problem solvers: problem Sherri Farmer and Signe E. Kastberg Decoding dimensions of the Dakota SHERRI FARMER places that they can visit or may have visited encourages them to build links between the mathematics they are exploring and the site. Ask, How many of you have been to New York and seen the large buildings? Have you seen large buildings like this one in your own city or travels? How are the buildings you have seen like the Dakota? To see the interior courtyard, use a mapping tool, such as Google maps, that looks down on the Dakota. Find another drawing that includes space for the courtyard on the architectural site. Discuss the shape of the building from the top view: Math is everywhere. These authors found mathematical architectural elements in a famous New York City landmark. On a recent trip to New York City, Mrs. Farmer took a picture of the historic Dakota apartment building. Built in the late 1800s, this landmark has had many famous residents, including singer John Lennon. After she got home, Mrs. Farmer was curious about the Dakota. She had some questions based on the observations she had made while she was in New York City. Problem scenario Dragons and Victorian heads are spaced evenly on a fence surrounding the famous Dakota apartment building in New York City. Many people call these heads the gargoyles of the Dakota. There are 18 heads along each side of the building. They are 12 feet apart. The Dakota is a perfect square. What is the perimeter of the building? In the center of the building is an open courtyard. This courtyard measures 105 feet long by 65 feet across. What is the area of the base of the building? Activity sheets on pages 75 76 have additional questions related to this problem scenario. Classroom setup As a lead in to the problem, visit an architectural site that has pictures of the Dakota, such as http://bit.ly/10cetpv. Showing students What do you notice about the shape of the building? What other observations can you make? What questions do you have? On the basis of their observations of the pictures, have students draw the building's shape from the top and from a side view. Make sure they observe the courtyard (which would be visible from above). As students are sharing their drawings, you might talk about the perimeter of the building (and connect the concept of perimeter to the idea of walking around the building). Provide time for students to wrestle with the perimeter portion of the problem. The activity sheets include a prompt to draw the gargoyles locations on one side of the building. As you circulate around the classroom, ask students to show you how they are marking the gargoyles. When you see that students have finished this task, call them back together to talk about their assumptions and calculations. Ask those who have different positions for the gargoyles to present their answers. For example, some students may place the gargoyles at the corners and mark only 17 segments of 12 feet each; others may have placed 18 gargoyles between corners, leaving a 6-foot distance to each corner 72 September 2013 teaching children mathematics Vol. 20, No. 2 www.nctm.org Copyright 2013 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
on the end of each wall. Be aware that some students may simply multiply 18 12 without a drawing that aligns to this thinking. Be sure to ask students to illustrate how the numbers correlate to their drawing. Asking the class how the presenters arrived at different answers is important, as is highlighting the role of assumptions in arriving at these answers. The discussion of the students exploration of the area of the base of the building should include a focus on assumptions. Help students to understand that making assumptions is reasonable when (1) clear descriptions of assumptions are shared and (2) when some information may be missing. As students explore the area problem, be sure to ask them to label their diagram with the measurements provided. As they work, identify different assumptions they are making. Figure 1 depicts one representation of the courtyard. To simplify their computations, some students may choose to assume that the interior courtyard is a rectangle. Of these students, some may elect to subtract too much for the courtyard when they use the rectangle ILKJ to represent the courtyard. Other students may use the rectangle NORS and thereby subtract too little area for the courtyard. Good questions for these students are, Did you overestimate or underestimate the area of the courtyard? What impact does this have on the overall area calculation for the building? Still other students will try to apply the given measurements using the relationships between segments AB, IL, and MT. If they find from the perimeter problem that AB is 204 feet and they know that IL is 105 feet, they may take the difference of about 100 feet and assume that the dimensions of MTON are about 20 20 feet. This approach illustrates sophisticated reasoning. Students may also reason using the symmetry of the figures they create. As students complete this part of the task, it is acceptable for their solutions to include assumptions and approximations as long as they are accompanied by articulated reasoning. When students have finished their approximations, select presenters to share at least two different approaches and their thinking, and ask the other students to compare their own thinking to those of the presenters. As a part of your summary, ask students to share one insight they have gained from a peer, another they Figure 1 Where s the math? The problem builds from an understanding of perimeter, using fencing, toward connections between three-dimensional objects and twodimensional representations of those objects. Additionally, the problem involves thinking