Ohio Standards Connection Geometry and Spatial Sense Benchmark E Use proportions to express relationships among corresponding parts of similar figures. Indicator 1 Use proportional reasoning to describe and express relationships between parts and attributes of similar and congruent figures. Benchmark G Describe and use properties of triangles to solve problems involving angle measures and side lengths of right triangles. Indicator 5 Apply properties of congruent or similar triangles to solve problems involving missing lengths and angle measures. Benchmark J Apply properties of equality and proportionality to solve problems involving congruent or similar figures; e.g., create a scale drawing. missing lengths and angle measures. Indicator 5 Apply properties of congruent or similar triangles to solve problems involving missing lengths and angle measures. Lesson Summary: Students explore pairs of triangles to develop a conceptual understanding for similar figures. Students use this definition to explore the proportional relationships involved in similar triangles and to find missing lengths and angle measures. Estimated Duration: One and one-half to two hours Commentary: The concept of similarity is not new to students at this grade level. However, there are aspects that require further investigation. One is the use of an asymmetrical triangle in the study of similar triangles. This takes some students out of their comfort zone but provides a deeper understanding of the students knowledge of similarity. The second focus is related to the orientation of two similar objects. Identifying corresponding angles and sides becomes more of a challenge to students when the orientation is changed from one object to the other. Research has also shown that computer assisted drawing tools help develop conceptual understanding of scale factor and proportionality among similar figures. These tools allow students to easily make changes to a figure by a given scale factor, change the orientation and compare lengths and angle measures of corresponding parts quickly and accurately to determine if objects are similar. Pre-Assessment: Distribute Geometric Figures, Attachment A, to each student. Direct the students to match each figure on the page to a similar figure or group of figures on the page. Pair the students and direct the pairs to share their ideas and to write explanations as to what makes the figures similar. Group two pairs and have each student draw an additional similar figure that belongs in this group. Have them include an explanation to justify why the figures are similar. Observe students as they work. Use the Pre-Assessment Checklist, Attachment B, to determine the students prior knowledge and to inform instructional decisions. 1
Ohio Standards Connection Mathematical Processes Benchmark B. Apply and adapt problem-solving strategies to solve a variety of problems, including unfamiliar and non-routine problem situations. Scoring Guidelines: Use Pre-Assessment Checklist, Attachment B, during the observation to assess student understanding of the Geometric Shapes activity, discussion and written responses. Evaluate the methods used by students to determine similar shapes. Strategies and discussion may include: identifying the number of sides and angles, finding corresponding sides and angles, distinguishing types of lines and angles and comparing sizes of figures using words such as shrunk, blown up, twice as large or three times bigger. Post-Assessment: Allow students to choose one of the following postassessment options. Have students design squares for a patchwork quilt or stained-glass window on grid paper by using six to eight types of polygons. Include two sets of congruent and two sets of similar, non-congruent figures to create the square using construction paper. Use proportional reasoning to prove that the figures are similar or congruent. Have students explain the reasoning used to prove the figures are similar or congruent. Instruct students to create a blueprint of a gymnasium floor or football field. Include the lines painted on the field or floor. Use measurements, ratios and proportions to show that the blueprint is similar to the real floor or field. Have students explain the reasoning used to prove the blueprint is similar to the real gymnasium floor or football field. Use Lost Lengths, Attachment C, to determine if the students can use their knowledge of similar figures to find the missing lengths. Scoring Guidelines: Check each drawing for correct proportional reasoning to explain how the figures are similar or congruent. Students having difficulty in creating congruent or similar figures may need additional instruction prior to completing this lesson. Instructional Procedures: Part One 1. Distribute one copy of Similar Triangles, Attachment D, to each student. Have students work with partners. 2. Tell students that the pairs of triangles are similar. 2
3. Instruct the students to measure all the angles and side lengths of each triangle. 4. Direct the students to compare their findings for the pair of similar triangles. Ask questions such as: What did you notice about the angle measures? What did you notice about the measures of the sides? Is there a relationship? Describe the relationship. Instructional Tip: Assist students who may have difficulty in seeing the relationships among the corresponding angle measures of similar triangles and among the corresponding side lengths by asking guiding questions. 5. Ask partners to make conjectures about the properties of similar triangles based on their findings, such as: Similar triangles have congruent corresponding angle measures, but the lengths of the sides are different. If proportional relationships are discussed, include in the conjecture. 6. Have students share conjectures and record them on the board or chart paper. 7. Have the class determine if each conjecture holds for all similar triangles by looking for examples that support or contradict the conjecture. 8. Facilitate a class discussion to create a general definition for similar triangles. Use the correct conjectures that the class agreed upon to develop the definition. Instructional Tip: The definition should include the fact that the corresponding angle measures are congruent, but the side lengths do not have to be congruent. Develop the fact that corresponding side lengths are to be proportional throughout this lesson. 