Geometry VNPS Lesson: Linear Measure

Last summer I attended the Vertical Nonpermanent Surfaces morning session at Twitter Math Camp by Jennifer Fairbanks and Kathy Campbell. One thing I’m trying to incorporate more this year is having my students work at the white boards. We created white board norms for each class and then went to work today on a lesson. Today’s lesson was pretty basic. I’m still working on trying to create low ceiling, high floor problems. I thought the lesson went pretty well. See what you think…

Geometry: Linear Measure

I.)  At the board:

First Marker:

1.) Draw line segment RT and label it.

    • Label a point S between R and T.
    • Label the distance between S and T 1.6 cm.
    • Label the distance between R and T 4 cm.
    • Find the length of RS. (2.4 cm)
    • Erase.

Second Marker:

2.) Draw line segment CD and label it.

    • Label a point B between C and D.
    • Label CB 2x.
    • Label BD 4x.
    • Label CD 12.
    • Find x.    (x = 2)
    • Find BC.     (BC = 4) How did you find BC?
    • Erase.

Third Marker:

3.)  Draw a line segment CD and label it.

    • Label a point B between C and D.
    • Label CB.  4x – 9
    • Label BD. 3x + 5
    • Label CD. 17
    • Find x.    (x = 3)
    • Find BC.    (BC = 3) How did you find BC?
    • Erase.

Fourth Marker:

4.)  Draw segment PQ and label it.

  • Draw a point Z such that segment PZ is congruent to segment QZ.
  • What is this point called? (midpoint)
  • Do not erase.

Fifth Marker:

5.)  Draw segment LM congruent to segment MN such that M is not a midpoint.

  • What word must be included in the definition of a midpoint? (collinear)
  • Erase.

Sixth Marker:

6.) Point K is between points J and L.

  • JK = x^2 – 4x , KL = 3x – 2, JL = 28, Find JK and KL   (JK = 12, KL = 16)

AP Calculus First Week Activities

The first week is under way and one of my goals this year is to blog more about AP Calculus. I have a wonderful colleague who also teaches the course and he shares a lot with me. So I will be sharing some resources and ideas out with you.

I know there are some different opinions on whether or not to review Precalculus content before starting Limits. We use the James Stewart textbook, Single Variable Calculus: Concepts and Contexts and spend the first 3 weeks reviewing Precalculus content. I have attached three activities we do during the first week of school. I make one copy per pair of students and reuse them each year.

This one is a good review of transformations. Be sure to discuss the differences between |f(x)| and f(|x|).

This one gives fun descriptions of distance time graphs and then students make up a story for the fourth graph.

Finally, this graph situation worksheet is also pretty fun. The students find it easy, but what is more engaging is that we discuss what the “wrong graphs” would depict. We’ve got time travel and kids launching off of a swing into space! Good times.

I hope everyone is having a great school year so far!


TMC18 Recap and my #1TMCthing

I’ve had five days now to reflect on TMC18 and I still haven’t processed the whole event. Life is back to normal at home. I’m a SAHM in the summers so it is hard to find time to do any work (which is not necessarily a bad thing). I have an amazing husband who made it possible for me to attend three TMCs in a row. This evening he was outside with a telescope with my two girls watching Mars rise along with Jupiter, Saturn, and the full moon. I was inside reassembling and re-ironing my daughter’s Perler bead hibiscus flower I had bumped earlier in the evening. Never a dull moment.IMG_9865

The morning session I attended was Discovering Geometry through Drawing and Discussion from Verbal Cues at Whiteboards by Jennifer Fairbanks and Kathy Campbell.  They used to assign us to visible random groups. We played the student role the first day and worked problems at the whiteboard while Jennifer and Kathy gave us instructions from a script. It is important to include in the script when students should switch markers and when to erase in order for the lesson to run smoothly. Students can take photos of their work before erasing if they choose.  Some Geometry topics that lend itself well to this format are: Pythagorean triples, trigonometric ratios, geometric mean in right triangles, special right triangles, and the Law of Sines. The second day we analyzed this method of learning. Doing problems at the whiteboards allows all students to look around the room at other work. If a teacher spots a mistake, try pausing and see if a student can spot the error. Erasing mistakes on a whiteboard is easier than on paper so students are more likely to dive right into the mathematics. On day 3 we worked with a small group to create our own script. Here was our group’s lesson on Area of a Regular Polygon.  My #1TMCthing is to implement vertical nonpermanent surfaces (#VNPS) in my Geometry classroom this year. Last year I tried it, but only had students do routine practice problems. I’m looking forward to my students doing discovery activities at the whiteboards as well. And to have them use a string to draw circles!

I would like to incorporate more playing with math this year. This includes games like Set and tiling puzzles. I was blown away by Amie Albrecht’s session on The Game of SET. I just had to go home and have my girls play with the cards and see how they would sort them.  Games For Young Minds is a great site to get ideas for parents to help their kids develop a love for mathematics. There are not many places I go where people think math is fun. I truly think the world of the #mtbos. There is nothing like it. Your students are incredibly lucky to have such caring and intelligent teachers. I hope to help share the Games For Young Minds and Talking Math With Your Kids sites with other parents so they can help their child develop a love for math too.

