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Youngstown City Schools MATH: GEOMETRY Unit 1D: THEOREMS ABOUT TRIANGLES / PARALLELOGRAMS / CONSTRUCTIONS (2 WEEKS) 2013-2014 SYNOPSIS: In this unit, students continue their work with theorems and work specifically with triangles and parallelograms. Students make constructions of triangles and parallelograms and relate these to real-life experiences. Finally, students apply the concepts in the unit by plotting three con-collinear points and finding the point that is the shortest distance from the three points; then find a fourth point that will form a parallelogram with the three non-collinear points. STANDARDS G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining the midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point G.CO. 11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. G.CO. 12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.) Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines; including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on a line G.CO. 13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. MATH PRACTICES: 1. 2. 3. 4. 5. 6. 7. 8. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning LITERACY L.2 Communicate using correct mathematical terminology L.4 Listen to and critique peer explanations of reasoning L.5 Justify orally and in writing mathematical reasoning L.6 Represent and interpret data with an without technology L.8 Read appropriate text, providing explanation for mathematical concepts, reason or procedures L.9 Apply details of math readings/use information found in texts to support reasoning and develop a “works cited document” for research done to solve a problem 6/30/2013 YCS Geometry: Unit 1D: Theorems: Triangles/Parallelograms/Constructions 2013-2014 1 MOTIVATION TEACHER NOTES 1. Teacher sets up connection of this unit to the previous ones on use of Theorems. 2. Teacher gives some type of pre-assessment on vocabulary terms to be sure students have understanding of terms needed for the unit 3. Tom Reardon- TI Inspire activity. Can be done with TI Inspire Player or TI Inspire Calculator http://education.ti.com/calculators/timathnspired/US/Activities/Detail?sa=5024&t=9123&id=13138 4. Preview expectations for end of Unit. 5. Have students set both personal and academic goals for this Unit or grading period. TEACHING-LEARNING Vocabulary Terms: Midpoint Median Circumscribe Centroid TEACHER NOTES Parallelogram Isosceles Triangle Quadrilateral Equilateral Triangle Inscribe Bisector Angle 1. Teacher asks students to draw different size triangles (on TI-Nspire, Geometer SketchPad, by hand, or cut out triangles); find the measure of all 3 angles to discover sum will always be 180°. (G.CO.10; G.CO.12) (MP-4; MP-6; MP-8) 2. Teacher reminds students that if 2 sides of an isosceles triangle are congruent, its opposite angles are also congruent; Teacher gives students a ruler and paper to construct an isosceles triangle; students fold it to “prove” the theorem (p. 216 of Geometry text); students discuss reasoning and proof of the theorem. (G.CO.10) (MP-1; MP-7; MP-8) (L.2; L.4) 3. Teacher gives examples for students to work out in their notebooks/grid paper/grid board, for the following: (G.CO.10)(MP-2; MP-4; MP-6) (L.5) An example for students to find the distance between two points: Distance Formula (round to the nearest tenth) ( x2 x1 )2 ( y2 y1 )2 An example for students to find the midpoint: Midpoint Formula x1 x2 y1 y2 , 2 2 An example for students to find the slope: Slope Formula m y2 y1 rise x2 x1 run 4. Teacher provides three points to plot on a coordinate plane, connecting to form a triangle; Students will follow along in notebooks or on grid paper/grid boards to draw a segment joining two sides of the triangle, at their midpoint. (G.CO.10) (MP-5; MP-7) (L.5) Students should use the midpoint formula, to find the midpoint of the two sides of the triangle. Students will be instructed to connect the joining segment Using the slope formula, students should recognize the slopes are the same and thus proving 6/30/2013 YCS Geometry: Unit 1D: Theorems: Triangles/Parallelograms/Constructions 2013-2014 2 TEACHING-LEARNING TEACHER NOTES that they are parallel; Question to the students: How do we prove parallel? 