Computer Math

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Discrete Mathematics

COURSE: DESCRIPTIVE GEOMETRY

TOPIC: Points, Lines, Angles and Triangles

STAGE 1: IDENTIFY DESIRED RESULTS
Content Standard(s) Generalizations about what students should know and be able to do
Established Goals: CT Frameworks: a) Geometric constructions either with drawing tools or software will be completed. b) Geometric theorems will be proved using deductive reasoning. c) The Pythagorean Theorem can be used to find an unknown length. NCTM Standards: a) Draw and construct representations of two-dimensional geometric objects. b) Establish the validity of geometric conjectures using deduction and proving theorems.
Enduring Understandings Insights earned from exploring generalizations via the essential questions (Students will understand THAT…)	Essential Questions Inquiry used to explore generalizations
The student will understand that: 1. There are four basic constructions. 2. Definitions, postulates, and theorems are necessary to prove theorems with deductive reasoning. 3. There are many classifications of angles. 4. There are many classifications of triangles. 5. Triangles can be proved to be congruent or similar.	1. What are the four basic constructions? 2. How are definitions, postulates, and theorems used in proving theorems? 3. What are the classifications and relationships between different kinds of angles? 4. How can triangles be classified? 5. How can given information be used to prove that two triangles are congruent or similar?
Knowledge and Skills What students are expected to know and be able to do
Students will know… 1. The four basic geometric constructions. 2. Definitions, postulates and theorems as they relate to angles, triangles, parallel lines and similar triangles. 3. The definition of congruent triangles. 4. The definition of similar triangles. 5. The classification of triangles by their sides or their angles. 7. The Pythagorean Theorem.	Students will be able to… 1. Using only a compass and straightedge construct the four basic constructions. 2. State the definitions, postulates and theorems and apply them by doing proofs with deductive reasoning. 3. Prove triangles congruent using the definitions, postulates and theorems from #2 above. 4. Prove triangles similar using the definitions, postulates and theorems from #2 above. 5. Classify triangles according to angle measure and length of sides. 6. Apply the Pythagorean Theorem and its converse to selected problems.
STAGE 2: DETERMINE ACCEPTABLE EVIDENCE
Performance Task(s) Authentic application in new context to evaluate student achievement of desired results designed according to GRASPS (Goal, Role, Audience, Setting Performance, Standards)	Other Evidence Application that is functional in a classroom context only to evaluate student achievement of desired results
Fred and Mary were studying for their Geometry test and Fred said to Mary, “ I’m going to use AAA to prove the triangles in problem #4 are congruent.” Mary replied, “Fred you must have been sleeping through class all semester? AAA is not a valid postulate or theorem to prove triangles are congruent!” You must decide who is correct, Fred or Mary? Then you must show by example why they are correct. Make sure you include a diagram with your explanation.	In addition to tests and quizzes, one or more of the following will be used: Cooperative learning Graphing calculator activities Informal and formal checks: Homework checks, problem of the day, and review worksheets.
STAGE 3: DEVELOP LEARNING PLAN
Learning Activities: Cooperative learning activities Graphing calculator activities Homework and review sheets Geometer's Sketchpad activities

