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Lesson plan: Review Common Core Geometry Lesson question: What are constructions? How do I create different constructions? Common Core Standards Addressed: Make geometric constructions: o CCSS.MATH.CONTENT.HSG.CO.D.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 the line. Lesson objective: Assessment: Students will correctly construct angles/lines/ angle and line bisectors. Students will have correct constructions on scrap paper. Teacher will be collecting student’s construction Multiple Choice. Teacher will observe students and ask for the level of difficulty for each construction. Opening: Students will discuss what types of constructions they remember preforming throughout the class year. This warm up will be a review of what they have been learning over the year. Students will use background knowledge to have a class discussion on the topic of constructions. Procedure: 1. 2. 3. 4. 5. 6. 7. 8. Students will complete warm up Teacher will discuss and go over warm up Teacher will ask if anyone has any questions about the warm up. Teacher will instruct students to open up the review packet handed out to them. Teacher will discuss topic of the day Topic that will be discussed will be how to construct a line and an angle Teacher will show a video on how to construct a line segment and an angle Teacher will discuss with class the steps that are necessary for them to create a line segment and an angle 9. Students will create their own line segments and angles. 10. Teacher will go through example constructions with students 11. Students will independently work on additional examples. 12. Students will discuss example constructions 13. Students will be asked about how they got their answers and why 14. Teacher and students will discuss example constructions 15. Teacher will introduce homework. 16. Teacher will discuss directions. Tiered by complexity: Each student will have separate homework assignments that will challenge each student and also allow for them to see success. Gabby- Angle Bisector Kenneth- Line Bisector Lenny- Line Bisector Closure: Teacher will discuss with students the overall lesson. Teacher will ask for student input on whether or not they liked how the lesson played out or if I could teach it better to meet all of their needs. We also discuss the different homework assignments. I tell the students that their homework is not all the same and they will need to attempt it on their own while providing me with two example constructions. They are to read the steps and create 2 constructions while using the steps to assist them. Materials: Notes Pen/pencil Calculator Promethean Board Internet Regnetsprep.org Scrap sheet for constructions Scrap sheets for homework Ruler compass Construction directions Review Geometry Copy a line segment Given: (Line segment) Task: To construct a line segment congruent to (line segment) . Directions: 1. If a reference line does not already exist, draw a reference line with your straightedge upon which you will make your construction. Place a starting point on the reference line. 2. Place the point of the compass on point A. 3. Stretch the compass so that the pencil is exactly on B. 4. Without changing the span of the compass, place the compass point on the starting point on the reference line and swing the pencil so that it crosses the reference line. Label your copy. Your copy and (line segment) are congruent. Congruent means equal in length. Explanation of construction: The two line segments are the same length, therefore they are congruent. Copy an angle Given: Task: To construct an angle congruent to . Directions: 1. If a reference line does not already exist, draw a reference line with your straightedge upon which you will make your construction. Place a starting point on the reference line. 2. Place the point of the compass on the vertex of (point A). 3. Stretch the compass to any length so long as it stays ON the angle. 4. Swing an arc with the pencil that crosses both sides of . 5. Without changing the span of the compass, place the compass point on the starting point of the reference line and swing an arc that will intersect the reference line and go above the reference line. 6. Go back to and measure the width (span) of the arc from where it crosses one side of the angle to where it crosses the other side of the angle. 7. With this width, place the compass point on the reference line where your new arc crosses the reference line and mark off this width on your new arc. 8. Connect this new intersection point to the starting point on the reference line. Your new angle is congruent to . Explanation of construction: When this construction is finished, draw a line segment connecting where the arcs cross the sides of the angles. You now have two triangles that have 3 sets of congruent (equal) sides. SSS is sufficient to prove triangles congruent. Since the triangles are congruent, any leftover corresponding parts are also congruent - thus, the angle on the reference line and are congruent. Bisect a line segment (Also know as Construct a Perpendicular Bisector of a segment) Given: (Line segment) Task: Bisect Directions: . 1. Place your compass point on A and stretch the compass MORE THAN half way to point B, but not beyond B. 2. With this length, swing a large arc that will go BOTH above and below . (If you do not wish to make one large continuous arc, you may simply place one small arc above and one small arc below .) 3. Without changing the span on the compass, place the compass point on B and swing the arc again. The two arcs you have created should intersect. 4. With your straightedge, connect the two points of intersection. 5. This new straight line bisects . Label the point where the new line and cross as C. has now been bisected and AC = CB. (It could also be said that the segments are congruent, .) (It may be advantageous to instruct students in the use of the "large arc method" because it creates a "crayfish" looking creature which students easily remember and which reinforces the circle concept needed in the explanation of the construction.) Explanation of construction: To understand the explanation you will need to label the point of intersection of the arcs above segment as D and below segment as E. Draw segments , , and . All four of these segments are of the same length since they are radii of two congruent circles. More specifically, DA = DB and EA = EB. Now, remember a locus theorem: The locus of points equidistant from two points, is the perpendicular bisector of the line segment determined by the two points. Hence, is the perpendicular bisector of . The fact that the bisector is also perpendicular to the segment is actually MORE than we needed for a simple "bisect" construction. Isn't this great! Free stuff!!! Bisect an angle Given: Task: Bisect . Directions: 1. Place the point of the compass on the vertex of (point A). 2. Stretch the compass to any length so long as it stays ON the angle. 3. Swing an arc so the pencil crosses both sides of . This will create two intersection points with the sides (rays) of the angle. 4. Place the compass point on one of these new intersection points on the sides of . If needed, stretch your compass to a sufficient length to place your pencil well into the interior of the angle. Stay between the sides (rays) of the angle. Place an arc in this interior - you do not need to cross the sides of the angle. 