Ruth and the Taco Cart

I love Dan Meyer's Taco Cart if you have never heard of it, here is the task it is great it is wonderful.

http://threeacts.mrmeyer.com/tacocart/

We were recently doing this activity in Algebra 2, because I love how it uses the Pythagorean Theorem and one of my students asks, "what if you split the angle in half and walked that, because that is what I would do." I thought that would be a wonderful geometry question.

Ruth and the Taco Cart

I posed this question to my geometry students, we weren't really learning about angle bisectors, but we had already learned it as well as trigonometry. It took them about 15 minutes to use pythagorean theorem and find the time it would take. However, Ruth was much more difficult.

After students completed the time section I asked what they needed to find the angle and the distance Ruth would have to go.

There were lots of good questions. I gave the rest of it as homework and only a few were able to do it completely.

For students that did the work, I had them work in a group and work on a couple of extension problems:

• If Ruth didn't want to talk alone, how long would she have to wait for the other to catch up?
• What if she took a different angle?
• Does changing the angle effect the time it takes for her to get to her destination?

The other students continued to work on the problem and were given some of these extension problems as they got the answer.

I loved the use of geometry, I will definitely use this when we are learning trigonometry.

3 ACT Task: Beat The Freeze: Circumference

Midseason form 😎❄️ pic.twitter.com/1i9N8ykeDf

— The Freeze (@BeatTheFreeze) March 29, 2018

The Situation: 
During Atlanta Braves games, one fan has a chance to race "The Freeze." Who is the first to reach the end?

Act 1: First 10 Second of the Video:

The video shows the contestant running the warning track in the outfield. It shows the lead the contestant has over "The Freeze."

Have the students discuss who they think will win.

How can we prove who will win or lose? What would we have to know in order to solve this problem? Are there properties of a baseball field that we need to know before beginning?

Act 2:

At the beginning of the video it shows the distance to the left field wall. The distance to the right field wall is 325. You can use 335 or round to 330. 

The Freeze runs at 22.5 ft/sec

The contestant runs at 19.1 ft/sec

The contestant is given an 70 foot head start.

Act 3: 

Show the full video. 

Extension Questions:

How long can The Freeze wait and still win?

What if the rates changed at the midway point?

What is the biggest lead you could give the fan and win?

If you were the contestant what strategy might you use to win?

Transversal Tag 🏃

One of my favorite geometry activities I did this year was Transversal Tag. I set up the gym so it had the pattern of a transversal, like the picture below:

Students were randomly assigned a number 1-8 and then a tagger was randomly chosen. I am sure this game would have been much better to play with 3-4 taggers and play freeze tag, but we played that if you were tagged you became a tagger as well. Also if you went to the wrong area you could be tagged and become a tagger as well.

The taggers had to decide which angle congruency to say to get the most amount of people. For example, a favorite to choose was alternate exterior angles, because half of them had to run to the other side of the gym. The taggers also had to be smart about choosing ones where the runner will be going. 

Especially closer to the end of the game the taggers had to come together to talk about which one would move the most amount of people and get a specific person out.

Students used the following:

Alternate Exterior

Alternate Interior

Corresponding

Consecutive Interior

Vertical

Linear Pair

It was a quick fun game that would have lasted longer if it was freeze tag, but the students had fun, used vocabulary, and had fun running around the gym for 20 minutes instead of being in class.

Head's Up: Review Game 🗣️

Head's Up! is a game of charades where the person puts their iPad or iPhone on their head and the audience performs and nods down for correct and backwards for pass. For students this was one of the best review games we used for vocabulary for geometry.

Students stood up in front of half the class and used my phone to show the students. This was excellent for students coming up with moving actions for such words as vertical angles and supplementary angles. Students were better able to self define vocabulary words better than previous, plus students had more fun reviewing.

It costs .99$ for the app and an additional .99$ to make your own cards. Well worth the money.

Artist Sol LeWitt and Points, Lines, Angles

Sol LeWitt was an artist born in Hartford, Connecticut in 1928 he was most known for his conceptual art, however in this overview we are going to focus on his Instructables. Instructables are wall art where the artist has to follow a particular set of instructions. Sol LeWitt came up with a large number of different instructions, some he never did himself.