COURSE: DESCRIPTIVE GEOMETRY

TOPIC: Polygons, Circles and Applications

STAGE 1: IDENTIFY DESIRED RESULTS
Content Standard(s) Generalizations about what students should know and be able to do
Established Goals: CT Framework: a) Relationships exist among sides, angles, perimeters and areas of congruent and similar polygons. b) Geometric Theorems and relationships may be used to solve a variety of problems. NCTM Standards: a) Analyze properties and determine attributes of two- and three-dimensional objects. b) Explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them. c) Establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others. d) Use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations. e) Investigate conjectures and solve problems involving two- and three-dimensional objects represented with Cartesian coordinates.
Enduring Understandings Insights earned from exploring generalizations via the essential questions (Students will understand THAT…)	Essential Questions Inquiry used to explore generalizations
The student will understand that: Circles have many properties. Polygons have many properties. Coordinate geometry can be used to investigate geometric relationships graphically.	What are the parts of a circle and what are the formulas for circumference and area? What are the different types of polygons and how do you find their areas and perimeters? How do the ideas of slope, midpoint, parallel and perpendicular apply to a graph?
Knowledge and Skills What students are expected to know and be able to do
Students will know… 1. The properties of circles. 2. The properties of polygons. 3. The formulas used to solve problems using coordinate geometry.	Students will be able to… Find tangents, chords, radii, arc length, central and inscribed angles, angles formed by secants and tangents, circumference and area of circles. Determine if a polygon is convex or concave, find the sum of the interior and exterior angles, determine the measures of the interior and exterior angles and find the perimeter and area. Use the distance formula, find the equation of a line, use the midpoint formula, and use the slope formula to determine perpendicular and parallel lines.
STAGE 2: DETERMINE ACCEPTABLE EVIDENCE
Performance Task(s) Authentic application in new context to evaluate student achievement of desired results designed according to GRASPS (Goal, Role, Audience, Setting Performance, Standards)	Other Evidence Application that is functional in a classroom context only to evaluate student achievement of desired results
You are working for a landscaping firm and need to create a garden to meet your client's specifications. The garden will have a square interior section circumscribed by a circle of radius 10. The diagonal of the square region is a diameter of the circle. You need to cover the square region with red cedar woodchips and the rest of the area within the circle with white crushed stone. How many square feet of area will be covered in wood chips and how much area will be covered in crushed stone? Write down all your formulas and show all work.	In addition to tests and quizzes, one or more of the following will be used: Cooperative learning activities Graphing calculator activities Informal and formal checks: homework checks, problem of the day, and review sheets.
STAGE 3: DEVELOP LEARNING PLAN
Learning Activities: Cooperative learning activities. Graphing calculator activities. Homework and review worksheets.

COURSE: DESCRIPTIVE GEOMETRY

TOPIC: Three-Dimensional Figures

STAGE 1: IDENTIFY DESIRED RESULTS
Content Standard(s) Generalizations about what students should know and be able to do
Established Goals: CT Framework: a) Relationships exist among sides, angles, perimeters, areas and surface areas of solids. b) Indirect measures of volume for some solids can be found through the use of formulas. NCTM Standards: a) Draw and represent three-dimensional geometric figures. b) Visualize three-dimensional objects and spaces from different perspectives and analyze their cross sections
Enduring Understandings Insights earned from exploring generalizations via the essential questions (Students will understand THAT…)	Essential Questions Inquiry used to explore generalizations
The student will understand that: Solids can be classified by the shape of the base they have or the number of bases they have. Formulas are a useful tool when working with solids.	Name the solids that have 2 bases, 1 base and no bases. What are area and volume formulas for prisms, cylinders, pyramids, cones and spheres?
Knowledge and Skills What students are expected to know and be able to do
Students will know… 1. That there are three classifications for solids. 2. That there are different formulas for area and volume for each solid.	Students will be able to… Identify the names of solids that have one base, two bases, and no bases. Calculate the lateral area, surface area, and volume for each solid.
STAGE 2: DETERMINE ACCEPTABLE EVIDENCE
Performance Task(s) Authentic application in new context to evaluate student achievement of desired results designed according to GRASPS (Goal, Role, Audience, Setting Performance, Standards)	Other Evidence Application that is functional in a classroom context only to evaluate student achievement of desired results
Continuing the landscaping problem: Find the number of cubic yards of woodchips and crushed stone that must be ordered to complete the gardens with a thickness of 4 inches. Remember that there are 27 cubic feet in one cubic yard. Write down all your formulas and show all of your work.	In addition to test and quizzes, one or more of the following will be used: 1. Cooperative learning activities 2. Graphing calculator activities 3. Informal and formal checks: homework checks, problem of the day, and review worksheets.
STAGE 3: DEVELOP LEARNING PLAN
Learning Activities: 1. Cooperative learning activities. 2. Graphing calculator activities. 3. Homework and review worksheets.