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.

Quadrilateral Unit

Quadrilaterals are a big topic in geometry. There are so many things to know that it tends to get confusing for students. Students have misconceptions from their middle school math classes that are hard to overcome such as that a square is a square and only a square. A square is no way, no how a RECTANGLE! OH My!

I created a quadrilateral unit where I begin with a card sort activity. The cards have different shapes on them and the students are asked to separate them into parallelograms, trapezoids and other major shapes. This year, I decided that they should have a “for sure” pile and also make a “not sure” pile. I love listening to the conversations. Below is a pic of some slides I show:

The next thing that I like to do is discuss the Venn Diagram for Quadrilaterals. For some students, this is a breeze but for others, they are totally confused on why I’m using ovals in a diagram to represent groups of quadrilaterals. It’s best to make sure your students remember what a Venn Diagram is. I like to give an example of a region with math students overlapping a region of biology students to show that the overlap means all students taking both math and biology. Look at the Venn Diagram below. Can you figure out what quadrilaterals go in each region? Can some go in more than one spot?

This unit is the best place to use always, sometimes and never questions and if the students understand the Venn Diagram, then the always, sometimes and never questions are pretty obvious. It’s also a good time to talk about what does opposite and consecutive mean? Many of the definitions and properties use this terminology, so I spend time helping them understand where opposite sides and angles are versus consecutive sides and angles.

I like to get the kite and trapezoid out of the way first, so I can spend most of my time on the parallelograms. Students are not familiar with the kite, so this is usually a brand new topic for them. They think they know what a kite is but usually they are getting a rhombus confused with a kite. Each time I present a new quadrilateral, I give the definition and then we try to find other things that are always true about the shape. This is cool, because you get to talk about the diagonals and how they create congruent triangles. I also try to put proofs into the lesson as much as possible.

During the trapezoid part of the lesson, there is a discussion on isosceles trapezoids, midsegments of triangles and medians of trapezoids. A good reminder at this time is how trapezoids are related to parallel lines cut by a transversal, so that they can understand that there are some same side interiors that will be supplementary. Again, there is so much information, that its hard to know when to stop. Trapezoids could be a two week lesson if you let it, but I keep it to two pages. After the trapezoid lesson and the kite lesson, I give the students some practice on finding various parts of the shapes.

The rest of this unit is spent on parallelograms. Each time that I get to a new shape, I call it a “Parallelogram Study” or “Rectangle Study” etc. I let the students work through the definitions, properties and proofs. The other aspect of this lesson is discussing the coverses of the definitions and properties. This helps the students realize that if you see a shape and you are not sure what it is, then what is the least information you need to decide it is a rectangle for example.

The lesson concludes with practice on the parallelograms. There is a page of work where some major algebra topics are practiced. For instance, there is a rectangle problem where the students have to set up and solve a system. There is a rhombus problem where the students have to solve a quadratic. There is a square where the students find the length of the diagonal using the variable “s” for a side. This problem is a lead up to 45-45-90 triangles. I usually have to help the students with this whole page, but I don’t mind. Since I’m an Algebra II teacher as well, I like my geometry students to see as much algebra as time allows.

There is a set of task cards that act as a review for the Quadrilateral Test at the end of the unit. The test is only two pages long, but it is pretty involved. There is a major problem where the students have to find quite a few things. The picture of the problem is seen below:

Finally, there is another quick assessment that I use as a retest. All answer keys are included. It usually takes me about two weeks to get through all the work plus a couple of extra days to review and take the test. I love this unit. The information is extensive and I love how it hits on previous geometry and algebra topics. If you are interested, please check it out in my store. Click the pic below to go see the Quadrilateral Unit. If you would like to read more about my geometry curriculum, I have a blog post that you can read here: Geometry Curriculum for the Year.

Geometry Curriculum for the Year

It took me a year and a half to complete my geometry curriculum, but I finally did it. I’m constantly updating and adding to it. I guess you could call it a living and growing curriculum. I’m using the curriculum myself, so I find things I want to update or make better all of the time.

(By the way, don’t forget to look for a special right triangle freebie at the bottom of this post!)

On the main page of my Geometry Curriculum on TpT, you will see every resource in the curriculum. I also sell each one of these resources separately in my store. A third way that I have the curriculum broken down is by units which I also sell in my store.

Why did I feel the need to write this curriculum? Many of my students come to me after taking Geometry over the summer. In my district, students have the opportunity to get ahead in math. If the student does not pass Geometry during the summer, they end up using the exact same curriculum again. They get to keep their book from the summer with all of the work already done. It doesn’t make sense. Also, if you don’t know about slader.com, you should check it out. All of the answers to the math book that we use are here! So frustrating! Kids know about this site. You should see if your geometry book is here.

You can find me on Instagram @timefliesedu . I like to show student work in action on my Instagram posts. Check it out if you have time. It will help you get a feel for some of the lessons. To see the curriculum in my TpT store, click the picture below. The picture shows one of my favorite conditional statement activities. As you can see, we do lots of cutting and pasting.

To see each unit, click on the unit picture below:

I really like a hands-on curriculum. I use patty paper, compasses, protractors, tape and scissors on a regular basis. I also like to give Google Form tests as well as activities on Google Slides and Boom Cards.

Many of the activities are discovery lessons. I make sure there is plenty of algebra involved to keep these skills alive for the next math class. If you have any specific questions about this Geometry Curriculum, please leave a comment below, or ask a question in my TpT store.

Here is a freebie from one of my special right triangle lessons. CLICK HERE!

To read more about the units, I’m slowly but surely writing posts about them so teachers have a better understanding of what they contain.

Unit 1 – Diving Into the First Unit of Geometry Headfirst

Unit 2 – Two-Column Proofs and Reasoning

Unit 3 – Parallel Lines, Transversals, Perpendicular Lines

Unit 4 – Transformations

Unit 6 – Quadrilateral Unit