Perimeter and Area
- by Tara Beattie, Evonne Der, Kanchan Dudani, Annie Graton, and Molly Lowder (UCIrvine, August 2009)
Cognition and Learning Background: Elementary Algebra
Perimeter is measured in units of length. Length is a one-dimensional measure. The perimeter of a polygon or object is the sum of the length of all its sides. When measuring area, one measures how much surface it takes up. Area is measured in a two-dimensional measure and is typically measured in sqaure units appropriate to what is being measure. (Rectanus, 1997, p. 2-3) Although most people are familar and capable solving for area and perimeter using the formula, more experience and opportunity needs to be given to comparing areas and perimeter of shapes.
- 2009 NCTM Final Report Recommendation:Geometry and Measurement: Teachers should recognize that from early childhood through the elementary schools years, the spatial visualization skills needed for learning geometry have begun to develop. In contrast to the claims of Piagetian theory, young children appear to possess at least an implicit understanding of basic facets of Euclidean concepts. However, formal instruction is necessary to ensure that children build upon this knowledge to learn geometry. (U.S. Department of Education, 2008, p. 29)
- Piagetian View: “Children will only develop logical measuring operations when they have established adequate conservation of length, and that socially engendered learning did little to facilitate that development”(Barrett, Clements, Klanderman, Pennisi, & Polaki, 2006, p. 188).
- It is important for students to learn and understand geometry and measurement as it helps them make sense of everyday life, solve problems, and use pictorial representations to view abstract symbols. However, in today's classrooms, students are memorizing geometric properties and formulas rather than taking part in hands-on activities which would build conceptual understanding of the topics. (Strutchens, Harris & Martin, 2001)
- The incentives to work on more complex content and to press for higher performance are not necessarily present. Teachers often do not have time to plan and organize rich experiences for pupils, and they cannot afford the looseness of more exploratory curricula. They feel pressure to make sure that pupils master required content. For example, the time required for students to “get inside” a topic like measurement may seem to conflict with the time needed to ensure that students also get to everything else. The pull toward neat, routinized instruction is very strong. Teaching measurement by giving out formulas—l x w = some number of square units and l x w x h = some number of cubic units—may seem much more efficient than hauling out containers, blocks, and rulers and having students explore the different ways to answer questions of “how big” or “how much.” With focused, bounded tasks, students get the right answers, and everyone can think that they are successful (Ball, Lubienski, & Mewborn, 2001, p. 436)
- When presented with two lengths perpendicular to one another starting from the same point, the pupil must have a concept of the area as a matrix consisting of “an infinite set of lines infinitesimally close to one another” (Piaget et al., p. 350) to make sense of length x width which is the area of rectangle. In order to provide these, it may require the teachers to have significant content and pedagogical content knowledge of area and perimeter (Yeo, 2008, p. 622).
Van Hiele believed that mathematics is taught as “parrot math” in many classrooms today, which is when the teacher shows an example and then students complete many similar types of problems on their own. However, current research shows that students learn through interaction with the knowledge and through social interaction. In the past, geometry has been taught by having students develop and reproduce proofs, but this has been problematic for teachers and students. The van Hiele model strives to change these negative feelings about geometry that students have due to their lack of success.
Characteristics of van Hiele levels: 1. The levels are sequential 2. The levels are not age-dependent 3. Geometric experience is the single most important factor in students moving to the next level 4. If instruction is given at a higher level than where a student is at, the student will not be able to learn
The van Hiele model of student geometric thinking: (from Wikipedia.com and also in Van de Walle)
Level 0. Visualization: students view figures holistically without analyzing their properties
Level 1. Analysis: children can discuss the properties of the basic figures and recognize them by these properties, but might still insist that "a square is not a rectangle." Children do not see the relationships between the properties. They might reason inductively from several examples, but not deductively.
Level 2. Abstraction: students begin to reason deductively. They understand the relationships between properties and can reason with simple arguments about geometric figures. Learners recognize relationships between types of shapes. They recognize that all squares are rectangles, but not all rectangles are squares, and they understand why. They can tell whether it is possible or not to have a rectangle that is, for example, also a rhombus.
Level 3. Deduction: learners can construct geometric proofs at a high school level. They understand the place of undefined terms, definitions, axioms and theorems.
Level 4. Rigor: learners understand axiomatic systems and can study non-Euclidean geometries.
Examples of Student Understanding
Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding of area and perimeter know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which is it useful to determine the area and perimeter of an object. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know (Kilpatrick, et al., 2001, p. 118).
