Intro to Trigonometry / What is Similarity?
Lesson 1 of 9
Objective: SWBAT build background knowledge for defining the trigonometric ratios as properties of the angles in a right triangle.
In order to help signify that there's a distinction between this semester and last, today's agenda starts by saying "Sit somewhere new, with people you can work with." I don't make a big deal out of this, but students like changing things up, and they appreciate the elegance of the seating policy. I'm meeting about 1/3 of the students in this class for the first time today, and this sort of clean, crisp, and reasonable policy makes a good first impression on new students.
I do make a big deal out of The Opening Problem. I circulate throughout the room looking to see who can follow the directions precisely. It's formative assessment within the first few moments of the class.
There are essentially three benchmarks I'm looking for:
- Can each student sketch and accurately label the diagram described in this problem? If so...
- Can they also connect points E and C and accurately identify which angle we're talking about in the problem? If so...
- Can they use their knowledge of geometry to determine the measure of this angle?
After giving students time to grapple with the problem and discuss it at their tables, I lead a brief discussion about it. To begin, I ask what shape everyone just drew. Invariably, someone shouts, "a house!" to which I say, "Yes -- and if you were thinking that but not saying it, I want you to know that informal language is going to pretty important in this class. So even though there isn't really a geometric object called a house, it's fine to use such a word as we begin a conversation about a new topic." I also point out that some students have drawn houses standing up, on their sides, or upside down, it all depends on where they placed points A and B.
I press students to tell me what they know about this diagram. I pay special attention to involving students who were able to draw the diagram but didn't know what to do next. I say that even though we might not know the measurement of the angle described in the problem, there are some angles we can know without using any tools. At this point, it's becoming pretty clear that, at the very least, we know that there are four right angles in a square. We can label those. "What about the angles in an equilateral triangle?" I ask. Many students can recall that each of these angles will measure 60 degrees. I ask why this is the case, and the details come rolling out - veiled in informal language though they may be. "There are 180 degrees in a triangle," someone might say. Someone else will say that all the angles are equal. I'm not too worried about precise language, because we're just getting started, so I just listen and annotate my diagram on the board.
If the class needs my prodding, I'll ask which line segment is longer: EB or BC? It's not obvious to everyone that they're actually the same length, so we go through some reasons as to why they are. I take some colored chalk and trace the isosceles triangle that holds the key to this problem. I ask students what they know about isosceles triangle, and just to foreshadow a little, I say that isosceles triangles will turn out to be a pretty important tool in this class.
Then, I simply say that I'll give everyone a few minutes to finish this problem up, and I leave it at that.
I shut off the lights and show this video:
The full context here is that this is the former English teacher for many of my students, having a little fun in her "etymology corner." I recommend talking your favorite ELA-teaching colleague into doing the same.
Etymology is a big deal in this course, and by showing this video I am both providing a starting point for our study of trigonometry and setting a tone that we'll be looking deeply at words over the course of the semester. Precise use of vocabulary is part of attending to precision (MP6), and our study of trigonometry will certainly demonstrate that. (It's funny to note, of course, that this course will turn the etymological definition of trigonometry on its head, as students will find out that circles have at least as much to do with trig as triangles do, but in class, I'll save that conversation for a later date.)
When the video is over, I put the lights back on and hand out the Course Syllabus. I take a few minutes to read it over with students. I place the most emphasis on how there are two kinds learning targets: the Mathematical Habits (direct from the CCS Math Practices) and Math Content targets that we'll study in each unit.
The back of the Syllabus lists the kinds of assignments students will have in the class, provides a brief overview of our four units of study, and gives some notes on how students are graded. For more information on the types of assignments I employ and how I grade my students, please refer to my strategy folder.
Kicking off Unit 1
Now it's time to get to work. I write the first learning target of Unit 1 on the board, and I ask for a volunteer to read it out loud. Then I give students a chance to say what they think are the most important vocabulary words in this SLT. This is how I'll introduce every content target over the course of the semester.
I distribute today's handout which includes a few parts. There are three purposes to this handout:
- To introduce the word "similarity" and provide a space for us to discuss it.
- To give students a chance to use the word "properties" in context.
- To introduce the first task of the Similar Triangles Project
Discussion of Similarity (Side 1 of handout)
The handout begins with a categorization problem. There is a set of shapes, and the question, "How many similar shapes can you find?" We're going to use this as a starting place for constructing a definition of similarity. I give students as much time as they need to discuss this prompt with each other - if there are great conversations happening, I don't want to stop them.