about two-dimensional views of a three-dimensional object a critical link for children to develop. Also incorporated in this work is the role of assumptions in problem solving. Because the Dakota problem does not tell us where the gargoyles of the Dakota are positioned or what shape the interior courtyard might be, students must make assumptions. These will impact the answers that students generate. Assumptions are an essential part of problem solving. Sometimes we use assumptions to simplify our work, so that a problem that seems unsolvable using mathematics can reveal something important about the problem situation. The Dakota problem also focuses on differences and relationships between perimeter and area. Traditionally, some students initially reason that a given area such as 24 square feet might have a fixed perimeter and likewise that a fixed perimeter might be linked to just one area. In this problem, students can begin to see components of perimeter, such as side length and how it may be related to the generation of areas. A C have gained about mathematical assumptions, and yet another that they want to share with you. These important insights allow you to see how the students are reasoning about the ideas of assumptions. I J M n S p t o r Q L K B D www.nctm.org Vol. 20, No. 2 teaching children mathematics September 2013 73
problem solvers: problem Variation To make the ideas in this problem accessible to young children, explore the following questions: What is the area of the base of the Dakota, including the courtyard? What is the perimeter of the H-shaped courtyard? These questions focus on what may be new ideas for students but provide a chance for them to explore rather than simply calculate. Share your students work Try this problem in your classroom. We are interested in how your students responded to the problem, what problem-solving strategies they used, and how they explained or justified their reasoning. Send your thoughts and reflections including information about how you posed the problem, samples of students work, and photographs showing your problem solvers in action by December 1, 2013, to Problem Solvers department editor Erin Moss, Department of Mathematics, Millersvile University, P. O. Box 1002, Millersville, PA 17551-0302, or e-mail her at Erin.Moss@millersville.edu. Selected submissions will be published in a subsequent issue of TCM and acknowledged by name, grade level, and school name unless you indicate otherwise. Sherri Farmer, sfarmer@paramountindy.org, teaches fourth grade at Paramount School of Excellence in Indianapolis, Indiana. She is interested in using photographs of her travels to support the growth of students mathematical thinking. Signe E. Kastberg, skastber@purdue.edu, teaches preservice elementary teachers at Purdue University in West Lafayette, Indiana. She is interested in the development of constructivist teaching and enjoys working with teachers as they support children s problem solving. Edited by Erin Moss, an assistant professor in the mathematics department at Millersville University of Pennsylvania. NCTM s Mathematics Education Trust: Supporting Teachers Reaching Students Building Futures NCTM s Mathematics Education Trust (MET) channels the generosity of contributors through the creation and funding of grants, awards, honors, and other projects that support the improvement of mathematics teaching and learning. MET provides funds to support classroom teachers in the areas of improving classroom practices and increasing mathematical knowledge. MET also sponsors activities for prospective teachers and NCTM s Affiliates, as well as recognizing the lifetime achievement of leaders in mathematics education. If you are a teacher, prospective teacher, or school administrator and would like more information about MET grants, scholarships, and awards, please: Visit our Web site, www.nctm.org/met Call us at (703) 620-9840, ext. 2112 E-mail us at exec@nctm.org Please help us help teachers! Send your tax-deductible gift to MET, c/o NCTM, 1906 Association Drive, Reston, VA 20191-1502. Your gift, no matter its size, will help us reach our goal of providing a high-quality mathematics learning experience for all students. The Mathematics Education Trust was established in 1976 by the National Council of Teachers of Mathematics (NCTM). 74 September 2013 teaching children mathematics Vol. 20, No. 2 www.nctm.org
problem solvers activity sheet (page 1 of 2) Name Decoding Dimensions of the Dakota Dragons and Victorian heads are spaced evenly on a fence surrounding the famous Dakota apartment building in New York City. Many people call these heads the gargoyles of the Dakota. There are 18 heads along each side of the building. They are 12 feet apart. The Dakota is a perfect square. What is the perimeter of the building? In the center of the building is an open courtyard. This courtyard measures 105 feet long by 65 feet across. What is the area of the base of the building? Part 1. Draw a picture of the Dakota from the top view. The following line segment represents one side of the Dakota. Place a point on the line where each of the 18 gargoyles of the Dakota might be located. Use your drawings to find the length of each side of the Dakota and the perimeter. Be sure to explain how you know your answer is correct. From the September 2013 issue of
problem solvers activity sheet (page 2 of 2) Name Decoding Dimensions of the Dakota Part 2. Draw a picture of the Dakota using the top view. Be sure to draw the interior courtyard as well as the exterior perimeter (see part 1). Using your drawing and the information in the problem, find the area of the base of the Dakota. Remember, the courtyard is not part of the area. Be sure to explain how you found your answer. From the September 2013 issue of