Conceptual understanding also supports retention. Because facts and methods learned with understanding are connected, they are easier to remember and use, and they can be reconstructed when forgotten. If students understand a method or technique for finding the area and perimeter of an object, they are unlikely to remember it incorrectly. They monitor what they remember and try to figure out whether it makes sense. They also may attempt to explain the method to themselves and correct it if necessary. Although teachers often look for evidence of conceptual understanding in students’ ability to verbalize connections among concepts and representations, conceptual understanding need not be explicit. Students often understand before they can verbalize that understanding (Kilpatrick, et al., 2001, p. 118). The points below are some examples of what students need to be presented with when learning about area and perimeter:
- The typical paper and pencil environment does not work for students, they need to be given open tasks to provide a variety of representations and make meaningful links to those representations. Students need to be given and give numerical representations, visual representations (using grids, etc.) and the symbolic representation such as the area formula.
- Students need to examine perimeter and area simultaneously to clearly distinguish between them. Students need to explore and use materials to determine area and perimeter, not just apply the formula to a picture. "They need to construct visual representations of figures with given areas and perimeters, generate related word problems, and justify properties" (Chappell & Thompson, 1999).
- Students need to explain their drawing and representations so that their true understanding can be discussed and they can see whether they have made an error (i.e. to create an object with a perimeter of 24 units and they draw an object that is 8x3 which is really 24 square units)...had they explained it, they could have seen their error when checking with the formula.
- Children develop measurement knowledge partly through education experiences that afford structure frameworks for indentifying, re-presenting, and interpreting aspects of spatial and geometric knowledge relevant to the growth of their measurement ideas and practices. Students need to engage in realistic comparisons among objects with measurable characteristics. (Barrett et al, 2006, p.189)
Procedural understanding refers to a tendency to apply a procedure without being able to explain why or how the procedure works (Muir, 2008, p. 229). Many elementary school teachers that lack pedagogical content knowledge in mathematics will most likely teach their students mathematical procedures without facilitating opportunities for them to conceptually understand the concepts. While learning procedural math is important to have success on exams, research shows that this is not always the best way to ensure student understanding and interest in math (Muir, 2008, p. 229). Thus, when teaching area and perimeter, in order to promote procedural understanding successfully with conceptual understanding, it would be beneficial to have students develop conceptual understanding first to have a firm foundation of their understanding as to dodge the presented problems that come with procedural understanding:
- Elementary textbooks are often inadequately developed, usually consisting of one or two examples and “hints and reminders” to students about what to do. In terms of area and perimeter, they are presented in terms of the formulas—l x w and 2 x l divided 2 x w—with, perhaps, some pictures to illustrate. (Ball, Lubienski, & Mewborn, 2001, p. 436)
- Most primary school pupils have a good understanding of perimeter as a special application of length that measures the distance around a figure. Pupils are so accustomed to finding perimeters where the length of every part of a figure is given and they just had to add all the given numbers. Pupils who do not have an adequate understanding of perimeter will find it difficult to deduce the length of the side when it was not stated explicitly. (Yeo, 2008, p. 622).
CA Content Standards for Geometry and Measurement in Elementary School
- 1.0 Students understand the concept of time and units to measure it; they understand that objects have properties, such as length, weight, and capacity, and that comparisons maybe made by referring to those properties:
- 1.1 Compare the length, weight, and capacity of objects by making direct comparisons with reference objects (e.g., note which object is shorter, longer, taller, lighter, heavier, or holds more.).
- 1.0 Students use direct comparison and nonstandard units to describe the measurements of objects:
- 1.1 Compare the length, weight, and volume of two or more objects by using direct comparison or a nonstandard unit.
- 1.0 Students understand that measurement is accomplished by identifying a unit of measure, iterating (repeating) that unit, and comparing it to the item to be measured:
- 1.1 Measure the length of objects by iterating (repeating) a nonstandard or standard unit.
- 1.2 Use different units to measure the same object and predict whether the measure will be greater or smaller when a different unit is used.
- 1.3 Measure the length of an object to the nearest inch and/or centimeter.
- 1.0 Students choose and use appropriate units and measurement tools to quantify the properties of objects:
- 1.1 Choose the appropriate tools and units (metric and U.S.) and estimate and measure the length, liquid volume, and weight/mass of given objects.
- 1.2 Estimate or determine the area and volume of solid figures by covering them with squares or by counting the number of cubes that would fill them.
