Classroom Case Studies, 6-8 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

If a dart has an equal chance of landing at any point on a circular target, is it more likely to land closer to the center or closer to the edge?

Problem C5

On a geoboard, you can make different shapes.

1. Make at least five shapes with four boundary pegs and no pegs inside. Find the areas of each of your figures.
2. Make at least five shapes with four boundary pegs and one peg inside. Find the areas of each of your figures.
3. Continue investigating other cases using different numbers of boundary and inside pegs. Can you find a rule for the areas of the figures?
4. Think of a way to explain why your rule works. Hint: What happens when you add a boundary point? How much area is added? What happens when you add an interior point? How much area is added?

Note 7

Problem C6

Fernando’s Frames offers a low-cost, do-it-yourself picture-framing option. To save money, you determine the shape of the frame and select and purchase the side pieces. Side pieces of various lengths are available. Corner fittings are free. Fernando helps you assemble the frame.

One day, Fernando’s friend Fred came into the frame shop. He asked for six side pieces, 2, 3, 4, 5, 6, and 7 inches long. He said he could take any three of those side pieces and make a triangular frame.

“Don’t be so sure of that!” said Fernando.

What is the probability that any three of these side pieces will form a triangular frame?

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

To make a shape on a geoboard, think about using a rubber band around pegs. You may want to get a real geoboard and work with your colleagues to experiment and gather some data. Remember, you don’t want to just think about particular cases such as only examining squares or only examining shapes where the sides are parallel to the sides of the geoboard.

Problem C1

1. The content in this problem covers congruence, properties of figures, and also ideas like counterexample and testing conjectures. Students will be able to reason through different properties and examine which one will be sufficient to assure congruence. Unlike parts a-c, in part (d) they will discover that having the same area does not ensure congruence of the shapes. In other words, areas can be the same for non-congruent shapes, even if they belong to the same class of shapes.
2. To work through this problem, students should already understand the idea of congruence well and should be familiar with all of the shapes and properties named. In this course, we looked at some triangle congruence and similarity tests, and this is a similar activity for other kinds of figures. This prepares students both for moving into the triangle congruence tests and for working more with proof and counterexample. It helps them test conjectures because they have to come up with cases to test themselves.
3. This requires a combination of level 1 (thinking about properties of figures, reasoning about same class of shapes) and level 2 (thinking about minimal information necessary to define and test congruence; also, “if-then” thinking — e.g., if two shapes have the same area, they may or may not be the same shape, and consequently may or may not be congruent).
4. To extend students’ thinking, you could ask them to come up with their own congruence tests for some shapes (for example, circles or cubes) and to think creatively about different tests that would work. For example, if you had two boxes, and you had sand (to fill them), paper (to wrap around them), and string, what are some ways you could test the two boxes for congruence? What would be enough to convince you one way or the other?
5. One possible lesson: Begin with a short activity that reviews congruence as “shapes fitting on top of each other.” For example, give each student a sheet with three different pictures, and ask them to find their “buddy” — the student with three pictures on their page congruent to the ones on yours. Each student will need a sheet with three pictures. You can vary from eight or so pictures, changing the orientations and size of the pictures so that only two students (or three students, or however many you choose) have matching sheets. You can use this to form groups for the day, or just as a warm-up. Review the idea of congruence as “same size and shape,” but explain that sometimes it’s difficult or impossible to try to fit the shapes on top of each other to check for congruence. Pass out sheets with the congruence tests. Ask students to create examples for each congruence test. (For example, use grid paper to draw several different rectangles with the same area.) Then decide if all the examples they drew are indeed congruent. End the activity by discussing each test, asking students to provide their test cases, and talking about how to come up with good test cases. (Slanted orientations, long and skinny rectangles, and other “extreme cases” should be the focus.)

