A B C

Solutions for Session 10, Part C

See solutions for Problems: C1 | C2 | C3 | C4 | C5 | C6

Problem C1

Problem C2

a.

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.

b.

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).

c.

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.

d.

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."

e.

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

 a. The content in this problem covers properties of figures and congruence. It also requires visualization and thinking through possible cases. b. To engage in the task, students should understand congruence and what a rectangle is. c. 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. d. 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. e. 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

Problem C5

a.

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.

b.

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.

c.

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.

d.

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.)

e.

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