# Review Day with QR code Stations…Points of Concurrency!

Today was my first time really using QR codes in class. So far in class students have found the circumcenter, orthocenter, and centroid separately, but never on the same triangle. In order to review for our test, I wanted students to practice finding points of concurrency, but I also didn’t want them calling me over every minute asking if their equations were correct.

I created three stations, one for each point of concurrency. However, I didn’t write what point of concurrency they were finding. Each sheet started with a part 1. In part 1, students had to find the equations of the sides of triangle, given the three points. No matter what station the student started at they only had to do part 1 ONCE. Students were then given instructions on what I wanted them to do. Each station was a different color and told students to keep track of their colors. When they finished the directions they would scan the QR code and it would take them to a Desmos graph of their specific station. I had the graph organized and labeled and in the order of which they found the information. When students got a difference answer than me this made the process of figuring out where they went wrong. Questions like… did you have the same midpoints, did you have the same slope, are your equations the same… really helped them and myself pinpoint where they were making the most mistakes. Students had to find where the point of concurrency was located in every station and then tell me which point they found. I liked this part because it had them go back and reflect on the process.

In my first class of the day, I only gave each station the instructions and told them to show their work. For the second class, I made an organizer for each station. Step 1 had its own page because that was the same no matter what station they started at. I asked my students if they felt the organizer was helpful. They told me that it was confusing at first, but they felt that it helped them keep their equations organized and that it was a good tool to help them study. I’m definitely making changes to it for next year, but here it is if you want to check it out!

Also here are my Desmos graphs for the circumcenter, orthocenter and centroid answers!

# Teaching Points of Concurrency Algebraically using white boarding, Peardeck and Desmos Activities!

Let me just start off by saying I LOVE Desmos Activity Builder. I’ve used already made ones for my algebra 1 class, but I decided to make my own for this unit in geometry. We just finished up our proving triangle congruence unit and started talking about points of concurrency. My students were introduced to the points of concurrency in our construction unit, but now we were learning how to find them algebraically.  Also if you click on the links it will take you directly to my activities for Desmos and Peardeck!

Day 1 ( 90 minute )

I’ve thrown in algebra concepts all year through weekly problem sets using Delta Math. They did a lot of practice identifying slopes of lines and perpendicular lines by looking at an equation in standard form. So the first day was a refresher of identifying slope in standard form, and writing equations in standard form when given only the slope. We also practiced finding slope and then putting it into stardard form and solving for c.

I do a lot of partner white boarding in my class to do practice problems. In parter white boarding each table of two has 1 white board and one marker.  Partners have to work together and talk about the problem to complete it. This also gives me a great chance to walk around and help students who are struggling.  Students practiced finding equations of lines in standard form when given two points. I embedded a desmos link into my peardeck so students could check their answers with their partner. Students also practiced finding perpendicular lines.

The last problem of the class asked students to plot three coordinate points in their peardeck. My students were confused at first on why I was having them graph three points. Students quickly noticed that the three points create a triangle. With their partners students worked together to find the equations of the lines of the triangle and the perpendicular bisector of each side. I didn’t introduce that the point of intersection was  the circumcenter quite yet.

When I do this next year I would give them more time to practice writing equations of lines in standard form and introduce perpendicular bisectors the next day. Here is a link to my PearDeck for Day 1.

Day 2 (55 min)

Day 2 started with my students checking their homework. I ususally have my homework answers in a google doc that I embed in my peardeck. This allows students to check their homework as soon as they walk in, and have the answers on their computer so they can check at their own pace.

On the next slide, I had a link embedded  student.desmos.com, so they were directly taken to the next activity. I gave them the code to start their Desmos Activity. This is the first Desmos Activity that I created! It was awesome. I also had the activity (and my peardeck) pulled up on my ipad. This allows me see all of my students work while I walked around the classroom. I could also control my Peardeck through my Ipad, so I can control my slides and be anywhere in the room.

This Desmos Activity focused on finding perpendicular bisectors on every side of the triangle and discussed how they were finding the circumcenter of the triangle. At the end of the activity students were asked to check their equations again mine and then they imedietely were taken back to PearDeck. Then on Peardeck we received solving systems of equations to find where our circumcenter is located. I really wanted my students to nail down another form besides slope-intercept and solve systems other than substitution, so I had my students keep their equations in standard form and solve systems only using elimination. It was challenging at first, but my started are really starting to understand using standard form.

