Teaching with and through Maps

Why Scale Matters: Borges and 1:1

In 6^th grade, students are introduced to the concept of ratio in the form a: b. They learn that a ratio is a comparison of two or more quantities and how to read the colon as “to” to describe the relationship between the quantities. In 7^th grade, they discover that the ratio notation can be used to represent scale. The meaning of the word “to” in 1:1 as relates to scale is not as intuitive as it seems. Given their familiarity with ratio notation, when presented with scale written 1:1 and asked to interpret it, students tend to write an equivalent ratio, 2:2, 3:3 and so on, but the greater point in such a ratio, as relates to scale, is that the representation is the same size as the actual object being represented. Inevitably, because of their experience with creating part-to-whole ratios from part-to-part ratios, a few students will create the ratio 1:2. For instance, given the part-to-part ratio, 1:1, to describe the relationship between apples and bananas in a fruit basket, a student can create the part-to-whole ratio 1:2 to represent that there is 1 apple out of every basket in this scenario containing 2 pieces of fruit. This partial understanding can lead students to deduce that they can also create part-to-whole ratios when they are presented with scale written in familiar notation. 1 cm: 10 km, however, cannot be rewritten as 1:11; there is not 1 centimeter for every 11 centimeter-kilometers.  Students must transfer their knowledge of operating with comparing objects to operating with comparing space. Coming to an understanding that 1 cm on a drawing represents 10 km in the real world, they are moving between two different dimensions, from 2D to 3D space and back again.

The surrealistic short story, On Exactitude in Science, by Jorge Luis Borges can help to illustrate the need for scale in moving between dimensions. While the structure of the one-paragraph short story may be difficult for some students to decipher given the sporadic capitalization and diction, and may require some adaptation for accessibility, tasking students with making the map to which Borges refers will be a useful exercise in understanding the necessity for scale. Borges describes a ridiculously huge impractical map that resulted from cartographers’ exactitude in representing 3D space on a 2D map. To help students uncover the flaw in the Borges map, the teacher should have students consider the following questions: What materials would be needed to make a map that follows a 1:1 scale?  How much paper would be needed to map the length and width of a 2 ft by 2 ft tabletop in the classroom if the map created uses a 1:1 scale? Consider an amusement park map that is the same size as the amusement park. How could you plan which rides you would like to visit in order in advance of occupying the actual space? Is a map of a place useful if it takes up the same amount of space as the actual place? Students will recognize the purpose served by representing something big with a smaller unit, in that scale drawings make it possible to navigate a space in advance of occupying the actual space.

Further, students will uncover the importance of maintaining the integrity of placement and size of objects in relation to one another in a particular space. In mapping the classroom, for instance, should the bookshelf take up more of less space than a student’s desk? And how do you determine where how far they should be from each other on a map?  The necessity of a scale becomes apparent for students after they experience failure with the Borges task. They discover how a bird’s eye view makes it possible for a map user to see how locations within a space are connected and can plan their future maps accordingly. The context of this exercise better situates students to make assertions when presented with other representations of scale.