Teaching with and through Maps

Mapping , The Constant of Proportionality

Having practiced naming points and recognizing their relation to one another, students will be prepared to recognize proportional relationships given a graph and extend that recognition to table representations. Key criterion of a proportional graph includes a line that passes through the origin, (0,0). Even when there is no line drawn on the graph, students will be expected to make generalizations about the pattern that plotted points follow and determine whether the points would make up a line that passes through (0,0). Students will test the coordinates plotted to determine whether they demonstrate proportionality. They make use of the ^new⁄_old structure they employed to find scale factor and derive the formula ^y⁄_x to determine if the points plotted on the graph have the same constant of proportionality, k

We have discussed how working with CPOs will strengthen ordered pair location skills and assist students in finding the actual distance between points given a scale and map. There are, however, some limitations to using points plotted on CPOs to demonstrate a proportional relationship outside of scale factor because doing so requires the values on the x and y axis to be related to one another. Drawing a parallel between CPOs and graphs that appear on coordinate planes, students will recognize a graph as a map itself as it maps relationships between two values. In creating a CPO, students drew the coordinate plane help make sense of the map and describe locations. In creating graphs, however, students are tasked with creating a coordinate plane that provides context itself—by graphing on the plane, students are mapping a relationship to help viewers navigate the association between the two values presented on the x and y axes.  With CPOs, the base layer maps and drawings offer context, but without a map base layer, students must rely on appropriate axis labeling to describe the relationship.  For instance, labeling time on the x-axis to demonstrate its relationship to distance on the y-axis.  

Tasking students with mapping data on a coordinate plane necessitates having them consider what kind of relational data can be used to create a table and graph. This type of consideration is implementation of MLR 4- Information Gap wherein some information is excluded from a task to prompt students to recognize and request more information.  By co-crafting questions, in alignment with MLR 5, students will consider what kinds of values could be derived from the maps they study and whether relationships between those values demonstrate a proportional relationship. With access to online interactive ARC GIS census data maps, students can observe how the scale, and thereby, constant of proportionality changes as they zoom in or out.  For instance, in a zoomed-out view, 1 dot represents 92 people but upon zooming in, the scale gets closer to 1:1. In their investigation, students will choose 2 different scale views, create a table for each, map the 2 relationships between dots and people on a graph, and write equations for both lines in the form  .  

Students can transfer their experience with creating tables, graphs, and equations in this context to determine , write an equation, and interpret a situation when presented with a single coordinate on a graph that does not include the simplified constant as its  value. For instance, given the coordinate (4, 42) on an urban planning graph with x-axis labeled distance, in miles, and y-axis labeled number of fire hydrants, students will determine that k= ⁴²⁄₄ or 10.5, produce a representative equation y = 10.5x or and that the point (4, 42) represents that there are 42 fire hydrants every mile. They will reason about the meaning of 10.5 in this context. Are there half hydrants? Should we round up, down, or does “between 10 and 11 hydrants per mile” accurately describe this real-world situation. Students must experience this discomfort with ambiguity, improving their abstract reasoning skills (MP2) to demonstrate sense-making and learning transfer in real-world contexts. Students who remain adamant about needing a definitive answer should be provided with Open Data DC (ODDC) maps showing fire hydrant placement in various DC neighborhoods, allowing them to realize that there is variation in the real world.

In the opening of a subsequent assignment, students will be presented with a map and a graph with one point plotted and untitled axes. Using data from the map, students should reason about what quantities could be represented by the x and y axes. In the next activity, students will be given a table showing time, x, and distance from school, y and a map with stars placed at the school and at various locations on the map with time stamps. After investigating with their group, students should co-craft questions (MLR5) to make connections between the two representations. What does the table clarify that is missing from the map? How can a graph be created to contextualize the information? What is the constant of proportionality for the relationship between time and distance? Students should deduce that each star represents a new distance from the school after a certain amount of time passes. Further, they should label the axes appropriately, plot the points on the graph, and write the equation of the line. Students who struggle with seeing the connection between the table and map should be given the sentence, “Baptiste was at school and walked at a steady pace from the school to the park.” Teachers should invite students to write down and share questions they have about this situation and to investigate if they observe answers to one another’s questions using the map and/or table. Teacher should introduce the following questions if students did not produce them: How long did it take to get from the school to the next star? Did it take the same amount of time to get to the next stop? What does steady pace imply?

Finally, students should investigate data maps and create graphs in their groups with less scaffolding. ODDC makes available many ARC GIS maps suitable for demonstrating how math and mapping are used to inform citizens and bolster civic engagement. The My Out of School Time DC (MOST- DC) map can be used to investigate the number of afterschool programs in each ward. Students can discuss if there is a proportional relationship between the number of miles and afterschool programs or population of children to programs, or number of wards to programs.