Better Questions: Special Right Triangles

This is my third year teaching Geometry and every year, students have a hard time with special right triangles. Last year, I had a student ask me, "Mrs. Newell, why is the hypotenuse in a 45-45-90 triangle always √2 times longer than the legs? Why isn't it only 2 times the leg, since 45 is half of 90?" This question made me reflect on my teaching and it made me realize that I was not teaching the "why" in special right triangles.

So, I had students discover the rules for 45-45-90 and 30-60-90 triangles by splitting a square in half and an equilateral triangle in half. They had to use the Pythagorean Theorem to solve for the length of the diagonal (square) and the altitude (equilateral triangle). I told students that each side of the square was 1 cm and they had to use the Pythagorean Theorem to find the length of the diagonal. After students used the Pythagorean Theorem to find the length of the diagonal of the square, I asked them to tell me the relationship between the legs and the hypotenuse of a 45-45-90 triangle. Many students told me that the hypotenuse is always √2 times longer than the leg "because the Pythagorean Theorem says so."

On the 30 60 90 paper, I told students that each side of the equilateral triangle is 2 inches and that they need to find the length of the altitude.




After those notes, I had students fill out this chart to see the pattern of (x, x, x√2). I love this chart because students can find a pattern between the numbers before using it in a 45 45 90 triangle. I split the chart into "three levels". The first level are just basic numbers where students do not have to do any math to fill out the chart. Once I feel that all students have "conquered level one", we move on to level 2. In level 2, I add radicals in the "x" column and students have to multiply the radicals to get the x√2 column. When students have "conquered level two", we move on to level 3. Level 3 is the column where I put a whole number in the x√2 column and they have to rationalize the denominator to get the value of 'x'. I find this chart very helpful for my special education students because they reference this page MORE than their foldable. It also helps remind students what to do when given each situation. 

After the chart, I introduce this Frayer-Model foldable for 45-45-90 triangle practice.  All of the Geometry teachers at my school use the "tic-tac-toe" method to solve for missing side lengths in special right triangles. The next image is what students will see in their notebook when they open it up. I have students highlight the angles and their side lengths. There are a total of 6 practice problems in this foldable and there is at least one problem from "each of the levels." 






The next day, we learned about 30-60-90 triangles. We used Pythagorean Theorem on an equilateral triangle (shown above). After those notes, I had students fill out the following chart to see the pattern of (x, x√3, 2x). I split the chart into "three levels" again. The first level are basic numbers where students do not have to do any math to find the rest of the row. For example, I put a 6 in the x column and they had to find x√3 and 2x. Another example would be putting a 14 in the 2x column and trying to find x and x√3. Once I feel that all students have "conquered level one", we move on to level 2 where I add in radicals in the "x" column. Students have to multiply the radicals to get the x√3 column. When students have "conquered level two", we move on to level 3. Level 3 is the column where I will put a whole number in the x√3 column and they have to rationalize the denominator to get the value of 'x'.


After the chart, I introduce this Frayer-Model foldable for 30-60-90 practice.  This has 6 practice problems. 





Overall, I feel like this lesson went GREAT. My special education students are ROCKIN' special right triangles. I feel like the discovery and the "why" helped them the most. They will now tell other students that in a 45-45-90 triangle, the "hypotenuse is always √2 times longer than the leg".

On the third day, we practiced some more with a "Stands Up-Hands Up-Pair Up" activity.

Directions: I hand out the Special Right Triangles: Hands Up-Pair Up to every student. The first thing that I tell students to do, is to "Write their name at the top of the paper." Next, I tell students to choose only ONE out of the 12 problems to complete. Once they complete the problem, I have them sign their name at the bottom of the box. When they are finished, they stand up, raise their hand ,and look for somebody else who has their hand raised. Students will switch papers with the person they pair up with and choose a problem on the other persons paper to complete. They will sign their name in the box that they just completed. Students need to get back their ORIGINAL paper so they can check over the persons work. Once they are finished checking work, they will repeat "Stand Up-Hands Up-Pair Up" and find a total of 11 different people to switch papers with.

