Why I hang orange plastic golf balls from the ceiling of my classroom. A favorite teaching moment.
If you walk into room 219 at South High School you may want to duck if you are my height (5’11”) or taller. Hanging from the ceiling all school year have been 3 orange plastic golf balls. (note you can take a video tour my classroom from this fall HERE). It is not until March that my student find out why they’ve been hanging there all school year. When I start talking about them in class I am always surprised to find about half of the class has never noticed them all school year. During the fall and winter I do have several students ask me why they are there, and each time I say….”Wait until later this school year, you will find out”. A couple of students asked me several times this year when we would find out – they could not handle not knowing.
I wait until March to reveal the reason behind the orange plastic golf balls hanging because this is when my districts Algebra 2 Advanced Solving Unit falls in the calendar. In this unit we solve systems with more than 2 variables as well as systems of inequalities. Since I teach a large number of students with interrupted educations I do take a day or two (and some homework assignments) to review solving systems with 2 variables (for many they are learning this for the first time). One of the goals of this review is for students to describe what the solution to a system of equations looks like graphically. The homework the night prior to this lesson is this one – 6M What do solutions look like in 2D homework. This homework refers to a ‘GREEN SHEET’ – every unit I give students 1-3 pieces of cardstock with reference material on them – this is entirely different post I need to do sometime -but the green sheet referenced in this homework is found in this document I give at the start of the unit 6a Daily Practice 2017 v2 if you are interested.
I start this unit with this visual made up from lots of pictures I’ll use in the unit and ask students to predict what we will learn about this unit. I do this in every unit to get students predicting what we will do. Several students notice the orange golf ball and associate it with the ones we’ve had in the classroom all year.
OK – that is enough background – my lesson with my suspended golf balls around my room has resulted in some of my favorite classroom moments ever. I have 5 questions I use in my personal planning of lessons. 2 of them are related to why I use golf balls…..
During a Algebra 2 team meeting a couple of years ago I was wondering aloud how I could have students think about graphing in 3 dimensional space. In the past I’ve done a pretty decent job using these 3D models from this post – but then my colleague – Rob Rumppe (he teaches pre-calculus to our students) – said, “Well, I hang a ping pong ball from my ceiling and ask students to describe where it is located”. I looked at my friend Morgan and this is the lesson we put together from his short sentence to us….
THE GOLF BALL LESSON
This was the prompt for my students as they walked into my classroom.
I waited for every student to write something. While they were writing I was watching what they were doing – where they where looking. I was looking for students to call on later in our discussion. Some students struggle to begin writing, but I just keep saying “You can describe it however you like”. During this time they also were looking at solutions to the homework linked above.
I then do a 60-90 second review of previous work and remind them what solutions in 2D look like. Then I ask them what they think a solution in 3D space looks like? I use a system we have already solved in class. We had started solving systems with 3 variables at this point, but had not yet talked about what the solutions we were finding looked like on a graph. Students shout out some answers, but for the most part they have no idea. The most common thing they shout out is the intersection of 3 lines. I don’t let this discussion go too long…..then…
I take 4 minutes to define ‘dimensions’. The conversation goes something like this…
- I stand on a point (circle of paper) I’ve attached to a line on the floor of my classroom. “Imagine I have no height, width or length. Where could I move if all I had in my world is this circle?” Students say ‘no-where’. “Could I reach Jose (seated in the back of the room) and give him a high five?” Students say ‘no’.
- I then show the next visual and tell my students I now live in a 1 dimensional world like the line on the floor. I model shifting left and right across the line. I say “I am inordinately short – what directions can I move?” “Can I give Jose a high-five yet?” I hold up a meter stick as a representation of a line. We talk about the fact to be a line it would need to go on forever on both directions. We talk about the first dimension is doubling the size of the previous dimension and connecting vertices.
- I quickly move on to the next visual. “I’ve added a 2nd dimension. This is the dimension you’ve done most of your mathematics in before today” I hold up a piece of paper and we talk about it being a representation of a plane (though a plane would go on infinitely in all directions – we talk about the paper cutting through the ceiling and floor and through the core of the earth….) Students notice again that we doubled the line segment from 1 dimension to 2 dimensions and connected the vertices. “What directions can I now go?”, I say as I shuffle left and right across the line and jump up and down? “Can I give Jose a high-five?” – Nope, not yet.
- On to 3 dimensions…. If we take two 2D shapes and connect all 4 vertices we get a 3D shape. I say ‘What directions can I do now?” “Can I reach Jose?” YES! I move left right – up down and forward and back. I say “this is the 3D world we live in in real life….but most of the math you’ve done has existed in books or a piece of paper – in math you’ve become use to only 2D world and have yet to visualize algebra in 3D. That is what we will be doing today.”
- I then have students visualize the fourth dimension. We talk about a hypercube (teseract). I tell them that the movie that freaked me out the most in life was one that tried to use computers to visualize the 4th and 5th dimension. My 3D mind could not wrap its head around it (I always have some of my top student and most creative students ask me at the end of class about this video – that I have no idea where it is). Here is a copy of this visual that students glue into their notebook. 4 DIMENSIONS
- I then show them the 4 dimensions a second time with a few GIFS like these….
