After our similarity unit, we move on to right triangles. Right triangles is one of my favorite units of Geometry to teach. It is the one unit that I have taught every single year that I have been teaching. Take a peek inside of my Geometry Interactive Notebook Right Triangles unit.
Pythagorean Theorem
We start our right triangles unit with the Pythagorean Theorem. My classes are mixed; some students are accelerated a year ahead, and the other students are not. The students that are accelerated essentially skip eighth grade math, where they initially learn about Pythagorean Theorem. So I always teach Pythagorean Theorem like all of my students are seeing Pythagorean Theorem for the first time. (Also, let’s be real, the students that have seen it before have not applied it in at least a year.)
With Pythagorean Theorem, we emphasize applying the theorem to find the length of a missing side and to determine if a triangle is a right triangle. Additionally, we discuss the most common Pythagorean Triples, and I encourage my students to memorize them.
Special Right Triangles
Using Pythagorean Theorem, we discover the relationships between the legs and hypotenuses of special right triangles. We look at 45-45-90 triangles as an isosceles triangles, and at 30-60-90 triangles as an equilateral triangle with an angle bisector. After taking our time to discover the relationships in each triangle, we practice some simple problems, as well as a problem relating the triangles to squares and rectangles. In our geometry interactive notebooks, this lesson is taught using a foldable so students can focus on the practice and discovery portion for each triangle.
Similar Right Triangles
Similar Right Triangles is a difficult concept for students to grasp. My preferred method of teaching similar right triangles is to use formulas to find the missing length. The formulas I use are based on formulas I found on Math Bits Notebook. Throughout the lesson, I explain that we are able to set up an equation using a proportion because the triangles are similar. I also point out to students that we need the altitude rule when we have a number or variable on the altitude, and that we use the leg rule when there is nothing on the altitude. After the lesson, students practice with a card sort that includes solving the problems.
Geometry Interactive Notebook Page: Trigonometric Ratios
When teaching trigonometric functions, I start with the vocabulary of all three sides of a right triangle. Students frequently mix up the opposite and adjacent sides. Especially during this lesson, where we find the three trig ratios for both complementary angles. To help students, I recommend finding the sides in order: Hypotenuse first, Opposite next, and Adjacent last. (The acronym HOA for Home Owners Association would be a great memory device to use if that is a concept your students are familiar with.)
Next, we focus on using the sides to create the trigonometric ratios. We talk about the acronym SOH CAH TOA, and how we can use it to remember the trig ratios. We practice finding the trigonometric ratios for both complementary angles, and then we use a card sort to practice determining which function to use when one side of the triangle is missing.
Cofunctions
Out of the entire unit, cofunctions is one of my favorite topics to teach. (This topic is also referred to as the Sine and Cosine of Complementary Angles.) It follows so naturally from determining the three trig functions of both complementary angles. Oftentimes, students notice the pattern immediately during our trigonometric functions lesson.
To begin this lesson, I start with the last example we completed on the previous day to reiterate the relationship that exists between the sine and cosine of the complmentary angles. Then, we talk about how the two complementary angles sum to 90 degrees. After a few guided practice problems, students work on a short task card activity.
Right Triangle Trigonometry – Solving Sides
One of my other favorite lessons in the unit, solving for missing sides, is when we string everything together. I love sharing the steps to solving for sides with my students because they already know how to do the first three steps. Usually, I try to remind students how to solve an equation, emphasizing that the trig function and angle are just a number. However, the function is so different for my students, that they usually need a little help. I teach them that they can put the trig function over one, and then cross multiply to solve, and they usually do better with this perspective. In our interactive notebooks, we complete nine practice problems.
Right Triangle Trigonometry – Solving Angles
After solving for sides, we move on to solving for angles. Again, the great thing here is that students already know the steps. What we have to build on in this lesson is using the inverse function. I remind students that we would divide to solve a simple equation like 2x = 6 because division is the inverse operation of multiplication. Then, I introduce students to the inverse trig functions. We complete nine practice problems in our geometry interactive notebooks.
Right Triangle Trigonometry Applications Geometry Interactive Notebook
Our final lesson of the unit is on right triangle trig applications. We apply trigonometry to word problems. Here, I stress using common sense when setting up the problem. Many times students need to draw their own diagram of a right triangle, and we typically draw it with vertical and horizontal legs. As students add values from the problem to the triangle, I ask questions like, “which side should be the ladder?” “If the ladder is laying flat on the ground, it will not do you any good. If the ladder is straight against the wall (and not anchored), the ladder will fall over as you climb it.” This way students understand that the ladder is the hypotenuse of their diagram. Our practice in our interactive notebooks is short for this lesson. After the lesson, we practice with questions from our state exam.
Geometry Interactive Notebook: Right Triangles Resources
Are you looking for materials to teach this unit? Look no further! All of the pages you see in my Geometry Interactive Notebook: Right Triangles are now available. Prior to uploading these pages for your use, I taught each lesson as described above. Some of the pages may not look exactly as they do in this post because they have all been edited and updated. Also, a full answer key for each page is included.
Looking for more resources? Check out my interactive notebook resources page!
