Document Sample

8th Grade Mathematics: Unit 4: Geometry and Measurement Ascension Parish Comprehensive Curriculum Concept Correlation Unit 4: Geometry and Measurement Time Frame: 4 Weeks Big Picture: (Taken from Unit Description and Student Understanding) Geometric figures can change position and maintain the same attributes on a coordinate plane. Geometric figures can change size and/or position while maintaining proportional attributes. The Pythagorean Theorem can be used to solve problems involving right triangles. Constructions are based on properties of geometric figures. Activities Documented GLEs Guiding Questions Essential Activities are denoted GLE’s GLEs Date and Method of with an asterisk GLEs Bloom’s Level Assessment Concept 1: Angle *Activity 46: The Define and apply the terms Relationships Bisection! measure, distance, midpoint, bisect, 23 23 GQ 14 bisector, and perpendicular bisector 14. Can students define and DOCUMENTATION (G-2-M) (Application) apply the terms measure, Activity 47: Chords and Demonstrate conceptual and distance, bisector, angle Triangles! 24, 28 practical understanding of bisector, and GQ 14 symmetry, similarity, and perpendicular bisector congruence and identify similar and 24 appropriately and use Activity 48: Folding congruent figures (G-2-M) them in discussing figures squares 23, 28 (Analysis) synthetically and with GQ 14 reference to coordinates as Predict, draw, and discuss the well? *Activity 49: Angle resulting changes in lengths, Relationships orientation, angle measures, and 23, 28 GQ 14 coordinates when figures are translated, reflected across 25 Concept 2: Transformations *Activity 50: horizontal or vertical lines, and Transformations! rotated on a grid (G-3-M) (G-6-M) 23, 24, 15. Can students use GQ 15, 16 (Application) 25 transformations (reflections, translations, 8th Grade Mathematics: Unit 4: Geometry and Measurement 8th Grade Mathematics: Unit 4: Geometry and Measurement rotations) to match figures Predict, draw, and discuss the and note the properties of Activity 51: Transform resulting changes in lengths, the figures that remain Me! 23, 24, orientation, and angle measures that invariant under GQ 15, 16 25 occur in figures under a similarity 26 transformations? transformation (dilation) (G-3-M) (G-6-M) (Analysis) 16. Can students use the *Activity 52: Dilations coordinate plane to GQ 15 Apply concepts, properties, and represent models of real- 24, 26 relationships of adjacent, life problems? corresponding, vertical, alternate Concept 3: Pythagorean *Activity 53: Developing interior, complementary, and 28 Theorem the Theorem 31 supplementary angles (G-5-M) GQ 17 (Analysis) 17. Can students state and *Activity 54: The Theorem apply the Pythagorean GQ 17 31 Construct, interpret, and use scale Theorem and its converse drawings in real-life situations (G- 30 in finding the lengths of Activity 55: The Converse 5-M) (M-6-M) (N-8-M) (Analysis) missing sides of right of the Pythagorean triangles and showing 31 Theorem Use area to justify the Pythagorean triangles are right GQ 17 theorem and apply the Pythagorean respectively? Activity 56: Rectangles theorem and its converse in real-life and Diagonals! 26, 30 problems (G-5-M) (G-7-M) 31 GQ 17 (Analysis) *Activity 57: Is This Table a Rectangle? 30, 31 GQ 17 Reflections: Concept 4: Scale Drawings *Activity 58: How Big is This Room Anyway? 30 18. Can students discuss GQ 18 similar and congruent Activity 59: Scale figures, and make and Drawings 30 interpret scale drawings of GQ 18 figures? Activity 60: Mapping my Way! 30 GQ 18 8th Grade Mathematics: Unit 4: Geometry and Measurement 8th Grade Mathematics: Unit 4: Geometry and Measurement Activity 61: How Big was it Anyway? 30 GQ 18 8th Grade Mathematics: Unit 4: Geometry and Measurement 8th Grade Mathematics: Unit 4: Geometry and Measurement Unit 4 Concept 1: Angle Relationships GLEs *Bolded GLEs are assessed in this unit. 23 Define and apply the terms measure, distance, midpoint, bisect, bisector, and perpendicular bisector (G-2-M) (Application) 24 Demonstrate conceptual and practical understanding of symmetry, similarity, and congruence and identify similar and congruent figures (G-2- M) (Analysis) 28 Apply concepts, properties, and relationships of adjacent, corresponding, vertical, alternate interior, complementary, and supplementary angles (G-5- M) (Analysis) Guiding Questions: Vocabulary: 14. Can students define and apply the terms Acute Angle measure, distance, bisector, angle Adjacent Angles bisector, and perpendicular bisector Alternate Exterior Angles appropriately and use them in discussing Alternate Interior Angles figures synthetically and with reference Angle Bisector to coordinates as well? Bisect Bisector Key Concepts: Complementary Angles Demonstrate conceptual and practical Congruent understanding of symmetry, Corresponding Angles similarity, and congruence, and Distance identify similar and congruent Measure figures (for example, recognize Midpoint Obtuse Angle reductions and expansions in similar Perpendicular Bisector figures in two and three dimensions). Right Angle Understand the terms distance Segment Bisector (between two points, two lines, or Similar from a point to a line) and midpoint, Straight Angle bisect and bisector in regard to lines. Supplemental Angles Apply concepts, properties and Transversal relationships of points, lines and line Vertex segments, rays, planes, diagonals; Vertical Angles right, acute, obtuse, supplementary, complementary, corresponding, vertical, and alternate interior angles; cube and rectangular prism. 