- 1.3 Find the perimeter of a polygon with integer sides.
- 1.0 Students understand perimeter and area:
- 1.1 Measure the area of rectangular shapes by using appropriate units, such as square centimeter (cm2), square meter (m 2), square kilometer (km 2), square inch (in 2), square yard (yd2), or square mile (mi 2).
- 1.2 Recognize that rectangles that have the same area can have different perimeters.
- 1.3 Understand that rectangles that have the same perimeter can have different areas.
- 1.4 Understand and use formulas to solve problems involving perimeters and areas of rectangles and squares. Use those formulas to find the areas of more complex figures by dividing the figures into basic shapes.
- 1.0 Students understand and compute the volumes and areas of simple objects:
- 1.1 Derive and use the formula for the area of a triangle and of a parallelogram by comparing it with the formula for the area of a rectangle (i.e., two of the same triangles make a parallelogram with twice the area; a parallelogram is compared with a rectangle of the same area by cutting and pasting a right triangle on the parallelogram).
- 1.2 Construct a cube and rectangular box from two-dimensional patterns and use these patterns to compute the surface area for these objects.
- 1.4 Differentiate between, and use appropriate units of measures for, two- and three-dimensional objects (i.e., find the perimeter, area, volume).
- 1.0 Students deepen their understanding of the measurement of plane and solid shapes and use this understanding to solve problems:
- 1.1 Understand the concept of a constant such as π; know the formulas for the circumference and area of a circle.
- 1.2 Know common estimates of π (3.14; 22⁄7) and use these values to estimate and calculate the circumference and the area of circles; compare with actual measurements.
Grades K-2 Measurement:
Understand measurable attributes of objects and the units, systems, and processes of measurement
- recognize the attributes of length, volume, weight, area, and time
- compare and order objects according to these attributes
- understand how to measure using nonstandard and standard units
- select an appropriate unit and tool for the attribute being measured
Apply appropriate techniques, tools, and formulas to determine measurements
- use repetition of a single unit to measure something larger than the unit, for instance, measuring the length of a room with a single meterstick
- use tools to measure
- develop common referents for measures to make comparisons and estimates
Grades 3-5 Measurement:
Understand measurable attributes of objects and the units, systems, and processes of measurement
- understand such attributes as length, area, weight, volume, and size of angle and select the appropriate type of unit for measuring each attribute
- understand the need for measuring with standard units and become familiar with standard units in the customary and metric systems
- understand that measurements are approximations and how differences in units affect precision
- explore what happens to measurements of a two-dimensional shape such as its perimeter and area when the shape is changed in some way
Apply appropriate techniques, tools, and formulas to determine measurements
- develop strategies for estimating the perimeters, areas, and volumes of irregular shapes
- select and apply appropriate standard units and tools to measure length, area, volume, weight, time, temperature, and the size of *select and use benchmarks to estimate measurements
- develop, understand, and use formulas to find the area of rectangles and related triangles and parallelograms
Cognitive Obstacles and Common Misconceptions
Real World Scenarios
- A difficulty with area and perimeter is that students don't understand how these measurements relate to their life. Academic language such as area and perimeter are not words commonly used in every day life. Instruction needs to be put into real world context to relate students to its uses. Connect perimeter and area to students interests/lives by using examples such as sodding a golf course or back yard, tiling a house, measuring a playground, or measuring their own foot.
- According to Malloy (1999) students tend to be confused between all of the formulas dealing with area and perimeter because they have not “fully conceptualized the meanings of these words” (p. 87). Memorizing procedural skills without understanding the idea of perimeter or area conceptually sets students up for confusion.
- Kenney and Silver (1997) discuss the Sixth Assessment of the National Assessment of Educational Progress showed the following results with respect to 8th graders:
- 69% were able to choose the correct missing length of one side of a geometric figure when given its perimeter
- 19 % were able to estimate the perimeter of a given geometric figure
- 32 % could match or identify the correct geometric shape when given a perimeter
- 66 % were able to draw a rectangle with an area of 12 units using a grid
Students' inability to estimate, match or draw in relation to perimeter and area show their lack of conceptual understanding. Knowing the formula for perimeter and area is far different than understanding what it means. Yeo (2008) explains that although the pupils were able to articulate the area of rectangle and square well, it seemed that the pupils were not able to define area. In the first lesson, pupils were asked to define area, and promptly answered, “length times breadth!” They understand area as a formula rather than as a concept – the amount of space covered by the boundaries of a two-dimensional figure (p. 623).