With this lesson, I joined in on the fun and switched papers with students too. I was able to gain a better understanding of how well students understand the problems and to see which problems were giving them the hardest time.

All of the foldables, activities, and homework can be found when you CLICK HERE.







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My Favorite: Converse of the Pythagorean Theorem

Today was a GREAT teaching day (on a Friday)! As students walked in the classroom, this "Who Wants to be a Millionaire" question was posted on the board.



I told students to take out a piece of scratch paper and write down their answer. Students were engaged the moment they walked in the classroom. After about 3 minutes, I collected all of the answers and played the video. Students were so surprised that this was an actual question/video on "Who Wants to be a Millionaire?" I definitely wish there were more Geometry questions like this out there. If anybody has any, please share :)

After the video, we went over the Converse of the Pythagorean Theorem through this foldable/sorting activity. I completed 2-4 problems with the whole class (depending on their level) and I showed them how to get through each step to determine the type of triangle the given lengths will make.



After we completed the examples as a class, students finished the rest of the cards by themselves. Students were engaged and by the end of the lesson, they understand the Converse of the Pythagorean Theorem. When students asked for help, I sort of "sing-songed" the steps to help them remember it. Some loved and some hated this, but I think it stuck with them :). Once they were finished, they raised their hand, and I gave them this homework assignment.


Overall, I am VERY pleased with this activity, because over the past two years, my students seemed to struggle with this topic. I felt like this day went by EXTREMELY fast, so I will definitely be using this lesson again in the future. Here are all of the files that I used in my lesson: CLICK HERE

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Creating Task Cards with QR Codes

Task cards with QR Codes is great to use in the classroom. This works with BYOD since students can have their phones out and use the free QR code reader app to check their answers.

Here are the directions to create your own QR coded task cards:

Step 1: Open up my task cards template here: Click Here for Task Cards Template

Step 2: Click on "Insert" in the title bar and click on "Text box." A text box will appear to allow you to decide where you want your question at.


Step 3: Type your question that you want on the task card into the text box. 



Step 4: Go to your browser and type in "QR code generator". I normally click on the first/second link to create my QR code.



Step 5: Click on "Text" on your QR Code Generator website and type in the answer for your question. For example, my question was "Solve for 'x': 3x - 2 = 10" so I will type in "x = 4" on the website since that is the answer to my question.  When you are finished typing in your answer, press "Create QR Code" and then when your QR code is generated, press "Download".
Step 6: Your QR code will be in your downloads file or it may pop up on your screen if you have a MAC. Next, right click on the code and press "copy" and then press "paste" it onto your task card (wherever you prefer it). Now, your task card is complete and you can repeat until you have your desired amount of task cards.

I hope you found this tutorials helpful, if you have any questions, please comment on this post.

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Surface Area of Prisms


I use task cards at least once every two weeks in my classroom. I never use them the same way twice since there are a variety of ways to implement them into the class. Students can download a FREE "QR Code Reader" App to check their answers. Here is a list of my favorite ways to use task cards in the classroom:

1) Speed Dating: Arrange desks in two rows and each student receives a card. They will work the problems on the worksheet and scan the QR code to check their answers. Then, the students trade cards with the person they are sitting across from and they will work the problem on the new card that they just received. The "expert" to the problem is the person who originally received the card. After a few minutes, the students receive their original card back and only one row will move down a seat. This will be repeated until all cards are finished.

2) Stations: Place 3-4 cards at a designated station and have students move in groups until they complete all stations.

3) Stand up, Hand up, Pair Up: Give a card to each student and have them work out the problem on their answer sheet. Once a student finishes, they must raise their hand and find another person with their hand raised and switch cards with that person. They will solve the new problem on their worksheet and will continue to "stand up, hand up, pair up" until they are finished with all task cards.

