Investigating Quadrilaterals Grade Four


 Gilbert Garrett
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1 Ohio Standards Connection Geometry and Spatial Sense Benchmark A Provide rationale for groupings and comparisons of twodimensional figures and threedimensional objects. Indicator 3 Identify similarities and differences of quadrilaterals; e.g., squares, rect, parallelograms and trapezoids. Mathematical Processes Benchmarks H. Recognize basic valid examples and counter examples, models, number relationships, and logic to support or refute. I. Represent problem situations in a variety of forms (physical model, diagram, in words or symbols), and recognize when some ways of representing a problem may be more helpful than others. J. Read, interpret, discuss and write about mathematical ideas and concepts using both everyday and mathematical language. Lesson Summary: In this lesson, students learn to compare and analyze the attributes of quadrilaterals. Students use attributes to sort and identify specific classes and hierarchy of quadrilaterals. Graphic organizers assist students in categorizing and comparing quadrilaterals. Working with partners, students have opportunities to discuss concepts and use mathematical vocabulary to communicate understanding. Estimated Duration: Three hours Commentary: Students typically enter fourth grade able to identify several shapes. Identifying shapes, however, does not provide sufficient foundations for the higher levels of reasoning required in later grades. Instruction in identifying specific classes of quadrilaterals and in understanding the hierarchy of quadrilaterals does help students move to a higher level of reasoning about twodimensional figures (NRC, 2001). There are two viewpoints for how a trapezoid is defined. The commonly accepted definition is a quadrilateral with exactly one set of parallel lines. The alternative definition is a quadrilateral with at least one set of parallel lines. This lesson uses the commonly accepted definition of a trapezoid. If using the alternative definition, make appropriate adjustments throughout the lesson. PreAssessment: Have students complete a quickwrite exercise in a journal explaining what they know about quadrilaterals. Select students to share what they know about quadrilaterals with the class and record the information on the board or chart paper. Ask students what questions they have about quadrilaterals and record those on chart paper. Instructional Tip: Use a KWL chart as an alternative. 1
2 Scoring Guidelines: Informally assess the students responses. Possible responses include: names of quadrilaterals (square, rectangle, rhombus, parallelogram and trapezoid) and if they know quantitative descriptions such as the number of sides and vertices qualitative descriptions such as types of (acute, right, obtuse) and line relationships (parallel or perpendicular) PostAssessment: Use Attachment A, Quadrilateral PostAssessment. Given a word bank of quadrilaterals, students select two figures to compare and contrast in a Venn diagram. Students then select two different quadrilaterals to compare and contrast in a table. Scoring Guidelines: Assess the diagram and chart for accuracy. Responses should indicate relationships among the quadrilaterals using attributes and recognizing hierarchy. For example, if the student chooses to compare the square and the rhombus with a Venn diagram, all attributes of the square should be contained in the intersection of the two squares. Attributes of squares should not be written outside of the intersection, because a square is a rhombus with four right. Instructional Procedures: Part One 1. Distribute Attributes of Quadrilaterals, Attachment B and Quadrilateral Cards, Attachment C to pairs of students and have students cut out the twelve cards. 2. Use the information about quadrilaterals in the preassessment and observations students make using the cards to complete the first column, Quadrilaterals, on Attributes of Quadrilaterals, Attachment B. Depending on depth of prior knowledge, as revealed in preassessment, choose to have partners complete the column or complete the column as a class. 3. Have students sort the cards by the shapes at the top of Attributes of Quadrilaterals, Attachment B. Observe students as they sort the cards and provide assistance as needed. 4. Have students complete the chart together, using what they know about the shapes and the cards. 5. Complete a class chart on the overhead and allow students to make changes to their own charts. An example of a completed chart is provided on page two of Attributes of Quadrilaterals, Attachment B. 6. Have partners sort the shapes using different attributes. They may choose to use the attributes listed on Attributes of Quadrilaterals, Attachment B. Then, select students to share the attribute they used to sort the shapes and present the sorting to the class. 7. Have students store their shape cards in resealable bags for the next lesson. Part Two 8. Have students take out the shape cards and Attributes of Quadrilaterals, Attachment B. 9. Have the students use the chart to compare the quadrilaterals. Have them create a list of attributes shapes share. For example: All sides of the square and rhombus are. 10. Select students to share their comparison statements with the class. 2
3 11. Explain to students that by sorting shapes additional comparisons can be made and relationships among the shapes can be revealed. 12. Have the class sort the shapes into two categories, shapes with parallel sides and shapes without parallel sides. 13. Observe students as they sort the shapes and assist as necessary. Reinforce the concept of parallel lines. 14. Select students to present the sorting and allow other students to provide feedback. 15. Direct the students to the pile of shapes with parallel sides. Have pairs sort the shapes into two piles, shapes with one set of parallel sides and shapes with two sets of parallel sides. Observe students as they sort and provide assistance as necessary. 16. Select students to present the sorted shapes. Have students identify the names of the shapes in each pile. 17. Explain to the students that trapezoids have at least one set of parallel sides and that parallelograms have two sets of parallel sides. Ask students to identify the shapes in the parallelogram pile. Tell students that squares, rect and rhombi are special parallelograms. 18. Have partners sort the parallelograms by angle measure, shapes with right and shapes without right. Observe how pairs sort the shapes and provide assistance as needed. 19. Have students identify the shapes in each pile. Ask students questions about the relationships. What do squares and rect have in common? (four right ) How are the rect and squares on the cards different? (lengths of sides) Can a square be described as a rectangle? Why? (A square is a special rectangle with four sides.) 