8th Grade Mathematics: Unit 4 -Geometry and Measurement 59 8th Grade Mathematics: Unit 4: Geometry and Measurement Assessment Ideas: Resources: See end of Unit 4 Chords and Triangles Handout Folding Squares Handout Activity Specific Assessments: Angle Relationship Handout Activity 48, 49 Square sheets of paper Protractor Graph Paper Teacher-Made Supplemental Resources Writing Strategies See the Teacher-Made Supplemental Resources for Daily Problems to use with journal writing. Instructional Activities Note: The essential activities are denoted by an asterisk and are key to the development of student understandings of each concept. Any activities that are substituted for essential activities must cover the same GLEs to the same Bloom’s level. *Activity 46: The Bisection! (LCC Unit 3 Activity 3) (GLE: 23) Materials list: grid paper, ruler, pencil, math learning log Provide students with the coordinates of the end points of a horizontal line segment and have them draw the line segment on a coordinate system. Next, have students determine the coordinates of the point that bisects the line segment. Discuss the length of the line and have the students determine how the coordinates can be used to determine the length of the segment. After the midpoint is determined, discuss the coordinates of the midpoint and how these coordinates relate to the coordinates for the endpoints of the segment. Have students draw a line perpendicular to the line segment through the midpoint, thus illustrating a perpendicular bisector. Repeat this activity with a vertical line segment and then line segments of positive or negative slope. Have students verbalize a method for finding the coordinates of the midpoint of a segment if the endpoints are known. (Average the x-coordinates and average the y-coordinates to find the x and y coordinate of the midpoint). As a real-life connection, have the students design a tile pattern for a rectangular room with dimensions of 10 feet x 13 feet. The owner of the house has one request: a design in the floor tiles should be in the center of the room. Students should use their understanding of finding midpoint to determine where to place the design with the tile. Students should record their method of finding the coordinates of the midpoint of a segment in their math learning log (view literacy strategy descriptions). Remind the students that their math learning log should reflect how they are thinking about the procedure so that they can use their thinking later when reviewing the concept. Note: Activity matches Textbook activity on page 271 Glencoe Course 3 (Eighth Grade) Mathematics textbook. 8th Grade Mathematics: Unit 4 -Geometry and Measurement 60 8th Grade Mathematics: Unit 4: Geometry and Measurement Activity 47: Chords and Triangles! (CC Activity 8) (GLEs: 24, 28) Have students construct a circle on their paper. Instruct students to draw two chords through their circle that intersect. Have students then connect endpoints of the two chords so that they form two triangles. Challenge the students to use proportional measurements to determine whether the triangles formed are similar. Repeat the activity with different chords to determine whether the conjectures from the first circle hold true. Lead a discussion about properties and relationships of angles formed with these chords. (See Teacher-Made Supplemental Resources) Activity 48: Folding squares (LCC Unit 3 Activity 11) (GLEs: 23, 28) Materials list: paper cut into squares for each student, pencil, paper Provide square sheets of paper to each student. Have the students fold the paper in half with a horizontal fold (fold 1), make a good crease, and open the paper up again. Then instruct students to fold each half in half again fol ds # 2 using a second horizontal fold (fold 2), make a good crease, and open the fol d # 1 paper up again. Have students make a vertical fold (fold 3), make a good crease, and open the paper up again. Ask students to make observations about the relationships of length of the line segments formed by the folds. fol d # 3 Have students identify these as a bisector and a perpendicular bisector. Instruct students to take the top right corner and fold it so that the vertex meets the intersection of their center folds, rotate their paper 180 and repeat this fold with the opposite corner. Have them open their paper and in their groups determine the measures of all angles formed by the different folds. Have students outline the hexagon that is formed after the folds have been made (see diagram) and use what they know about angle measures to determine the number of degrees in the angles of a hexagon. Have groups prepare a presentation to the class and justify their angle measurements (e.g. complementary, supplementary, vertical angles, etc). (See Teacher-Made Supplemental Resources) Assessment Have students turn in folded square with every angle measured and labeled. Assessment The teacher will provide the student with a list of vocabulary (bisector, perpendicular bisector, complementary angles, supplementary angles, vertical angles, adjacent angles, corresponding angles, corresponding angles, etc.) used in the unit. The student will write the vocabulary on the folded square used in the activity. *Activity 49: Angle Relationships (LCC Unit 3 Activity 10) (GLEs: 23, 28) Materials list: paper, pencil, protractor 8th Grade Mathematics: Unit 4 -Geometry and Measurement 61 8th Grade Mathematics: Unit 4: Geometry and Measurement Have students investigate the relationship among the angles that are formed by intersecting two parallel line segments with a transversal. Have