Ann Stevenson's CU

Learning One Thing Well: Perimeter

Ann Stevenson

Academic Setting

This unit was designed after I taught for one year at a relatively small middle school of 600 students in Albuquerque’s north valley. The population is approximately sixty percent Hispanic and forty percent Anglo, with a small minority of Asian, American Indian, and African-American students. Just under 40% of the students receive free or reduced lunch. The diversity of income levels is high, as the neighborhood is very diverse.

I taught sixth and seventh grade math. The math department tests incoming students and all current students in math and the high-scoring students are segregated into enriched math classes. My classes, then, were called “regular,” meaning a high percentage of the high-achieving students had tested into enriched, leaving me with a higher percentage of low-achieving students. However, there were many, hard-working, high achieving students in each class. Many of my students had very low math skills and performed poorly on the Terra Nova and in the previous year’s math class. As most teachers do, I needed to design lessons to meet the needs of a wide variety of abilities.

         It became clear that my students had difficulty solving problems on perimeter. For instance, a figure was drawn with perpendicular angles and the length of some of the sides given, and they were expected to find the length of the missing sides. The majority of the class had difficulty reasoning the answer from the information given. I decided they had not had enough experience with actual measuring of objects and that this would help. However, they also needed to be exposed to the kind of activity that would promote abstract thinking. This unit was designed with these goals in mind.

The solution is not to simply move the curriculum down from an upper grade to a lower one. Instead, we need to build knowledge carefully by teaching concepts thoroughly and at depth in a coherent and cohesive way. The transition that students make from simple to more complex should be the focus of the curriculum. By demanding that students learn the material at one grade, knowing they will need to apply that knowledge in the years to come, shows respect for the students intelligence and their ability to progress.

Context and Background

The Need for Change

The Third International Mathematics and Science Study, (TIMSS) supplied evidence that children in the United States are not improving in their mathematical ability as they progress through school. At fourth grade, they scored above the international mean, but by the eighth grade the scores began to slip, and by twelfth grade, scores were well below average, with 20 of the 41 participating countries scoring better than U.S. students (NAEP). The TIMSS is a more accurate assessment than, for example, the Terra Nova achievement tests, because it includes a study of classroom practices and curriculum to determine the reason for discrepancies in scores. Many interesting things were recognized about the way students are taught mathematics in the U.S.‚ and suggestions made as to how to improve. This paper will examine the way geometry is taught in middle school along with how it is learned according to one model, the von Heile model of geometric thought. In this model, students must go through a series of steps, from simple to complex, or from concrete to abstract, in order to conceptually understand geometry. This paper will also attempt to demonstrate how the historical progress of human knowledge concerning geometry followed a progression that built from one concept to the next.

One of the numerous things the TIMSS pointed out is that while many topics were introduced in the early grades, by middle school the number of new topics was greatly reduced in U.S. schools. In fact, only one new topic was introduced with the amount of focused attention that other countries spend on 15-20 topics. Instead, U.S. middle schools spend as much as 75% of their time reviewing basic arithmetic skills (Schmidt). It’s been suggested that the practice of spiraling the curriculum, that of revisiting material before advancing, has instead become a vicious cycle of repetition (Schmidt). Teachers are not presenting material in a way that students will retain and be able to apply at a higher level. Instead, we’re simply, and rather, degradingly, asking students to learn basic arithmetic facts, the same ones they were supposed to learn last year, with no purpose. Why should students bother to learn math facts and concepts when they know they will be taught the same thing again next year? The TIMSS found that students in the U.S. are being taught math that is a mile wide and an inch deep (Cogan). We have thicker, more colorful textbooks than the countries who outperformed us, but the material is covered superficially, with much time spent on old topics and little focus given to new topics.