20. Have students compare a rhombus and a square. Ask questions and allow pairs to discuss before selecting students to respond. Questions for discussion include: How are the square and rhombus alike? How are the rhombi and squares on the cards different? Can a square be described as a rhombus? Why? (A square is a special rhombus with four right.) 21. Summarize the relationships visually using a graphic organizer. Quadrilateral Relationships, Attachment D is an example of an appropriate organizer. Distribute to the students and complete the organizer together. Encourage students to use the shape cards to resort the shapes with the attributes used during the lesson. Instructional Tips: In the following days, have students share the graphic organizer and write comparisons of the quadrilaterals. For morning work or problem of the day, present prompts for students to respond, about quadrilateral relationships. Prompts may include: 1. All squares are rect, but not all rect are squares. Why? 2. All squares are rhombi, but not all rhombi are squares. Why? 3. Squares, rect and rhombi are parallelograms. Why? 3
4 Differentiated Instructional Support: Instruction is differentiated according to learner needs, to help all learners either meet the intent of the specified indicator(s) or, if the indicator is already met, to advance beyond the specified indicator(s). Use twodimensional manipulatives or geoboards to investigate the properties, make conjectures and draw conclusions on quadrilaterals. Provide students with a dichotomous key to use as a guide in classifying the shapes. Have students create a game board using the twodimensional shapes with game cards asking questions identifying the shapes, and stating questions with answers on their similarities and differences. Such questions may be: How is a square similar to a rectangle? How is a rhombus like a parallelogram? Why does a trapezoid not fit in with the parallelogram, rectangle, rhombus, and square? Extensions: Make a class dictionary on the quadrilaterals and the vocabulary terms studied. Provide students with shapes that include polygons other than quadrilaterals such as pentagons, hexagons and different kinds of tri. Place shapes in an envelope. Have students sort them into 2 groups and explain why some shapes fit in one group and why others are left out of that group. Have students sort two to three times. Home Connections: Assign an interdisciplinary activity. Assign homework where the student will draw a design or creature with the shapes: trapezoid, parallelogram, rectangle, rhombus, and a square. When the student returns his creative quadrilateral creature, provide them with an overhead transparency to draw a habitat for the creature. Place the overhead transparency over top of the creature or animal and share during Science class. Have the students communicate with their families the similarities and differences of quadrilaterals: trapezoid, parallelogram, rectangle, the rhombus and square. Have students build models of quadrilaterals out of household materials such as toothpicks, cotton swabs, spaghetti, and pieces of yarn or string. Include a writing portion of the assignment in which students describe the quadrilateral. Materials and Resources: The inclusion of a specific resource in any lesson formulated by the Ohio Department of Education should not be interpreted as an endorsement of that particular resource, or any of its contents, by the Ohio Department of Education. The Ohio Department of Education does not endorse any particular resource. The Web addresses listed are for a given site s main page, therefore, it may be necessary to search within that site to find the specific information required for a given lesson. Please note that information published on the Internet changes over time, therefore the links provided may no longer contain the specific information related to a given lesson. Teachers are advised to preview all sites before using them with students. For the teacher: chart paper, markers, overhead transparencies For the student: resealable plastic bags 4
5 Vocabulary: parallel lines parallelogram quadrilateral rhombus trapezoid Technology Connection: Use geometry software to draw and explore characteristics of quadrilaterals. Research Connections: Almy, M., E. Chittenden, & P. Miller. Young Children s Thinking: Studies of Some Aspects of Piaget s Theory. New York: Teachers College Press, National Research Council. Adding It Up: Helping Children Learn Mathematics. J. Kilpatrick, J. Swafford, and B. Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, D.C.: National Academy Press, Attachments: Attachment A, Quadrilateral PostAssessment Attachment B, Attributes of Quadrilaterals Attachment C, Quadrilateral Shapes 5
6 Attachment A Quadrilateral PostAssessment Name Date Word Bank Parallelogram Rectangle Rhombus Square Trapezoid Directions: From the list of quadrilaterals, select two. Use the Venn Diagram to compare and contrast them. List at least two similarities and two differences for each. Name of Quadrilateral 1 Name of Quadrilateral 2 Directions: Complete this activity again using two different quadrilaterals. Use the table to compare and contrast them. List at least two similarities and two differences for each. Name of Quadrilateral 3 Name of Quadrilateral 4 Common Characteristics Different Characteristics 6
7 Attributes of Shapes Number of sides Number of Congruent sides Congruent Right Parallel sides Symmetry Congruent Investigating Quadrilaterals Grade Four Attachment B Attributes of Quadrilaterals Quadrilateral Square Rectangle Rhombus Trapezoid Parallelogram 7
8 Attributes of Shapes Number of sides Attachment B (continued) Attributes of Quadrilaterals Quadrilateral Square Rectangle Rhombus Trapezoid Parallelogram Number of Congruent sides have sides Congruent have Right have a right angle Parallel sides have parallel sides Symmetry have symmetry Congruent have All sides are All are All are right Opposite sides are parallel Has four lines of symmetry All are Opposite sides are All are All are right Opposite sides are parallel Has at least two lines of symmetry All are All sides are Opposite are Does not have to have right Opposite sides are parallel Has at least two lines of symmetry Opposite are have sides have have right At least one set of opposite sides are parallel have a line of symmetry have Opposite sides are Opposite are have right Opposite sides are parallel have symmetry Opposite are 8
9 Attachment C Quadrilateral Shapes 9
10 Attachment C (continued) Quadrilateral Cards 10
11 Attachment D Quadrilateral Relationships 11
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