students determine pairs of angles that are complementary, supplementary, congruent, corresponding, adjacent, and alternate interior. Using a protractor, have students determine the measure of each of these pairs of angles. As an application, pose the following problem to students: As a class project, you are going to build a picnic table with legs that form an “X.” Of course, the top of the table must be parallel to the floor. If one of the legs is attached so that it forms a 40o angle with the top of the table, what measure should the leg form with the ground to ensure the tabletop is parallel to the floor? Ask students to explain their reasoning. Note: Teacher could choose to use this activity for assessment of Activity 48. (See Teacher-Made Supplemental Resources) Assessment Provide the student with a sketch of an ironing board. In a math learning log entry, the student will explain the relationships of the angles formed by the legs of the ironing board. The teacher will provide the student with a sketch of an ironing board. In a journal entry, the student will explain the relationships of the angles formed by the legs of the ironing board. 8th Grade Mathematics: Unit 4 -Geometry and Measurement 62 8th Grade Mathematics: Unit 4: Geometry and Measurement Unit 4 Concept 2: Transformations GLEs *Bolded GLEs are assessed in this unit. 23 Define and apply the terms measure, distance, midpoint, bisect, bisector, and perpendicular bisector (G-2-M) (Application) 24 Demonstrate conceptual and practical understanding of symmetry, similarity, and congruence and identify similar and congruent figures (G-2-M) (Analysis) 25 Predict, draw, and discuss the resulting changes in lengths, orientation, angle measures, and coordinates when figures are translated, reflected across horizontal or vertical lines, and rotated on a grid (G-3-M) (G-6-M) (Application) 26 Predict, draw, and discuss the resulting changes in lengths, orientation, and angle measures that occur in figures under a similarity transformation (dilation) (G-3-M) (G-6-M) (Analysis) 28 Apply concepts, properties, and relationships of adjacent, corresponding, vertical, alternate interior, complementary, and supplementary angles (G-5- M) (Analysis) Guiding Questions: Vocabulary: 15. Can students use transformations Center of Rotation (reflections, translations, rotations) to Congruent match figures and note the properties of Dilation the figures that remain invariant under Distance transformations? Line of Reflection 16. Can students use the coordinate plane to Reflection represent models of real-life problems? Rotation Symmetry Key Concepts: Similar Translation Vertex Assessment Ideas: Resources: See end of Unit 4 Transformations Handout Transform Me Handout Activity Specific Assessments: Graph paper Index Cards Rulers Post-it Graph Paper Teacher-Made Supplemental Resources Writing Strategies See the Teacher-Made Supplemental Resources for Daily Problems to use with journal writing. 8th Grade Mathematics: Unit 4 -Geometry and Measurement 63 8th Grade Mathematics: Unit 4: Geometry and Measurement Instructional Activities Note: The essential activities are denoted by an asterisk and are key to the development of student understandings of each concept. Any activities that are substituted for essential activities must cover the same GLEs to the same Bloom’s level. *Activity 50: Transformations! (LCC Unit 3 Activity 1) (GLEs: 23, 24, 25) Materials list: One Inch Grid BLM, Index Card Shapes BLM, ¼ Inch Grid BLM, Transformations BLM, Transformation Review BLM, pencils, paper, scissors, ruler, unlined 3” x 5” index cards, large sheet of newsprint Have students work in cooperative groups of 4. Give each student in the group a copy of One Inch Grid BLM. Have students in each group cut off the edges around their grid paper and tape the four sheets together to form a large coordinate grid. Tell students to draw the x and y axes in the center of the large coordinate plane. Each sheet will represent one quadrant of the coordinate plane. Have students label the point at which all four sheets meet as the origin. Ask them to label both the x- and y-axes, indicating the locations of –10 to 10 on each axis. Distribute four 3” x 5” index cards to each group. Make sure the students have assigned tasks as they prepare these cards. Have students follow the steps listed below, the results of which are shown on the Index Card Shapes BLM: A B 1. Index card #1 - Label the vertices of the index card with A, B, C, and D. C D 2. Index card #2 - Mark the midpoint of one of the 3” sides. Draw segments connecting this midpoint to each of the vertices on the opposite side. Cut out the isosceles triangle that is formed. Label the vertices of the triangle E, F, and G. 3. Index card #3 - Put points on one of the 5” sides at 2” and 4” (i.e., 2 inches from one vertex and 1 inch from the other vertex). The segment between these two points forms the top of a trapezoid. Connect these points to the vertices the on the opposite side. Cut out the trapezoid. Label the vertices of the trapezoid H, I, J, and K. 4. Index card #4 – Measure 2 inches along one of the 5” sides and mark a 2 in point. Connect this point to the vertex on the opposite side to form an isosceles right triangle. Cut out this triangle. Label the vertices of the triangle formed L, M, and N. After students have created shapes, the teacher could choose to have the students either do the entire project after teaching each concept or complete each column of the chart immediately after that specific concept is taught. Post a large sheet of newsprint on the wall for the new vocabulary used. As each new geometry term is discussed, have a student add the word to the word wall poster. 