Personal Experience

I taught middle school after spending 8 years teaching fourth and fifth grades. In the elementary school, teachers seemed to spend more time on forming lessons that progressed through a unit of teaching and required students to apply their learning. Time was spent discussing teaching and learning models and methods, theory, and comparing techniques and ideas. I believe this is evident in the TIMSS‚ where scores are higher in fourth grade. However, the middle school where I taught spent little attention on pedagogy and what’s best for children. Although our mission statement acknowledged that students at this age need to interact with peers and that tracking is harmful, for example, little time or attention was given to focused instruction around a common goal, and students were tracked, for convenience sake. Math teachers spent most of the year on review, and though it was a never-ending complaint that kids didn’t know their skills, no real progress was being made in improving their level of achievement. Finishing the book was one goal for teachers, and this exemplifies the problem, according to the TIMSS. Students don’t learn because they are not expected to and they know they will be taught the same material next year. They do not retain material because they are not asked to use and apply it to challenging, realistic problems. They do not grow mathematically because they are not taught how to use their knowledge to solve increasingly complex problems. In addition to having low expectations for real learning progress, my middle school gave a test to all students that assessed visual and spatial reasoning. If this was not taught during the school year, what role would the test serve except to track students based on some innate ability?

The problem as I see it is that there is no coherence to the mathematics curriculum. Students in elementary schools are taught using manipulatives to describe how math works, but there is no coherent instruction to help them bridge over to more abstract thinking. Then, when they hit middle school, the use of manipulatives all but vanishes, and the ones who did not form those bridges or connections begin to fail. This failure, along with repetitive skill practice without careful instruction (i.e. there are no adjustments made for the different levels of students) leads to a bad attitude about math. Even hard working students become frustrated with the lack of meaningfulness in their math classes. The way math is taught in middle school is the key to student success in high school mathematics, and if we care about that, we have to address the middle school curriculum. This is a daunting task, considering the turtle-like and paper thin “reforms” that have occurred so far. But we as teachers can create lessons and adopt teaching ideologies that address students learning in a progressive manner that takes them from where they are forward to higher levels of genuine mathematical understanding.

A Look at Solutions

One intriguing model for advancing thinking and learning in geometry is the Van Haeile model of geometric thinking. It was developed in 1959 by two Dutch educators, Pierre van Heile and Dina van Heile-Geldof. The model proposes that there are five different levels of geometric thought. The levels are sequential but not age dependent; that is, a first grader and a tenth grader could both be at level one, after having moved on from level zero, and will progress to level two, given the opportunity. However, no matter what the age, a student can’t be expected to skip a level. If the teacher or textbook presents material that a student is not ready to accomodate to his present knowledge, rote learning may occur, but genuine understanding will not (Van deWalle). The experiences that the students have will determine whether they progress to the next level of geometric thinking. As they explore and interact with the content of the next level, their ability to think at a higher and more abstract level is increased.

            The Van Heile model makes a good deal of logical sense to me. Upon examining a mathematics book called Mathematics From the Birth of Numbers, its description of elementary geometry is clearly sequential. Certain words and their meaning must become a part of a learner’s vocabulary, such as perpendicular, point, and line. Learning the symbols that represent these words and the concept that the words represent, must happen in order to understand material later in the chapter. However, I don’t think I could understand the definitions themselves if I didn’t have some experience with the concepts. For example, recognizing the planar surfaces on prisms would be hard if I had never seen one of the solid objects that were pictured in the book. A teaching tool I made to construct a cone made the descriptions of oblique and circular cones understandable. So from my own personal experience, the lessons from a textbook on geometry must be sequential and must be accompanied by some concrete experience that I can connect to.

The Van Heile method received limited attention in the United States (It was widely accepted in the Soviet Union in about 1959.) (Van de Walle 223). However, some studies were done to verify the validity of the model in the U.S.   Students were given a pre-test to determine their Van Heile level before a one-year high school geometry course, then re-tested using the same test to see if they had moved on to the next Van Heile level. The study found that many of the students tested at level 1 or 2 and this was found to coincide with a lower acheivement during the year. It could be said that the pre-test level could predict achievement in geometry. This study made some conclusive statements but I came to realize this would not be an easy task to undertake. How can an instructor teach a heterogeneous group of Van Heile levels? Students will not learn the material if it is presented at a too-high level, but what about those students who are beyond that level? I can see that this would require some thoughtful and careful planning on the part of the teacher.