For example Wall Drawing #65 in colored pencil is of follows:

Lines are not short, not straight, crossing and touching, drawn at random using four colors, uniformly dispersed with maximum density, covering the entire surface of the wall.

This is what Sol LeWitt came up with:

This is bad example, because it does not take in the sheer size of the piece. Since it is a wall piece it is so large that you could not fully see it from one spot.

So how does this relate to math?

Sol LeWitt has hundreds of these instructions were he takes shapes such as squares, circles, and triangles. He also loves lines, some straight some not, and vertical and perpendicular angles. So to introduce and apply the first section of geometry points, lines, and planes. We attempted our own Sol LeWitt.

Our instructions were: On a wall surface, any continuous stretch of wall, using a hard pencil, place fifty points at random. The points should be evenly distributed over the area of the wall. All of the points should be connected by straight lines.

I assigned all students a letter and then had them connect to each other, so we only really had 26 points, but our artwork was just as amazing.



It did take a little bit more time than I was planning, but the picture at the top took 8 days to make.

We talked about lines and line segments and this brought up a good conversation about how we name lines. I would ask a student which one is the longest line, but would not let them get out of their seat. So it was easier for the student to name the line segment than point.

I love using art in the classroom and Sol LeWitt's Instructables are an easy way to get art in the geometry classroom.

Below is a PDF with some Instructions to do you own.

Geometry Project Runway

One of my favorites lesson this year was Project Runway. When I think of teaching and learning geometry I think of the most boring lessons, some of the theorems can be hands-on, but to me patty paper is not hands on. It's visual, not interactive or engaging.

We were finishing the very first lesson in circles. Students just learned about chords, radius, tangents, centers, and secants. We reviewed to start the lesson what each vocabulary word was and what it meant.

I put students in groups, you can have them choose their own groups, this class couldn't handle that though. Plus, I was still in my visibly random groups phase (#VRG).

I played the introduction to Project Runway on YouTube. https://youtu.be/cqQrdkv-WTo

I told the students you are given 20 minutes to put together a piece of clothing for a fashion show on Project Runway. You will need to include different vocabulary words from our circles unit as your creation makes its way down the runway. You need to include: secant, tangent, chord, radius.

The second part was that they needed to select a model from their group to model down the runway. The two other designers will point out each of the properties of a circle. At the end they needed upload a picture of model on Seesaw and label each of the parts.

 All of the students finished in the 20 minutes. They worked extremely hard the entire time and even though they had trouble finding a path to go down and work as a group, it was a learning experience for everyone.

Here are some of the models as they go down the runway. The students convinced me to play Beyonce softly in the background.

 

One of the things I need to work on this activity for next year is to give students planning time in the beginning for them to gather their thoughts. Secondly, tell students that they can not write on their project runway clothing.

Geometrical Dodgeball

I am not the worlds greatest geometry teacher. In fact I am not the best in the school that I teach at.

One thing I dislike about teach geometry are the number of theorems and how old school geometry is in the way of teaching it. I know I hear it coming already about patty paper and all the things you can do with it, but it's not the same. There is no real life situation when you have to know the Central angle theorem. 

One thing I have been trying to do more of is getting students outside and playing games. Right now we are learning about polyhedra and their surface area and volume. Right now we are trying to remember the names of different polyhedra with different sides, triangles, quadrilaterals, pentagons, etc. Some students remember it from 7th and 8th grade, but those have become high school standards now in Nebraska. The way we learned this was by playing two groups of geometrical dodgeball, we played this outside, but would work much better in a gym.

The way the game is played is everyone is assigned a number and stands in a circle, one person throws the ball up into the air and calls out a number, when that person touches it they yell stop. All players stop and they get 2 free jumps to get closer. If they get hit, they are out, if it is caught or misses the thrower is out.

The way I mathematized it was at the beginning of the round they had to name the polygon they formed if someone was out and named it first they got to jump back in. I played it with two groups outside and worked well since there were shortened periods.

5 Math Projects for Project Based Learning

At Schuyler Central High School next year we will have a new course in every subject area under what we call "Project Succeed." This will be a credit recovery class where students will undertake 5 projects throughout the year to earn those 5 credits.  If students do 3 of the 5 projects they earn 3 of the 5 credits.  Since this is the first year of the course, I am going to do 5 projects that someone else has done before.  Next year I will try to incorporate different projects that are more authentic.