8th Grade Mathematics: Unit 4 -Geometry and Measurement 64 8th Grade Mathematics: Unit 4: Geometry and Measurement Ask students to create a chart as shown below: Shape Original Translation Rotation Reflection Reflection Coordinates Coordinates Coordinates coordinates coordinates x-axis y-axis rectangle A: A: A: A: A: B: B: B: B: B: C: C: C: C: C: D: D: D: D: D: isosceles E: E: E: E: E: triangle F: F: F: F: F: G: G: G: G: G: trapezoid H: H: H: H: H: I: I: I: I: I: J: J: J: J: J: K: K: K: K: K: isosceles L: L: L: L: L: right M: M: M: M: M: triangle N: N: N: N: N: Distribute Transformations BLM and ¼ Inch Grid BLM Have students place the rectangle in the first quadrant and record the coordinates of all four vertices of the rectangle in its original position in column one of the table. Have the students translate the rectangle up (or down) and right (or left), and then record the new coordinates in column two. Have students return the rectangle to its original 8 location and record coordinates of each vertex after B C 6 a 180 clockwise rotation. Discuss rotational original symmetry as students begin to rotate their shapes. (If 4 this is new to the students, it works well if students A D 2 put a small piece of tape on the rectangle to hold the rectangle in its original place on the grid, trace the -10 -5 tracing paper 5 10 figure, and then rotate the traced figure 180 -2 around the origin.) Have students discuss the new coordinates and identify the quadrant in which the -4 rotation rotated rectangle lies. -6 Have students return the rectangle to its original -8 location and then perform a reflection of the rectangle across the x-axis. Be sure to discuss line of symmetry as the rectangle is reflected. Model lifting the rectangle from the plane and flipping the triangle over the x- axis, if needed. Have students record coordinates of the four vertices. Have students return the rectangle to its original position, perform a reflection across the y-axis, and then record the new coordinates. Have the students complete the same actions using their trapezoid, right triangle, and isosceles triangle, recording all of the new coordinates on the chart. Remind them always to return their shapes to the original position before making a transformation. After the class has had time to complete the transformations of all four shapes, have the groups make some conjectures about how they might be able to determine the positions of polygons after a transformation from the information in the chart. Have the groups share their conjectures with 8th Grade Mathematics: Unit 4 -Geometry and Measurement 65 8th Grade Mathematics: Unit 4: Geometry and Measurement the class. (See Teacher-Made Supplemental Resources)Have the groups share their conjectures with the class by using the professor know-it all (view literacy strategy descriptions). The group that is sharing conjectures will be selected by the teacher; therefore, all groups should be ready to go first. The group will go to the front of the class, and using its conjectures, justify its thinking and answer questions from the class about one of its conjectures. The teacher will then select a second group to share another conjecture and continue until all conjectures and thinking are clearly understood by the class. Have the students use the Transformation Review BLM as a graphic organizer (view literacy strategy descriptions) to guide them as they review the results of the different transformations. Go through the example as a class. Allow students to discuss the answer with a partner. Students should write that the result of reflecting a polygon across the y-axis is that the x-coordinates are opposites of the originals and the y-coordinates stay the same. The BLM gives them either the initial position with the transformation used or the result of a transformation, and the student should give the other. As a result, some bridges have more than one solution. Problem 4 presents a new situation for students. Activity 51: Transform Me! (CC Activity 2) (GLEs: 23, 24, 25) Have students chose one of the four polygons from Activity 50. Students will find the measure of each angle, locate the midpoint of each side, and find the distance from vertex to vertex (i.e., length) for each side. (Teacher Note: Have students use rulers to measure lengths of sides which are not vertical or horizontal.) Discuss which (if any) of the properties of the polygon changed and which (if any) remained the same from Activity 50. *Activity 52: Dilations (LCC Unit 2 Activity 2) (GLEs: 24, 26) Materials list: Dilations BLM, Quadrant I Grid BLM, protractor, pencil, paper, ruler Discuss dilations as another transformation. Ask if anyone has an idea about what a dilation might be. Students will relate to the eye doctor dilating their eyes, but very few of them relate a dilation to being an enlargement or a reduction. Provide students with copies of the Quadrant I Grid BLM and the Dilations BLM. Have students plot the vertices of the polygon given on the Dilations BLM on a coordinate grid and then connect the points to form the polygon. Have students find the measure of each angle, and find the distance from vertex to vertex (i.e., length) for each side. (Teacher Note: Have students use rulers to measure lengths of sides which are not vertical or horizontal.) Next, have students use a ruler and draw a dotted line from the origin and extend the line through Vertex A of the polygon, continue to do this by drawing 35 lines from the origin through each of the other four vertices 30 A' B' C' (see diagram). 