Tell students, “Distance can be discussed and explored via incorporation of a variety of mathematical topics. Such a mathematical objective can be presented in the form of a cumulative problem. A cumulative problem simply takes a broad view of a topic and incorporates several different ideas in a very coherent and connected manner. Below is an example of such a cumulative problem:”
Activity 1
Read aloud the following scenario:
Robert is in charge of building a sun roof/awning. The width must be 9 feet. The area covered by the roof should be 108 ft². Approximately 20% of the roof should include a window or windows. Robert can decide upon the shape of the window or windows. He must determine the following:
- the necessary length of the sun roof.
- the number and shape of the windows and dimensions for the given area percentage.
(Answers may vary because there are multiple configurations possible for the windows; 20% of 108 square feet is approximately 22 square feet. That may range from one square window approximately 4.7 feet wide to 22 windows, 1 foot wide each.)
Activity 2
Tell students: “Create a cumulative problem, in which data will be gathered in the examination of distance. You must incorporate percent and unit rate into the problem. Write an article, suitable for a scholarly journal, which outlines your approach to the particular distance topic you are discussing.”
“The plotting of distances on grids and maps is quite commonplace. Some examples might include a primitive video game, archeology dig site grid, the layout of cubicles in a large office workspace, and local storms on a county-sized weather map.”
“Investigate possible needs for plotting distances on a coordinate grid. Look into various professional fields.
1. Make a list of these needs and provide accompanying illustrations, superimposed on an excerpt of a coordinate grid.
2. Find at least one distance per need and explain this distance as related to the context of the scenario.
3. Discuss any difficulties with plotting any distances that you envisioned. For example, if you wished to plot and examine distances, using latitude and longitude coordinates, what areas of concern are there? (horizontal and vertical distances and how to measure them) Does distance between the coordinate points truly represent distances between locations? Why or why not? (The actual distance is related to the coordinate points by the scale of the drawing.)
The components of this activity should be presented with both visual and auditory formats. You will narrate and illustrate your findings, using the animation and record features of PowerPoint, Jing, or Camtasia, depending on the resources available to you.”
In groups of three to four, students will apply understanding, related to distance and measurement conversions, to develop a self-awareness of the reasonableness and appropriateness of the use of estimation in measurement. In other words, as a group, students critically explore and debate when the use of exact and estimated measurements are more appropriate, reasonable, and even desired. Create a list of a minimum of 10 examples. Code each example as either Estimation Preference or Exact Measurement Preference. Students group examples together, according to any categories they might see. Students also provide illustrations for at least three of the examples and a justification for each example. A table can be created, using either Word or Excel. Each group will present its table and provide a brief 5 minute presentation on the topic.
Attend to the idea that humans make preferences every day, depending on certain criteria. For example, a person may have preferences regarding the purchase of a computer. If a person simply wants a processor with Internet capabilities, s/he would likely choose a Notebook™ or iPad™. However, if the person wants a processor with ample memory and ability to read and write DVDs, s/he would likely choose a laptop.
Note: An example of an Estimation Preference would be the desire to estimate the number of light years away a star is from the Earth or another star. An example of an Exact Measurement Preference would be the distance an artery is from a surgery site.
Using the information on the nearest star to the Earth, Proxima Centauri, found at http://heasarc.gsfc.nasa.gov/docs/cosmic/nearest_star_info.html, convert the star’s distance from the Earth to millimeters. Show your process and solution; discuss any difficulties.
Activity 3
Debate the prominence/need for the metric system or the customary system. (Students should prepare a debate similar to the famous e vs. pi debate.) In other words, a student must choose either the metric system or the customary system. The student must provide thorough and detailed reasons to support the choice. It is agreeable that knowledge of both systems is most beneficial. This activity is an exercise in looking at the applicability of each system in the world around us. Where do we see such measurements? What conversions do we often see? and so on. Students should include at least three conversion examples within the debate. Students must prepare a 5-minute presentation to debate their position with other classmates.
For review of the lesson, provide an open forum, whereby students can discuss ideas, questions, and difficulties related to the activities.
Extension:
- Have students create a visual to accompany Activity 1. The visual will include a labeled drawing with appropriate units. The drawing should also be scaled appropriately to show relative sizes.