I will now describe the levels and include some examples of activites that would be appropriate for students at that level. It is important to reiterate that the level is not age dependent. Also, a student who has achieved one level of geometric thinking cannot be expected to leap suddenly into the next. The quality and amount of experiences should introduce the next level and challenge the student to recognize the additional complexity based on what he or she already knows. Supplying experiences that help students make connections between levels are the most valuable; supplying encouragement and challenge help students move toward a higher level of thinking. This is precisely what the TIMSS found we don’t do.

Level 0: Visualization. The student recognizes shapes. Activities should include the use of as large of a variety as possible of models that students can manipulate. Many sizes and types of shapes should be examined by the students. Sorting and describing of shapes along with building, drawing, putting together and taking apart are activites that are important. EXAMPLE: Make collections of shapes from tagboard and have students sort them. Notice how they have chosen how to categorize them and encourage them to label the categories. Discuss why some shapes belong in more than one category and if any shapes do not fall within any of the categories.

Students at this level are usually thought to be in the early primary grades. However, it is important to assess students at higher grades, too. Otherwise, if they have not achieved Level 0 thinking, instruction will not result in understanding.

Examples of middle school activities that would help students who are at this level to move on to level 1 including lessons using tangrams, quilting, mobiles, pattern blocks, color tiles, tessellations, and geoboards. Lessons should be carefully designed so students recognize objectives, i.e. they don’t just make a pretty pattern but instead identify the math concept, such as similarity and congruency or symmetry. Problems should include plenty of opportunity for students to reason spatially. Time and opportunity are the key. Students should be allowed the time to work with manipulatives and reason through the relationships, such as recognizing patterns. Direct instruction about the mathematics involved ensures that students do make those desired connections between manipulation/observation and expected knowledge about geometric concepts.

Level 1: Analysis. The students can identify the properties of shapes, such as the existence of right angles and parallel sides. They can name the properties of a shape at this level; however, they do not identify the relationship between classes of shapes, such as that a square is a rhombus. Many of the same materials as those used for Level 0 instruction can be used, which means that several investigations can be going on at the same time at different levels. Sorting of shapes and the use of Venn diagrams, Guess My Rule, and using words like all, none, and some should occur. Geoboards and dot or grid paper can be used for students to draw, build, and measure figures. Using direct comparisons and informal units is better than sophisticated measurements, even in middle school. Develop fluency of vocabulary by having students construct shapes with right angles, parallel sides, lines of symmetry, etc. Use Miras, mirrors, straws, popsicle sticks, etc. and have students discover properties of shapes and the relationships between them, such as how all quadrilaterals have 360 degrees and what conditions are necessary for a quadrilateral to be a parallelogram. Escher-type tesselations and measuring the circumference and diameter of cylinders (cans) are activites that develop understanding of relationships.

Level 2: Informal deduction. The student can construct relationships between classes of figures.   Proofs are not understood. Models are used as a tool for thinking and verification rather than for exploration. Have students decide if if-then statements are true or false. For example if two rectangles have the same area, then they are congruent (Van de Walle 299). At this stage, informal proofs can be introduced, such as an area interpretation of the Pythagorean theorem. Compass constructions and constructing figures on a computer using logo allow students to see relationships of lines, angles, and arcs. Working with solids, students can recognize properties of 3-dimensional objects.

Level 3: Deduction. The student understands deduction and the role of theorems and postulates. Proofs are understood. This is the level that it is assumed all students are at when they enter a tenth grade geometry class. If they have not had the experiences necessary, according to the Van Heile model, many students will not succeed in gaining genuine geometric knowledge. They may undergo rote learning, memorizing the material to do well on tests, but the level of learning will not be as high as it could be. Can you see how incomplete understanding of geometry at this age might deter students from choosing to take advanced math classes? Even the best students need to feel mastery of the subject in order to feel successful and capable of continuing to higher levels.

Level 4: Rigor. The student can make abstract deductions. Dina Van Heile did not believe students in secondary schools can achieve this level so I will not discuss it here.