25 Instruct the students to follow the steps on the Dilations 20 E' D' BLM and then discuss their conjectures about dilations and A B 15 C their effect upon angle measures, side lengths, and 10 E D coordinates of the original figure. Make sure the students 5 understand that the dilation is different from the reflections, 10 20 30 40 50 translations and rotations because it is the only one that produces similar figures – the other transformations produce congruent figures. 8th Grade Mathematics: Unit 4 -Geometry and Measurement 66 8th Grade Mathematics: Unit 4: Geometry and Measurement As a real-life connection, lead a discussion about when dilations that are used in everyday life: using a projector to show an image to an entire class, enlarging a picture from the image stored in a digital camera, projecting a video on large screens at sporting events, or making a scale drawing of a large object. 8th Grade Mathematics: Unit 4 -Geometry and Measurement 67 8th Grade Mathematics: Unit 4: Geometry and Measurement Unit 4 Concept 3: Pythagorean Theorem GLEs *Bolded GLEs are assessed in this unit. 26 Predict, draw, and discuss the resulting changes in lengths, orientation, and angle measures that occur in figures under a similarity transformation (dilation) (G-3-M) (G-6-M) (Analysis) 30 Construct, interpret, and use scale drawings in real-life situations (G-5-M) (M-6-M) (N-8-M) (Analysis) 31 Use area to justify the Pythagorean theorem and apply the Pythagorean theorem and its converse in real-life problems (G-5-M) (G-7-M) (Analysis) Guiding Questions: Vocabulary: 17. Can students state and apply the Area Pythagorean Theorem and its converse in Hypotenuse finding the lengths of missing sides of Legs right triangles and showing triangles are Perpendicular right respectively? Pythagorean Theorem Right Triangle Key Concepts: Square Root Graph on the coordinate plane to represent real-world problems, including graphing ordered pairs in all four quadrants. Predict the results of and perform transformations (translations, reflections, rotations) in problems set in a real-world context. Construct or use scale drawings. Understand and use the Pythagorean Theorem, including recognizing situations in which the theorem is relevant (with pictorial illustration). Assessment Ideas: Resources: See end of Unit 4 Developing the Theorem Picture CC Sample Assessment with Activity Specific Assessments: Developing the Theorem Activity 53, 54, 55 CC Sample Assessment with The Theorem Rectangles and Diagonals Handout Is this table a Rectangle Handout Graph Paper Teacher-Made Supplemental Resources 8th Grade Mathematics: Unit 4 -Geometry and Measurement 68 8th Grade Mathematics: Unit 4: Geometry and Measurement Writing Strategies See the Teacher-Made Supplemental Resources for Daily Problems to use with journal writing. Instructional Activities Note: The essential activities are denoted by an asterisk and are key to the development of student understandings of each concept. Any activities that are substituted for essential activities must cover the same GLEs to the same Bloom’s level. *Activity 53: Developing the Theorem (LCC Unit 3 Activity 4) (GLE: 31) Materials list: grid paper, straight edge, scissors, paper, pencil Have students draw a right triangle on grid paper with the two perpendicular sides having lengths of 3 and 4 units. Have students draw a square using one of the legs of the triangle as the side of the square (i.e., draw a 3 x 3 square). Repeat using the other leg as a side of a square (i.e., draw a 4 x 4 square). Have students find the area of each square. Ask students to determine a method for finding the area of the square of the hypotenuse of their right triangle and how the areas of the three squares relate to one another. (Some students may remember the Pythagorean Theorem from previous years and use that information to determine the length of the hypotenuse. Others may compare the length of the hypotenuse to the units on the grid paper. The process used is not important, but all students should eventually see that the hypotenuse length is 5 and the area of the corresponding square is 25 square units.) Have students show that the sum of the areas of the two smaller squares is the same as the area of the square formed by the hypotenuse by cutting and rearranging the small squares inside the larger squares. (Many texts and websites show how to do this. Two websites which use animations to develop the Pythagorean Theorem are: http://www.nadn.navy.mil/MathDept/mdm/pyth.html http://www.pbs.org/wgbh/nova/proof/puzzle/theorem.html.) Have students draw a triangle on the grid that is not a right triangle and determine whether they get the same results. Discuss conjectures that students develop about the results of their explorations. (See Teacher-Made Supplemental Resources for a picture) Using a modified version of reciprocal teaching (view literacy strategy descriptions), have students brainstorm predictions as to whether or not the Pythagorean theorem will work when finding side lengths of triangles that do not have a right angle. Reciprocal teaching is used to move instruction from delivery to discovery. Have groups write their predictions about the use of the theorem in these other triangles on paper. The prediction is the first part of a reciprocal teaching lesson. Assign the roles of questioner, clarifier, predictor and conjecturer to groups of four students as they experiment with these other triangles. The „questioner‟ will begin by asking the group to restate how it thinks its prediction relates to the triangles without right angles. The clarifier should make sure that the answers that the questioner gets to the questions are clear and understood by all group members. Have students draw a triangle on the grid that is not a right triangle, and have the questioner ask the group questions that will help it determine whether it gets the same results. The „clarifier‟ will offer input, and the group will then work with the „conjecturer‟ to write its summary statement. The predictor might make other predictions as other triangles are drawn to test the conjectures made by the conjecturer. As a class, discuss conjectures that students develop about the results of their explorations. 