After approximately 10 minutes, I start by asking if anyone wants to share their answer to question 3: "Can you name any shapes that are always similar?" This question serves as a check-in that allows everyone - both myself and my students - to know where everyone else is at. Circles and squares are pretty unanimously agreed upon, even though elegant reasons why this is the case might be few. Rectangles and triangles get a little air time, as do any number of other shapes. I make sure not to judge what anyone says here. I want to get the ideas out there. There will be time to assess the accuracy of our claims soon.
I ask for a volunteer to record some notes on the board about the definition of similarity. Students share out their definitions, characteristics, synonyms, and other ideas, and as a group we build an understanding. Students have seen this word before, but they need a little time to review it.
Invariably, someone says that congruent is another word for similar. After students have had a chance to share, I draw attention to this word. I ask if congruence and similarity are the same thing. This jogs students memories as they recall from a prior class that they're not the same thing. So what's the difference? It doesn't take long for someone to remember that if two shapes are congruent, they are exactly the same - size and all. Two shapes can be similar without being the same size. The imprecision of this statement is ripe for conversation. "So what exactly is it that makes two shapes similar?" I ask. Students respond with something along these lines:
- Similar shapes must be the same shape, ie have the same number of sides.
- They do not have to be the same size.
- They must have the same angles.
And this is where we get a little stuck. At this point, a little more conversation brings us to the point where we get to this question: are all squares and rectangles similar? According to the definition we've established so far, squares and rectanlges might be similar, and some students believe this to be the case. Others are troubled that something just doesn't seem to be right, but for these students to prove their point, they'll have to add something to the definition we're constructing, and we're just not sure what that thing is. This is a question we'll return to, and a tremendously exciting place to leave off.
Discussion of Properties (Side 2)
After similarity, another key vocabulary word that's going to help make sense of the learning target is properties. I give students some time to answer the questions here. This is also a review of some conventions for naming points, sides, and angles on a geometric figure. I make sure students understand that in this class, points will be represented by uppercase letters, and the shorthand for sides will be lowercase letters. I point out that it is definitely correct to refer to line segment AB as a side of the rectangle in this diagram, but that it is a shortcut and just as appropriate to refer to it as side a.
I really try to let students answer these questions for themselves or with the help of their classmates. As they work, I circulate and check in to see what they understand. What properties of this shape are they naming? Can they identify congruent angles? Are they skeptical of this diagram -- after all, nowhere is all of the information given that would confirm this shape as a rectangle with two pairs of parallel sides, right angles and the like? If they ask this question, I commend them for their attention to detail, and I say that for now, we're going to be ok with this being a rectangle and all that entails.
Completion of those questions is prerequisite to receieving a piece of colored card-stock that will be used to complete the first part of the Similar Triangles Project.
To launch this project, I have prepared 8 different Colored Rectangles. There are two of each of four colors: red, green, yellow, and orange. There is no significance to the colors, it's just an identifier. On each rectangle, I write the letter A or B, to further differentiate between them. Therefore we can refer to "Rectangle ID's" like Red A, Red B, Green A, Yellow B, etc.
At each table of up to six students, I distribute a different rectangle for each member of the table. I try to distribute equal numbers of each different rectangle throughout the class, because later students will regroup by which rectangle they recieve.
The initial task is for each student to measure the length of each side and the diagonal of their rectangle, and to record these values, then to calculate the ratios of the lengths of these measurements.
We're using our first tool of the semester: rulers, and I want to make sure that everyone is clear on how to get a measurement to the nearest tenth of a centimeter. I ask students if its possible to measure these rectangles to the nearest hundredth of a centimeter, and we realize that it would be if we had the right tools, but our plastic school-issued measuring sticks are only marked as far as 0.1 cm. The lesson we can only be as precise (MP6) as the tools that we're using (MP5).
With that in mind, what is the proper way to round the ratios that we calculate after making these measurements. We have a brief review discussion of significant digits, and we realize that with one significant in each measurement, we can round the ratio to two decimal places.
Homework: Problem Set 1
With five minutes left in class, I distribute the first problem set of the semester. Students usually have approximately a week to complete each problem set, although for this one I just give them the couple of days between now and our second class.
Once the semester gets going, a new problem set is handed out each week. Click here to see my notes on weekly problem sets.