Responses to the have a heart problem. (Problem Solvers).
1. Predict which number will be larger-the heart's area or its perimeter. Explain why you believe that this value will be larger.
2. What other shapes can you see in this heart? Will these shapes help you determine its perimeter or area?
3. Do you need to use formulas to determine the heart's perimeter and area? Can the perimeter and area be measured directly? What "tools" would help you make these measurements?
4. When you are finished, write down and label the heart's perimeter and area. Then be sure to explain the techniques that you used to find these values.
* Try to make a heart with a perimeter that is exactly twice that of the heart pictured here. Determine its area. How does the area of the bigger heart compare with the original heart's area?
* Create a rectangle with the same perimeter as the pictured heart. What are the length and width of this rectangle, and how does its area compare with the heart's area?
* Create a rectangle with the same area as the heart pictured here. What is the perimeter of your rectangle? How does this number compare with the heart's perimeter?
* Create a square with the same area as the pictured heart. What is the length of each side? How does that length compare with the area of the heart?
We received several responses to this problem that illustrate that there are not only several ways to find the solutions but also different, but correct, solutions.
This variation occurred because classes used hearts of different sizes and many teachers adapted the problem by asking students to measure varying sizes of hearts. Keep this variability in mind as you read through the following responses, and remember that the methods and reasoning that students use are more important than the actual answers.
Students found several ways to determine the perimeter and area of their hearts. Using string to find the perimeter and grid paper to find the area was a common approach. Angel, a sixth grader in Deane Cook's Churchill Elementary School classroom in Galesburg, Illinois, approached the problem as shown in figure 1. Angel's string method for finding perimeter proved successful; several students tried this method and got similar results. The grid-paper method for area, however, produced more variability in the answers because students were not always sure how to handle the portions of squares inside the heart. Some chose to "throw them away," while others chose to count them all. Still, many students developed reasonable strategies to deal with the dilemma. Many of them, like Rachel from Mrs. Cook's class, combined the parts by using "the ones that weren't squares to try to make squares." Others even counted the parts as fractions of squares and determined the total area by adding all the fractional parts. Lydi a from Mrs. Cook's class used this method to determine an area of 15 2/3 or 15.6666667 square centimeters for her heart.
Mary Jo Gonsiorowski of Dutch Neck School in Princeton, New Jersey, pointed out that many students in her class at first tried to use pattern blocks to fill the heart, but after realizing that the pattern-block shapes had different areas, they resorted to other available manipulatives. Some of the tools that they used were string, tape, rulers, grid paper, centimeter cubes, and base-ten blocks. Aneesha demonstrated a solid connection between measurement and number sense in her description: "I had 20 base-ten blocks together and it equaled 200 blocks. Then there were 41 ones blocks, so it was 241 blocks altogether." Kieya, who used centimeter cubes to determine both the perimeter and the area, simply explained, "I put the cubes around the heart to find the perimeter. I put the cubes inside the heart to find the area." That seemingly simple statement reflects a profound understanding of both perimeter and area.
Ashley, a student in Joannie Nedwreski's fourth-grade class at East Side Elementary School in Johnstown, Pennsylvania, used the symmetry of the heart to expedite her perimeter measurements and solution. Her unique description is seen in figure 2. The notion of symmetry also is present in the work of Morgan and Olivia from Rocky Mount, North Carolina, who saved time computing the area of their heart by constructing centimeter squares on only half the diagram, then multiplying by two to get their final answer.
Morgan and Olivia's classmates in Karen Boone's fifth-grade class at Winstead Avenue Elementary School also used a variety of methods to solve the heart problem. One pair, Brandon and Alex, combined formulas, precise measurements, and centimeter grid paper to determine their solutions. Their diagram and explanation appear in figure 3. The boys' work shows that they had excellent measurement skills but were confused by how to determine area. Like many who used formulas, these students were limited by the "length times width" definition of area and tried to adapt that to the heart figure.
