## Trigonometry for the Geometry Classroom

It’s finally trig time! Yay! I love trig. Students tend to enjoy it too because it is so different from everything they have been taught so far. Over the years, I’ve tried different approaches to teaching trig. I know what kids struggle on and I finally feel like I’ve got a good way of teaching it. I see my trig unit broken into these parts:

• Intro to Trig
• Practice Finding Opposite, Hypotenuse and Adjacent
• Setting Up Problems and Solving Them
• Practice
• Review
• Assessment
• More Assessment

Trig can be simple but to some students it can be complicated. They actually love it once they get the hang of it and how fun is it to use the calculator this much?! (When I first learned trig, we used charts to find the answers. We did not have calculators that would do the calculations. Yes, I’m old!)

When I created this unit, I knew what the two main issues were with teaching trig: 1) Teaching them which trig function to use 2) Teaching them how to solve the different types of problems

I decided to work backwards a little. In my introduction, I just tell them (As Bill and Ted would say) we are about to embark on an excellent adventure called Trig. I introduce a right triangle and tell them to visualize that they are in a right triangular room. They are sitting in one of the corners (not the right angle). I go on to talk about where opposite is and how when you are sitting in the corner, you can touch the hypotenuse and adjacent sides at the same time, but you can’t reach the opposite side. There are some notes we take and then we play a dice game.

For the dice game, I usually get my first class to cut out and put together the dice. Now I have the dice for the rest of the day. I put students into groups of 3 or 4 and they are competing against the rest of the class. There are three dice. One with triangles, one with dots and one with the words, hypotenuse, opposite and adjacent. Click the link below to watch the dice game which practices knowing the different sides with respect to a certain angle. Dice Game Short Video.

Before going any further, I teach kids SOH CAH TOA and we do some practice on finding those ratios. That part is normal progression, but here is the part that might seem a little backwards: I teach them how to solve trig equations next! The students do not know how to set them up yet, but I have figured out that if I go ahead and teach them how to solve the equations, then once they start setting them up, solving is a breeze. I teach them how to solve these three types of problems:

• Looking for an angle
• Looking for a side and the x is in the numerator
• Looking for a side and the x is in the denominator

By the way, when teaching them how to solve these problem, get them to completely solve for x before typing anything into the calculator. Don’t let them find the sin of an angle, then multiply by the side. Let them type the whole thing in: 12 sin(36). I like this method because then the students aren’t rounding answers until the end of the problem. You can see that I did that in the examples above in problems 5 & 6.

Next is the PowerPoint. In the picture to the right, you can see one of the slides in the PowerPoint. Only the triangle with the sun, and the two arrows appear and students have to name which trig function is being referenced. I don’t use degrees for a while, I’ll just use symbols. I don’t want the variables and numbers to get in the way. Toward the end of the PowerPoint, the students are asked to set up the problems and then at the end, they go back to solve them.

Now it’s time to practice. I have 3 worksheets that help students find missing sides and angles. The first one places only an x on one side, a number on a side and gives one angle. This makes it easy to determine the trig function and it is like the PowerPoint. The next worksheet gives the students two sides and asks them to find the missing angle. The last worksheet is the toughest because now the students have to find x, y and z… two sides and an angle. This is much more difficult because it will not be obvious from the start which trig function to use. Students need to see that they actually have a choice sometimes and they need to decide where to start and ignore the extra info. I also throw in some special right triangles and an right triangle altitude problem to see if they remember those rules. The PowerPoint from earlier brings up that there might be more than one way to solve a problem, so hopefully when they get to the worksheet, they will use a quick special right triangle rule instead of trig, but if they can find the answer either way, I’m happy.

I have another resource that is not in this trig unit that I do at this point. It’s the Trig Maze. The students really get into it and work at it. It’s cool to work a problem and then see your answer on the paper (they are thinking, “YAY, I did it right!”) and it’s even cooler that it leads you to the next problem you are supposed to work. The maze comes with an answer document, so you can see all of their work!

Finally, I like to do some task cards with some real-life situations. Some of the task cards contain a ladder against a building, finding a flagpole height, finding the diagonal in a rectangle etc. There are 12 of these problems.

I end the unit with what I call the “Poodle Problem”. It is a group of 5 triangles that have been put together to look like a poodle. Go back and look at the very first picture at the top of this blog. That’s the Poodle Problem! The students find all the answers, then total them for one final answer. How fast is this to grade? Super fast! It’s a great quiz and a great end to the unit.