Connections to Other Learning Theorists

Two other educational theorists developed widely accepted philosophies about children’s learning that deserve some attention related to this subject. Jean Piaget believed children must be developmentally ready to accomodate, or integrate information before learning actually occurs. (Elkind 99). Maria Montessori developed a curriculum based on children’s development level. (Van de Walle 343). The similar premises are that learning does not occur until a)the child is developmentally ready to receive new information and can adapt it, either through assimilation or accomodation‚ to his or her way of thinking and, b)that the experiences a child has, past and present, affect the way he or she accepts and learns the new information. The Van Heile model follows this same line of thinking, but the levels of geometric thought are not age dependent, as in the child development models. All of these models address the constructivist idea of conceptual change; that is, that the student makes sense of new information by using his or her existing knowledge (Hewson 131). To address the earlier part of this paper, our goal as educators should be to bring about long-term conceptual change. The learning that occurs should be thorough, with no gaps is understanding, so that the students are prepared to apply their knowledge to more advanced lessons as they progress through school.

The Van Heile model is an example of a method that examines the developmental level of each student and takes that information into account when designing lessons for students. Its appeal for me is that it provides a theory of learning that teachers should be aware of: that students cannot genuinely understand until the geometric thought sequence is complete. It accents the importance of preparing students during middle school for the abstract geometrical thinking they are expected to do in high school.

The importance of informal geometry and students’ perceptions and knowledge of space and shape is recognized as an important premise for understanding abstract geometry (Van de Walle, Lindquist). The NCTM Principles and Standards for School Mathematics outlines curriculum that would address different levels of understanding, but do not stress the importance that learning must occur sequentially. The new Navigation Series addresses “Prior Knowledge” as part of a formal lesson plan, but it does not state that success of the lesson is based on the ability of students to accomodate the new material with their existing knowledge. It also strongly assumes that students have had adequate school experiences and have “informal knowledge about points, lines, planes, and a variety of two- and three-dimensional shapes; with experience in visualizing and drawing lines, angles, triangles, and other polygons; and with intuitive notions about shapes built from years of interacting with objects in their daily lives” (Standards 233). How do teachers assess if students have indeed had this kind of informal learning, and what do we do if they haven’t? This does not seem to be addressed in the Standards, which may be beyond the scope of the document. It implies a trust that learning occurred that may not have. Also, it ignores an important premise of the learning theories outlined earlier. One more thing: does sequential and experiential learning in elementary school justify ignoring the importance of this type of learning in middle school? With all the other things early adolescents have on their minds, the concepts they learned in elementary school could easily be forgotten if not reinforced and built upon during the middle school years.

Another problem that may occur, even if elementary geometry has been taught in a sequential and experiential manner, is that of middle school students making the connection between informal and manipulative learning experiences and more formal, abstract mathematical ideas (Hart). According to Hart, this has not been addressed very thoroughly. Statements in math teaching manuals imply or even state that “children will come to realize” math formulas after exploring with manipulatives. However, few children actually make this formal connection. For example, a teacher who knew the rule or formula and devised a set of lessons using, manipulatives that she believed would lead students to “see” the rule may not provide a clear direction of where the lesson was leading.    Her students might complete the task but not connect the task with the learning objective. Often, incorrect or incomplete concepts are formed. Let me use myself as an example in the lesson that was done at the Wemagination Center. We were asked to create a model, using aerosol can lids, using the rule, a green plus a green make a green, green plus purple make purple, and purple plus purple make purple. The result was a lovely Pascalian triangle with an obvious pattern. My group had some very fast thinkers, and my role was just to hand them the color they asked for. Learning on my part was very minimal. Then my group worked on coloring a triangle using Mod 3, and the result was a colorful image of greens, purples, and blues in a pattern. So...what. I can see the result and intuitively know that some mathematical principle is at work here, but I wouldn’t use this activity without myself understanding the purpose and the broader implications of the exercise. I do know that mathematics is the study of patterns, but do not see how this particular activity can be used without the teacher knowing the mathematical significance of the pattern. Which brings me to the topic of the actual curriculum unit.