5. Without changing the width of the compass, place the point of the compass on the other intersection point on the side of the angle and make the same arc. Your two small arcs in the interior of the angle should be crossing. 6. Connect the point where the two small arcs cross to the vertex A of the angle. You have now created two new angles that are of equal measure (and are each 1/2 the measure of .) Explanation of construction: To understand the explanation, some additional labeling will be needed. Label the point where the arc crosses side as D. Label the point where the arc crosses side as E. And label the intersection of the two small arcs in the interior as F. Draw segments and . By the construction, AD = AE (radii of same circle) and DF = EF (arcs of equal length). Of course AF = AF. All of these sets of equal length segments are also congruent. We have congruent triangles by SSS. Since the triangles are congruent, any of their leftover corresponding parts are congruent which makes equal (or congruent) to . Parallel -through a point Given: Point P is off a given line Task: Construct a line through P parallel to the given line. Directions: 1. With your straightedge, draw a transversal through point P. This is simply a straight line which runs through P and intersects the given line. 2. Using your knowledge of the construction COPY AN ANGLE, construct a copy of the angle formed by the transversal and the given line such that the copy is located UP at point P. The vertex of your copied angle will be point P. 3. When the copy of the angle is complete, you will have two parallel lines. This new line is parallel to the given line. Explanation of construction: Since we used the construction to copy an angle, we now have two angles of equal measure in our diagram. In relation to parallel lines, these two equal angles are positioned in such a manner that they are called corresponding angles. A theorem relating to parallel lines tells us that if two lines are cut by a transversal and the corresponding angles are congruent (equal), then the lines are parallel. Perpendicular - lines (or segments) which meet to form right angles. Perpendicular from a point ON a line Given: Point P is on a given line Task: Construct a line through P perpendicular to the given line. Directions: 1. Place your compass point on P and sweep an arc of any size that crosses the line twice (below the line). You will be creating (at least) a semicircle. (Actually, you may draw this arc above OR below the line.) 2. STRETCH THE COMPASS LARGER!! 3. Place the compass point where the arc crossed the line on one side and make a small arc below the line. (The small arc could be above the line if you prefer.) 4. Without changing the span on the compass, place the compass point where the arc crossed the line on the OTHER side and make another arc. Your two small arcs should be crossing. 5. With your straightedge, connect the intersection of the two small arcs to point P. This new line is perpendicular to the given line. Explanation of construction: Remember the construction for bisect an angle? In this construction, you have bisected the straight angle P. Since a straight angle contains 180 degrees, you have just created two angles of 90 degrees each. Since two right angles have been formed, a perpendicular exists. Given: Point P is off a given line Task: Construct a line through P perpendicular to the given line. Directions: 1. Place your compass point on P and sweep an arc of any size that crosses the line twice. 2. Place the compass point where the arc crossed the line on one side and make an arc ON THE OPPOSITE SIDE OF THE LINE. 3. Without changing the span on the compass, place the compass point where the arc crossed the line on the OTHER side and make another arc. Your two new arcs should be crossing on the opposite side of the line. 4. With your straightedge, connect the intersection of the two new arcs to point P. This new line is perpendicular to the given line. Explanation of construction: To understand the explanation, some additional labeling will be needed. Label the point where the arc crosses the line as points C and D. Label the intersection of the new arcs on the opposite side as point E. Draw segments , , , and . By the construction, PC = PD and EC = ED. Now, remember a locus theorem: The locus of points equidistant from two points (C and D), is the perpendicular bisector of the line segment determined by the two points. Hence, is the perpendicular bisector of . The fact that we created a bisector, as well as a perpendicular, is actually MORE than we needed - we only needed to create a perpendicular. Yea, free stuff!!! Construct an isosceles triangle whose legs and Construct an Isosceles Triangle base are of the predetermined lengths Using Given Segment Lengths: given. Construct the new triangle on the reference line. When constructing an isosceles triangle, you may Using your compass, measure the length of the given "base". be given pre-determined segment lengths to use for the triangle (such as in this example), or you may be allowed to determine your own segment lengths. Either way, the construction process will be the same. Do not change the size Using your compass, measure Without changing the You now have three of the compass. Place the length of the given "leg". size of the compass, points which will define your compass point on Place the compass point where move the compass point the isosceles triangle. the reference line point the previous arc crosses the to the point on the and scribe a small arc reference line and scribe reference line. Scribe which will cross the another arc above the reference an arc above the line line. line such that it intersects with the previous arc. Construct an Equilateral Triangle Using a Given Segment Length: Construct an equilateral triangle whose sides are of given length "a". Construct the new triangle on the reference line. When constructing an equilateral triangle, you may be given a pre-determined segment length to use for the triangle (such as in this example), or you may be allowed to determine your own segment length. Either way, the construction process will be the same. Using your Do not change the size of Do not change the size You now have three compass, the compass. Place your of the compass. Place points which will define measure the compass point on the the compass point the equilateral triangle. length of the reference line point and where the arc crosses given segment, scribe an arc which will the reference line and "a". cross the line and will rise scribe another arc which above the line. crosses the previous arc. Alternate Method for Constructing an Equilateral Triangle: Draw a circle and place a point on the circle. Do not change the size of the An equilateral triangle can be easily constructed from a circle. The secret to this method is to remember to keep the compass set at the same length as the radius of the original circle. compass. With the compass still set at the same size as the radius of the circle, place the compass point on the point on the circle and mark off a small arc on the circle. You now have a circle with six equally Connect every other point on divided sections on its the circle to form the equilateral circumference. triangle. Now, move the compass point to this new arc and mark off another arc. Continue around the circle.