1. Conceptual Art Project
This project is putting students in the drivers seat of incorporating art and math together.  Students will have the opportunity to put one concept of mathematics into a work of art.  They can create a watercolor, painting, or sculpture.  Students will then write a one page paper on what math concept was described and how it came out in the artwork.

I was thinking of using the timeline of this other Calculus Artwork Project.

2. Making a Math Treasure Map
This project has students using Google Maps to create a mathematical treasure map where students are given clues that lead them to the buried treasure.  Students will use coordinate geometry, and equations and lines to make the map.

3. Paper Folding 
This project has an "art-y" feel exactly like the first one.  Students will use origami to make a 3 Dimensional shape.  You will explore the relationship between surface area and volume. Is one always bigger than the other? Can you make cubes with the same surface area but different volumes?  Students will work in pairs to create either a cube or tetrahedron and then derive the surface area and volume formulas. 

4. Nutritional Math
In the launch activity students use unit rates and proportional reasoning to calculate how long they'd have to exercise to burn off different McDonald's menu items.  They then discuss which they think is a better representation of nutritional information.  Students would then create their own infographic for number of minutes in each activity to burn off a particular McDonalds item. 

Students would then create a personalized menu tailored to their own bodies, diets, and exercise routines.  

5. Three Shots
This last one I want to be more of a data based project where they are collecting and analyzing datat, we will see what I eventually get to.  In Three Shots, students will compute the probabilities of a Memphis basketball team win, loss, or tie when fouled at the buzzer and explore this even further in two project tasks, To Foul or Not to Foul and That Is The Question.

Paper Doll Math

I have recently blogged about Ada Lovelace and her paper doll which could be used as a history piece in a history of mathematics center that students can learn about mathematics and how math was developed back in the day.  Today, are some activities that you can use in your classroom that revolve around paper dolls and mathematics.

Patterns: Figures alternate, for example: right arm up, left arm up, right arm, left arm.

Reflections: Students love that every doll is a flipped copy of the one next to it.  Technically, that's called a reflection, one of three kinds of geometry transformation students study in elementary school, the other two being rotation and translation.

Powers of Two: Fold the paper twice, you get four figures.  Fold the paper three times, you get 8 figures.  Fold four times, you get 16 figures.  Every fold is a power of two of the figures.

Multiplying fractions: Every time you fold, the dolls become half as wide.  That's a visual illustration of what it means to multiply a fraction, in this case x 1/2.  Or, if the first fold divides the paper in half, and the second fold divides it in quarters.

Learn more here: Paper Doll Math

View other activities and information here:
Math Manipulatives
Geometry of Folding
Probability

Forgetting Proofs

I was reading Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity by Loren R. Graham and I came across a great little quote, but we will get back to that later. The book was like Paul Erdos book The Man Who Loved Only Numbers style of quick writing.  It was a fascinating book with history of some of the most famous Russian mathematicians of the 19th- 20th Century.  This book reminded me of a professor that I had at the University of Nebraska at Omaha.

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A quick quote gave me inspiration to get students up to the board.

He would begin a proof at the blackboard, pause, and then say, "I cannot recall the proof; perhaps one of my colleagues could remind me."  This was a challenge that the class felt obligated to meet.  One student would jump up, go to the blackboard, attempt the proof, fail, and then sit down with a red face.  Another would get up, perhaps a 17 year old, successfully write the proof on the blackboard while the entire class stared enviously, and then sit down.  Professor Luzin would turn to that student, bow slightly, and say "Thank you, my colleague."  Luzin treated the students as intellectual equals, and his teaching led them to prepare for and anticipate coming lectures.

One of them later ask, "Had Luzin [really] forgotten the proof, or was it a well-constructed game, a method of arousing activity and independence?" They never knew.

This small process of accidentally forgetting the proof or answer to an example is a great way to get students up to the board and motivated to do mathematics.  I especially love the part where the instructor bows to the student and offers a sincere Thank you and recognizes the student as an equal, in mathematics you are always trying to get students to enjoy math and approve of their mathematics.

How could you incorporate this small idea of forgetting proofs in to your teaching?  What benefit do you think would this have in your classroom?  How do you do this now in your room?