Getting From Here to There: Progressing Through the Levels of Geometric Thought in Middle School Lessons

The human race has relied on what mathematical knowledge was created, expanded upon, and passed down throughout its prehistory and history. The word geometry comes from the Greek ge, Earth, and, metria, measurement (Gullberg 364). The Babylonians recognized the importance of a circle and developed a numerical system and calendar based on equal divisions of a circle. The Egyptians used geometry to accurately measure land; of course, one cannot ignore the geometry needed to construct a pyramid. The Greeks, Plato and Aristotle, began to formally study geometry as an abstract science. Euclid’s collection of Elementary Geometry, written about 250 bc‚ used logical reasoning based on a central core of postulates, or axioms. He used an unmarked compass and assumed planar and polyhedral surfaces (that is, constructions of a plane surface and solids bounded by regions). Ptolomey used descriptive geometry in his astronomical calculations, about 150 bc and Hipparchus is credited with inventing trigonometry as a way to study astronomy in about 120 bc. Although, Ptolomey’s calculations were based on an incorrect assumption, that the earth is the center of the universe, his ideas were accepted and taught for over a thousand years, until Copernicus proved that the planets revolve around the sun. Geometric knowledge has continued to grow from the Greek/Arab world to include levels of geometry past the scope of this paper. Knowledge continues to evolve with the creation of Chaos Theory.

Learning occurs when our existing knowledge is found to be deficient. In Piagetian philosophy, our thoughts must be thrown into disequilibrium (Elkind). In constructivist language, we must undergo conceptual change to make sense of new information by using existing knowledge (Hewson). The study of astronomy resulted in man’s ability to navigate the seas, and our understanding of the solar system and our place in the universe is due to geometrical understanding. Ptolomy’s explanation of the apparent retrograde motion of planets seems illogical to us now based on knowledge accumulated since then. Copernicus’s explanation got him in trouble with the church, but he revised a popular concept that made no sense to him. The Greek constellations, lines drawn from star to star based on a planar sky, gave way to the knowledge that, for example, the alignment of the three stars of Orion’s belt are actually an accident; they are light years apart in a three dimensional sky (Stewart 24). Newton’s “discovery” of gravity was based on Kepler’s observation of the elliptical motion of the planets, and he used inductive reasoning to develop his inverse-square law and prove that the attractive force of the sun can be applied to any two bodies.

More “down to earth” applications of geometry were discovered by Euler, who developed the “wave equation” in about 1748. He studied the vibrations and movement of a violin string and determined that the deformation of the string, no matter what it’s starting point, moves along a predictable path (Stewart 66). The Bushoong people in Africa used wave theory informally to create patterns in the sand to aid in telling stories. Children of the Bushoong use the same concept to play games. From a starting point, children trace around a number of points without lifting their finger or stick from the sand. Decorations on pottery and clothing also reveal the same geometrical concept. This use of graph theory and symmetry, without formally understanding the mathematical principles behind them, are used to create useful and aesthetic designs in many cultures around the world (Ascher 45). Whether crossing the bridges of Konigsberg, studying the science of music, or playing games in the sand, the study of geometry that could be applied to more formal thinking creates those intuitive pathways that may be necessary for geometrical thinking to advance.

However, the purpose of this curriculum unit is to identify what must occur before geometrical thinking can advance in middle school-aged children. The unit must proceed from identifying the geometric thought level of each student, introducing lessons that are appropriate for each level and that challenge the student to advance to the next level, then validate that indeed the next level has been reached. So there are three parts to this unit: assessing entry level, designing appropriate experiences, and assessing exit level. This necessitates choosing a particular geometric concept. One study, done on high school students in Austra�lia, limited the topic to parallel lines. Students sorted pairs of lines into parallel, not parallel, and not sure. A set of eight lessons was then taught:

1)      Work in groups to sort pairs of lines and construct a class definition of parallel.

2)      Use edges of boxes to define the need for parallel lines.

3)      Use file cards to measue the distance between lines and perpendicularity.

4)      Identify sets of parallel lines in tesselation patterns that students constructed

5)      Identify ladder and zigzag patterns in real-life examples and discover through comparison that corresponding and alternate angles are equal.

6)      Develop an argument that the angle sum of a triangle is 180 degrees.

7)      Using knowledge of angle properties, solve problems involving parallel lines.