- Students rely too much on their visual perception to make comparisons of area; therefore, leading to problems in understanding the true meaning of area when shapes are represented in different forms. Students need opportunities to see and manipulate a variety of shapes. This will enable them to become familiar with various shapes, perimeters and areas. Students also need to manipulate these shapes to build conceptualization. Refer to Pedagogical Strategies and Tools for specific activities.
- When dealing with area, the final answer is written as X units², however, it is not said X units squared, because that would be implying you are squaring the expression. The correct way to say it is X square units, even though the square is written after the units.
- In finding the area and perimeter of triangles and parallelograms height becomes an obstacle with students. They have been taught Length X Width in relation to squares and rectangles and thus need to be introduced to the concept of height. When height enters the picture there is confusion as to what that is for many differing shapes. Make sure students understand height and can identify the height of a shape before they calculate the area.
Confusing Perimeter and Area
- Research states that students possess difficulty in explaining and illustrating ideas of perimeter and area. Students confuse perimeter and area because the topics are usually learned as a set of procedures and formulas instead of using rich contextualized problems....thus leading to misunderstanding the importance of the measurement behind them. Students don't know what the answer they found represents and thus don't understand when to use units and when to use square units. the process of conserving an area and simultaneously studying it in relation to the perimeter of its figure is significant since students confuse these concepts and use them alternately. See strategies section for the notecard activity that conserves area but changes perimeter.
- Teachers confuse the concepts of area and perimeter frequently assuming that there is a constant and direct relationship between area and perimeter. Further, teachers often do not use appropriate units when computing area and perimeter, commonly failing to use square units when reporting measures of area (Mewborn, 2001, p. 30).
- Many people (including teachers) believe that shapes with a fixed perimeter have the same area because the length is always the same. Students can be challenged to make different shapes using fixed length and then solving for the area of each shape to address this misconception. Give students a piece of string (fixed length/perimeter) and have them create different sized shapes and find the area of each shape. You can manipulate the sides of a fat square into a long skinny rectangle with the same perimeter but the area would be much less in the long rectangle.
- Students may believe that the area of a shape will always be larger than the perimeter because area uses multiplication and perimeter uses addition. A way to help students alleviate this misconception is to give them a number of shapes some of which may have a larger area than perimeter and some with a larger perimeter than area. Have students find the perimeter and area of the shapes and make discoveries. Examples of squares and rectangles with larger perimeter than area (or equal perimeter and area): 4x3 rectangle, 4x4 square, 8x2 rectangle, 5x3 rectangle. Also, students can be taught another way to solve for perimeter that also uses multiplication: P = 2(L + W).
- Students believe the area will be 12X7 - not taking into consideration that there is a missing piece. Give students the problem along with graph or centimeter-squared paper and have them pretend they are tiling a room. How many tiles will they need. They should notice the area will be smaller because of the missing piece and that has to be taken into consideration when finding area.
Pedagogical Tools and Strategies
The perimeter of a shape is the total length of its boundary. Perimeter is measured in units of length. Length is a one-dimensional measure; common to all measurements of length is that they tell how far something stretches.
The area of a shape is how much surface it takes up. It is measured in two dimensions and in square units.
Shapes with the same length perimeter can have different areas. Shapes with the same area can have different length perimeter.
Perimeter of a square, P = 4s
Perimeter of a rectangle, P = 2l + 2w
Perimeter of a triangle with sides a, b, c, P = a+b+c
Perimeter of a regular polygon, P = ns
Area of a square, A = s²
Area of a rectangle, A = lw
Area of a parallelogram, A = bh
Area of a triangle, A = ½bh
Van de Walle Activities (Van de Walle, 2007)
- Two-piece shapes - cut out many different size rectangles and give a few to each group. Students cut rectangles into two equal triangles and see how many shapes they can make when putting the triangles back together again. Note: to create new shapes two equal sides of the triangles must be placed together.
- Tangrams areas - outline several tangram shapes and decide which are larger and which are smaller.
- Rectangle Compare - give students pairs of rectangles with same the area but different dimensions, and some with different areas and different dimensions. Ask students which rectangle is bigger than the other. They may cut and fold rectangles, and they must explain and justify their answers. They next step would be to give them rectangles with units and a ruler. Students are not allowed to cut or fold the rectangles this time, and must instead use the units given to them.