Day 3

Day three started the same way with students checking their homework on peardeck and answering homework questions. Day 3 was focused on finding the centroid. We worked on finding the midpoints and I had them check their midpoints compared to mine before moving on. Then I introduced creating medians from the midpoint to the opposite vertex. Student’s were able to practice with their partners and then were able to check their answers on the embedded Desmos.

Day 4

Day 4 was probably my favorite day. After checking homework and answering questions, I briefly introduced how to find the orthocenter via peardeck and then sent students to a desmos activity. This activity scaffoled the entire proccess. Students were able to go through each step and check their work against mine on each slide. Here is a link to my Desmos Activity!

I really like the activity builder for finding points of concurrency algebraically because students can check their work by graphing, can work at their own pace, and I can see their classwork during and after class! Although tricky, students are really grasping writing equations in standard form and finding the different points of concurrency. I loved this unit. It’s so fun to see how geometric shapes work algebraically !

# Radical Jail … Prison Break Edition!

We just finished up our triangle congruence proofs and were moving on to exploring the pythagorean theorem and special right triangles. I noticed that my students heavily relay on their calculators for EVERYTHING, so I decided to include a lesson that explored roots (which I realized they never learned?) and simplifying radicals.

I started the class by giving students a root exploration activity created by Julie Reulbach. I told my students that they could work with a partner and each at least one person needed a graphing calculator. This activity explored the relationships between squauring and taking the square root, and cubing and taking the cubbed root of a number. When my students first saw a cubed root I was asked “Do we multiple 3 by the square root of 5?” It did not occur to be that this was the first time that they ever saw a cubbed root. They worked through the activity and then we discussed the answers.

Next, I told my students to get one whiteboard and one marker for each group of 2. I told them today they were going to learn how to break out of radical jail.  I started out by putting the square root of nine and asking them how they knew it was 3. They told me because 3 times 3 was 9. I then gave them the square root of 8 and asked how we could put this is simpliest form without using a calculator. My students knew that we had to use prime factorization!

I told my students to think of the radical symbol as a jail cell and anything inside the radial is in jail. When we are simplifying radials we want to them escape radial jail. When we are helping them escape we use prime factorization to break up the inmates into prime people. I had my students circle every prime person. We talked about how a prime person could NEVER escape radical jail alone. The index tells the students how difficult it is to break out of the jail and how many prime people there need to be escape. When we were taking the square root I told my students that they need two prime people who are the exact same in order to escape radical jail. I had students put a box around the groups of two. If a prime person didnt have a partner in crime then they were stuck in jail. If they did have a person only one of them made it out alive but they needed each other to make it out of jail.

My students loved this! I put problems up on the board and told them to help them get out of radical jail! I had them work with the person next to  them on their whiteboard. I love only giving each group one marker. It makes sure that they are talking about how to figure out the answer and not just working indivually.

I used this lesson to lead into using the pythagorean theorem to find the relationships between the sides of special right triangles! Overall, it was a pretty fun and successful lesson! Let me know if you have any questions about escaping radical jail!

# Survival Guide for Proving Triangle Congruence

I am constantly AMAZED on how create my students are.  I occasionally try and throw in an assessment that is NOT a test because I have a lot of students who don’t test well.  We just finished up our unit on proving triangle congruence, and I decided that my students would make a proof survival guide for their assessment. I gave them two and a half class periods (the same amount of time I would have spent reviewing and testing)  to work on the project. Many students also worked on it over the weekend and at home.

• Introduction:
• Introduce your project. Talk about your experience with proofs (if you like them, dislike them, easy, hard).
• Discuss what you are going to talk about in your guide.
• Model Proofs (with helpful hints)
• Prove triangle congruence by the following.
• SSS
• SAS
• ASA
• AAS
• CPCTC
• For the “helpful hints” students had to provide an insight into their thought process when solving proofs. If someone was trying to solve their proof what type of things would they have to think about?
• Practice Problems
• Students had to provide 5 practice problems  for proving triangle congruence. Students also had to provide an answer key for the practice problems.
• What doesn’t work?!
• In this section, students had to explain what pieces of information wouldn’t work for proving triangle congruence (SSA and AAA). They were required to provide pictures to help their explanation.
• Triangles in the Real world
• Students had to research where triangles were used in the real world. They also had to research which type of jobs would require knowing properties of triangles and triangles congruence.
• Conclusion/Reflection
• In this section, students had to provide a thoughtful reflection about the proof process and their project.