8)      Work in groups to summarize properties of parallel lines and of angle relationships found in parallel lines.

A posttest was given immediately after the unit which included developing a concept map. This test showed most students had constructed new knowledge, but the real test of the unit was a test given six months after the instruction. Most of the students did show that their understanding of the concept of parallel lines had made permanent modifications. A control group received lower scores, showing that the set of lessons was effective in achieving conceptual change (Mansfield).

Implementation

Seventh graders seem to have a difficult time with area and perimeter. In particular, why are those two words so hard for them to remember? They constantly get them mixed up. What experiences would enable them to gain conceptual understanding of the meaning of the terms? How could lessons proceed sequentially so that they progress from their level of thinking at the start of the unit to the next level, and retain that knowledge indefinitely? The above unit and others could serve as models in developing the lessons.

I decided to use only the concept of perimeter to limit the breadth and instead focus on depth of learning. Most lessons teach the two concepts of area and perimeter together, which seems logical, except when one notices the confusion that students have. A complete knowledge of perimeter would prevent them from confusing it with the concept of area. The Van Heile Geometry test, though it was designed for tenth graders, could provide examples of items to include on the pre- and post-tests to determine thought level. Lesson ideas came from various sources but were adapted to fit my teaching style. Several of them are my original ideas. The unit will serve as an example of how to develop curriculum based on learning theory, in that students are given the opportunity and encouragement to construct thorough knowledge about perimeter that they can apply to other areas of geometry.

Albuquerque Public Schools Standards Met By the Unit

Standard 1: Number Sense and Operations

1.4.3 Construct and demonstrate more than one strategy to solve real-life problem situations involving rations, proportions, and percents.

Standard 2: Geometry, Spatial Sense, and Measurement

2.2    Transform geometric figures using reflections, translations, and rotations.

2.4 Describe, analyze, and reason informally about the properites of polygons.

2.4.2        Investigate properties and classifications of triangles and quadrilaterals.

2.5 Extend problem solving involving measurement to include 3-dimennsional figures.

2.5.2 Create and test strategies for finding volume and surface area of cylinders and cones.

2.5.4 Select and apply appropriate formulas in problem solving.

Pre-test (Refer to Figure 1.)

1. A farmer’s field looks like this. If you had to put a fence all the way around it, how many feet of fence would you need? Write how you know this. What is this measurement called?

2.      Draw a line that divides each of these figures into two equal pieces (congruent). Is there another line you could draw that would also make 2 congruent figures? What is this line called?

3.      What is the perimeter of the circle? (how far around) How do you know this? Is there another name for the perimeter of the circle?

4.      What is the perimeter of the circle?

Lesson Plans

One way to accommodate the various levels of students’ geometric thoughts levels and abilities is to do a committee cycle using some or all of the lessons given here. Students at similar levels could work together, but groups would cycle through each activity. The committee could be modified to meet the needs of each group. At the end of the committee cycle, the class could discuss each committee together. Each of the lessons could also be taught as a whole class lesson.

A. Toothpick measuring

Objective: Students will explore what shapes can be made when a figure with a perimeter of 6 toothpicks is expanded to a perimeter of 12 toothpicks.

Procedure: Each student gets 6 toothpicks. What different shapes can be made, using only right angles? (Provide cards or blocks to check for perpendicular angles.) Now make a hexagon and pretend its a pasture where you keep a horse. You are given some more land and 6 more toothpicks (pieces of fence). Add them to your hexagon so that 2 squares jut out from 2 of the hexagons sides. What is the perimeter of the corral now? Moving the inside toothpicks so that the horse can’t get out (that is, no holes are ever in the fence), move the inside toothpicks to enlarge the pasture. The result should be a dodecagon (12 sided figure).   Transition to isometric grid paper and draw the 6-sided horse corral and the 12 sided one.

Assessment: Have students write on their paper the perimeter of each figure.

Lesson PlansOne way to accomodate the various levels of students’ geometric thoughts

B. Konigsberg paths

            Objective: Students will explore movement and drawing on a single path (a 2-dimensional medium) to determine the distance covered and to use critical thinking skills to solve a puzzle.