5. Use OGT Released Test 2006, question 40 as a practice problem. Teacher gives more examples, for students to work on. (G.CO.10; G.CO.12) (L.5; L.6) 6. Have students use Geometer SketchPad or TI-Nspire to create a triangle, its midpoints, and then connect the vertex with the midpoint to show all 3 medians meet at a point. Have the students choose a point and make new triangle. (G.CO.10; G.CO.12) (MP-5; MP-7) 7. Students will use coordinate grid paper to draw their own individual triangle. Instruct the students to find the midpoints, using midpoint formula; connect the vertex to the midpoint. The three medians will meet at a point. (G.CO. 10, G.CO. 12) (MP-4; MP-8) 8. Teacher introduces students to two different terms: parallelogram and quadrilateral. Give examples, students take notes and offer practical examples. Have students look for figures in the classroom, and discuss the attributes. (G.CO.11) (MP-4; L.2, L.4) 9. Give students coordinates for two different parallelograms; have students draw on grid paper. Have students prove that the opposite sides are congruent. Ask the students: How do you prove two lengths are congruent? Students construct and prove the two opposite sides are congruent. (Students should use the distance formula to solve these problems). Discuss results with students. (G.CO.11; G.CO.12) (MP-6; MP-8) (L.2; L.4) 10. Ask the students: What do we know about the angles A and B in parallelogram ABCD? Students should recall consecutive interior, parallel, transversal, etc. leading them to the sum of the interior angles of a parallelogram and the idea that angles of the line segment equal 180°…proving opposite sides are congruent, opposite angles are congruent. Demonstrate for students that if angle A = 30°, then angle B = 150°. Since angle B is 150°, then angle C must = 30°. Hence, Angles A and C are congruent, each being 30°. If angle A + angle B = 180°, and angle B + angle C = 180°, then angle A + angle B = angle B + angle C (subtraction); and angle A = angle C. (G.CO.11) (MP-3; MP-4) (L.5; L.6) B C A D 11. Students will use Geometer SketchPad or TI-Nspire to construct a visual of proving opposite sides and opposite angles of a parallelogram are congruent. Also spend time working on the converses of these theorems. (G.CO.11; G.CO.12) (MP-5; MP-7) (L.5; L.9) 12. Use coordinate grid paper/grid board to create parallelogram. Draw the diagonals with E being the intersection of the diagonals. Use distance formula to prove that line segment AE is congruent to line segment EC and line segment DE is congruent to line segment EB. …proving diagonals bisect each other. (G.CO.11; G.CO.12) (MP-3; MP-8) (L.5) 13. Extend Example to rectangle and prove diagonals are congruent by using the distance formula. Use Geometer SketchPad or TI-Nspire to prove. (G.CO.11; G.CO.12) (MP-2; MP-8) (L.2) 14. Review the vocabulary terms: inscribe and circumscribed. Give examples, students take notes, include practical examples. (G.CO.12; G.CO.13) (MP-1) 15. Using Geometer SketchPad or TI-Nspire, have students construct an equilateral triangle inscribed in a circle. Then have students inscribe a square and regular hexagon in a circle. (G.CO.12; G.CO.13) (MP-2; MP-3) 6/30/2013 YCS Geometry: Unit 1D: Theorems: Triangles/Parallelograms/Constructions 2013-2014 3 TEACHING-LEARNING TEACHER NOTES 16. Students create construction book or are familiar with all basic constructions listed below: (G.CO.12, G.CO.13, MP.4, MP.5, MP.6, MP.7, L.2, L.5) Copy a segment Copy an angle Angle bisector Perpendicular bisector Line perpendicular to a given line through a point on the given line Line perpendicular to a given line through a point not on the given line Line parallel to a given line through a point not on the given line Divide a segment into a given number of congruent segments Equilateral triangle Regular hexagon Square Equilateral triangle, square, and regular hexagon inscribed in a circle TRADITIONAL ASSESSMENT TEACHER NOTES 1. Paper-pencil test with M-C questions and 2-and 4-point questions TEACHER CLASSROOM ASSESSMENT TEACHER NOTES 1. Quizzes 2. In-class participation and practice problems for each concept AUTHENTIC ASSESSMENT TEACHER NOTES 1. Students evaluate goals they set at beginning of unit or on a weekly basis. 2. Students solve a series of real-life geometry problems where they apply the distance formula, work with perpendicular bisectors, mid-point and properties of parallelograms (attached on page 5 and rubric on page 6) (G.CO.10; G.CO.11; G.CO.12; G.CO.13) (L.8; L.9) 6/30/2013 YCS Geometry: Unit 1D: Theorems: Triangles/Parallelograms/Constructions 2013-2014 4 AUTHENTIC ASSESSMENT GEOMETRY UNIT 1D THEOREMS ABOUT TRIANGLES, PARALLELOGRAMS, CONSTRUCTIONS STANDARDS: G.CO.10, G.CO.11 You have been hired by a landscaper to create a drawing of a garden on graph paper using coordinate geometry. The garden must have the following: 1. The garden is in the shape of a parallelogram with a possible area of 120 to 156 square boxes. Draw the parallelogram and assign capital letters to the vertices and then assign coordinates to the vertices. 2. Use the distance formula to find the length of the sides of the parallelogram. 3. Determine if the drawing is truly a parallelogram. Support your answer with a theorem. 4. The garden is to have four walkways that are formed by connecting the midpoints of the adjacent sides of the parallelogram. Find the midpoints and label with capital letters and coordinates. Then connect them to form the walkways. 