Special Right Triangles
Lesson 1 of 6
Objective: SWBAT derive special right triangle relationships, memorize them, and apply them to solve problems.
In this section students Construct Special Right Triangles. This is an authentic way for students to practice and apply constructions we have learned and also lays a key conceptual foundation for later sections in the lesson. Namely that special right triangles are derivatives of equilateral triangles and squares. (Note: I have also taught this section having students use a protractor and straightedge to create the triangles rather than incorporating constructions)
To begin, I explain to my students that they will be creating triangles to specifications (language with which they are familiar from previous experience). I emphasize the importance of them interpreting the specifications on their own and not trying to make their triangles like their neighbors' triangles.
This section of the lesson is semi-independent. I encourage students to share ideas and help each other, only as needed. I also walk around providing hints and nudges. Mainly, I want to ensure that we get a nice range of different sized triangles so that it will become clear that the triangles are similar, and that their corresponding pairs of side lengths are proportional. What I don't want is students copying because this will reduce the variety of triangles we get.
Invariably, there are students who have no idea how to perform the constructions. After the majority of the class has had time to forge ahead, I demonstrate the constructions for these (and all) students under the document camera using classic tools or else I demonstrate the constructions using Geometer's Sketchpad (the Sketchpad constructions show students the basic approach but leave students to figure out exactly how they'll perform the constructions using classic tools). In any case, I want to make sure that the constructions are not a barrier to students getting through this lesson.
As students are starting to complete their constructions and take the prescribed measurements, I make a huge table on the whiteboard with columns: Lengths- AB, BC, and AC and rows: Students 1, 2, 3...n
I announce "Students, when you have completed and checked your constructions and measurements, please come to the board and record your data in the table"
As students are recording their data in the table, I (or possibly a designated student) will be entering the data into the Data Spreadsheets (See reflection for ideas on enhancing this and similar lessons using spreadsheets)
Once enough data are in the table, I get the students' attention and explain,
We are about to do some inductive reasoning. In other words, we are going to observe our data and try to see patterns and/or relationships that might be generally true. Our statements expressing these things that we think might generally be true are called conjectures. For a simple example of a conjecture, you might, after seeing me sneeze on three different occasions when there has been a cat nearby, make the conjecture that I am allergic to cats...or at least that they make me sneeze.
Similarly, I would like for you to look at the data we've compiled and use inductive reasoning to make at least one conjecture for 30-60-90 triangles and at least one for 45-45-90 triangles.
At first I have the students think privately for a couple of minutes about their conjectures. Then I have them share their reasoning and conjectures with each other. Finally, I have several students share the reasoning process that led to their conjecture. At this point, I don't feel a need to confirm or disprove the conjectures. That will occur naturally later in the lesson. I really just want students to engage in the process of formulating the conjectures. So instead of saying "Right!" or "Wrong" my posture is more like "Thanks for sharing; let's see how true that turns out to be."
In this section of the lesson, I sort the students into groups of four students each. Each student receives . I then ask students to read and follow directions 1 through 4 on the hand out.
First the students must each choose a different even number between 6 and 30. Then they must draw special right triangles on the provided equilateral triangle and square. Then they must wait for instructions.
Once students have completed the first three steps and are waiting for further instructions, I inform them that the even number they have chosen will be the value of AB on the equilateral triangle and on the square. This is why it was important for each group member to have a different number.
Next I explain that the students will be working independently at first to complete the first two pages of the packet. I ask that they work independently as much as possible but I also tell them that they are all working on similar problems so if any of them need help, they can look to their group members for guidance on how to approach the problem.
As students are working, I walk around to the groups and make sure things are on track. I also ask each group to show me the short leg, long leg, and hypotenuse of a right triangle. I am merely trying to introduce this new vocabulary and have students use it.
When I see that a group has finished the first two pages of the handout, I direct them to the third page and explain that they will be compiling their results in the table before breaking off into pairs in to make conjectures based on the patterns and relationships they observe in the table. I remind them to follow the protocol closely as it is written on the handout.
When a group has finished making the conjectures, I check them for correctness and precision. I also ask "So, have you proven these conjectures?" to see how well they've understood the statement at the bottom of page 3. Then I send them forward to the last page of the handout. For students who finish early, I may ask them to go back and see how far off our original data in the table (from last section) were, on average, from what they theoretically should have been.
When I assess that it is time to start wrapping things up, I have the students transition back to row seating. I start by briefly recapping what has taken place so far. How we started with special right triangles with messy side lengths but nonetheless were able to observe some relationships and make some conjectures. How then we chose integer side lengths and chose to express non-integer side lengths in simplest radical form in order to be able to see the relationships better and make stronger, more complete conjectures. And finally how we show that the relationship is true generally.
This last bit, since students may not have finished or understood it, I model it for them. I also explain that they will have a quiz on which they will have to show exactly what I am about to demonstrate. Within that modeling, I explain, for example, why 2x was a practically good choice for the side length of the square as opposed to x. I also explain how 2x^2 is not the same as (2x)^2. So in this last part, I basically tell them everything they need to know in order to do well on the quiz. No excuses.
Mnemonics and Number Sense
Now that students know the relationships between the sides of special right triangle and are hopefully convinced they are true, it's time to get these relationships into long term memory in a way that will be useful to students when they need to call up this information.
My approach for myself and for my students is to always try to compress whatever we are trying to remember into the smallest chunk of memory needed to trigger the conceptual understanding that will completely flesh out what is being remembered. The approach for this particular lesson relies on remembering a few key numbers and then relies on number sense to flesh out the rest.
Check out Mnemonics, Number Sense and Decision-Making to get a feel for how help students to remember these all-important special right triangle relationships.
For this section, I have students work special right triangle problems from the text book. I'm careful to select problems of varying difficulty. For example, I make sure to include problems that have the short leg of a 30-60-90 triangle with a radical expression, 6 rad 3 for example. This ensures that students are not going about things mindlessly.
I allow students to use the schematic to help them, but I just ask that they gradually try to wean themselves off of it. For example, I might say, try thinking through the problem without looking at the schematic and then look at the schematic to see if your first thought was correct.
In any case, I want them to get lots of practice with this so that they feel confident about these types of problems. When we get to later units on areas of regular polygons and areas and volumes of solids, these things will need to be second nature.