8th Grade Mathematics: Unit 4 -Geometry and Measurement 69 8th Grade Mathematics: Unit 4: Geometry and Measurement Assessment The student will work these problems as journal prompts and explain the answers. a) Washington, DC, is 494 miles east of Indianapolis, Indiana. Birmingham, Alabama is 433 miles south of Indianapolis. Determine the distance from Birmingham to Washington D.C. b) The ladder of a water slide is 8 ft. high, and the length of the slide is 17 ft. Determine the length of the horizontal base of the slide. Justify all of your thinking using valid mathematical reasoning. (See Teacher-Made Supplemental Resources) *Activity 54: The Theorem (LCC Unit 3 Activity 5) (GLE: 31) Materials list: The Theorem BLM, pencils, paper, calculators, graph paper Provide students with the side lengths of several right triangles. Have students compute the square of each measure and then add the squared lengths of the two smaller sides and compare this sum with the square of the longest side (the hypotenuse). Make sure that students understand that in a right triangle, the sum of the squares of the two perpendicular sides is the same as the square of the hypotenuse. Extend this activity to include real-life situations that require students to find the length of one of the sides of a right triangle with situations like the ones that follow: James has a circular trampoline with a diameter of 16 feet. Will this trampoline fit through a doorway that is 10 feet high and 6 feet wide? Explain your answer. A carpenter measured the length of a rectangular tabletop he was building to be 26 inches, the width to be 12 inches and the diagonal to be 30 inches. Explain whether or not the carpenter can use this information to determine if the corners of the tabletop are right angles. Provide students with the side lengths of several right triangles missing the length of one of the sides. Discuss the use of the formula as it applies to the missing lengths in the triangles. Extend this activity to include real-life situations that require students to find the length of one of the sides of a right triangle with situations by distributing The Theorem BLM. Have students verify their solutions to the BLM by comparing answers with another student and discussing any results that differ. Assessment Assign students these problems as journal prompts and have them explain the answers. a) Washington, DC, is 494 miles east of Indianapolis, Indiana. Birmingham, Alabama is 433 miles south of Indianapolis. Determine the distance from Birmingham to Washington D.C. b) The ladder of a water slide is 8 ft. high, and the length of the slide is 17 ft. Determine the length of the horizontal base of the slide. Justify all of your thinking using valid mathematical reasoning. The teacher will provide the student with a list of number triples that represent the side lengths for triangles. The student will determine which triples represent the side lengths of a right triangle. 8th Grade Mathematics: Unit 4 -Geometry and Measurement 70 8th Grade Mathematics: Unit 4: Geometry and Measurement Activity 55: The Converse of the Pythagorean Theorem (LCC Unit 3 Activity 9) (GLE: 31) Materials list: grid paper, protractors, pencil, paper Have student pairs cut out squares from grid paper that are 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and 169 square units. Then have them create triangles using the sides of any three squares. Have students use a protractor to determine the measures of each angle in the triangles formed. Next, have students determine the relationship between the sum of the areas of the two smaller squares and the area of the largest square (i.e., are they the same or different?) Have students make a conjecture about the relationship between the areas of the squares when one of the angle measures of the triangle is 90 degrees. Remind students that these relationships are those of the Pythagorean Theorem and its converse (studied in earlier activities). Lead a discussion of applications of the converse of the Pythagorean Theorem to real-life situations. For example, a carpenter goes to the corner of a frame wall that he is building and marks off a 3 foot length on one board and a 4 foot length on the adjacent board. He then nails a 5 foot brace to connect the two marks. What is the purpose of his work? (He is making sure that the two boards are perpendicular (that his wall is ‘ square’) because a triangle with sides of 3-4-5 is a right triangle.) Assessment Provide the student with a list of number triples that represent the side lengths for triangles. Challenge students to determine which triples represent the side lengths of a right triangle. Activity 56: Rectangles and Diagonals! (CC Activity 4) (GLEs: 26, 30) Provide students with grid paper and straight edges. Have the students use the grid paper to