Speaking of games–I had enough courage to attend my first TMC Game Night this year and it was really fun. I watched a few teachers play four-dimensional set:


I was also pretty proud of figuring out this tiling:IMG_9930

I also attended Trivia night, and then went to watch the sun set on Lake Erie–it was a beautiful view!


Julie’s keynote, Teacher Leaders, touched me quite a bit. She gave us prompts to tweet out using #TeacherLeader. Why is it so hard to complement someone else but not ourselves? She also put stickers under all our chairs that read,  “YOU ARE NOT AN IMPOSTER…YOU ARE ENOUGH”. I know I’m a good teacher and that there are things that I do that are just as good as the “celebrities” on Twitter. After attending three TMCs, I do feel like I am worthy of having an equal voice on a blog just like the other teachers in the MTBoS. Last year my AP Calculus students’ scores were above the national average. My Geometry A students gave me excellent teacher report cards with several telling me I was the best math teacher they ever had. I had several Computer Science students thank me for giving them the opportunity to learn how to code. A goal of mine this year is to blog more of my classroom ideas.

Screen Shot 2018-07-27 at 10.57.23 PM

Finally, Marian’s keynote, Measures of Center, was a crucial message about equity in the classroom. I was touched by her personal stories and am striving to do better to create equity in my classroom. I’ve already got several questions ready for my students’ back to school survey. I plan on reading For White Folks Who Teach in the Hood by Chris Emdin, and participate in the MTBoS Book Club–Exploring Math and Social Justice. Finally, I was just given the opportunity to attend a workshop at Iowa State University on Equity in the Mathematics Classroom next month. I’m hoping to share all that I learn with my department.

Before I sign off, a special thanks to Megan (@MeganHeine), a former colleague, who attended the conference with me. I’m one of those who is too shy to even wear one of Sam’s “You seem nice…” buttons. We basically talked the entire ten hour drive to and from Cleveland and it was fantastic. We talked about how it is great to just have the unstructured time to sit down with other math teachers and talk about how we teach math. Thanks again! 🙂

A Trigonometry Unit for Geometry

The 2016-2017 school year is about complete, so as I sit here waiting for students to take final exams I am reflecting on the most significant changes I made this school year. The first one that comes to mind is how I reorganized the Trigonometry unit in Geometry. I have taught Geometry for 10 years and I finally feel like my students are finding success during this unit. I even gave a survey at the end of the school year and it was the second most enjoyed topic of the year! That was a major win in my opinion.

Our Glencoe textbook organizes the chapter in this order:

8-1 Geometric Mean

8-2 Pythagorean Theorem and Its Converse

8-3 Special Right Triangles

8-4 Trigonometry

8-5 Angles of Elevation and Depression

8-6 The Law of Sines and Law of Cosines

8-7 Vectors

This unit is quite overwhelming for students. First, I eliminated Section 8-1. Students find it confusing, and I find it unnecessary for the students’ success in future math courses. Next, I split up the unit into two different chapters. The first unit consists of sections 8-2, 8-4, and 8-5. The second unit consists of 8-6, 8-7, and 8-3.

One reason for this change was that in the past once students learned the shortcuts for 45-45-90 and 30-60-90 triangles, they assumed these rules applied to all triangles that looked like special triangles. This posed a major problem when teaching the sine, cosine, and tangent ratios. Once I split the units up, the mistake was virtually eliminated. One thing I would do in the future is to give more practice time on the special right triangles before testing. We derived them, but never spend enough time practicing. Our curriculum requires students to apply these rules later when finding area, surface area, and volume.

A big thank-you goes out to Kate Nowak. I used just about everything on this post to teach trigonometric ratios.  It is an amazing lesson plan so check it out. One modification I made is that when students found the ratio for their angle in the Geogebra applet, they entered it into a Google Sheet that was shared with the entire class.

Kate’s Row Game for Radical Review worked well to start the chapter.

Labeling Triangles Worksheet: I did not make this worksheet, but if you teach Geometry you must have students practice labeling the opposite side, adjacent side, and hypotenuse of a right triangle at least one day before you teach trigonometric ratios. It makes all the difference. I had students use a marker to color the marked angle. Next, label the side not colored the “O”. Then find the longest side opposite the right angle and label with “H”. Finally, the side not labeled is “A”. Students used this method to label triangles throughout the entire chapter, and the process of finding trig ratios was so much smoother. Page 1 of this file was sufficient practice.

I used NCTM’s Illuminations activities to derive both the Law of Sines and the Law of Cosines. Most students can get through the Law of Sines activity without much help, but my students needed an algebra review before tackling the Law of Cosines. I created this review to help. Non-right triangles are still difficult for students, and they still needed a lot of practice. While we were reviewing for the final exam I had one student swear he had never seen the Law of Sines! On the other hand, I had several students use the Law of Sines to solve all triangles, including right triangles. I feel like I could still improve a lot on this topic.

