Getting Started 
Misconception/Error The student is unable to write the ratio of the area of the object in the scale drawing to its actual area as a unit fraction. 
Examples of Student Work at this Level The student:
 Finds the areas but does not write a ratio.
Â
 Assumes that the ratio of areas is the same as the ratio of lengths (i.e., 1:8).
Â
 Writes the ratio of an area to a length or a length to a length (e.g., 39:112).
Â
 Writes the ratio of the areas but not as a unit fraction (e.g., 68.25:4368).
Â

Questions Eliciting Thinking What is the area of the drawing of the solar array wing? What is the area of the real solar array wing? What is a ratio?
What kind of ratio are you asked to write in the first question?
What does it mean for a ratio to be written as a unit fraction? How is that different from the ratio 68.25:4368? 
Instructional Implications Explain the distinction between the scale, the scale factor (the ratio of corresponding lengths), and a ratio of areas. Ask the student to calculate the area of the figure in the drawing and the figureâ€™s actual area. Guide the student to carefully attend to the description of the ratio in the problem and then write the ratio in the manner described. Provide additional opportunities to calculate actual areas of figures and their areas in drawings, and write ratios of areas as fractions in lowest terms.
Remind the student that a unit fraction is a fraction with a numerator of one. Guide the student through the process of converting a ratio to a unit fraction and provide additional practice opportunities. 
Making Progress 
Misconception/Error The student is not able to recognize the relationship between the scale factor and the area ratio. 
Examples of Student Work at this Level The student correctly writes the ratio of the areas as the unit fraction 1:64 but is unable to relate this ratio to the scale. Instead the student provides an interpretation of the ratio.
Â Â Â
Â 
Questions Eliciting Thinking Can you write the ratio and the scale factor side by side? How are they the same or different? Can you find a numerical relationship between them? 
Instructional Implications Guide the student to recognize that the area ratio can be thought of as a scale for area. Just as the scale is a ratio that compares a number of inches in the drawing to a number of feet on the actual solar array wing, the area ratio compares the number of square inches of area in the drawing to the number of square feet of area on the actual solar array wing.
Provide the student with pairs of similar polygons (e.g., a pair of similar rectangles, triangles, or parallelograms) along with their dimensions. Describe one as a scale drawing of the other and ask the student to write the scale factor for each pair. Ask the student to calculate the area of each polygon and then write the scale factors and the ratios of areas sidebyside (e.g., 1:3, 1:9; 1:6, 1:36; 2:7, 4:49). Then, ask the student to look for patterns. Guide the student to observe that the ratio of areas is the square of the scale factor. Next provide scale factors and ask the student to determine the ratio of areas and vice versa. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student correctly identifies the ratio of areas as 1 Â to 64 . The student explains that the area ratio is the square of the scale, 1 inÂ to 8 feet.
Â 
Questions Eliciting Thinking Do you think a scale that relates lengths and an area ratio will always be related in this way?
What if the ratio of the area in the scale drawing to the actual area is 4:9? Can you find the scale factor for the drawing?
Suppose the ratio of the area of a square to the area of a rectangle is 16:25. Does this mean that the ratio of lengths is 4:5? 
Instructional Implications Provide additional opportunities to calculate areas of figures in scale drawings and areas of actual figures. Ask the student to relate the ratio of areas to the scale factor and generalize the result (e.g., explain that the ratio of areas is always the square of the scale factor). Then provide scale factors and ask the student to determine the ratio of areas and vice versa.
Provide the student with problems in which the ratio of areas must be determined from the scale factor in order to find the solution. For example, pose a problem such as:
In a drawing of a classroom, 1 cm corresponds to an actual length of 2 feet. If the area of the classroom in the drawing is 300 square centimeters, what is the area of the actual classroom?
Ask the student to imagine a square and a rectangle with areas in the ratio 16:25. Challenge the student to determine if the ratio of lengths must be 4:5. Discuss with the student the circumstances under which the ratio of lengths can be determined from the ratio of areas (and vice versa). 