sketch a scale model of a 10 foot by 16 foot room. Have students draw a diagonal through the scale model of the room. Ask the students to draw a smaller rectangle inside the first rectangle, using the diagonal drawn through the scale model of the room as the diagonal of the new rectangle. Have students work in small groups to determine whether the rectangles are similar and repeat the actions with different sized rectangles to determine if their conjectures hold true. Have students find the actual dimensions of the new rooms, using the scale established using the original rectangle. Have groups share and justify their conjectures. Note: The two rectangles should share the diagonal. *Activity 57: Is This Table a Rectangle? (CC Activity 7) (GLEs: 30, 31) Present the following scenario to students: Jason‟s dad is a carpenter. He asked Jason to find out if the rectangular table he was building had square corners. Jason said he had learned something in math class that would help him find out. The table had measurements of 14 feet x 10 feet. Jason measured the diagonal and found the diagonal to be 16 feet. Have the students determine if the table has square corners, explain how they know, and provide a scale drawing as part of the explanation. The website http://www.tpub.com/builder2n3/65.htm is an architectural website that gives some examples of using the Pythagorean Theorem to make sure beams are perpendicular. 8th Grade Mathematics: Unit 4 -Geometry and Measurement 71 8th Grade Mathematics: Unit 4: Geometry and Measurement Unit 4 Concept 4: Scale Drawings GLEs *Bolded GLEs are assessed in this unit. 30 Construct, interpret, and use scale drawings in real-life situations (G-5-M) (M-6-M) (N-8-M) (Analysis) Purpose/Guiding Questions: Vocabulary: 18. Can students discuss similar and Blueprint congruent figures, and make and Diagonal interpret scale drawings of figures? Dimensions Length Model Scale Width Assessment Ideas: Resources: See end of Unit 4 Graph Papers How Big is this Room Anyway grid Key Concepts: Mapping My Way Handout Construct or use scale drawings Poster Board Ruler / Straight Edge Index Cards Teacher-Made Supplemental Resources Writing Strategies See the Teacher-Made Supplemental Resources for Daily Problems to use with journal writing. 8th Grade Mathematics: Unit 4 -Geometry and Measurement 72 8th Grade Mathematics: Unit 4: Geometry and Measurement Instructional Activities Note: The essential activities are denoted by an asterisk and are key to the development of student understandings of each concept. Any activities that are substituted for essential activities must cover the same GLEs to the same Bloom’s level. *Activity 58: How Big is This Room Anyway? (LCC Unit 3 Activity 6) (GLE: 30) Materials list: meter sticks or tape measures, newsprint or other large paper for blueprint, rulers, scissors, pencil, paper Assign different groups of students the task of measuring the classroom dimensions. Have the class determine a scale that would fit on a piece of newsprint or poster board and then have someone draw the room dimensions to scale on the poster. Tell students that the class will make a classroom blueprint. Divide students into groups of three to five. Assign each group a different object in the classroom to measure (file cabinets, book shelves, trash can, etc. - remember only length and width of the top of the object is needed for the blueprint). Have students convert actual measurements using the scale measurements determined earlier. Instruct students to measure, draw and cut out models from an index card. Have each student measure his/her own desktop and make a scale model for the classroom blueprint. Remind students to write their names on the desktop model. Ask, “What is the actual area of your desktop? What is the scale area of your desktop? What comparisons do you see as you make observations of the areas of your room and desktop?” List your observations. Have groups submit their scale models of the classroom objects (not desks at this time) for the blueprint. Discuss methods used to determine the measurements of the models and then glue the models in the correct position on the classroom blueprint. Have students, one group at a time place their desktop models on the classroom blueprint, working so that those who sit in the center of the room can add their models first. Post blueprints/scale models on the wall for all classes to compare. Using the class scale model of the classroom, have students make predictions about distance from various points in the room (i.e., If the distance from the teacher’s desk to the board is 5 inches on the scale model and the scale is 1 inch represents 4 feet, then the that the actual distance is 20 feet.) Have student measure the actual distance(s) to check for accuracy of the scale model of the classroom. Note: Possible modifications including having each group create a blueprint for the room. (See Teacher-Made Supplemental Resources) Activity 59: Scale Drawings (LCC Unit 3 Activity 12) (GLE: 30) Materials list: Scale Drawings BLM, pencil, paper Provide the students with the problems to practice scale drawing problems by distributing Scale Drawing BLM. Give students time to work through these situations and then divide students into groups of four to discuss these situations. Have students in groups come to consensus on the solutions to these problems and then have them prepare for a discussion using professor know-it- 8th Grade Mathematics: Unit 4 -Geometry and Measurement 73 8th Grade Mathematics: Unit 4: Geometry and Measurement all (view literacy strategy