A hyperboloid is a mathematical surface with a surprising property. It is curved like an hourglass but can be made entirely out of straight lines. A hyperboloid can be understood by older students as a rotated hyperbola that has a simple quadratic formula. The form has important real-world architectural and engineering applications. In this workshop, students create a fun, flexible model of the hyperboloid from skewers and rubber bands, which will help them understand some of its mathematical properties.

This activity can be done by students of all levels if the mathematics is explained age-appropriately. It is an exciting way to make math visible in the classroom. Furthermore, it works as a team-building project, encouraging collaboration and mathematical communication. It is valuable for developing patterning skills and careful observation. In addition, it helps younger students develop fine motor skills.

The skewer hyperboloids are portable models for students to take home. After mastering the construction principles, the class as a whole can optionally make a single giant hyperboloid to leave on display in the school.

This activity provides rich classroom material for teachers following the Common Core Standards for Mathematical Practice. This lesson also offers cross-curricular connections to art, architecture, and higher-level strands of mathematics.

**1 to 1.5 hours for individual hyperboloids****0.5 hour for optional large group hyperboloid**

**24 Shish kebab skewers per hyperboloid (12 inch long works best)**

**approximately 100 small rubber bands per hyperboloid (e.g., for pony tails or Rainbow Loom, http://www.amazon.com/Dream-Rubber-Bands-300s-Black/dp/B0014CCNJ4)**

**24 or more bamboo sticks, dowels, or broomsticks (or in a large class, two per pair of students)**

**approximately 100 larger rubber bands**

- Chop sticks also work well, but
are slightly shorter than the skewers so result in a smaller
model.

- If you are concerned about sharp tips on the skewers, you can choose to clip the tips off ahead of time with a saw.
- Watch out for splinters from some
brands of skewers.

- This activity works best when students are in pairs. It can be adapted for students to work individually, but would require extra time.
- The instructions are based on
using 24 skewers and seven rows of rubber bands, but you could
experiment with other numbers. Using more skewers will
require more time. The number of rows of rubber bands
should be approximately one third of the number of
skewers. The photo at the top of this page shows 32
skewers and eleven rows of rubber bands.

2. Talk to students about the fact that mathematics is the study of patterns. Various branches of mathematics explore the patterns in different subject matter. The commonality is the desire for mathematicians to discover, understand, and extend patterns. Students at all levels are familiar with many kinds of patterns. For example, they are able to extend numerical patterns when we ask what is the next number in the series 3, 6, 9, 12, ... They are able to extend geometric patterns, rhythmic patterns, etc. Give students a few grade-appropriate examples as a minds-on puzzle activity. Here is an example of a fractal generation pattern:

3. Tell students that they will be creating a simple visual geometric pattern using lines, represented by skewers. Draw on the board to demonstrate the following pattern with two families of sticks. Point out that the ones slanting in one direction are always behind the ones slanting in the other direction. This seems like a simple idea, but we find it is counter-intuitive for many students who naturally want to weave the sticks with an over-under-over-under pattern or are not careful and make random crossings. Being able to visualize and reproduce this pattern is crucial for a successful hyperboloid construction.

4. Ask
students to draw the pattern using two different colors being
careful to indicate how one family of sticks is always behind
the other family of sticks. Ensure each stick crosses
three other sticks, with one crossing at the midpoint and one
near each end. Point out how the pattern can be seen as a
series of overlapping X's.

**Part B****.
Individual Construct****ions
**

1.
Pair up students into teams of two. Hand out 24 skewers
for each pair. Place small piles of rubber bands on each
table.

2. Ask
students to position some of their skewers to represent the
pattern they have drawn. Students should check that one family of sticks is always
behind the other family of sticks.

3. Show
students how to take two skewers and wrap a rubber band around
them three times. Slide the rubber band to the middle of
the skewers and open the pair up to make an X. Demonstrate with a scissoring
motion that you can switch from one way of slanting to the other
way of slanting.

4. Ask a volunteer to come to the front of the class and make
another X.

5. Ensure
both X's look identical. Their slants must go the same
way. Scissor one if necessary if they do not initially
match.

6. Ask the volunteer to hold both X's beside each other with the ends overlapping as in their drawing. Point out that the tips might be in front or behind, but only one way gives the consistent pattern we are looking for. Wrap a rubber band (three times) around the top crossing. Wrap another around the bottom crossing. At this stage you should have a double X as in this image:

7. Tell students to work in pairs to make their own double
X's. Check the crossing on each for consistency.
Recommend that the rubber bands be placed an inch or more from
the ends so they do not easily pop off.

8. Students
may now extend the pattern to twelve X's using the twenty-four
skewers. There should be three straight rows of rubber
bands: one at the centers of the X's, one near the tops, and one
near the bottoms. Ask students to double-check that the
over-under pattern is consistent.

9. When a group has completed a row of twelve connected X's,
double-check the over under pattern and use their model to
demonstrate how to close it into a cylinder. An easy way
to do this is for one team member to hold it upright on the
table and curve it around while the other person places the two
rubber bands that close the cycle. Ensure that those last
two skewer crossings follow the consistent pattern. Ask
all students to do the same, making a cycle of twelve X's.

10. Demonstrate how the top and bottom rows of rubber bands can be rolled towards the center row ("the equator"), making room for another row of rubber bands at the top and bottom of the cylinder. When you do this, new X's appear near the tips, inviting you to continue the pattern with an additional row of rubber bands near each end. At the end of this stage, there should be five straight rows of rubber bands.

11. Continue
to roll the rubber bands towards the center so rows six and
seven can be added. (Students have discovered that if the
structure is worn like a hat by one partner, the other can add
the rubber bands more easily.)

12. The
structure is complete with seven rows of rubber bands.
These rows should be adjusted to be straight and balanced for
optimal aesthetic effect. By this point, students will
discover how their hyperboloid can be flexed to create different
shapes. If it is flattened down to a disk and let go
quickly, it will pop up in the air.

13. Show** ****this
video** to explain some of the mathematical ideas and
suggest a larger construction.

**Part C****.
Group Construction
**

1. As a fun
culminating activity, the class can make a large hyperboloid out
of broomsticks, dowels, bamboo, or any long sticks that may be
affordably available. You might plan ahead to have one
stick per person (rounded to an even number). This photo
shows one made from twenty-four 48-inch dowels that can be
purchased at a hardware store.

2. Begin by telling students how many sticks you have and asking
them to think of a strategy for making a large-scale classroom
hyperboloid. Two natural ideas that might emerge are (1)
to lay the sticks on the floor making the flat X pattern as they
did with the skewers; or (2) to have them stand in a circle and
begin by making the cylindrical pattern of X's. Half the
students can each hold two sticks in an X formation, while the
other half of the students stand between them applying the
rubber bands.

3. After the
class agrees on a strategy, let them take over and build.
Let students decide how many rows of rubber bands make for the
best hyperboloid.

**Further
Explorations**

Variations.

Applications.

Formulas.