5. Find the lengths of the walkways using the distance formula. 6. Determine if the quadrilateral formed by connecting the four midpoints is a special type of quadrilateral? If so, what type? 7. Next to the garden will be a triangular area for roses. Draw an isosceles triangle using one side of the parallelogram as a side of the triangle. Label any points that have not been previously labeled on the triangle with capital letters and coordinates. 8. Use the distance formula to prove that your triangle is isosceles. 9. The triangular area for roses is to be divided in half to accommodate two different types of roses. Draw the perpendicular bisector to the base of the isosceles triangle to divide the triangle into two equal parts. Label the point where the perpendicular bisector intersects the base of the triangle with a capital letter and coordinates. 10. Find the length of the perpendicular bisector using the distance formula. This is great. You now have helped your employer design a garden that he can build for his client. 6/30/2013 YCS Geometry: Unit 1D: Theorems: Triangles/Parallelograms/Constructions 2013-2014 5 RUBRIC ELEMENTS OF THE PROJECT 0 1 3 Labeled 2 of the 4 vertices with capital letters and coordinates Used the distance formula, but had errors in two of the calculations Determined if drawing is parallelogram; did not support with correct theorem Labeled 3 of the 4 vertices with capital letters and coordinates Used the distance formula, but had errors in one of the calculations Determined if drawing is parallelogram; had correct theorem but did not explain it correctly Found and labeled 3 midpoints with capital letters and coordinates; then connected midpoints incorrectly Drew parallelogram and labeled the vertices with capital letters and coordinates Did not attempt Used the Distance Formula to find the lengths of sides of parallelogram Did not attempt Determined if the drawing is truly a parallelogram and supported answer with a theorem Did not attempt Determined if drawing is parallelogram; did not support with theorem Found four walkways formed by connecting midpoints of adjacent sides of parallelogram; labeled midpoints with capital letters and coordinates Did not attempt Found and labeled 2 midpoints with capital letters and coordinates; then connected midpoints to form a walkway Used distance formula to find the lengths of walkways Did not attempt Found and labeled midpoints with capital letters or coordinates; did not connect midpoints of adjacent sides to form walkways Did not use distance formula to find length of walkways Determined if quadrilateral formed was a special type of quadrilateral and supported with theorem Did not attempt Attempted to determine if special quadrilateral was formed; no theorem Determined if special quadrilateral was formed, but could not cite a theorem Drew isosceles triangle and label points with letters and coordinates Did not attempt Drew isosceles triangle, and labeled points or coordinates Use Distance Formula to prove triangle is isosceles Did not attempt Drew isosceles triangle, but did not label points or coordinates Used wrong formula. Draw perpendicular bisector and label with capital letter and coordinates Did not attempt Use the distance formula to determine the length of the perpendicular bisector Did not attempt Drew perpendicular bisector, but did not label point or give coordinates Did not know distance formula, and attempted to use another strategy Drew perpendicular bisector, labeled either the point or give coordinates Used distance formula but with multiple calculation errors 6/30/2013 Labeled 1 of the 4 vertices with capital letters and coordinates Did not use the distance formula to find lengths of sides 2 Used distance formula but had multiple errors in calculating length of walkways Used distance formula but had error in calculations Used distance formula but had minor errors in calculating length of walkways Determined if quadrilateral was formed, and cited a theorem but there was an error NA Used distance formula, but did not connect result to isosceles triangle NA Used distance formula but had one calculation error YCS Geometry: Unit 1D: Theorems: Triangles/Parallelograms/Constructions 2013-2014 4 Labeled vertices with capital letters and coordinates Used the distance formula and found the length of sides correctly Determined if drawing is parallelogram; supported with correct theorem and explanation Found and labeled the 4 midpoints with capital letters and coordinates; then connected midpoints of adjacent sides to form walkways Used distance formula to find the lengths Determined if quadrilateral was formed and gave a theorem for support Drew isosceles triangle, and labeled points and coordinates Used distance formula correctly and proved triangle was isosceles Drew perpendicular bisector and labeled point correctly and listed coordinates Used distance formula correctly 6