In 1992 (yes, a long time ago) I was given the not yet published version of Interactive Mathematics Program (IMP) to teach. I credit this program to a lot of the teacher I am today. I received phenomenal training on the curriculum over 4 years and went on to train many teachers on the curriculum. My specialty was year 3 of the curriculum (kind of like Advanced Algebra level). In 2004 I left teaching HS to teach middle school and later led teachers at the district level. During the last 12 years I have often gone back and used the rich tasks and resources I used during my IMP years.
One of the units in IMP year 3 is titled ‘Meadows or Malls?’. In the unit students solve a system of inequalities involving 6 variables. In order to solve the unit problem students learn how to solve systems of equations involving more than 2 variables. During this time, someone (I am not sure who) taught me how to make a 3D model out of 6 sheets of graph paper.
I have found this model to be worth the time it takes to make so that my students can begin to visualize graphing equations in 3-dimensions. I have found if students can touch the 3 dimensions it seems to translate to their brain better than even the coolest animations I can now get online.
I kept a couple of these paper models in my middle school classroom because as I taught students to represent linear equations on a 2-dimensional coordinate plane I would always at some point get a student who would ask what would happen if we had a 3rd variable. When asked I would reach for my model and give them a preview to what they would be learning in the future. (of course a discussion of 3D always led to questions of 4D).
Currently in my class we are in a unit on solving systems of equations and inequalities This week we began solving equations with 3 variables with matrices. Before solving a system of 3 equations I found it really important for students to imagine what we were looking for (an intersection of planes vs. the lines or other graphs they are use to in the 2D world). I used several great online animations, but paper can be way helpful to many students – something that they can touch.
Since I have loved this model so much, I thought I would share with you how to make this model and the directions I use so students can make their own model.
DIRECTIONS FOR MAKING THE MODEL:
- Print 6 sheets of this ddocument using 3 different colors of paper. 3 dimensional graphing template Each student will need 2 copies of 3 colors for a total of 6 sheets of paper. I have almost always assigned the cutting of the squares out as a homework assignment since the amount of time it takes a student to cut paper varies. I tell students to cut out each square carefully. I show them what we will be making the next day so that they know what the end result will be. I make it optional to fold each square as HW, but I find about 1/3 of my students come to class with them folded already – especially if I have shown them what the folded up tents look like.
- The next day in class I start by having student fold each of the 6 squares to make ‘tents’. I have students help their neighbors. The better students are at making exact folds and nice creases, the better the end product will be.
I wait until (nearly) the entire class is done folding. Then I partner students up. It is much easier to make this if you are working with a partner. I have them decide whose model they will make first. I have found it most successful to have everyone stand as we do this. Here is the script I use with the class to make
the first model – I make everyone stay with me and not work ahead. I used two of my teaching peers for the photos in this post.
- Person A hold 2 of the PINK colored tents like this (I model making one with them using a student I know might struggle with others):
- Person B pick up one YELLOW tent and place it over 2 of the pink flaps.
- Person A rotate the pink tents so Person B can do the same thing on the opposite side with the 2nd YELLOW tent. (note: I have found it most successful to have every student working with the same colors as I am at the same time so I really structure this
- Person A, open up your pink and yellow model on one end. Both Person A & B – Ask yourselves, WHAT COLOR DO YOU SEE THE MOST OF?
In this case YELLOW – Our goal will be to cover up the yellow and tuck our renaming purple pieces inside of the pink flaps.
- Person B pick up one PURPLE tent. Cover 2 of the yellow flaps with the purple tent and tuck 2 of the purple flaps inside of 2 pink flaps.
- Person A flip the model over and Person B repeat this with the last PURPLE tent. Carefully make everything fit together tightly.
- Now repeat the entire process for the other person.
- Invariably there will be students who excel at putting these models together. I don’t lead the making of the 2nd model, I go around and help students and I assign students who excel to students to help. I have the entire process down to about 20-25 minutes if students arrive with models cut out and you follow a version of my script.
- It is not necessary, but some students will want to add a few staples to their model (or tape) to keep it together.
- Depending on my students, I give every student 3 meat skewers to serve as axes. These are sharp, so you need to model appropriate use with students if you do this. Most importantly I talk about how to use and not use the meat skewers outside of my classroom.
- I love this activity for any shortened class period due to auditoriums. The auditorium schedule at my school is 35 minutes of class time and this would be perfect. I have loved using this activity with my middle school students (especially if they ask about the 3rd dimension) the day before winter break.
- After we make these models I have students write on them by labeling the planes, and axis. We talk about the skewers and paper just being a model of axes and planes which have infinite measurements.
- I then have them put their finger at the origin and graph ordered triples. (note, this works best if you have everyone hold their 3D model using the same orientation, you will hear me say – hold your model so the purple plane is parallel to the floor and you are looking at the yellow plane – I then name these planes using X, Y and Z).
- Here is the list of ordered triples I have my students graph. I intentionally use points with at least one zero so that we can draw them on the model. I end with one that can’t be drawn . (0, 2, 3); (0, -1, 4), (-3, 1, 0), (-4, 0, 0), (0, 3, 0), (0, 0, -2), (2, 3, 4)… and so on. I use a meat skewer to represent the ‘floating points’ by putting 2 round stickers back to back on the sharp end of the skewer. I start the point at (0, 0, 0) and move it to (2, 3, 4).
- In the past I have followed up this activity with another activity where we graph the equation x+y+1/2z=4 in the first octant using a cardboard box covered in graph paper (inside on 3 sides), push pins to represent points and string to model the plane. We then talk about what a feasible solid would look like vs. a feasible region in 2 dimensions – but that is an entirely new post to describe.
A couple of final notes:
- I made this one quick for you with the colors of paper I had on hand. If I had my wish I would not use pink and purple at the same time as they look too similar to some students. The more contrast in colors of paper the better. I have often used white for one color as I have been at schools where colored paper is only available when I buy it. It was totallyy worth the purchase in the past if this were the case.
- Here is the graph paper I use to do this – feel free to change it how you like. 3 dimensional graphing template
- Here are the quick notes I gave to my accelerated 7th grade students last year before I made this model. We were in a unit on solving systems of 2 equations for 2 variables.
- By the way, when I tested my 7th grade students on solving 2 equations with 2 unknowns I threw in a 3 variable problem. I had never taught them how to solve systems like this before but 2/3 of my accelerated students found a solution. Here is one students solution. I was pretty impressed. I am not sure if visualizing what 3 dimensions looks like led to so many students being successful, but in my heart I think it helped.
- I’d love to know how you model 3D, 4D and other dimensions for students. Tweet me @saravdwerf or email me at email@example.com or comment below.