descriptions). With this strategy, the teacher selects a group to become the “experts” on scale drawing required in the situation that is selected. The group should be able to justify its thinking as it explains its proportions or solution strategies to the class. All groups must prepare to be the “experts” because they are not told prior to the beginning of the strategy, which group(s) will be the “experts” and ask questions about scale drawings. Have students measure the side lengths of their bedrooms, the length and width of their beds, and length and width of any other bedroom furniture. Give students a sheet of grid paper and have them construct a scale drawing of their bedrooms. Instruct students to determine the appropriate scale to make their rooms fit on the grid paper. Lead a discussion to generate questions that students can solve using their scale drawings (e.g., distance between objects, proportions involving the length of sides of their scale drawing to actual objects). Activity 60: Mapping my Way! (CC Activity 16) (GLE: 30) Provide the students with the following information: Sandy was given the assignment during a summer job to draw a map from the city recreational complex to the high school. Sandy started from the recreational complex and walked north 3.5 miles, west 10 miles, north 5.3 miles, and then east 3 miles. Sandy was given a space 3 1 inches x 4 inches to sketch the route on a brochure 2 being made by the staff at the complex. Determine a scale that Sandy will be able to use and draw a map that can be used in the space provided. Explain how the scale was determined. (See Teacher-Made Supplemental Resources) Activity 61: How Big was it Anyway? (CC Activity 17) (GLE: 30) Provide groups of four students with situations like the ones below. Lead the class in a discussion of these scale-drawing situations after the groups have had time to complete the problems. 1. Draw a diagram of a rectangular bedroom with dimensions of 24 feet by 15 feet. Use a scale of 1 inch = 6 feet. 2 2. The picture of the amoeba at the right shows a width of 2 centimeters. If the actual ameba‟s length is 0.005 millimeter, what is the scale of the drawing? 8th Grade Mathematics: Unit 4 -Geometry and Measurement 74 8th Grade Mathematics: Unit 4: Geometry and Measurement Unit 4 Assessment Options General Assessment Guidelines Whenever possible, the teacher will create extensions to an activity by increasing the difficulty or by asking “what if” questions. The student will create a portfolio containing samples of experiments and activities. The teacher will provide the student with unlined paper and rulers. The student will design a stained-glass window to show understanding of the terms midpoint, bisector, perpendicular bisector, symmetry, similar, complementary, supplementary, vertical angles, corresponding angles, and congruent angles. The student will label the different components of his/her stained-glass window to assure that examples have been included for each of the vocabulary words from the unit. The student will present his/her stained- glass sketch to his/her group and justify examples to the group members. The teacher will provide the student with a rubric to self-assess his/her work prior to presentations and teacher evaluation. The teacher will provide the student with a sketch of a baseball diamond showing that there are 90 feet between the bases. The student will prepare a presentation explaining how to determine the distance the catcher must throw the baseball to the 2nd baseman if he needs to get the runner on second base out. The teacher will give the student a piece of grid paper which shows a polygon and a transformation of the polygon (the second polygon). The student will determine a transformation or transformations that would produce the second polygon. The teacher will provide the student with several right triangles that have a missing side measure. The student will find the lengths of the missing sides. (See APCCSM) The teacher will provide the students with paper and the scale of 0.25 inches to represent 2 feet. The student will a) draw a model of a rectangular swimming pool measuring 16 feet by 36 feet; b) draw a 2 foot by 6 foot diving board so that it bisects one of the short ends of the pool; c) find the perimeter and area of the pool; and d) put a walk around the perimeter of the pool with a width of 4 feet and find the area and the outer perimeter of the walk. The student will create a scale drawing. A rubric that assesses the appropriateness of the scale factor, as well as the accuracy of the drawing, will be used to determine student understanding. Activity-Specific Assessments Concept 1 Activity 48, 49 Concept 3 Activity 53, 54, 55 8th Grade Mathematics: Unit 4 -Geometry and Measurement 75 8th Grade Mathematics: Unit 4: Geometry and Measurement Name/School_________________________________ Unit No.:______________ Grade ________________________________ Unit Name:________________ Feedback Form This form should be filled out as the unit is being taught and turned in to your teacher coach upon completion. Concern and/or Activity Changes needed* Justification for changes Number * If you suggest an activity substitution, please attach a copy of the activity narrative formatted like the activities in the APCC (i.e. GLEs, guiding questions, etc.). 8th Grade Mathematics: Unit 4 -Geometry and Measurement 76

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 11 |

posted: | 11/16/2011 |

language: | English |

pages: | 21 |

OTHER DOCS BY ynzl43

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.