Problem C2

1. This task is focused on different types of “if-then” thinking. Some of the statements are based on definitions and are fairly straightforward. Others are about families of figures, and still others require more logical deduction; e.g., “If two rectangles have the same area, then they are congruent.”The geometric content is different depending on the statement students investigate. For example, the statement “If two triangles have the same perimeter, then they are congruent” is about congruence of triangles. The statement is false. The counterexample is that it is possible to have two triangles of the same perimeter that are not congruent — for example, triangles whose sides are 5, 5, 2 and 4, 4, 4. These are isosceles and equilateral triangles, respectively, and they are not congruent. A true statement may be the following: “If two triangles have all three sides the same length, then they are congruent.” A false statement may be as follows: “If two triangles have the same perimeters, then their areas are also the same.” Notice the content area is no longer the congruence of triangles, but rather the concept of area and perimeter of triangles.
2. To work through this task, students should be familiar with all of the terms used so that they do not struggle when coming up with examples to test. They should also have experience in other activities of coming up with examples and non-examples to fit definitions. This activity prepares students for work on mathematical proof. Here, students must decide on the veracity of a statement (a key first step in proof), and come up with counterexamples to false statements (an essential proof technique).
3. This problem is similar to the definition activities we did in Session 3. This is a level 2 task, with a focus on “if-then” thinking and simple deductions. Students come up with their own test cases and thus make generalizations, which they compare to the given statements and make further deductions.
4. To extend students’ thinking, you can vary the types of “if-then” statements used. You can also use activities like this to introduce and explore the idea of a converse (switching the “if” and “then” parts of a statement). Sometimes the converse of a statement is true, and sometimes not. Examples:
   • Statement: “If a polygon has three sides, then it is a triangle.”
     True converse: “If a polygon is a triangle, then it has three sides.”
   • Statement: “If a quadrilateral is a square, then all of its sides are the same length.”
     False converse: “If a quadrilateral has all sides the same length, then it is a square.”

   You can use non-mathematical situations to explore the idea of converse further, and perhaps make it even clearer. Examples:

   • Statement: “If you live in New York City, then you live in New York state.”
     False converse: “If you live in New York state, then you live in New York City.”
   • Statement: “If an animal has feathers, then it is a bird.”
     True converse: “If an animal is a bird, then it has feathers.”
5. One possible lesson: Choose one of the false statements; for example, “If two triangles have the same perimeter, then they are congruent.” Tell students they have five minutes to decide if it is true or false and to find a way to convince you and their classmates that they are right. Encourage them to draw or write down their ideas. When it seems most students have decided, ask one or two students to answer true or false and explain how they decided. Introduce or review the word “counterexample” when a student shows an example of two triangles with the same perimeters that are not congruent. Then ask students to, on their own, come up with a statement like that one, but that they are sure is true. Ask a few students to share their statements, and how they know they are right. (For example, students might say, “If two triangles have the exact same side lengths, then they are congruent,” because that is a triangle congruence test that they know. Or they might create the converse of the given statement, “If two triangles are congruent, then they have the same perimeter,” explaining that if they’re congruent, each of the three sides have the same length, so the total is the same.) Then pass out several sheets of paper. Each sheet should have just one “if-then” statement on the top, with room for students to draw examples and to write their conclusion. They are to decide if the statement is true or false and explain why.

Problem C3

1. The content in this problem covers properties of figures and congruence. It also requires visualization and thinking through possible cases.
2. To engage in the task, students should understand congruence and what a rectangle is.
3. This is a level 1 problem, with the possibility for extensions to level 2 thinking. The students begin by thinking about and analyzing all types of rectangles, not just one. They then move into “if-then” type of thinking. For example, if the two new shapes are to be congruent, then they must have all side lengths the same. It would be interesting here to explore why cutting a rectangle into two shapes with the same area would not be enough to ensure they are congruent shapes. Writing up a careful solution of all possible cases (including starting with special rectangles) and explaining why it’s a complete list would be a level 3 task.
4. The problem could be further extended by asking questions like, “I cut my rectangle into two congruent squares; what can you say about my original rectangle?” You could also think about equal area rather than congruence, and cutting into more than just two pieces.
5. One possible lesson: You could use this as an introduction to a longer activity. Hold up a regular sheet of paper and say: “I want to divide this into two congruent pieces. What can I do?” Take some suggestions from students, try each one, and test (by fitting the figures on top of each other) if it results in congruent pieces. Tell students that you want to divide the sheet of paper into two congruent squares and get their reactions. When they decide it is impossible, ask them on their own to come up with a rectangle that can be divided into two congruent squares. Then pass out several sheets of paper and scissors to each student. Their goal is to find every possible way to divide the rectangle into two, three, and then four congruent pieces. You could also have different students or groups work with different starting shapes to see if some figures give more options for results than others.