9. Have students identify the ratio or scale factor between the corresponding side lengths for each pair of similar triangles. 10. Have students show their understanding of similar figures by writing a lesson summary in their mathematics journals. They should include a definition using mathematical terms to explain the characteristics. Read through student summaries to identify level of understanding. Part Two 11. Draw two similar triangles on the board. Label the sides with the lengths 6 inches, 8 inches and 12 inches. Label the side of the second triangle corresponding to 6 inches in the first triangle with the measurement of 4 inches. If proportions are not understood, use grid paper to draw accurate triangles and the grid squares to estimate the missing side length. 12. Ask students to determine the ratio of the two triangles (6:4). Have partners find the lengths of the other two sides of the triangle. Observe the strategies students use to find the lengths of the sides. Students who have proportional reasoning will use the ratio 6 to 4 and make equivalent ratios and proportions. 3
13. Have the partners share their methods for finding the solution. Modify the class definition to include the fact that the side lengths of similar triangles are proportional. 14. Provide practice to reinforce the concepts of similarity and strengthen the evidence to support the conjecture. a. Draw two similar triangles on the board with two side lengths missing. Ask students to write explanations of how to find these missing side lengths. Instruct students to use correct vocabulary words such as similar, proportion, congruent and corresponding. b. Have students create triangles or pairs of triangles to give to partners. Have the partners create similar triangles or prove that two triangles are similar and provide mathematical evidence for their solutions. 15. Pose the question, Think of the minimum requirements you need to know if two triangles are congruent, such as angle, side and angle. Can these minimum requirements be used to determine if two angles are similar? 16. Have small groups test the minimum requirements and make conjectures. Select students to share the groups discoveries. 17. Have students play the following game as a final review. a. Arrange the class in groups of three or four. b. Tell the students that there is a triangle drawn on the index card in your hand. c. Instruct the students to come up with the criteria that they would need to know in order to draw a triangle that was similar to your triangle and one and a half times larger than your triangle. 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). Have students find and draw similar shapes other than triangles. Use easier or more difficult measurements depending on the students. Extensions: Ask students to determine the minimal information required to prove that two triangles are similar. Extend the content to include objects that physically cannot be measured. (e.g., use shadows and other indirect measurements to find the height of a tall tree, a building, a flagpole.) Using what the students learned about similar triangles, take them outside to use their shadows to find the height of a very tall tree or building. Have students stand outside in the sun. Have them measure their heights and the length of their shadows. (Do this activity in small groups.) Have the students sketch this in the form of a right triangle with their heights and the shadows as the legs of the triangles. Select a tall tree or building and sketch it with its shadow as a right triangle. Have students use proportional reasoning to measure the length of the shadow and use it to find the height of the tall object. 4
Home Connection: Instruct students to list examples of objects that are similar or congruent; measurements should be included to verify their conclusion. For example, they could compare two different boxes, two different size cans, (e.g., coffee and soup) or windows in their house. 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: attachments, chalkboard or overhead projector For the student: paper, rulers, protractors Vocabulary: congruent corresponding equivalent fraction proportion similar Technology Connections: Use geometry software to create two figures and verify that they are similar or congruent. Research Connections: Marzano, Robert J., Jane E. Pollock and Debra Pickering. Classroom Instruction that Works: Research-Based Strategies for Increasing Student Achievement, Alexandria, Va.: Association for Supervision and Curriculum Development, 2001. Attachments: Attachment A, Geometric Figures Attachment B, Pre-Assessment Checklist Attachment C, Lost Lengths Attachment D, Similar Triangles 5
Attachment A Geometric Figures Name_ Date_ Directions: Cut out each of the figures and organize them into groups of similar figures. 6
Attachment B Pre-Assessment Checklist Name_ Date_ Directions: Use this checklist to pre-assess students for the content of this lesson. 1. Figures are appropriately grouped. 2. Figures are divided by more criteria than just shape. 3. There is understanding of the meaning of similar, as seen by the grouping of figures. 4. Similar attributes have been identified for different groups of figures. 5. Correct explanations are given as to why the figures in each group are similar. 6. Additional similar figures have been drawn that fit into each category. 7
Attachment C Lost Lengths Name_ Date_ Directions: Complete each of the following problems. 1. Emily needs to create a smaller version of her club s symbol seen on this poster. The new drawing needs to fit on a T-shirt. Draw two smaller, similar triangles that she could use. Hint: 1ft = 12 inches 2 ft Good Book Club 2 ft 3 ft 2. The following triangles are similar triangles taken from the roofline of this home. Find the missing side lengths labeled x and y. 10 ft 10 ft 14 ft 7 ft y x 3. Describe what you know about the angle measures and side lengths of two similar triangles. 8
Attachment D Similar Triangles Name_ Date Directions: Four pairs of similar triangles are shown. Determine the measurements of the lengths of the sides and angles. Compare the measurements of the corresponding sides and angles. 9
Attachment D (continued) Similar Triangles Name_ Date_ 10
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