Getting Started 
Misconception/Error The student does not understand the concept of population density. 
Examples of Student Work at this Level The student does not use an appropriate strategy to estimate the state’s population. For example, the student:
 Multiplies the perimeter of the state’s boundary by the population density.
 Correctly calculates the area of the state but divides the area by the population density to calculate the population.

Questions Eliciting Thinking What is population density? What does it mean?
What are you asked to find in this problem? What important information are you given?
You are given the population density as the number of people per square mile. So, approximately how many people live in one square mile in Utah?
How did you determine the population? How did you decide what operation to use to calculate the population? 
Instructional Implications Review with the student the concept of population density. Explain that population density is the average number of people in one unit of area (in this case, a square mile). Ask the student to use the population density to determine the number of people who (on average) live on 2 square miles of land, 10 square miles of land, 100 square miles of land, and n square miles of land. When the student understands that the population can be estimated by multiplying the density by the land area, guide the student to calculate the area of Utah using the diagram. Then ask the student to estimate the population of Utah.
Provide additional opportunities to apply concepts of density in modeling situations.
Consider using the MFAS task How Many Trees? (GMG.1.2) if not previously used. 
Moving Forward 
Misconception/Error The student uses the wrong unit of measure when labeling the estimate of the population. 
Examples of Student Work at this Level The student numerically calculates a population estimate but uses the wrong unit of measure when labelling the estimate.

Questions Eliciting Thinking What were you trying to calculate? What is its unit of measure?
How did you determine the unit of measure of your estimate? 
Instructional Implications Ask the student to reread the problem introduction and review what is to be calculated (i.e., an estimate of the state’s population). Be sure the student understands that population density is given as a rate and the unit of measure for population is people. Then ask the student to include units in his or her calculations. Assist the student in understanding how to determine the unit of the final answer by using a technique such as dimensional analysis.
Consider using MFAS task How Many Trees? (GMG.1.2) if not previously used. 
Almost There 
Misconception/Error The student makes a calculation error. 
Examples of Student Work at this Level The student shows an understanding of the concept of population density but makes an error in some aspect of his or her work. For example, the student:
 Determines the correct area for the state but sets up a proportion incorrectly when solving for the population.
 Calculates the correct population but does not round off to the nearest whole person.
 Makes a computational error when multiplying.

Questions Eliciting Thinking Can you explain your procedure for finding the population? How will this procedure result in the population?
I think you made a calculation error in your work. Can you find it? How can you correct it?
Is it correct to write a state’s population as a decimal? Explain why or why not. 
Instructional Implications Provide feedback to the student concerning any errors made and allow the student to revise his or her work. If needed, clarify that population density is a unit rate that compares a number of people to one unit of land area (in this case, number of people per square mile). If the student incorrectly wrote a proportion, assist the student in rewriting the proportion correctly.
Remind the student that the population density is written as a decimal because it represents a calculation that results in an average number of people per square mile. However, the state’s population should be rounded to the nearest whole person. Provide additional opportunities to apply concepts of density in modeling situations. Consider using MFAS task How Many Trees? (GMG.1.2) if not previously used. 
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 uses the diagram to estimate the area of the state (84,111 square miles) and multiplies this area by the population density, 34.489 getting a population estimate of approximately 2,900,904 or 2.9 million people.
The student shows adequate work to support the calculation.

Questions Eliciting Thinking How much confidence do you have in your estimate? What is likely to contribute to error in your estimate?
To what decimal place should you report the population estimate? What level of precision makes sense?
Is there another way you could have solved this problem? 
Instructional Implications Ask the student to consider how the model of Utah could be improved and how this would impact the student’s estimation. Ask the student to model the shape of another state such as Kansas, research the current population, and determine the population density.
Consider using the MFAS tasks How Many Trees? (GMG.1.2) or Mudslide (GMG.1.2) if not previously used.
Consider using the Illustrative Mathematics tasks A Ton of Snow (https://www.illustrativemathematics.org/illustrations/1794), How Many Leaves on a Tree? (https://www.illustrativemathematics.org/illustrations/1137) or How Thick is a Soda Can? (https://www.illustrativemathematics.org/illustrations/1173). 