Procedures: Create a maze with the desks in the room so that students must get from the front of the room to the back without backtracking or walking on the same path twice. Count the number of steps it took; try to find a shorter path that uses less steps. Provide students with some paths or mazes, like the Konigsberg paths, where students can cross each bridge once and only once. (See Figure 2.)

Assessment: Using the handout provided, students could follow the directions and get from one point to another without crossing the same bridge more than once.

C. Geoboards

Objective: Students will copy given figures on a geoboard and determine the perimeter. They will then draw figures of a given perimeter on square grid paper.

Procedure: Make sure students know that between each nail is one unit; however, on the diagonal the distance is more than one. Copy given figures (horizontal and vertical lines only) that the teacher makes on the overhead,geoboard‚ find the perimeter. Have student volunteers make figures to copy and find the perimeter. Students make their own geoboard figure and have a partner find the perimeter. Draw figures on square dot or graph paper with a given perimeter. Example: The teacher says, “Make a five-sided figure with a perimeter of 12 units,” or “Make a 6-sided figure with a perimeter of 10.” Students label the figures and turn it in. A handout could be made with 10-20 similar problems for homework.

Assessment: Students made figures of a given perimeter on the geoboard. They could figure out the perimeter of their partner’s geoboard figure. Students could draw the figures with the correct number of sides and the correct perimeter.

D. What if it’s a triangle?

            Objective: Students will make and measure the perimeter of right triangles.

Procedure: Make a right angle that is 3 units by 4 units on a geoboard. Introduce the algorithm of the Pythagorean theorem. If a squared plus b squared equals c squared, then 9 plus 16 should give you the value of c squared, or 25. Measure the distance of c and see if it’s the same as 5 units. (A non-standard unit of measurement, such as a mark on a pencil, can be used to determine if the length of the hypotenuse is the same as 5 units.) Students make several right triangles and make a chart. Look for a pattern that verifies the theorem. Draw right triangles on square dot paper and determine the perimeter. Use a calculator for values that are not whole numbers.

Enrichment: Use equilateral and other triangles and have students use what they learned about the perimeter of right triangles to find the perimeters of these triangles. Encourage them to make rectangles and then right triangles from parts of the triangles.

Assessment: Students drew right triangles and could calculate the perimeter accurately.

E.      What if it’s a circle?

Objective: Students will measure the circumference and diameter of circles in order to see the ratio as the value of pi.

Procedure: Provide pairs of students with different sized cylinders, such as cans, lids, cups, etc., a tape measurer and a ruler. Make a chart with three columns titled, diameter, circumference, and difference. Students list the measurement of both the circumference and the diameter of the objects. Compute the difference between the two measurements and include that in the chart. Students should construct knowledge of pi when they see that the circumference is always about 3 times the diameter. Use this to estimate the circumference of circles drawn on graph paper.

Assessment: At my school, there are several cement circles of various sizes outside. Have students determine the circumference of the circles. Observe which method they use; ideally, they would find the diameter and multiply it by three to demonstrate their understanding of pi.   Students should be able to tell you what another name for circumference is (perimeter).

F. African huts.

Objective: Students will make a model of an African “beehive” hut to explore how other cultures use measuring in a useful and practical way. (See Figure 3.)

            Procedures: Students will work in heterogeneuos cooperative groups. A task card and materials will be provided for each group. Task Card will read: You will construct a model of an African dwelling called a “beehive hut.”   You will draw it to scale on graph paper, then make a model using the materials provided. (Materials: construction paper, raffa paper, popsicle sticks, rulers, tape measurer, scissors, glue, tape, sidewalk chalk) 1. The size of the diameter of the hut is the iaa (a unit of length of the arm span from fingertip to fingertip). Measure this using the tallest person in the group and a measuring tape. The African people would not use a measuring tape. They use a rope that is the length of the iaa. Draw a line that is that length outside on the sidewalk, using chalk. Using a string that is the same length as the iaa, draw a circle whose diameter is the iaa. (Hint: Fold the string in half to make the radius of the circle. Put the chalk in the fold of the string, and keeping the ends at the center and the string taught, make a circle.) Decide on an appropriate scale and using a proportionately smaller string or a compass, make a scale drawing of the circle on graph paper.

