 1/2 as long, the total perimeter must be 1/2 as much.
5. One possible lesson would involve starting with a review of the ideas of congruence and getting a definition from students (for example, two shapes that are identical, so you can fit one on top of the other, etc.). Next, hold up a standard sheet of paper and ask for several ways to cut that sheet into two congruent pieces. Follow through with each suggestion (cut parallel to the short sides through the midpoints, parallel to the long sides through the midpoint, and along a diagonal to form two congruent triangles). In each case, use the “fit on top of each other” test to make sure the result is two congruent figures. Give students equilateral and L-shaped figures, and ask them to cut those into two congruent pieces. Let them share their results. Finally, pose the challenge problem of cutting the figures as described, with students making a poster or display of their solution.

Problem C3

1. This problem requires building a definition from examples and then applying the definition to other cases. Here students will be working with the definition of equilateral polygons, or more specifically, quadrilaterals. (Polygons that have equal sides are called equilateral.) Next, they will look at quadrilaterals that do not have equal sides, and finally, they will extend the definition onto other polygons that may or may not share that property. For example, regular pentagons or regular triangles would share the property of having equal sides.
2. This requires focusing on properties of figures and reasoning about them, and in this way it is similar to the polygon classification in Venn diagrams from Session 3. Creating definitions and understanding definitions that have been created by others is an essential part of learning and doing mathematics. A key part to understanding a mathematical definition is to create both examples and non-examples for yourself. (See Session 3 for more on creating and understanding mathematical definitions.) Activities like this encourage students to play with mathematical ideas and develop these important habits for later work.
3. This problem represents level 1 thinking (the focus is on properties rather than specifics of a figure; students must consider all the shapes in a class of shapes, rather than a single one) and some level 2 thinking (if this shape has the property, then it can’t be this property; if I need to make decisions about triangles and pentagons, then the property can’t depend on number of sides).
4. Ways to extend students’ thinking include similar activities with more complicated (or less familiar) shared properties; asking students to give a name (either standard or invented) to the property and write a definition of it; and providing students with a written definition for which they create examples and non-examples.
5. One lesson might go like this: Choose an attribute that several but not all students in the class have in common; for example, wearing sneakers or having long hair. Select several but not all of the students with that attribute to come to the front of the room. Tell the class that their job is to be detectives; that all of these students have something in common, and their job is to figure it out. Next, bring up several students who do not fit the profile, and announce that all of those students do not have the attribute in question. Finally bring up five or so students, some with the attribute and some without. Ask students to silently write on their papers the names of students they think “belong” to the original group, and those who do not. Have the group return to their seats, where they can make their guesses as well. Finally, ask for a show of hands for who thought each student belonged, and why — what was the property in question? When this introductory activity is finished, tell students that they will now play detective with shapes. Pass out several worksheets like the one provided. You might add places where students can do the following:
   • Write down a name or description of the property they detected
   • Draw two other examples and non-examples of their own
   • Create their own detective worksheet to share with a partner

Problem C4

1. This problem focuses on properties and thinking about classes of figures. Students also have the opportunity to develop their thinking about the relationship between the number of sides and angles within a shape. For example, in part (c) they will discover that the only shape possible, if all the angles are 90°, will be one with four sides. This is similar to other classification problems and activities, but differs in that the properties (or the combinations of properties) are less familiar, so they require more creative thinking on the part of students.
2. To work through this lesson, students should already know all of the vocabulary. The challenge should be creating figures with the properties, not understanding what the properties are. This prepares students for thinking about some difficult mathematical ideas, like how you create examples to show something is possible and how you convince yourself and others that something is impossible (a much harder task). Both of these are key parts of mathematical proof. The first is an example of proof by construction (proving that something exists by creating it). The second has a long and famous history in mathematics, including impossibilities like creating a formula like the quadratic formula to solve 5th-degree polynomials (equations with an x5 in them); trisecting a general angle with straightedge and compass; and “squaring the circle” — creating a square with the same area as a given circle, again using straightedge and compass.
3. This is a level 1-focused problem (it requires students to examine a particular property within an entire class of shapes). The addition of combinations of properties that are impossible, and asking students to justify how they know it is impossible, bridges this into a level 2 task as well.
4. To extend students’ thinking, you can ask them to explain why some of the constructions are impossible — what about a pentagon prevents you from making one with five right angles? This is essentially creating a proof that the construction is impossible. You can also ask students to salvage the bad constructions — if you can’t make a pentagon with five right angles, what’s the best you can do? Can you make one with two right angles? Three? Four?
5. One possible lesson: Hand out the materials students will be using for construction. (Possibilities include toothpicks and gumdrops, linkage strips, or other construction materials with sticks and connectors.) Allow students a few minutes to explore the materials, then pose the first construction challenge to the whole class. Ask several students to hold up their solutions and explain why they meet the construction criteria. Pose a second construction challenge. (Make sure the first two use familiar vocabulary and are both possible to do.) Finally, pose one of the impossible challenges and give students several minutes to work on it. Eventually, someone will exclaim that it can’t be done. Ask the class who thinks it is possible and just hard to do, and who thinks it is impossible. Ask students to explain their opinions and try to convince each other. When the class (with some help from you) understands that sometimes there’s no shape that satisfies the requirements, hand out several more construction challenges. Tell them that some are possible and some are impossible. Their job is to solve the possible ones, and to draw one (or more) example of what they created. For the impossible ones, they should write that it’s impossible, and try to explain what goes wrong.