4) Card Game: Pair students up and give one person evens and on person odds. Have them work out the problem and then switch cards when both are finished. Have them compare their answers and repeat until all cards are finished.

5) Interactive Notebook: Print 8 task cards to a page and have students tape down one side of the card into their notebook and allow them to work out the problems underneath the task cards so they have an extra reference sheet/practice.

Helpful Hints: Print task cards on card stock and laminate for durability. I know some teachers laminate them so students can write on them with an expo marker for easy erasing. 



You can download these free color/black and white task cards when you CLICK HERE. Please let me know how these task cards worked in your classroom. My next blog post will be how to create your own task cards with a free template that you can use. 

UPDATE: I just uploaded my volume of surface area of prisms: CLICK HERE


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Special Right Triangles Color By Number

I LOVE coloring activities. I teach mostly on-level students with 5 out of my 6 classes being in class support (special education). Coloring is a stress reliever for both students and adults, so why not incorporate it into the classroom? I am currently working on a right triangles foldable, but it is hard since I really want to focus on where students have the most misconceptions. If you have any suggestions, please share because I will upload the foldable when it is complete. But, you can download this Special Right Triangles Color By Number Activity for FREE when you CLICK HERE. Please leave feedback on how the activity went in your classroom.







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Similar Triangles Foldable

I wanted to share a foldable that I made this past week that I am IN LOVE with :). Students will fill out the foldable for their notes and then after wards, they will sort the cards into the pockets. I am hoping that this lesson goes well, since students struggle with similar triangles. You can download this foldable and activity when you Click Here If you use, please give credit back to this blog and let me know how it goes in the classroom.





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Groups in the Math Classroom

This year I put my students in groups of 4 in the classroom. I will post a picture of my groups soon because I love the set-up. I have always been too afraid of putting my students in groups because of how students may affect my classroom management. However, I was wrong because I will never have my students in rows again. The first day, I allowed students to sit where they wanted (which made me very anxious since I like control in the classroom). I told students that they can sit where they would like but the moment they decided to talk while I am talking, I will stop teaching and move them immediately. All it took was one example from each class to show them that I was not messing around and students completely stopped talking while I was teaching since they did not want to be moved away from their friends.

Students think that teachers will not go through with what they claim they're going to do so they test it and once they know that you are not holding any empty threats, they will succumb to your rules :) (most of the time ;). Having my students in groups have also changed the way I do quizzes. For the past two years, I have used the quizzes that my team make which consists of two separate forms (form A and form B. I used to separate my desks in rows and hand out the quizzes and tells students there is absolutely no talking and no electronic devices. It was like a mini-test and students started to dread quizzes. This year, I came up with an idea last minute since I wanted to do an activity before a quiz and didn't want to separate my desks and put them back together in the same class period.

I decided to make a GROUP QUIZ, where I created 4 separate forms of the quiz (Form A- Form D). I told students they are not allowed to use any notes, only the help of their peers. The results were amazing due to the amount of talking among students about the material. They were willing to ask questions (when they would not ask me) about the material and how to do it.

Here is an example of one of my group quizzes over triangle basics that you are more than welcome to use and try in the classroom. Please do not upload to any websites and tell me how this works in your classroom.

Click Here for Triangle Congruence Quiz






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Algebra Proofs Cut and Paste Activity

I have an activity that I would love to share because of how successful it was in the classroom! I teach about 50+ special education students in Geometry and after this activity, students fully understood algebra proofs. This activity is differentiated since I gave my advanced students all the scrambled statements and reasons with the 10 problems, while I gave my lower students the specific statements and answers to each problem. I had four classes make posters and two classes make algebra proofs books. The results were awesome...





Feel free to download and use in the classroom.
You can find the Algebra Proofs Cut & Paste Activity when you Click Here


Please leave comments on how this activity went in your classroom.

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