Next, I hand out the Pythagorean Theorem practice problems. I ask the students to work together to solve the problems.
As we have used the Pythagorean Theorem in the past, I am confident the students can work through most, if not all, of this problem set independently. I have chosen these problems, however, not just to provide practice on the Pythagorean Theorem. The majority of them work out to be “nice, easy” numbers intentionally, because in the next lesson the students will revisit these problems, looking for common Pythagorean triplets. It is my belief that “fluency” in Geometry includes fluency with common Pythagorean triplets, 45o-45o-90o triangles, and 30o-60o-90o triangles. Therefore, I have designed the next several lessons with these special triangles in mind.
There are many opportunities in these problems to revisit previously learned concepts. I highlight many of these topics in the discussion of the problems found below. As I circulate around the room, I anticipate that there might be questions about some of these problems. I do not tell students how to do these problems, but I am prepared with questions that should help them to find their way:
Some students become stumped because it appears that they know the length of just one side of the triangle. I ask, “What do you know about the sum of the angles of a triangle?” Hopefully this is enough to set them on their way; if not, I might ask, “What kind of triangle is this?”
All of these problems require that the students draw in an auxiliary line, a skill specifically mentioned in the Common Core Standards for Mathematical Practice, MP7. As the students reach this point in the problem set, I take a moment to make sure that all of the students understand exactly what the height of a triangle is. For those students requiring scaffolding, the heights of the triangles could be drawn in explicitly in problems 6 and 7.
Problem 8 and several problems following it require that students be aware of a very basic feature of rectangles – that the opposite sides of a rectangle are congruent, a concept that they have definitely been exposed to in this and in previous courses. A question like, “What can you tell me about the opposite sides of a rectangle?” should be sufficient.
Problems 9 - 10
These problems require that the students understand that the two triangles in each diagram are congruent. This provides a nice opportunity to revisit the five methods for proving triangles congruent, and to help struggling students to see that, in this case, we can use Hypotenuse-Leg. I might ask, “What angles or segments do we know are congruent in these two triangles? What information does this give us about these triangles, and why? What does it mean when we say triangles are congruent?”
A possible extension of problems 6 through 11 is to ask the students for the area of these figures. The only formulas required would be the area of a triangle and a rectangle, and students should already be familiar with these.
The students have not studied quadrilaterals yet this year, so they may not know that the diagonals of a rhombus bisect each other. However, this is a wonderful opportunity to ask questions about congruent triangles, in order to help students come to the conclusion that all four triangles contained in the diagram are congruent and that the diagonals must, therefore, divide each other into congruent segments.
In this problem, students must first find the length of the altitude using the triangle on the left, in order to find x in the right-hand triangle. Questions like, “To use the Pythagorean Theorem, how many sides of a triangle do you need to know?” and “How many side lengths do you know in the left-hand triangle? And in the triangle on the right?” might help a student to work his or her way to the solution.
It is also very possible in this problem that some students might try to use their previously learned knowledge about the altitude to the hypotenuse of a right triangle to find the length of the altitude; if so, this would be a great opportunity to ask the entire class why this method won’t work in this problem.
In Problem 14 students may try to begin with the triangle with side x. This is another opportunity to stress that two sides of a triangle must be known in order to solve for the third side.
As I circulate the room, I keep an eye on students’ answers. Despite the detailed notes above, I do not expect to go over each and every problem. I am careful to redirect students to any problems that have been solved incorrectly. I have found that having students work in groups aids greatly in this process, as they tend to check each other’s work as they go.
With a few minutes left, I ask that the students stop working. I give them an opportunity to ask questions or share comments about the problems covered today. If the issue is not raised in a question or comment, I will say to the class:
Any observations about the side lengths of the triangles you’ve been working with? Are you seeing any patterns in the numbers or specific triangles more often than others?
It is my hope that students are looking for patterns and beginning to recognize the more common Pythagorean Triplets. I leave them pondering this question as I let them know that tonight's homework is to complete the Pythagorean Theorem practice problems.