Problem C5

1. This task is focused on definitions and the equivalence of different sets of properties. For example, some properties listed may include the following:
   • Rectangles are quadrilaterals; they have four sides.
   • All angles must be 90°.
   • Opposite sides are parallel.
   • The sum of the angles is 360°.

   None of these alone is sufficient to define a rectangle; to do that, the statement must be true of every rectangle, and not true of anything that’s not a rectangle. All of the statements above are true of every rectangle, but you can also create non-rectangular shapes that satisfy them.

2. To tackle this task, students must first be very comfortable with the shape in question (here it’s a rectangle, but the activity is easily adapted to anything else). They should also have some experience with activities like the one in Problem C3, since in this case they will have to create their own examples and non-examples to construct an adequate definition. This activity prepares students for more formal mathematical reasoning, both in thinking about definitions and also in proof, where careful (and flexible) use of equivalent statements is crucial.
3. This activity requires level 1-type of thinking (analyzing and reasoning about different properties for a whole class of shapes; i.e., rectangles). Moving into level 2 would involve directing students toward thinking about the equivalence of the listed properties and focusing on the information necessary to define a rectangle. The definition of a rectangle as a quadrilateral with all four angles equal to 90°. would be an example. This problem is very similar to the definition activities we did in Session 3.
4. To extend students’ thinking, you could do a similar activity with a more complicated figure. (Defining a circle, for example, is a more challenging task, as is defining any sort of three-dimensional figure.) You can also ask students to focus on the minimal information necessary. (For example, “a quadrilateral with three right angles” is enough — the fourth angle is forced to be 90°. as well. Similarly, “a parallelogram with one right angle” is enough.)
5. One possible lesson: As a whole class, brainstorm several properties of rectangles. Generate a long list on the board, and really push students to offer obvious (four sides) and not as obvious (congruent diagonals) properties. You can ask leading questions (“What about the diagonals?”) and give them time to think and sketch as you generate the list. When the list is fairly long, go item by item and put stars next to things that students think are true only of rectangles and nothing else. (Don’t worry if they are not correct; refining their ideas is part of the activity.) Explain that a good definition will describe every rectangle in the world and nothing else. Ask students to spend several minutes drawing figures that fit the properties with stars, trying specifically to draw things that are not rectangles. End this part of the activity by refining which statements have stars by them based on what students have found. You can end the activity by assigning groups of students to come up with alternate definitions based on combining the remaining statements. Their job is to combine two or three of the non-starred statements into something that describes only rectangles. You can end the activity by writing several of the definitions on the board. You might want to talk about why we want alternate definitions (sometimes it’s convenient to think about shapes in different ways, and it’s useful to have definitions that relate to angles, definitions that relate to sides, definitions that combine these, to make our work easier). You can also provide some standard definitions from textbooks and dictionaries, for students to see how their definitions compare, and to see that even the “real” definitions will vary from source to source.

Problem C6

1. On the face of it, this seems like a straightforward task of relating inputs (number of sides of a polygon) and outputs (the total number of sides of the resulting two figures) of a function (drawing a segment). Once you investigate it, though, you find that it is more complicated than it appears on the surface. This is a problem in examining extreme cases (minimum and maximum number of sides possible as a result), and in considering what causes the extreme cases (connecting vertex to vertex, side to side), and so on. Further examination reveals that there are only three possible cases (connecting vertex to vertex, side to vertex, and side to side), and the results begin to emerge into a pattern.
2. There is no prerequisite knowledge necessary other than the ability for students to identify and count the number of segments on a polygon. To be successful, though, students should have some experience with open-ended problems, creating and testing cases, creating tables to track data, and working with non-obvious number patterns.This task prepares students for thinking through a complicated situation, for relating different areas of mathematics (functions, number patterns, and geometry), and for explaining their results in informal arguments.
3. In this case, students will be able to explore the problem and come up with their own generalizations. As a result, they will be able to construct an informal argument that supports their explorations. On the surface, this appears to be a level 1 task, with an easy entry for students who might be struggling to move towards deduction. It does, however, bridge to level 2 thinking.
4. To extend student thinking, you could lead into activities like the cross sections of three-dimensional figures (see Session 9). Questions here could include both what cross-sectional shapes are possible and the total number of faces of the resulting two figures. (This is a much more difficult task than the one presented in two dimensions!) Another extension is thinking about how segments can split regions apart more generally. For example, start with a circle: One line can split it into two regions. Two lines can split it into either three or four regions (how?). What are the possibilities for three, four, or five lines? This is an interesting problem, because if you focus on the maximum number of regions, there is a rather simple pattern at first, but it breaks down when you look at the case of six lines!
5. One possible lesson: Draw several triangles on the board. Select students to come up and draw a segment across each triangle. When they have finished, as a class, count up the total number of sides of the resulting two figures. (It needs to be clear to everyone that the drawn-in segment is counted twice — once for each of the new figures.) If students have not done so themselves, make sure that some of the segments go side to side and some go side to vertex. You can draw another triangle, say you have another way to create a segment, and demonstrate it. When the task is clear — finding all the possible ways to draw a segment and then counting up the total number of sides — pass out sheets to each group of students. The sheets should have several copies of each starting shape for students to work with and a place for them to keep track of their results. To wrap up the lesson, write the names of shapes on the board (triangle, quadrilateral, pentagon, hexagon) and, based on the students’ data, fill in the totals that were possible for each one. Allow for explanation and examples, for example, if one group found something that another group did not. Try to draw out of students a description of the three possible segments and how they affect the total number of sides.