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)
Examples of noticing and wondering
I notice that the data help me to understand the "crowdedness" of the two schools.
I notice that Mountain Heights Middle School is the more crowded of the two.
I notice that I can think of crowdedness as a rate (people per square foot or square feet per person).
I notice that there is more than one possible strategy for making the two schools equally crowded.
I notice that there are many things to consider (besides the unit rates) before making decisions about what to do about crowdedness.
I notice that it might help to imagine "putting the two schools together."
I wonder how the crowdedness of the two schools got to be so different.
I wonder how I can calculate the two unit rates.
I wonder if how the two unit rates relate to each other.
I wonder how the schools could make the crowdedness more equal between the schools.
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.
Creating Something New
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 build out 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.
This Creative Math Prompt leads to conversations and calculations around unit rates: people per square foot (population density) and square feet per person. Some helpful information:
- Students may choose the unit rate that they prefer.
- "More crowded" means either more people per square foot or fewer square feet per person.
- The two unit rates involve dividing in opposite orders (number of people ÷ square feet or square feet ÷ number of people).
- The two unit rates are reciprocals.
- Some students may be able to create and solve algebraic equations in order to solve the problem.
- Students may also solve the problem in other creative ways. For example, they may imagine physically "joining" the two schools and then spreading the population out uniformly throughout the available space.
- The overall unit rate that equalizes the crowdedness is not the average of the individual unit rates. Why not? Which is it closer to and why?