- Fixed Perimeter - give students a fixed length of string and have them create as many rectangles as they can using the entire perimeter.
- Fixed Area - give students a fixed number of square tiles and have them create as many rectangles (with different dimensions) as the can using all of the tiles as the area.
- Area of a Parallelogram - give students many parallelogram cutouts and drawings. Have them derive the area of a parallelogram from what they know about the area of a rectangle.
- Create a large shape drawn with tape on the floor and measure the area with tag board unit square cut-outs
- Design activities in which students are required to sort shapes by their attributes.
- Have students create lists of shape properties, including naming the fewest properties needed to describe a certain shape.
- Create family trees of shapes with students to determine the relationships among their properties. Discuss examples and non-examples of a particular shape. Have students compare and contrast the different shapes and from there create their own definitions, conjectures and descriptions.
- In order to build a deeper conceptual understanding of perimeter design activities in which students use string, measuring tape, and other measuring tools when learning about perimeter.
- Use graph paper and geoboards and allow students to count the units embedded in the figure.
- Give students a parallelogram cutout, and have them cut it up to arrange it in parts to determine its area more easily.
- Use a hundreds grid to show squared units. You can draw a shape on the grid, and then shade in the squares it covers in its area to show square units.
- After teaching the area of a square and/or rectangle, take one of these shapes (using construction paper) and cut down the diagonal, showing that you have created to equal triangles. Then arrive at the formula for area of a triangle to be half of the area of a rectangle and/or square.
- Have students measure the perimeter and area of a 5 in. by 5 in. index card. Then, students will cut the index card into two new shapes; allow them to use straight or curved lines. Use string and graph paper to measure the area and perimeter of the new shapes. The purpose of this activity is for students to realize that the area will stay the same but the perimeter will change and the longer the curves of the shape the larger the perimeter.
- Have students trace their foot (with no shoe) on a sheet of centimeter-squared paper. They need to figure out the area of their foot in square centimeters. After finding the area of their foot, they will cut a piece of string the length of the perimeter (outline) of their foot. Have students tape the string in a square shape on centimeter-squared paper. Then, they will find the area of the square. Students should notice that while the perimeter stays the same, the area is different.
- Perimeter Activity: Using Cuisenaire Rods - Give the students certain rods, such as one red, two light green and one purple, as well as a piece of centimeter-squared paper. The students must arrange the rods to make a shape that stays along the lines of the paper and will remain in tact if cut (if only the corners are touching, it is not allowed). They must trace the shape and record the perimeter for at least four shapes. Their goal is the make a shape with the longest possible perimeter and a shape with the shortest possible perimeter.
- Use the book, Spaghetti and Meatballs for All! written by Marilyn Burns. It is a story about Mr. and Mrs. Comfort hosting a family reunion meal. Before the guests even arrive, Mrs. Comfort designs a seating chart for thirty-two people to sit at eight square tables with four people at each table, but as the guests begin arriving, they decide to rearrange and combine the tables. As a result, they must break apart the tables and rearrange them to allow for more seating. In the end, they end up with the same configuration as Mrs. Comfort designed (Moyer, 2001). This book allows students to work with manipulatives, such as color tiles to represent the tables and centimeter cubes to represent the people, as a way to reorganize the tables and people as the book continues with its storyline. It gives students the opportunity to see that shapes can have the same area, but different perimeter as a basis for more conceptual understanding.
By the Unit or Square Unit (Ferrer, Hunter, Irwin, Sheldon, Thompson, & Vistro-Yu, 2001, p. 134)
Objectives: to clarify the meaning of area and perimeter, identify important relationships of measurement and area, and to put them in perspective to real world surroundings,
1. Students identify objects in classroom that have measurable areas: (ceiling, top of crayon box)
2. Students decide what units of measurement could be used for specific areas.
3. Extension Activity: students engage in area sort where they sort unit of measurements with small and large objects (estimation)
4. Extension Activity: Students use cut square meter of newspaper to find the number of square meters that would fit a large area.
Pentominoes: A Pentomino is five congruent sqaures joined together so that each sqaure shares a common side with its neighbor. Ask students to find as many different pentominoes as they can. Next have the students find the perimeter of all the different pentominoes. Note the differences in the number of sides that will be counted as the pentominoes change figure.