- We then look at these GIFS and pictures and connect them to solutions in mathematics. This looks something like this….2D is connected to their homework and finding a coordinate point that is the intersection of 2 lines in a plane. I then ask what they think the solution to system in 3D would look like? Students still struggle to answer this – though some are now thinking it might be intersections of planes.
- I say ‘Well we need to do some graphing in 3-dimensions. Let’s start from the beginning“. I say get out what you wrote at the start of class (10 minutes ago). I have them stand up (we do something called ‘stand and talks‘ everyday in my classroom) with their notebooks and read what they wrote out loud to a classmate. I then have them sit and give them 30 seconds to fix what they’ve written so far. I then call on students to read what they wrote. I point at the orange golf ball in the middle of the room and say, “Give me directions on how to get there“. I then act out what they read. Their descriptions are horrible. Students laugh together as I fail. I never reach the ball. I get close sometimes but what they write is not good. I’ve done this many times and not yet has anyone given me a good description of where to go the first time. I then call on the student who I noticed at the start of class doing something interesting. This student(s) is the one who either looked at the ceiling in my classroom. I ask this student to read what they wrote and it usually goes something like this. “Start in that corner of the room (pointing at one of the 4 corners of my room) and go 4 ceiling tiles forward, 2 left and 1 down….” I have yet to quite get there with these directions but I get really close. Students always express audible awe with this students idea.
- I then say “Let’s try again. This time I want all of us to write the same exact directions to get there. I will answer only 3 questions before you begin. Think carefully – what do you want to know?” Some student usually asks ‘Where are we going to start?”. I say “Great question – you mean the origin.” I whip out a yellow wiffle ball I’ve attached 3 meat skewers to and spin it around in a location. I say “What other questions do you have?” Students ask “What direction do the axes go?” (or what direction is positive x, y and z). I stop spinning the origin and define this. (Note, for ease sake – I point the y-axis up and down – we do talk about that traditionally in math text books the z-axis will point up and down) I have a student stand up and say “Look at Amber’s head. Her head is all positive – x, y and z are positive – this octant is where all 3 variables are positive. I then say Look at Isabel. What other questions do you have?” They then usually ask, “What unit will we measure in.” I then tell them to look at our classroom floor. Our floor is made up of 1 foot by 1 foot tiles. I hold up a 6 foot meter stick next to the origin and say, “OK, you’ve asked your 3 questions. Write where the orange ping-pong is located. Stand up if you need to.” Students get up and furiously write. I call on a student to read what they wrote. Usually what they write is an ordered triple (even though I did not tell them to do this). We define ‘ordered triple’ and start over another time with another ball in my room. (I have 3 hanging around the room).
- By this time students are feeling good about 3D and I have them get up with a partner and hold a 3D model for graphing I’ve made. You can read more about how to make this with resources HERE. I have students start with their pencils at the origin and graph the following ordered triples.
- We then move to talking about what the graph of x + 2y + 4z = 24 would look like. In the past I’ve used string, push pins and a cardboard box cut open to form an octant and covered in graph paper to do this – but in this class we just imagined this set of points forming plane. What I do does depend on time.
- We go back to the system from earlier in class and talk about the solution to a 3 variable system being the ordered triple – point where 3 planes intersect.
- Here are some visuals students glue into their notebook the next day. (and the word doc What do solutions look like for notebook
- I have LOVED the discussion with my students teaching this lesson. Everyone of them is engaged the entire time. I also have loved how much they understand about solving systems with 3 variables once we have this. They have a visual model to tie solutions to.
I filmed a quick video of my room so you can see my set-up in something closer to 3D than what I’ve talked about above.
Whew – that is a lot in this post. I am going to push send right now (6pm 4.8.18) and clean up mistakes or things I forgot later. Forgive me if this post is a bit messy. I am pushing send now though since I might not do it and I’ve not blogged since September and it is time to get back into it again. Until next time…..
Additional 3D Resources
- GeoGebra’s 3D applet
- Mathpix 3D online graphing
- CPM 3D Plotter
- Books: Flatland, A Romance of many dimensions of by Edwin A. Abbott by and Flatterland by Ian Stewart
- How to imagine the 10th dimension Youtube video
- Understanding 4D – the Tesseract Youtube video
- Imagining the Fourth Dimension Youtube video
- 4D world explained Youtube video
- A beginners guide to the 4th dimension Youtube video
- Matt Parker on Youtube Things to do and see in the 4th dimension and his book of the same title.
Note: If I taught middle school I would still do this lesson with a few tweaks when we learn to solve systems in 2 dimensions. I’ve had the best discussions about 3D and 4D graphing with middle schoolers in my career….even more than HS. They can handle this at that age and it sets them up beautifully for HS.
A final Note: I’d love to say I had intentionally hung up the 3 plastic golf balls all school year to add a sense of mathematical intrigue and curiosity into my classroom but in reality they were hanging up all year because I was too lazy to take them down from the previous school year. That said, I HIGHLY recommend you intentionally hang them up in your classroom at the start of the year – torturing the students with ambiguous answers for 1/2 the school year is well worth the enjoyment and created a sense of engagement when we did get to answering their questions I had not seen in previous years.