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Thought for the day: Respond with respect and interest to all "noticings and wonderings" that your students produce.  All ideas matter, whether they connect to personal experience, humor, the concepts that you are teaching, or other mathematical ideas.

Concepts: unit rates; reciprocals of unit rates; population density; proportional reasoning (possibly percentages and/or solving algebraic equations)



I notice that the data help me to understand the "crowdedness" of the two schools.
I notice that Mountain Heights Middle School is more crowded than North Star Middle School.
I wonder how the crowdedness of the two schools got to be so different.
I notice that I can think of crowdedness as a rate (people per square foot or square feet per person).
I wonder how I can calculate the two unit rates.


I notice that there are many things to consider (besides the unit rates) before making decisions about what to do about crowdedness.
I wonder how the two unit rates relate to each other.
I wonder how the schools could make the crowdedness more equal between the schools.
I notice that there is more than one possible strategy for making the two schools equally crowded.
I wonder how much space you would need to add to Mountain Heights School to make its crowdedness the same as at North Star.
I wonder how many students would need to move from one school to the other in order to equalize their crowdedness.
I notice that it might help to imagine "putting the two schools together."



Even though this prompt suggests a specific problem to solve, there is room for students to create and explore. For example, they may create different types of drawings to represent the crowdedness of the schools or to help them visualize a solution. They may create a story for the two schools, explaining how one became more crowded than the other. They may imagine actual floor plans for the schools and use them to inform their decisions about the best way solve the problem. They may also create and solve similar problems of their own—for example, in a school district that has more than two middle schools.

Reflecting and Extending

I notice that the district could solve the overcrowding problem by either building on to Mountain Heights or by moving students from Mountain Heights to North Star.
I notice that it helps to begin by estimating how much (the number of square feet) to build on or the number of students to move between schools.
I notice that “more crowded" means either more people per square foot or fewer square feet per person.
I notice that the number of square feet per person is the reciprocal of the population density (the number of people per square foot).
I notice that I can calculate the two unit rates by dividing the numbers in the opposite order.
I notice that I can solve the problem (of moving students) by imagining joining the two schools and spreading the students out uniformly.
i notice that I can write an algebraic equation to help me solve the problem.
I predict that, in the real world, it would probably not be important to make the crowdedness exactly the same.
I predict that there may be many more things to think about than crowdedness before taking any action to build or move students.
I wonder why the unit rate that equalizes the crowdedness is not the average of the individual unit rates.
I wonder why the unit rate that equalizes the crowdedness North Star’s unit rate than to Mountain Heights’ unit rate?
I wonder how much more complicated the problem would be if there were three schools.


This Creative Math Prompt leads to conversations and calculations around two related unit rates: people per square foot (population density) and square feet per person. Each measures crowdedness a different by related way. For a more structured version of the problem showing detailed solution strategies, see Exploration 7: Grab Bag in my book Advanced Common Core Explorations: Ratios, Proportions, and Similarity.

The unit rates for the two schools are approximately

North Star 163 square feet per person 0.0061 people per square foot
Mountain Heights 110 square feet per person 0.0091 people per square foot

In order to equalize the crowdedness, the district may either

(1) Add about 25,100 feet of floor space to Mountain Heights, or
(2) Move about 107 students from Mountain Heights to North Star.

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