I’m not finished yet! Now I like to test all of the right triangle content. I have a test that I call the Right Triangle Test that has 10 questions with the following problems:

• One Pythagorean Theorem Problem where they have to find the perimeter of the triangle.
• One Right Triangle Altitude Problem where they have to find the perimeter of the triangle.
• One 30-60-90 Problem where they have to find the area of the triangle.
• One 45-45-90 Problem – easy, they just find the hypotenuse
• Six Trig Problems – Just find a missing side, except for one problem is like the task cards, but a little tougher.

I had problems with cheating one year, so I went crazy and made 5 versions of the same test. You even have a choice of an answer bank or no answer bank. One of the 5 tests is a shortened version that I’ve used as a retest or a modified test. (It gives the students a little help on setting up some of the problems too.) I don’t like to give long tests. Students get enough testing. I like tests that are short and to the point. As long as I can tell they “get it”, why does it have to be super long?

I’m very happy with this unit. The only thing that it doesn’t contain right now is angle of elevation and depression problems. I’ll try to add this to the unit this summer. These problems were a big deal at one time, but it seems like we’ve gotten away from them in Geometry. I still think it’s good for students to see them.

Trig is fun and different and essential to future math classes. Below is all of my right triangle lessons including the Trig resource I’ve been talking about. What’s next on my agenda after right triangle trig? Law of Sines and Cosines of course! Law of Sines and Cosines is sold separately in my store, but it is also a part of Unit 7 below.

## STEM Activities in the Secondary Classroom

Teachers know the importance of incorporating science, technology, engineering and math. In today’s world, students need a set of skills for future jobs that we can’t even imagine, so we must give them a variety of experiences. My husband and I have several activities that we believe qualify as STEM activities. Our thought process is that any time you can get students to do math in a real-life situation, you are headed in the right direction. Throw in some collaboration and problem solving and you’ve got yourself an even better experience. Remember, technology does not always have to be digital. Any use of a tool could be technology, such as using tape measures or any measuring device. Technology can be as simple as collecting data. The activities that we have below are excellent and will provide your students with some valuable thinking and exploration.

Vitruvian Man

Vitruvius asserted that every structure must be strong, useful and beautiful. This assertion was one of many in his writings. He also made claims about the proportions of the human body. Leonardo Da Vinci’s famous Vitruvian Man was an attempt to take what Vitruvius claimed and capture it in a geometric drawing. This measurement lab gives the students a chance to learn some history, collect data, through measurement, set up some mathematical problems, collaborate and analyze. The students go through 6 stations to find the measurements they need, then it’s time to test the ratios to see if they prove or disprove Vitruvius. You’ll love the accountable talk and the thinking that takes place in this very interactive lab.

This is a self-paced project where students learn how to take information and plug it into an excel sheet then produce a graph of the data. I like to use it as a project grade that will replace the lowest test grade during my quadratic unit, but this does not have to be used in math classes only. This activity has a health aspect and can be used in science, health classes or even a medical class. Students create data tables for male and female blood pressures and then they are asked to take their own blood pressure to see if the data seems accurate. The school nurse will usually take blood pressure if you ask him/her or they can get it done at the machine at their local drug store. The final part of the activity is to write a short report on the findings. Below you will see part of a page that is explaining how to get started in the excel sheet:

My advice to the teacher before assigning this project is to go through the excel sheet yourself first, so that you can answer any questions from the students. As a math teacher, I love showing students how to use excel to create a table. This can come in handy any time they want to take a data table and create a graph. This skill could come in handy when writing future reports in any class such as science or social studies.

Nutrition Label Project

Dr. H developed this project because in the past the Texas Biology EOC would contain an analysis question using a nutrition label. This project is based on information found in nutrition labels but it has the students using many scientific processing skills. They must interpret and compare quantitative data using nutrition labels from cereals that they have chosen to analyze and investigate. Some parts of the assignment requires drawing inferences from the data and constructing comparative graphs to visually communicate information they have gathered.  The final step of the project allows them to analyze and interpret the data into a written article explaining and defending their conclusions. As the students move through the assignments, you will notice the project becomes something they take ownership of and will have discussions with other students about their results. (HELLO CRITICAL THINKING!) They develop an understanding of that label on their food which they had never thought about before. Students will begin bringing you food labels and discussing ingredients or questioning if this food is good for them. The project is broken up into 4 parts that get increasingly complex and it builds confidence for many of your students. Dr. H says, “I assign this in the spring and, every year, I will get surprising work from a few students that kind of underachieved all year.”