GEOMETRY 2013-2014

August 27, 2013 Toolbox Automaticity Geometry Tools August 27, 2013 Toolbox Automaticity

STANDARDS Trimester 1 Day 1-----G.CO.13 Equilateral Triangle & Regular Hexagon Day 2 & 3 -----FAL: Having Kittens Day 3 & 4---G.CO.12 Copy Segments & Bisect Segments Day 5 , 6, & 7 --G.CO.12 Copy Angles and Bisect Angles Day 8 -----G.CO.12 Perpendiculars and G.CO.13 Inscribed Square Day 9-----G.CO.12 Parallel Lines Day 10-----G.CO.12 & G.CO.13 Putting it to

G.CO.13 G-CO.13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

GEOMETRY Geo: earth Metres: measure Geometry began as the study of earth measure.

GEOMETRIC CONSTRUCTIONS "Construction" in Geometry means to draw shapes, angles or lines accurately. These constructions use only compass, straightedge (i.e. ruler) and a pencil. This is the "pure" form of geometric construction - no numbers involved!

Inscribed In: Inscribed Polygon Inscribed Circle A circle is inscribed in a polygon if the sides of the polygon are tangent to the circle. A polygon is inscribed in a circle if the vertices of the polygon are on the circle.

Equilateral Triangle: An equilateral triangle is a triangle whose sides are all congruent

REGULAR HEXAGON: A hexagon (six sided polygon) with congruent sides and angles

Creating an Inscribed Hexagon This is one of the easiest constructions ever. The radius of a circle can be struck around a circle exactly six times. Lets watch: http://www.mathopenref.com/constinhexagon.html Time to Try!!

Using Geometry in Design:

MANDALA

QUESTIONS

Having Kittens Work out whether this number of descendants is realistic. Here are some facts that you will need:

HAVING KITTENS Can you make a diagram or table to show what is happening? Can you now look systematically at what happens to her kittens? And their kittens? Do you think the first litter of kittens will have time to grow and have litters of their own? Then what about their kittens? What have you assumed here?

Collaborative Activity Work in groups of two. Produce a solution that is better than your individual solution. Take turns to explain how your did the task and how you now thing it could be improved, then put your individual work aside. Try to produce a joint solution to the problem.

Assessing Sample Student Responses Your task is to correct the work and write comments about its accuracy and organization. What has the student done correctly? What assumptions has he or she made? How could the solution be improved?

Sample Response: Alice

Sample Response: Wayne

Sample Response: Ben

Reviewing Work I have selected the important facts and used them to solve the problem. I am aware of the assumptions I have made and the effect these assumptions have on the result. I have used more than one method I have checked whether my results make sense and improved my method if need be. I have presented my results in a way that will make sense to others.

G.CO.12 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 the line.

TODAY WE WILL Copy a segment. Bisect a segment.

There are three undefined terms in geometry: Undefined terms are basic terms we need to describe the shape and size of objects. There are three undefined terms in geometry: Point, line, and plane

POINT Point (0-D): A location. Example: The following is a diagram of points A, B, C, and Q:

SPACE Space: the set of all points.

LINE Line (1-D): A series of points that extend in two directions without end. Example: The following is a diagram of two lines: line AB and line HG. The arrows signify that the lines drawn extend indefinitely in each direction.

PLANE Plane (2-D): a flat, two-dimensional object. A plane must continue infinitely in all directions and have no thickness at all. A plane can be defined by at least three non-collinear points or renamed by a script capital letter.

GEOMETRIC CONSTRUCTIONS "Construction" in Geometry means to draw shapes, angles or lines accurately. These constructions use only compass, straightedge (i.e. ruler) and a pencil. This is the "pure" form of geometric construction - no numbers involved!

SEGMENT Line Segment: part of a line containing two endpoints and all points between them. A B AB or BA

Getting Started In your notes draw two segments. Similar to the ones below.

Construct: 𝑿𝒀 so that 𝑿𝒀  𝑹𝑺 COPYING A SEGMENT Given: 𝑹𝑺 Construct: 𝑿𝒀 so that 𝑿𝒀  𝑹𝑺 http://www.mathopenref.com/constcopysegment.html

PRACTICE #1: Given: 𝑹𝑺 Construct: 𝑊𝑍 = 𝟑𝑹𝑺

PRACTICE #2: Given: 𝑹𝑺 & 𝑷𝑸 Construct: 𝐻𝐺 = 𝑃𝑄 + 2 𝑅𝑆 .