The extensions to the heart problem also generated several interesting responses. In particular, when students were asked to produce a heart with a perimeter twice the size of the original, many predicted that the area also would be twice that of the original. After measuring on grid paper, however, they were surprised to find that the area was indeed much larger. Interestingly, many students discovered that the area was four times as large as the original. Marisa, a student in Mrs. Cook's class, doubled the length of her original piece of string, then created a heart on grid paper using that doubled perimeter. When she counted the interior squares, she found an area of 64 square centimeters--exactly four times the area of her original heart (see fig. 4). Not all students constructed hearts that were exactly similar to the original, and not all students concluded that the area of the new heart was four times larger than the original one. Most students were surprised, however, when they discovered that doubli ng the heart's perimeter actually increased the area by a factor much greater than two. Although counterintuitive, this result seemed to provide new insights into the area ratios of similar figures.
The other extensions were intended to help students understand how figures with equal perimeters can have different areas, and vice versa. It turned out that these extensions also expanded the students' number sense. For example, Benjamin and Matt were trying to construct a rectangle with an area equal to their heart's area of 29 square centimeters. They had seen their classmates construct 6-centimeter-by-5-centimeter rectangles to get an area of 30 square centimeters and 4-centimeter-by-7-centimeter rectangles to get an area of 23 square centimeters, but they still were having difficulties. Matt declared that they could not construct their rectangle because 29 is a prime number. After some deliberation, however, he realized that he could use "1 x 29." He ran out of the room and returned with a piece of paper displaying a long, skinny, 1-by-29 rectangle. Classmates Holly and Allison had a similar experience while trying to create a square with an area that matched their heart's area of 20 square centimeters. They told their teacher, Mrs. Boone, that "10 is much too big, since 10 times 10 is 100" and "4 times 4 is 16 and 5 times 5 is 25." Mrs. Boone replied, "So what is in the middle?" The children said 4.5. They were now on their way to constructing a square with an area of 20 and had a visual foundation for square roots.
Overall, the Have a Heart problem produced many interesting responses and solutions. Much of the work demonstrated that a basic string-and-gridpaper approach to perimeter and area was more effective than formulas in solving the problem. With the use of manipulatives and some guidance, the problem was challenging yet accessible to most students. Students investigating the problem and its extensions demonstrated skills in mathematics vocabulary, measurement, computation, number sense, reasoning, and communication.
Unexpected Insights into Student Reasoning
Surprisingly, one seemingly insignificant question in the Have a Heart problem resulted in a myriad of varied responses and unexpected insights into how students think and learn: "Predict which number will be larger--the heart's area or its perimeter. Explain why you believe that this value will be larger." This question originally was intended only to get the students interested and personally invested in the problem. The responses to this question, however, were so thought-provoking, intriguing, and amusing that they merit their own analysis.
Below are student responses to the "Which is larger?" question, followed by explanations. The responses represent only a sampling of the many submissions received. They were chosen because they reflect the attitudes of many or are especially insightful or creative. The responses have been categorized by the students' methods of proof.
Proof by intuition
Note that the following students made estimates (guesses) based on what they saw in front of them:
Area. I just guess.--Luke, Illinois
Area. Because it looks larger.--Anessha, New Jersey
Perimeter. Because it looks bigger.--Kyle, Illinois
Area. The heart looks bigger in the middle than around.--An gel, Illinois
Proof by divine intuition
Some students stated their results with confidence:
Area. Because it is bigger.--Tom, Illinois
Proof by perspiration
Other students chose to not waste time with estimation and proceeded to complete the task at hand:
Area. Because I measured it.--Savannah, Illinois
This interesting response reveals a common student attitude toward estimation: "Why bother when I can just get the answer directly?" Teachers should consider addressing this attitude before planning units that focus on estimation. They may want to avoid handing out any measuring tools until students have completed the estimation process.