I will conclude by sharing that next year will be the first time in my 10 years of teaching high school math that I will not be teaching a regular Geometry course. I am very sad, but am looking forward to the new challenges of Algebra 1C and Geometry A. I will still be teaching AP Calculus, and I will also be piloting an Introduction to Computer Science course. I would be happy to share my ideas with you. You can tweet me @MAllmanAHS if you have any questions. Have a great summer!

My Love Letter to the MTBoS

I have been a lurker for resources on the #MTBoS (Math Twitter Blogosphere for my non-Twitter readers) since it’s inception. And let me tell you–this is an awesome community. Today I would like to contribute to the MTBoS with my first blog post. In my blog I will share resources from my own classroom, comment on how things are going in my classroom, give ways I incorporate technology, and describe some activities I do with my own daughters (6 and 4) to ignite their mathematical curiosity.

This week I attended my first Twitter Math Camp (#TMC16). For the past four years I didn’t feel I could talk to other math teachers online that I haven’t met in person. After attending TMC I know that I was being silly, but at the time it made me uncomfortable. Since I got home Tuesday I needed a few days to step outside of my teacher self to know my true feelings on what I just experienced. Was it just the energy of being around 200 other amazing, passionate math teachers? Would the clock strike midnight and I would be whisked away leaving all the magic in Minneapolis? Would I forget everything I promised myself I would pursue when I got home? Not a chance.

Around no other person or group of people do I feel so determined to a better math teacher. The positivity by everyone at Augsburg was infectious. Even though I was mentally tired I am now somehow reenergized to begin the new school year. Some organizers love this group so much they stayed up until 3:00am writing lyrics for the TMC song in order to express their love for the teachers in attendance at the closing session.

I am still trying to synthesize everything I have learned. I am overwhelmed, but in a good way. I learned that the other MTBoS teachers are just like me. They are not trying to impress anyone. They are simply sharing resources and discussing math in an attempt to improve our profession as a whole. But more importantly, the members are trying to find a connection with other teachers like themselves that they may not get from daily face-to-face encounters. Some days being a math teacher can be isolating. How many people get excited over a free radian-scale protractor?!?

Like many of you, I currently have dozens of tabs open in my browser with articles and resources waiting to be read. At the conference I figured out early on that my big takeaways from TMC16 would be from the three keynote speakers. Don’t get me wrong. This was a tough decision as I gained so much from ALL of my sessions, especially Demystifying Calculus where I sat for three mornings and we just did math with other math teachers–it was amazing and inspiring. Thanks Bruce Cohen (@mathcohen) for leading a valuable session.

Keynote 1: Jose Vilson (@TheJLV) I know now that I can and should be discussing race in my classroom. After the tragedies going on around our country, now more than ever we need to be discussing race and the discrimination going on in our country. We need to be open to talking about race and discrimination with our students. I am not quite sure how I will implement or approach this yet, but I am open to ideas.

Keynote 2: Tracy Zager  (@TracyZager) One of my goals this school year is to work with elementary and middle school math teachers in my district. Last year I sat in on one 5th grade teacher’s math lesson and it was an amazing experience. We have some outstanding teachers in our district and I am looking forward to reaching out to them. I learned from Tracy that we often times don’t get together because we are scared of each other. We need to step out of our comfort zone in order to grow. I don’t know how many teachers in my district I will get to join the MTBoS but I will continue to invite them.

Keynote 3: Dylan Kane (@math8_teacher) discussed something I’ve always pondered in the back of my mind, but after this speech the lightbulb came on. He shared that just because we find an amazing resource doesn’t mean it will translate to a good lesson in the classroom. We are obligated to plan a well thought out lesson plan with essential questions and formative assessments to follow up. The MTBoS has a ton of resources but it is my responsibility as an educator to combine the resources with sound pedagogy to the best of my ability. I have learned this the hard way as I have had an amazing card sort or other lesson that fell flat because I did not ask good questions or didn’t close the lesson. I will continue to try harder. We all are just trying to improve each day.


Lisa Henry (@lmhenry9): For telling us we are great. Because we are. And for being the lead organizer for Twitter Math Camp. TMC16 was an outstanding experience. Thank-you, Lisa.

Glenn Waddell (@gwaddellnvhs): Your speech was very moving. You said to yourself on your motorcycle trip, “You can give up right now. You can turn around and no one will ever care. And no one will ever know.” I have a similar, but less ambitious, story. I was new to TMC this year. I wanted to attend the newbie dinner, but instead of walking through the doors on Saturday night I cowered in my hotel room. I did not face my fear. However, the next day Alex Overwijk (@AlexOverwijk) asked me to join his Trivia Night team, so I did walk through the doors of Republic on Sunday night, and I had an awesome time. Thank-you, Glenn and Alex.

Finally, thank-you to all of you whose resources I have borrowed…ok, let’s be honest, stolen…over the past five years: You have truly influenced me and impacted my teaching and I am forever grateful.

Kate Nowak

Elissa Miller

Sam Shah

Sarah Carter

Julie Reulbach


Mimi Yang

Fawn Nguyen

Bob Lochel

Mary Bourassa

Andrew Stadel

Amy Gruen

Sean Sweeney

I am excited to start sharing my classroom experience with you. Follow me at @MAllmanAHS . I am looking forward to Atlanta next summer tweeps!