This lesson is the second part of Dig In! In the first part of the lesson, the class worked in groups of four to determine how many different shapes could be created using four isosceles right triangles, and then created categories for their shapes. In this lesson, the second part, the students begin by looking over the different categories.
I write my Do Now instructions on the board and hand each student a small piece of paper as he or she walks in the room. The instructions on the board read:
1. Silently, walk around the room and look at the posters that have created. Compile a list on your paper of all of the categories chosen. Return to your seats when finished.
2. In your groups, compare your lists and brainstorm other categories that might have been chosen.
When all the groups have finished their discussions, the students share out their ideas for other possible categories.
I ask one student from each group to remove their poster from the wall and to bring it back to their desks. I ask the students questions about perimeter:
I then repeat this process with area.
Next, I pose the questions: What are the dimensions of our triangles? If each square on the grid is a quarter-inch long, how long are the legs of the triangle? With just a little discussion, the students arrive at the fact that each leg is 3 inches long. I then ask
With a little discussion, the students remember the Pythagorean Theorem, and I ask them to coach me through this process as I write the steps on the board. I am careful to ask questions about the equation 3 squared + 3 squared = x squared, like, "Why is the x value all alone on the left?" Often I find that students will answer this question by saying that it is there because it is the unknown, and I think this is an important misconception to correct.
When we have arrived at x equals the square root of 18, I ask students what this value means and what kind of number it is. We talk about the fact that it is irrational and what exactly irrational means, and compare this to square roots that are rational (the square roots of perfect squares). I ask them to estimate the value of the square root of 18, and we discuss how we can arrive at an answer to this. I then ask what the square root of 18 is exactly, and we discuss the process needed to answer this question. Almost always, at least one student will volunteer, "Plug it in the calculator" and we discuss why the decimal answer given by a calculator is only an approximation, not an exact answer.
This discussion provides an opportunity to talk about precision (MP6). We discuss circumstances in which it is very important to be very accurate and precise, and contrast this with events in which being so precise would be inappropriate.
When we have finished discussing the contents of the the content of the students' posters, I ask the students to go back to their posters and to calculate the area and perimeter of each of their figures. When I inform the students of this task, they inevitably look at me with a resentful gleam in their eyes, like, “You’re kidding, right?” and a few brave souls might even voice this sentiment. However, I assure them that I’m not kidding and suggest that they get to work. (Here comes the perseverance required to promote MP1!)
When they look at their triangles mounted on the poster paper, it becomes immediately clear to them that each leg of their triangles measures 3 inches. This leaves them with the question of the length of the hypotenuse. Some students (unfortunately!) must be convinced that counting squares does not work for finding the length of the hypotenuse, and then the students must figure out a way to find this length, and these issues lead to great discussion in the groups. Often they will ask for a ruler and I tell them no, that I’d like them to find the exact length of the hypotenuse (MP6), in simplest radical form. In very little time, through collective head-scratching and brainstorming, I find that the students recall the Pythagorean Theorem and go to work to find the length of the missing side. They then find that this length is not a whole number and they must work together again to recall how to simplify radicals. Once this is accomplished, they are now ready to compute area and perimeter.
I never suggest that the students write the lengths of the sides on the poster, but I have found that almost every group does this. They discuss in their group exactly how to find perimeter and area, and most of the time the groups tend to divide up these tasks – “You two find the area; we’ll find the perimeter.” Within 10 minutes or so, I hear students begin to remark to each other in a puzzled tone, "Hey, these_Areas_are_all_the_same?" This is probably my favorite moment of this entire activity, this moment in which they discuss and explain to each other that the areas must be the same because each figure is composed of the same four triangles. This moment of discovery and understanding – totally developed by the students themselves – is so empowering for them!