Proof by past experience
While some students relied on intuition, others chose to rely on previous mathematical experiences:
Perimeter. Perimeter is always bigger.--Brooklyn, Illinois
Perimeter. Area is almost always less than perimeter.--Alexander, Illinois
Perimeter. Because the perimeter is usually larger than the area.--Benjamin, North Carolina
Perimeter. If you had low numbers, you would probably come up with a higher perimeter than area.--Hannah, Chelsea, and Carrie, North Carolina
Unlike students in the other categories, these students all agreed that perimeter should be larger because that is what they had seen in the past. This may suggest that students need a more diverse exposure to perimeter and area so that they will experience shapes that sometimes have larger areas than perimeters. Such exposure may lead to investigations that either support, extend, or refute Hannah, Chelsea, and Carrie's claim.
Proof by definition
Area. Perimeter is just the length around; area is the entire inside.--Joe, Illinois
Perimeter. Because perimeter goes around the heart and the area goes inside a heart.--Paul, New Jersey
Area. It has more space inside the heart than the distance around the heart.--Connor, New Jersey
Area. It looks like more centimeters than the outside of it.--Lydia, Illinois
Many students relied on their understandings of perimeter and area to justify their answers. Although all these students used the definitions and distinctions of area and perimeter quite clearly, they disagreed on which should be larger. Engaging discussions likely occurred in their classrooms as these students defended their answers.
Proof by properties
Some students moved beyond the definitions of area and perimeter to their properties, yet still disagreed on the final outcome:
Area. Because there's more blocks in the area than perimeter.--Kidya, New Jersey
Perimeter. A square centimeter takes up more space than a regular centimeter, and the perimeter is being measured in regular centimeters.--Ali, Illinois
Area. Because L X W, which is multiplying, is greater than adding.--Adam and Robbie, North Carolina
These responses demonstrate that students thought about area and perimeter in ways ranging from formulas to blocks to a comparison of square and linear centimeters.
Proof by inspiration
The following responses reveal unique and clever justifications of which value is the largest:
Area. Shapes are larger than string.--Tegveer, New Jersey
Perimeter. Because it curves in and has a lot of rails.--Tralisa, Illinois
Area. Because the perimeter is part of half and the area is a part of a whole, and a whole is bigger than a half.--Neelesh, New Jersey
Perimeter. It surrounds the area.--Hayley, Illinois
Perimeter. Because not all the squares will be whole.--Charmaine, Illinois
Area. Because it will take two times around.--AnaKate, Illinois
Perimeter. Because it will take more than double around.--Katie, Illinois
Proof by example
Finally, one pair of students supported their reasoning with the well-documented example in figure 5. Of course, Reid and Katie also needed to consider other examples to test their theory, but they were already beginning to think like mathematicians, that is, "I don't know about the heart shape, so I will start with something I do know--the rectangle." They also did an excellent job of showing their work and explaining their argument.
This sort of thinking made the responses to the estimation question of the Have a Heart problem a joy to read and ponder. The responding classrooms likely experienced lively discussions as debate ensued. The actual answer to which is larger depended on the particular heart shape used in the classroom. Therefore, perimeter was the correct answer in some classrooms and area was correct in others. In either case, students learned a lot about perimeter, area, estimation, mathematical communication, and proof. Meanwhile, this author learned that the unplanned journey often can be more rewarding than the expected one.
Thanks to all the students who worked on the Have a Heart problem and all the educators who helped attain the solutions and submitted them, especially:
Karen Boone's fifth graders, Winstead Avenue Elementary School, Rocky Mount, North Carolina
Deane Cook's sixth graders, Churchill Elementary School, Galesburg, Illinois
Mary Jo Gonsiorowski's class, Dutch Neck School, Princeton, New Jersey
Joannie Nedwreski's fourth graders, East Side Elementary School, Johnstown, Pennsylvania.
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|Publication:||Teaching Children Mathematics|
|Date:||Feb 1, 2003|
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