PRACTICE #3: Given: 𝑹𝑺 & 𝑷𝑸 Construct: 𝐻𝐺 = 3 𝑃𝑄 - 2 𝑅𝑆 .

Bisect To divide into two equal parts. You can bisect lines, angles, and more. The dividing line is called the "bisector

Midpoint Midpoint: The point of a line segment that divides it into two parts of the same length. 𝑨𝑪  𝑪𝑬

PERPENDICULAR BISECTORs Perpendicular Lines: two lines that intersect to form right angles. (SYMOBOL ) Perpendicular Bisector of a Segment: a line, segment, or ray that is perpendicular to the segment at its midpoint, thereby bisecting the segment into two congruent segments.

BISECTING A SEGMENT http://www.mathopenref.com/constbisectline.html

Practice #4: Given: 𝑨𝑩 Construct: 𝑿𝒀  𝑨𝑩 at the midpoint M of 𝑨𝑩 .

Practice #5: Given: 𝑨𝑩 Construct: Divide AB into 4 congruent segments.

QUESTIONS

G.CO.12 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 the line.

TODAY WE WILL Copy an angle. Bisect an angle

RAY Ray: part of a line that consists of one endpoint and all points of the line extending in one direction A B AB , not BA

ANGLES () An angle is two rays meeting at a common vertex.

Angle: a pair of rays that share a common endpoint. ANGLES () Angle: a pair of rays that share a common endpoint. The rays are called the sides of the angle. The common endpoint is called the vertex of the angle.

ANGLES () Obtuse angle 90 < mA < 180 Right angle m  A = 90 Acute angle 0 < mA < 90 A A A

COPYING AN ANGLE The angle RPQ has the same measure as BAC http://www.mathopenref.com/constcopyangle.html

Practice #1 Given Acute Angle: ABC Duplicate ABC, so that ABC = A’B’C’

Practice #2 Given Obtuse Angle: DEF Duplicate DEF, so that DEF = D’E’F’

Practice #3 Given Acute Angle: ABC Construct: XYZ = 2mABC.

Given: Two acute angles: Practice #4 Given: Two acute angles: MNO & QRS Construct MNO + QRS = TUV

A ray that divides that angle into two congruent coplanar angles. Angle Bisector A ray that divides that angle into two congruent coplanar angles. Its endpoint is at the angle vertex.

Bisecting an Angle Given: A Construct: 𝑨𝑿 , the bisector of A Step 1: Put the compass point on the vertex of A. Draw an arc that intersects the sides of A. Label the points of intersection B and C. Step 2: Put the compass on point C and draw an arc. With the same compass setting, draw an arc using point B. Be sure the arcs intersect. Label the point where the two arcs intersect as X. Step 3: Draw 𝐴𝑋 𝑨𝑿 , the bisector of A

Practice #5 Draw two angles (one obtuse and one acute) and bisect each one of them.

QUESTIONS

G.CO.12 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 the line.

G.CO.13 G-CO.13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

TODAY WE WILL Constructing perpendicular lines, including the perpendicular bisector of a line segment Construct a square inscribed in a circle.

Perpendicular Lines Lines that intersect at 90 Symbol: 

Constucting Perpendicular Lines http://www.mathopenref.com/constperplinepoint.html

Practice #1: Construct a line perpendicular to line l through point Q.

Constructing a Perpendicular to an external point http://www.mathopenref.com/constperplinepoint.html

Practice #2: Draw a figure like the given one. Then construct the line through point P and perpendicular to 𝑅𝑆

Practice #3: Draw a figure like the given one. Then construct the line through point P and perpendicular to 𝑅𝑆 .

Inscribed In: Inscribed Polygon Inscribed Circle A circle is inscribed in a polygon if the sides of the polygon are tangent to the circle. A polygon is inscribed in a circle if the vertices of the polygon are on the circle.

SQUARE A square is a parallelogram with four congruent sides and four right angles.

SQUARE In a square, look at the diagonals. What do you notice? How can this be used to construct a square?

SQUARE A square is a parallelogram with four congruent sides and four right angles. A few properties of a square: 1. The diagonals are congruent. 2. The diagonals are perpendicular.

Creating and Inscribed Square Given: Circle O. Construct: Square ABCD. Step 1: Label any point on the circle A. Step 2: Construct line AO. Step 3: Label the second intersection of line AO with the circle point C. Step 4: Construct segment AC. Step 5: Construct a  to AC through point O. Step 6: Label the intersections of the perpendicular with the circle points D and B. Step 7: Construct segment DB. Step 8: Construct segments AB, BC, CD, and DA. ABCD is the required square.

QUESTIONS