I provided my students with these guidelines and also a grading rubric with my expectations. They could present their survival guide in any way they wanted to. I provided poster board, markers, and construction paper during class. A bunch of students created their project on Keynote, pages, and powerpoint. Others created their survival guide in a small notebook, huge poster, FBI secret files, scrapbook pages, and many more!! I’m always so impressed when I have students do projects. I love giving my students the autonomy on assessments. Even the small the choice of choosing how they want to present it inspires so much creativity that I normally don’t see in the classroom!

Below are pictures of a few of the projects and here is my rubric and guidelines for the project! Let me know if you have any questions!

# Using Edpuzzle in the classroom

I was missing class last week for a conference, and I didn’t want to lose a day with my Geometry class by giving them busy work. We’ve been working on parallel line indirect proofs by working through Parallel Line Land mazes.  While I was gone I wanted to introduce them to parallel line direct proofs. This was a long block day (1:30 class), so they had plenty of time to be productive. Our schedule rotates, and it happened that I would be in class for 2 sections of geometry on Tuesday, but miss one section on Wednesday. I decided to try out my plan with my Tuesday classes and then tweak my lesson so my students on Wednesday could do it without my help.

I was slightly familiar with using EDpuzzle but never used it in class before. Edpuzzle is a tool that is typically used in a flipped classroom. You can upload videos, add voice overs, comments, and questions for students to answer.

I didn’t make my own video, but found a one that walked students through a formal proof. During the video I created scaffolding questions. I wanted my students to be engaged in the video. Just like with flipped classroom videos, the length of the video should be 7 minutes tops! If you need the video to be longer you should split it up into multiple videos. I noticed that students zone off if it the video is too long.

At the end of the video, I added a comment with directions to practice proofs on different websites. The first site had a proof and then had students fill in the last “reason” of the proof. The second site was a little more difficult.  Students were given “the givens” and what they were trying to prove. They were also given a list of statements. They had to click on the statement that should come first and then a list of givens popped up and they had to pick which one went with the statement. This really confused my students when I wasn’t there to help them, but once I explained the concept they understood.

After they finished working through the links,  I  added a place for students to tell me how they felt about the video/websites and if what questions they had.

For homework students had 3 proofs they needed to work through. I created another edpuzzle for the next class (I was out two days) of me explaining the homework, so students could watch it and ask me questions while I was out.

After seeing the questions by two classes had about the activities, I typed up a sheet answered all of the questions that I was asked.  I wanted to make the activity completely doable. I left a huge sub  folder with materials for my students. However, my sub somehow did not get my sub folder and my students were in a PANIC trying to figure out what to do. This activity also required students to try and really think about proofs, so students had a difficult time understanding that they could struggle. “Productive struggle” is a phrase we use daily in class, however, without my direction sheets and without me being there to help, students felt completely lost 😦 I would recommend not using this for a sub plan. It worked so much better when I was there!

However, I really do like edpuzzle. It was a great way to get feedback and give video lessons or just a review of homework. You can set due dates for the completions of the videos, but students can watch the video whenever and how many times they like!

# Dance Dance Transversal!

I really wanted to teach properties of parallel lines through investigation When I was working on my constructions unit using GeoGebra I found this great scaffolding worksheet for properties of parallel lines. I decided to try it so my students could discover and play around with parallel lines cut by any transversal. Some students figured out the sheet very quickly and others were extremely confused. I had students work on their own on GeoGebra but could work on the questions with a partner. This helped clear up a lot of confusion and created a lot of mathematical discussion.

After they completed the geogebra investigation, all of my students got onto Pear Deck. I started my Pear Deck lesson with the question “what is a transversal? ” All students answered and I was able to scroll through their answers to see who understood the activity. We then came up with a definition together. Next I was able to discuss corresponding angles, alternate interior, alternate exterior, same side interior and exterior, and vertical angles. I wanted to see if my students understood where these angles were from the previous activity so I had them shade in their angles on Pear Deck. This lead to the discussion on why certain angles were there and which ones were congruent or supplementary. At the end of the Pear Deck I had students rank their understanding of the lesson and ask me a question. I did student takeaways so the entire Pear Deck lesson was sent to each of the students google drive, and I was able to go in and enter comments to answer their questions. SO COOL.

Day two of this lesson was introduced to me by Julie Reulbach. She suggested that I do Dance Dance transversal. In order to this I had to make 10 dance floors around my classroom (two parallel lines cut by the transversal) using tape. I suggest using painters tape! SO much easier to take off!

I wanted to clear up any misconceptions before we danced so I had students work with a partner and find a dance floor. I gave them each scraps of paper and they had to use them to label each type of angle. This gave me a chance to walk around and help students who were struggling and to see who really had a firm grasp on the lesson.