2. How much material would you need to construct the wall? Measure the height and armspan of all the group members and look for a pattern. About how tall would the walls have to be so that the tallest person could stand up comfortably? About how much material would be needed to make walls of that height covering the circumference of the circle? Cut construction paper to fit the circle on the graph paper.

3. The doorway is as wide as the circumference of the beehive hut’s owner’s head. Guess how wide that would be. Measure the circumference of a group member’s head. Cut a rectangle of that width out of the wall to make the door.

4. The roof is cone-shaped. Make a center pole using straws and tape, then attach popsicle sticks from the top to make the roof. Use string or tape to join sticks at the center.

Assessment: Groups produced an African beehive hut model which has a circular shape, walls that go around it sufficiently, a door, and roof. They should include the scale of their model.

F Exploring symmetry and perimeter.

Objective: Students will create a symmetrical picture while focusing on the perimeter of each object in the picture. They will draw a wall or stained glass window which has at least one line of symmetry. They will be able to demonstrate how each side of the picture has congruent sides.

1. Show the students examples of carved doors, stained-glass windows, murals, tablets (as from ancient Babylonia or the Mayan civilization). Demonstrate the lines of symmetry using a mirror.

2. Use pattern blocks to create a design, then make the same design using mirror symmetry. Give the students the opportunity to use a mirror if they desire. Copy one of the patterns on isometric grid paper. Be sure the students copy it by looking at the design and reproducing it on the paper. Students often try to trace the blocks so that the design doesn’t fit within the grids on the paper, which looks sloppy. Help the students focus on perimeter by verifying that the lines (as opposed to the shapes) are congruent on each side of the line of symmetry. Of course, this activity could also be used when teaching about area.

3. Students use their imagination and examples to create a design on isometric grid paper. There must be at least one line of symmetry. Encourage students to use patterns and variety to create an interesting piece. Provide time to color the design, or assign it for homework.

Assessment: Students created a completed design with at least one line of symmetry. The design was neatly drawn and colored. Intricate detail and unusual shapes that are correctly drawn on each side of the line of symmetry would receive extra points

G.     Making connections

Objective: Students will identify when the word perimeter is used and what other clues can be used to remember the concept.

1. Peripheral vision is what you see at the edge of your vision. The periphery is the outside boundary. Those words contain the same root perimeter.

2. Make up a story to help them remember the term. For example, a boy named Perry went out to the fence to meet his girlfriend. Perry met her (peri-met-er) at the fence. Maybe students could think of better stories and word clues.

            Assessment: Give a vocabulary pop-quiz and have students write the definition of perimeter.

H. Drawing objects with a given perimeter.

Objective: Students will use grid paper to draw a variety of shapes with a given perimeter. For example, the teacher could give the perimeter as being 12 units. Students will draw as many different-shaped figures as they can with a perimeter of 12 units. This activity is similar to a popular lesson that uses pentominoes (figures made up of five squares) to teach perimeter and area. You might choose to use color tiles in the teacher demonstration and/or have students use color tiles to make figures of a given perimeter before drawing them on grid paper.

Procedure: Using a transparency of square grid paper, ask students to name a 2-digit number. Draw a figure with that perimeter. Ask volunteers to make a different figure of the same perimeter. Assign each student a number and have them draw as many figures with the number of units as the perimeter as they can. Have them color the figures and clearly mark the length of each side

Assessment: Students drew a variety of figures all with the same perimeter

I. Perimeter of letters

            Objective: Students will use block letters on square grid paper and write out the word “PERIMETER.” They will determine the perimeter of each letter. (See Figure 4.)

            Procedure: Supply each student with square grid paper. They will color block letters and make the word perimeter. They can choose to use diagonal lines, but they must incorporate the square root of 2 into their equation of the perimeter of the letter if they do. They will write the perimeter of each letter below the letter. Extension: students could prepare a bulletin board using cut-out block letters which have a patterned border around each letter. How much border is needed for each letter?

            Assessment: Letters are neatly made and perimeter accurately determined.