Constructions Day 1

Now for some vocabulary  Angle: the union of two distinct rays that have a common end point  Vertex: the common end point of an angle  Line segment: the part of a line between two points on the line  Ray: a ray begins at a point and goes on forever in one direction  Intersecting lines: two lines meet at a point. That point is the only point that belongs to both lines  Intersection: all the points 2 figures have in common  Parallel lines: two lines in a plane that do not intersect  End points: the points at the end of a line segment  Compass: instrument used to draw circles. It has two tips – center and pencil.  Bisector: divides a segment or angle into two equal parts  Perpendicular lines: two lines that intersect to form four right angles  Circle: all points in a plane at a constant distance, called the radius, from a fixed point, called the center  Radius: the length of a line segment between the center and circumference of a circle  Equidistant: equally distant from any two or more points  Arc: a continuous portion of a circle  Congruent: having the same size, shape and measure  Point: One of the basic undefined terms of geometry. No length, width, or thickness. Dot represents it.  Line: One of the basic undefined terms of geometry. A set of points whose length goes on forever in two opposite directions.  Plane: One of the basic undefined terms of geometry. Goes on forever in all directions; two- dimensional and flat.

Are you ready to move? Stand up, push your chair under and face me. We are going to use our arms to show our vocabulary terms we just learned.

Explore ► Please take some white paper, a straight edge and a compass. ► Practice making lines, circles, arcs, and shapes if you can. ► Make sure you get used to the compass.

What are Constructions?  The study of Geometry was born in Ancient Greece, where mathematics was thought to be embedded in everything from music to art to the governing of the universe. Plato, an ancient philosopher and teacher, had the statement, “Let no man ignorant of geometry enter here,” placed at the entrance of his school. This illustrates the importance of the study of shapes and logic during that era. Everyone who learned geometry was challenged to construct geometric objects using two simple tools, known as Euclidean tools:  A straight edge without any markings  A compass  The straight edge could be used to construct lines; the compass to construct circles. As geometry grew in popularity, math students and mathematicians would challenge each other to create constructions using only these two tools. Some constructions were fairly easy (Can you construct a square?), some more challenging, (Can you construct a regular pentagon?), and some impossible even for the greatest geometers (Can you trisect an angle? In other words, can you divide an angle into three equal angles?). Archimedes ( B.C.E.) came close to solving the trisection problem, but his solution used a marked straight edge.

Challenge  Your First Challenge: Can you construct a Target?  A very simple target consists of three circles.  The largest circle would have a radius that is three times the length of the radius of the smallest circle  The middle circle would have a radius two times the length of the radius of the smallest circle.  On a separate sheet of paper, construct a target with a straight edge and compass.  Then write a set of instructions that another student could use to create your target.  Then write a set of instructions that another student could use to create your target.

Constructions Day 2

Copying a Line Segment Begin with a line segment AB. Draw a line with a straightedge longer than segment AB below. Label the left endpoint C. Place the compass point on point A. Stretch the compass so that the pencil is exactly on point B. Without adjusting the compass span, place the compass point on point C and swing the pencil so that it intersects the line. Label the intersection point D. AB C C D

Copying an Angle Begin with  CAB. Draw another line and label the left endpoint A’. With the compass on point A, stretch its width to point B. Without adjusting the compass, place the compass point on A’ and draw a wide arc across the line. This establishes a new point B’. Place the compass point on B and stretch its width to point C. Without changing the compass span, place the compass point on point B’ and draw an arc across the previously drawn arc. The intersection of these two arcs becomes point C’. Using a straightedge, draw a line from point A’ to C’ where the arcs intersect.  C’A’B’ is  (equal in measure) to  CAB. C A B A` B` C`

Constructions Day 3

Bisecting a Segment Begin with a line segment AB. Place the compass point on point A. Stretch the compass along segment AB to a length greater than half the segment length. Construct a circle (or wide arc) with center at point A. Without adjusting the compass span, place the compass point on point B. Construct a circle (or wide arc) with center at point B. Mark and label the intersection points of the two circles as points C and D. Using a straightedge, draw a line through points C and D. Line CD intersects line AB at the midpoint, M. AB C D M

Perpendicular Bisector Student Investigation Draw a point on the perpendicular bisector. Measure the distance between the point on the perpendicular bisector and one endpoint of the segment. Compare this distance to the distance between the point on the perpendicular bisector and the other segment endpoint. Question for Students: What is true about the distances from a point on the perpendicular bisector to each endpoint of a segment?

Angle Bisector Begin with an angle. Draw a circle (or wide arc) at point O with an arbitrary radius, making certain the circle intersects both angle sides. Label the points on the angle sides as A and B. Draw a circle (or wide arc) at point A such that its radius is more than half the distance between A and B. Without adjusting the compass span, place the compass point on B and draw a circle (or wide arc). Mark and label the intersection point of the two arcs as point C. Using a straightedge, draw a line through points O and C. O A B C

Angle Bisector Student Investigation Question for Students: What is true about the distances from a point on an angle bisector to each side of the angle?

Constructions Day 4

Perpendicular to a Line Through a Point on a Line Begin by drawing line segment. Draw and label a point C anywhere on the line but not too close to either end. Construct a circle (wide arc) with center at point C that intersects the line. Where the circle intersects the line, label the intersection points as A and B. Place the compass point on point A and stretch the compass to a distance greater than AC. Draw an arc above the line. Without adjusting the compass span, place the compass point on B and draw an arc above the line, generating intersecting arcs. Label the point D. Construct the perpendicular bisector of segment AB. AB C D

Perpendicular to a Line Through an External Point Begin with a line and a point C not on the line. Construct a circle (or wide arc) with center at point C and radius greater than the distance from C to the line. Where the circle intersects the line, label the intersection points as A and B. Place the compass point on A and draw an arc below the line. Without adjusting the compass span, place the compass point on B and draw an arc, generating intersecting arcs. Label the intersecting point D. Construct a perpendicular bisector of segment AB. C AB D

Constructions Day 5

Parallel Line through a Given Point Begin with a line segment and a point A not on the line. Draw a line from A through the original line label the intersection X. Put the point of your compass on X and draw and arc that intersects both lines, label the intersection points B and C. Keep the compass width the same and make the same arc with the compass point at A label the intersection point D. With the compass point on B measure from B to C. Keep the same width, place the point of the compass on D and make an arc that intersects the present arc label the intersection E. Using a straightedge, connect points A and E with a line. The line that contains points AE is parallel to line XB A X B C D E