Quilt Activity:(Westegaard, 2008 p. 363) With quilt sqaures students can draw the quilt formations on dot paper. Students can engage in a series of problem solving activities. For example, the teacher can ask the following questions when using quilt squares:
1. Find the sqaure that has the most shaded area.
2. Find the length of one side of the sqaure.
3. Find the perimeter of shaded areas or the quilt sqaure.
4. Solving for the area of one shaded area and determining what fraction is of the entire quilt square.
- Keep in mind that two rectangles that have the same area do not necessarily have the same perimeter, and two rectangles with the same perimeter may not have the same area. This can be true for some other shapes as well.
- Moving students from one van Hiele level to the next:
1. Have students gather information by working with many examples
2. Create extended tasks that uses this initial information
3. Students should become aware of the relationships between shapes and be able to explain them
4. Students move onto more challenging and complex tasks, and then summarize and reflect on what they have learned (Malloy, 1999)
- The shared language the teacher and students is also important for student progression. The teacher should also be aware of the different levels their students may at in the van Hiele model. Students of differing levels can work together and solve similar problems, however they will use different strategies (Malloy, 1999).
- Advanced geometric thinking can be formed if students are given opportunities to work with geometric figures in complex settings and are able to visualize their properties. (Strutchens, Harris & Martin, 2001)
- Allow students to come up with the formulas for perimeter and area inductively.
- Questions to ask to begin discussion and introducing relationship of perimeter and area and engage in thinking that challenges students to think more conceptually:
1. If the area increases, will the perimeter always increase?
2. If the area decreases, will the perimeter always decrease?
3. If two areas are the same, are the perimeters always the same?
4. What statement can you make about the relationship of area and perimeter that is always true? (Ferrer et al., 2001, p. 134)
- Learning activities should be solved "in as many ways as possible" (Balomenou & Kordaki, 2006)
- Give students a ruler to measure the length of the sides for perimeter and to measure each one square unit to get the area
- There are different ways to measure perimeter and area: rulers, counting squares inside shapes and applying formulas.
- Studies show that creating a unit where students use perimeter and area to build something that relates to them and their culture (i.e. their shoe) ended in the students performing better on tests on those topics than students who learned the traditional way through procedures and textbooks.
Curricula and Technological Resources
Perimeter Explorer - http://www.shodor.org/interactive/activities/permarea/index.html
- Centimeter-squared paper
- Graph paper
- Counting squares
- Construction paper cutouts of various shapes
- Students used Microsoft Word Documents to create grids and use the autoshapes to determine area and perimeter of those shapes. This allows the students to not only have hands-on experience with creating their own objects to figure out area and perimeter, but it gives them a more explicit learning opportunity. (Yeo, 2008, pg. 621).
Books & References
- Mouse and Elephant: Measuring Growth - (Shroyer and Fitzgerald, 1986). This book has a chapter which focuses on area and perimeter. It focuses on a having a constant area while changing the perimeter, and on haivng a constant perimeter while changing the area.
- Measurement in the Middle Grades (Geddes, 1994) – assists teachers in implementing the NCTM’s Standards.
- Mathematics inthe Middle School (Burns, 1989) - Shows suggested activities that involve holding either the area or the perimeter constant while allowing the other quantity to change.
- Spaghetti and Meatballs for All (Burns, 1997) - This book presents a mathematical situation in which students can relate and make sense of math by manipulating objects as the book progresses though the story. It is about a family who is planning a meal for a family reunion and continues to move the tables around when a new person arrives. The students can connect their existing understanding with a pictorial model, indicating that the perimeter on the model is the "distance around the figure" and that the area is "the number of square units enclosed by the figure."
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Rectanus, Cheryl (1997). Math by all means: Area and perimeter grades 5-6. Sausalito: Math Solutions Publications.
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Rickard, A. (2005). Constant perimeter, varying area: A case study of teaching and learning mathematics to design. Journal of American Indian Education, 33(3), 80-100.
U.S. Department of Education. (2008). Foundations for success: The final report of the national mathematic advisory panel. Washington, D.C.: Author.
Van de Walle, John A. (2007). Elementary and middle school mathematics: Teaching developmentally. 6th ed. Boston: Pearson Education, Inc.
Westegaard. S.K. (2008). Using quilt blocks to construct understanding. Mathematics Teaching in the Middle School. 14(6), 361-365.
Whitin, P. (2004). Promoting problem-posing explorations. Teaching Children Mathematics, 11(4), 180-186.
Yeo, J. K. H. (2008). Teaching Area and Perimeter: Mathematics-Pedagogical-Content Knowledge-in-Action. National Institute of Education, pg. 621-626.