In this Warm-up, students examine different circles to assess which one most accurately represents a circle twice as big as another. This will help students when they draw a data representation using circles of proportional size in a later activity.

The purpose of the discussion is to address a common misconception when talking about the size of two-dimensional objects.

Poll the class on which circle they chose. Select students to explain why they chose circle and did not choose circle . Record and display the responses for all to see. If possible, display and refer to the image as students explain their reasoning.

Tell students that circle has a radius that is twice as large as circle 's radius and circle has an area that is twice as large as circle 's area. Then ask students to find the radius and area of each circle.

• Circle : radius of 2 units, area of square units
• Circle : radius of units, area of units

Ask students if they recall what happens to the area of a two-dimensional figure when all the lengths are scaled by a factor of . (The area is scaled by a factor of .)

If any students remain unconvinced that circle is a better representation of a circle that is twice as big and are convinced that using the radius is better, tell them that the effect is even more exaggerated with much greater differences. For example, this image shows circle with a radius 10 times as large as circle 's radius and circle has an area 10 times as large as circle 's area.

In this activity, students write a function to show the relationship between 10 pieces of data and parts of the circles that represent the data on a map. Because the area is proportional to the data, the relationship to the radius requires a square root. The structure of the relationship between a circle's area and radius is important for students to write their functions (MP7).

In this activity, students use data of their choosing to represent on a map using proportional circles. Students decide on some data that can be represented by location, scale the data so that circles can be drawn on a map with area proportional to the data, then draw the data on a map. This process involves many aspects of modeling including selecting variables and deciding how to represent the information (MP4).

A backline master of a blank US map is included with this lesson, but some students may wish to use a global map, state map, or local map to represent their data. Consider having a few of these options for students to use.

In the digital version of the activity, students use an applet to create their data display. The applet allows students to focus on their data and on writing a correct function relating the data to the radius of an appropriate circle. The digital version may be preferable if students would benefit from focusing on the material from this unit rather than on physical construction of circles.