When they all finished we trashed the scraps and had 10 rounds of dance dance transversal!! We had 10 rounds and partners switched every other round. This made properties of parallel lines so much fun!! Check out my instagram for videos of my students dancing!!!

# Making Math Stations Easy By Using Folders!

In Geometry, we have been working on even/odd proofs. Some students were picking it up extremely quickly, but others were struggling (mostly over thinking everything)! I wanted to create an activity where students were able to work at their own pace to practice even/odd proofs. I decided that I would try out creating stations. I created 9 even/odd proofs ranging in levels of difficulty. I allowed my students to work with a partner and use their notes from our class before.

I started class by handing everyone proof #1 and a list of algebraic properties and explained the rules of the day.

1. Work through proof #1with your partner. If you get stuck look at the examples in your notes. If you still don’t understand call me over for help.
2. When you finish find the folder with the same color as your proof labeled “#1” Open the folder and check your answers. If it doesn’t match up where did your proof go wrong?
3. Once proof #1 is perfected find the folder labeled #2. Paper clipped to this folder is your sheet for Proof #2. Go back to your seat to complete it.
4. Repeat until finished Proof #9.
5. When finished turn your proofs into a “proof book”
6. ALL proofs must be completed before class tomorrow.

This activity was GREAT. It took a lot of prep, but students were able to check their proofs right away and work at their own place. Some students finished all of the proofs in class while others still had a few left. The ones who finished in class walked around and helped students who were struggling. For the students who did not finish had to complete the proofs for homework. I posted the answers to the proofs online, so my students could check their answers. Each proof was written in a different color, so it was easy to decipher which proof students were struggling with. It also made for a colorful booklet 🙂

These stations were great! They weren’t the typical stations where students rotated from table to table to switch problems, but I think they really enjoyed getting up to check their answers and grab the next problem. I liked having  them work with a partner because they wouldn’t move on to the next proof until both of them understood it.  Although this required a lot of prep, I had to do very little in class. Students were extremely very self-directed and only called me over to ask specific questions. These were questions that they were not able to figure out from their notes the day before. Once students started finishing up and started helping their peers this also decreased my involvement with my students.  By having the answers in the folders the question of “is this right?!” was completed eliminated. This gave me time to walk around and answer essential questions and figure out which students were struggling.

I LOVED using folders for stations. I’ve had students in Algebra 1 make their own stations with folders and worked beautifully. Having a problem on the front of the folder and the answers and work on the inside eliminate the teachers work of having to discuss every single problem. I also liked the folders for stations because it gave students instant feedback and a created a place to keep the papers for each problem.  If you do math stations of any kind I highly recommend using folders! The classroom felt so alive, students could work at their own pace, receive instant feedback, and ask questions!

# Teaching Logic by using Pear Deck

So Yesterday Julie Ruelbach introduced me to Pear Deck. Pear Deck makes notes interactive and it is integrated with google apps. All students sign in through their google drive and all the data I receive from my students can be exported to my to my google drive.

After I saw it demonstrated in Julie’s Algebra 2 class yesterday, I decided I wanted to try it out in my Alg 1 class. I used it briefly in my algebra 1 class to look at graphing functions, and my students LOVED IT. I decided to try in out in my geometry class today. Students follow along and interact in the lesson on their computers or phone.

Today’s Agenda:

Formative Assesment:  What is deductive and inductive reasoning? Provide examples of each. Students responses showed up on the board. I had the Smart Board on student view so every students response showed up on the board, but no one could see who posted what. I could  hide students responses and wait to show them until I wanted them to show up on the board. I could also freeze student responses to keep students from changing their answers. THIS WAS GREAT! I was giving a quiz halfway through class and this was a great way for the whole class to review and for me to see what students understood. This also got students ASKING QUESTIONS and really participating in the review.

Other questions asked: Underline the hypothesis in blue and conclusion in red of the statement.  Students could draw on the statements and their responses showed up on the Smart board. So cool!!

Quiz: After we did a quick review of the lessons from the past few days we had a quiz. I quizzed not using pear deck, but when students finished I had them reopen their computers so we could finish our lesson.

New lesson: During this lesson I introduced Law of Contrapositive, Law of Syllogism. and Modus Ponens. After giving them these laws, we used our knowledge to create proofs on Pear Deck. It was great to see what they could do before we worked through a proof together!!

After they completed one proof on their own, I completed two with them. I then put more examples on my Pear Deck. However, these were not interactive slides. Students grabbed a white board to share with their partner and they worked through different proofs together on their whiteboards. During this time, I was able to walk around and see students understanding of the lesson.