Problem C4

1. The content here is areas of circles (and parts of circles) and probability. The nice thing is that it’s a vague statement that students need to clarify. What does “closer” mean? Answering this question requires calculating which area is larger and thus would have a larger chance of being hit.In this case, the total area can be divided into two areas — closer to the center and closer to the edge. The area closer to the edge will be larger and thus more likely to be hit. Here’s how it can be done. Draw a circle. Draw another one inside of it, with the same center and a radius half as big. If the dart lands inside this smaller circle, it’s closer to the center. If it lands outside the smaller circle, it lands closer to the edge. The area of the smaller circle is (r2)/4. So the area of the other shape (it’s called an annulus) is r2 minus the little circle, or (3/4)r2. So it’s three times as big as the little circle, and three times as likely to be hit.
2. This is very similar to the multi-step geometric problem solving that will be asked of students in high school and beyond. It helps them practice skills of reasoning through difficult problem statements, drawing pictures of situations, and applying knowledge (areas of circles, for example) in unfamiliar contexts.
3. This is a level 2 task; it asks students to use their knowledge of calculating areas to draw conclusions about probability. This problem also asks students to apply knowledge of probability to solve a geometric problem.
4. Ways to extend this task include using different-shaped dartboards, creating a dartboard where points are assigned based on difficulty of hitting the region, and creating fair probability games. (For example, you want a dartboard divided into two regions — a circle and an annulus like above — but you want an equal probability of hitting each piece. Where should you divide the circle? Does it depend on the size of your starting circle?)
5. One possible lesson: You can use a magnetic dartboard or similar toy to start a game. Divide the class into two teams and have each team send a representative who will take turns throwing the darts. The rule is, if the dart lands closer to the center, team A gets a point. If it lands closer to the edge, team B gets a point. If it doesn’t hit the board, it’s a do-over. Let each team take five to 10 tries. The class will have to decide how to measure distance to the center and distance to the edge, though you can lead them to the idea of measuring along a radius that passes through that point. Total up the points, and ask if you think it was a fair game. Get lots of responses, and take some notes of students’ ideas on the board. Ask students what calculations they could make to decide if it really is a fair game, and guide the discussion to the idea of comparing areas. Review the area formula for a circle, and then give students some time to make whatever calculations they think are appropriate. End with a discussion in which students share their ideas and in which someone (a student or you) shows a picture like the one above and explains how to calculate the relevant areas. You can end there, or with the question of how to reposition the dividing line to make the game fair.

Problem C5

1. This problem contains information about areas, asking students to develop a formula that depends not on the kind of shape (i.e., how many sides it has), but instead on its configuration on a geoboard. The relationship that they discover is between the interior points, exterior points and the area of a shape. The formula, known as Pick’s theorem, follows:Let P be a lattice polygon. Assume there are I(P) lattice points in the interior of P, and B(P) lattice points on its boundary. Let A(P) denote the area of A. Then A(P) = I(P) + B(P)/2 – 1.So, for example, using this formula, the area of a square that includes four boundary pegs and no pegs inside will be A = 0 + (4 / 2) – 1 = 1. This is just what you would expect.
2. To engage in the task, students should have a solid understanding of forming different shapes on a geoboard and calculating their areas. They should be able to find the areas of long skinny triangles and of tilted squares, as well as the areas of shapes in standard positions. Students should also have experience gathering data, organizing it in a table, and generalizing from patterns.
3. The initial investigation is a level 2 task, asking students to come up with cases to check and develop rules like, “If it has four boundary points and one interior point, then the area is 2 no matter what the shape is.” Using the induction-like argument suggested in part 4 of the problem moves this to a solid level 3 task, where students must think to divide a bigger shape into smaller components, about which they already know the areas.
4. This problem moves students toward thinking about more complicated numerical patterns and problem situations, since there are two variables (boundary pegs and interior pegs) that need to be dealt with separately. It introduces students to the very important idea of controlling variables, allowing only one thing to change at a time so you can see how different changes affect the situation. (Note that the problem itself suggests ways for extending students’ thinking, especially in part 4, which moves them toward the idea of proof by induction.)
5. One possible lesson: On a demonstration geoboard or on the blackboard, draw several shapes and explain how to count boundary pegs and interior pegs. Then ask students to tackle part 1 of the problem (making shapes with four boundary pegs and no interior pegs). Give them several minutes for the task; then bring the class together to share results. They should have found that all of the shapes — triangles, squares, and parallelograms — had an area of exactly one. Make sure to show several different examples of each type of shape, including strangely oriented ones, to get students thinking about those types of figures. Ask students to conjecture whether shapes with four boundary pegs and one peg inside will have more or less area than the ones with just four boundary pegs. Also, will they all have the same area? Put some conjectures on the board, and then again ask the students to investigate it. Bring the class back together and discuss their results. Start a table on the board like this:
   