To see this part of the lesson unfold, watch: Classroom Video: Perseverance
Finding the length of the hypotenuse requires that the students use the Pythagorean Theorem and that they simplify radicals. Finding the perimeter involves adding rational and irrational numbers, reminding the students that they can only add similar terms, and the computation of area requires adding either decimals or fractions. In all of these cases, important prior knowledge and skills are being recalled, and we will, therefore, be able to move on in the course with these skills readily available to the students, and without having to supplant precious learning time with the review of these concepts.
I ask the students to return to the piece of paper that I gave them at the start of the class and to draw an isosceles right triangle on the paper. Then I ask them to pick a number between 1 and 10, and label the legs of their triangle with that number. Then I ask them to calculate the length of their hypotenuse as an exact value.
When it looks like everyone has completed this task, I ask that those who chose 10 to give me the length of their hypotenuse, then those who chose 9, then 8, and so on. As I draw quick sketches of these triangles on the board, labeling the lengths of the sides, the students begin to observe a pattern: In all of the examples the measure of the hypotenuse is the length of the leg times the square root of 2.
As I get to legs of 3 and 2 and 1, the students anticipate the lengths of the hypotenuse, and I ask them what our conclusion might be. What Generalization can we make? When they state their conclusion, I ask, "Can we assume this works for all isosceles triangles? How might we prove this?"
I draw an isosceles right triangle on the board and label the legs a and a, and ask the groups to work on finding the length of the hypotenuse in this case. With a little bit of work and class discussion, we prove that the length of the hypotenuse will indeed be a times the square root of 2. This is our first use of MP8: Look for and express regularity in repeated reasoning and MP2: Reason abstractly and quantitatively.
For the last stage of this lesson, I pull the class together as a group for discussion. I first ask the students a series of questions about the content of the lesson:
I try to drive home to them the prior knowledge and the vocabulary, in particular, so that they know that this material should now be “at their fingertips,” and available for use in future lessons. I then write three questions on the board and ask the students to return to their groups to discuss them:
After giving the groups sufficient time to discuss these questions, I pull the class together and explain that we will be working in groups throughout the year in Geometry. I state that I think it is important that the class as a whole develop a set of Classroom_Expectations or norms. I explain that norms are a set of rules that will govern how we act and interact with each other, and that these rules will help to make it possible for every student to participate actively and comfortably in their group.
To see this part of the lesson unfold, watch: Classroom Video: Shared Expectations
I go around the room and ask each of the groups to offer a skill that they had discussed and thought was important to use when working in a group. I have a student create a list on the board as the groups report out. We repeat this process, with another student creating a second list on the board, this one a list of behaviors that make it difficult to work in a group. Working from these two lists, the class creates their own set of norms, which I write on the board as the students discuss it. At the end of the class, I save the list and ask for a volunteer (or volunteers) to create a poster on which their norms are listed (see an example in my reflection on Shared Expectations). I will post this list in a prominent place in the classroom, so that the students are always aware of and can reference this list of important skills and behaviors that they themselves worked to compile.
Perhaps it is also worth mentioning that I do not use this lesson as the basis for a grade. If a group does not finish calculating all of its perimeters or if all 14 shapes are not found, I do not make an issue of it. It is my hope that this lesson primarily drives home two important facts – that, contrary to popular opinion, math can be fun, and that working together is really beneficial to learning. I will strive to reinforce these two facts again and again, throughout the entire school year.
I have posted the Mathematical Practice Standards on a wall in my classroom. As we prepare to end the class, I pull the class together and talk about the Common Core. My Geometry students took the New York State Common Core 8th grade test last year, so they have heard a little about the Common Core. I talk about the fact that in Geometry, the content of the course has not changed all that much. However, I explain, what is revolutionary, in my opinion, is the Mathematical Practice Standards. The heading of my wall display is "Think like a mathematician..." and I think that this is exactly what the Mathematical Practices will accomplish.
To see this part of the lesson unfold, watch: Classroom Video: Standards Alignment
I briefly explain each standard, and then ask the students to identify all of the Mathematical Practice Standards that we used today.
As the bell rings, I hand out the homework assignment. It consists of four irregular polygons for which I ask the students to find the area and perimeter.