Teaching with and through Maps

Activity: What’s Puzzling about Scale?

Materials: Geometry toolkits should always be available to students and includes the following: tracing paper, rulers, index cards, colored pencils, erasers

FIGURE 1. Millionth Map of Hispanic America from Life Magazine Dec. 8, 1941 ¹⁸

Method

The teacher will display a portion of the school blueprint on the smartboard. Students will jot down and share things they notice and wonder about the blueprint. The teacher will question students to highlight features they may have missed and explain that a blueprint is a map. The teacher can emphasize to the utility of blueprints by calling attention to fire evacuation maps posted at classroom doorways.

In triads, students will receive a photocopy of the blueprint and square shaped pieces they will assemble to produce a picture that is a scaled up or scaled down copy. While puzzle assembly is a worthwhile cognitive skill, it is not the goal of the activity so the teacher may scaffold as necessary by numbering the back of puzzle pieces. Groups will receive 1 of 4 differently scaled puzzles for each of 3 rounds. The 4 different types should be distributed equally between groups in each round to deter students from relying on a neighboring group for answers. Students will notice whether the assembled picture increases or decreases in size and will use “scaled up” or “scaled down” in their descriptions. They should predict whether the scale factor that takes the photocopy to puzzle size is a whole number or fraction. Given access to their geometry toolkits, they will discuss how they intend to use tools to measure but should not be given a specific strategy for measurement by the teacher (MP5). Students will find scale factor by dividing the measure of any part of the puzzle by the measure of its corresponding part on the photocopy. Students will record their findings in a table and look for structure (MP7), explaining how to find scale factor (MP8) based on observations. Further, they should realize that a scale factor, k, of 2 doubles an image while k= ¹⁄₂ produces a copy half the original size; k=4 quadruples an image while k= ¹⁄₄ results in a copy that is a quarter of its original size.

After the last round, students will be provided with a copy of the blueprint for inclusion in their atlases. The teacher will ask students to choose a region on their individual blueprint to focus on and enclose the area with a rectangle or square. Students should consider how the space needed to display the region changed when the scale changed. How did the number of square puzzle pieces used to represent that region change when the scale factor changed? Students should express that all scale copies of the blueprint maintain the same features in the same locations, understanding that lines take to lines and angles take to angles in scale copies. Students will journal their reflections in their atlases. They should respond to the prompt: You want to make a blueprint puzzle that is scaled up even higher than the puzzles we investigated. Make a reasonable scale factor prediction to make a puzzle as big as 4 desks. How does the scale factor change if you make one almost as big as the floor?  How big should your puzzle pieces be? Explain why you chose this scale factor using data from the blueprint puzzle activity. (MP3)

In a follow-up activity teachers will provide each group with original copies of a map (a copy for each student) and 1 square piece of a scaled-up puzzle map. Groups will be tasked with determining the scale factor used to create the large puzzle map. Students may request 1 additional adjoining piece at a time as they try to match a region or features pictured in the puzzle pieces with the same region on the original. Once students determine a corresponding part on both the original map and the larger scale copy, they will measure, divide, and determine the scale factor that took the original to the puzzle map.  Students will reflect on their atlas entry containing predictions for  to create a floor-sized puzzle. Was your scale factor prediction reasonable? Why or why not?