# Eighteen Sides

How can you make a group of shapes that has a total of 18 sides?

Kayla drew 4 two-dimensional shapes. She counted a total of 18 sides. What 4 shapes might she have? How many answers can you find?

- What 2-dimensional shapes do you know? How many sides does each shape have?
- What 4 shapes might you start with? What changes can you make to your shapes so that the total number of sides is exactly 18?

- Is it possible to solve this problem using 4 matching shapes? Why or why not?
- Can you figure out a way to solve the problem with 3 of the 4 shapes matching each other?
- A decagon is a shape with 10 sides. Is it possible to draw 4 shapes with a total of 18 sides if one of the shapes is a decagon? How do you know?

In this open-ended activity, students use what they know about the attributes of two-dimensional shapes to make sets of shapes with a given total number of sides. Solving this problem requires students to represent or describe two-dimensional shapes with a focus on the number of sides and to show multiple ways to make a sum of 18 with 4 unknown addends. This activity also provides opportunities for students to make generalizations. For example, students may note that any polygon must have at least 3 sides or that they can take a side from one shape and give it to another shape while conserving the overall quantity of sides. This latter generalization is an exciting opportunity to connect geometric and algebraic reasoning.

There are multiple strategies that students could use to solve this problem. They might represent a set of 4 shapes, determine the total number of slides, and then make adjustments to the shapes to get exactly 18 sides. For example, a student may start with 4 quadrilaterals with a total of 16 slides, then adjust one of the quadrilaterals to add more sides to the set. Or students might start with only 2 shapes of the set, determine the number of sides required to get a total of 18 sides, and then carefully choose the 2 remaining shapes to complete the set. Once they have created one solution, they may make adjustments to that set to come up with alternative solutions. For example, if a student already has a solution set that includes 2 pentagons, they make “take away” a side from one pentagons and “give” it to the other pentagon; this will make a different set of shapes with the same total number number of sides (in this example, a quadrilateral and a hexagon).

Various apps may be used to represent the context of this problem.

- The Geoboard app allows students to make a set of shapes, as shown in the work-in-progress here. One useful feature of the Geoboard app is that it allows students to easily revise an existing shape. For example, a student can change a quadrilateral to a pentagon by pulling the center of a band to a different peg.
- The Number Lines app can be used to show 4 jumps to get to 18, as shown in the work-in-progress here. Only one number line can be shown at a time on this app, so students may stack two sets of jumps on the same number line or open the app more than once to show different solutions on different screens.

As students work, model the use of geometric shape names. For example, refer to “the hexagon” instead of “the shape with 6 sides.” Extend students’ thinking by asking questions such as these:

*How can you represent your thinking with an equation? Where do you see the 5? The 3? The 18?**Do you have any 2s in your work? Why not?**How can you adjust the solution you already have to make a different solution?*