# Friendly Fly Houses

- Geometry
- Measurement

What’s the smallest space a community of friendly flies needs?

Friendly flies like to live together. They live in groups of 3 in square apartments. They enjoy having neighbors in at least 2 apartments they share walls with, but they don't want neighbors on all 4 sides. That's just too much.

- What different rectangular apartment buildings could house 33 friendly flies?
- What is the smallest rectangular apartment building that could house 33 friendly flies?
- Friendly flies want flies in at least 1 apartment in every row and column of their building. What is the largest rectangular apartment building that could house 33 friendly flies?

- How many friendly flies can live in a 2 x 2 rectangle? A 3 x 3 rectangle? A 4 x 4 rectangle?
- What patterns do you see?

- Explore rectangles with exactly 2 rows. What do you notice about how many friendly flies can occupy these apartment buildings? What pattern do you see?
- Explore rectangles with exactly 3 rows. What do you notice about how many friendly flies can occupy these apartment buildings? What pattern do you see?
- There are 60 friendly flies in a colony. They like to live together in 1 rectangle (apartment building). They don’t want more than ¼ of the square units to be empty. What’s the smallest rectangle that could house a colony? What’s the largest?

In this task, students look for patterns to determine ways to arrange friendly flies in “apartments.” This task encourages the use of multiples or skip-counting procedures to identify the number of square units needed, and area models to build an apartment building large enough to hold the entire group of friendly flies. Students compare areas to check if they have found the smallest or largest apartment building.

Students may choose to approach this task in a variety of ways.

- If students first identify the number of units needed, they might start with a rectangle with exactly enough units and check it to see if it will work for the flies’ neighbor requirements. When they find that it doesn’t, they will change the dimensions of the rectangle or add additional units as needed to meet the requirements. Exploring patterns related to fly housing placement will help them determine whether an apartment building with fewer units exists.
- Students may also start with an overly large grid and start placing flies in a rectangular array. As they work, they will start to see which configurations are more helpful in housing flies and make adjustments until they find an array that works. They may have to attempt another arrangement to be certain they have found the smallest possible rectangle.

Students may use apps in different ways to explore this problem.

- In the Number Frames app, students may start first by seeing how they can arrange friendly flies in apartments. Skip-counting by 3s, they can place flies keeping track of how many they have housed. They may cross out apartments when they notice the unit has 4 neighbors (as shown here).
- Using the Geoboard app, students can follow the same approach described in connection with the Number Frames app. In either app, they can also start by calculating the minimum number of units necessary and drawing an array with at least that number of units. Next students draw flies in the units until they have housed all of the flies while also attending to the neighbor rules (as shown here). They’ll make adjustments by placing flies differently, trying a different arrangement, or enlarging the rectangle.
- Another approach students might use in the Geoboard app is to just begin the build out of a rectangle, using two perpendicular bands. As they identify patterns when they are placing flies, they will determine what the dimensions need to be (as shown here). They can adjust the original bands as needed.

To extend student thinking related to area, ask: *Is there another rectangle that has the same area as your apartment that would house 33 friendly flies?* To encourage students to generalize patterns of housing flies, ask: *How can you tell that you have the smallest apartment or largest apartment for your friendly flies? How can you tell if other apartments with the same area will work for your friendly flies?*