This was the most interactive day of taking notes!! I had students tell me to “do this again! This was fun!!” It was amazing on how engaged they were in the lesson. Students were able to take notes during the entire class, and I was able to see students progress the entire class!

Even better! I can go back and look at every class and every students’ work throughout the lesson! I can also export this onto my google drive! I’m definitely going to use this in my classes more often. Currently, I am using a 30-day free trial but I trying to convince my school to purchase a license. It’s amazing!

# Intro to Logic… Reasoning and Conditional Statements

Logic… OH BOY.  Teaching logic is like introducing an entire new language to my students. The language of math. I started out this unit with a lesson on inductive and deductive reason that was created by Julie Reulbach.  This lesson consisted of a SMART notebook and a foldable handout.

I had students discover the difference between inductive and deductive reasoning by using slips of paper with the following statements, pictures, and numbers on it. I told them to separate the slips into two piles. How are the piles the same? How are they different? Why did you group certain ones together? A common response I heard was “one pile is words and the other is shapes and numbers”. It wasn’t until i had them finish each statement that they started to see the pattern. At the end of the lesson we watched this clip from the Princess Bride. It was great to show them logic in a round about way and good wrap up for the lesson.

Next time I would give students inductive reasoning examples that did not have numbers or shapes. When they did their homework they had a difficult time deciding which type of reasoning was which. My students were also over thinking almost every statement.

So now day two…. Conditional Statements.

So first off… I am OBSESSED with Don’t Panic, The Answer is 42. I’ve found amazing things for my geometry class from her blog.

I used her lesson on Conditional Statements and decided to recreate it by adding my own little touch. I turned the note section of her lesson into a foldable and a SMART Notebook. I started the lesson out with students reading misleading newspaper article headlines to make them see that words can be misleading. We talked about how in math we need to come up with a universal way to get across what we want to say. Before even introducing Conditional statements I had my students write 3 if-then rules that they live by. I then had 3 students to come to the board to write their rules. When I finally did introduce conditional statements we used my students rules to point out the hypothesis and conclusion of each statement. This was good because the students felt by looking at their own rules. I also had students come up to the board and underline and label the hypothesis and conclusion. This got my students up and moving around the room. During a class that is mostly lecture it can be hard to keep students engaged. I tried my best to keep students moving and involved in the notes.

After students became comfortable with labeling conditional statements we moved on to using p and q to represent statements. I told students that mathematicians are “lazy” and wanted a quick way to write conditional statements that made sense to everyone. We worked through a few statements, making them conditional and then writing it in p q form.

Next we moved onto truth values and negations. I introduced truth values the day before but now they were being applied to conditional statements. Students seemed to understand negations pretty quickly. Although taking it one step at a time seemed a little show students seemed to truly understand what was going.

Students were finally introduced to truth tables by given a p and q and then multiple case scenarios. Students were able to see and figure out themselves that the only time the conditional statement was negative was when the hypothesis was false. I thought that this lesson went pretty well even though “math felt like an english class today”. The intro into logic can be tough and confusing, so I hope that my students continue to go into it with a solid foundation!

# Brain Dumping… Test Review

It’s finally time for the first test! I wanted to create a review that was more than a practice test or worksheet . We had an long block today ((1hr30min), so I had plenty of time to do a good review. I started class with a 35 minute brain dump. I handed out large blank pieces of computer and told my students to dump their brains on to the paper with everything we’ve done in geometry. I told them to try and write everything down without their notes before turning to them. Some students used different colors and highlighting to organize their dump sheet.

After the 35 minutes I had my students turn to the person next to them. They were supposed to look and talk about their sheets. The purpose of this was to talk about what they wrote and see if their partner had something they may have forgetting. I gave them a quick 5 minutes break after they traded breaks and when they returned to class they had a new piece of paper on their desk.

This sheet was a graphic organizer that I put together. The first section of the sheet were vocabulary words that I expected them to know and recognize. Students then had to define each vocab word and represent it with a picture of symbol. The next section was a list of constructions they should know how to create on geoegebra. I had them create the constructions and explain to me what they did. The last section was all about points of concurrency. In words they had to explain to me how to create each point of concurrency and tell me what happens to the point in different types of triangles. I felt like this sheet was a guide to helps students study, but it wasn’t as direct as a practice test.

I think that I am going to use dump sheets as a review before every test. Not only does it help students organize their notes on to one page, but it also helps students see what they know and don’t know. Another bonus to our brain dumping is that students can save these sheets throughout the year and use them to study for midterms and finals.