   B (Boundary Pegs)	I (Interior Pegs)	A (Area)
   4	0	1
   4	1	2

   Tell students that their job is to continue the table, create several examples of each case, and come up with a formula for the area based on boundary and interior pegs. Let students work for a long time on this task. Depending on how much structure they need organizing their work, you may want to provide them with a table in which numbers of boundary and interior pegs are filled in, so that it is set for them to only change one variable at a time. Alternately, you may just want to make that important strategy clear at the outset and leave it to students to organize the work themselves. Wrap up the activity by extending the table on the board (with students’ help) to several other cases, and writing a formula. Depending on the class, you may want to work with them on parts of the explanation suggested in part 4 of the activity.

Problem C6

1. The underlying content here is the triangle inequality, which states that if the sum of any two lengths is greater than the third length, then the three lengths will make a triangle. Again, there is some elementary probability involved. The problem requires students to think systematically about sets of three lengths and to find a way to know they have considered all of the possibilities. The total number of possible cases is 20. Seven combinations will not make a triangle, so the probability there is 7/20. Therefore, 13 combinations will make a triangle (for a probability of 13/20, or 65%).
2. To engage in the task, students should already know the triangle inequality (though they may have forgotten its exact statement), and they should know how to calculate probabilities.
3. The thinking skills necessary to look at this problem move it into a level 2 task. Students explore the relationships between different side lengths and produce conjectures about probability. Again, they utilize the “if-then” type of thinking; e.g., if the sum of two sides is greater than a third one, it will make a triangle. This task prepares students for later work in which they must bring different areas of mathematical knowledge together; for example, using their algebra skills in a geometry problem, and here combining geometry and probability. It is also good for them to practice longer word problems, where they must decipher what the problem is and what the relevant information is.
4. Questions that could extend students’ thinking include the following: Come up with a set of six lengths so that any three do in fact make a triangle. Come up with a set of six different lengths so that any three do in fact make a triangle. Come up with a set of six lengths so that no set of three of them will make a triangle.
5. One possible lesson: Pass out the problem; give students a moment to read it over, and encourage them to underline anything they think is particularly important to the problem. Then ask one or more students to summarize the problem: What are we trying to answer? Ask students to help you formulate a question to write on the board and to decide on the information that will help you answer it. Pass out materials such as linkage strips or snap cubes, and ask students to work on the problem. After several minutes, bring the class together and ask them what they’ve found. Did anyone find a set of three lengths that didn’t work? What went wrong? Now is the time to review the triangle inequality and to write down the statement on the board. Review what students will need to know to answer the problem — how many combinations of three lengths are possible and how many of them do in fact make a triangle. Then ask students to resume the task, keeping these two goals in mind. Complete the activity by asking students to share their probabilities. It’s likely that students missed some of the total combinations as well as some of the working combinations, so you may want to model for them how to make an organized list of possibilities, ensuring you don’t miss any, and then how to check each one for satisfying the triangle inequality.