Mathematical Topics

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Pre Algebra

 

COURSE:  PLANE GEOMETRY

TOPIC: Language and Notation

 

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 geometic 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 applications for

triangles. 

 

 

      

1.     What are the four basic constructions?

2.     How are definitions, postulates, and theorems used in proving theorems?

3.     How are classifications, congruence relationships, similar relationships and the Pythagorean Thorem apply to triangles?

Knowledge and Skills

What students are expected to know and be able to do

Students will know…

1.     there are 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 postulates and theorems concerning

medians, altitudes and bisectors

6.     that triangles can be classified.

7.     what the Pythagorean Theorem states.

 

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.     do problems concerning medians,

altitudes and angle bisectors.

6.     classify triangles according to angle

measure and length of sides.

7.     apply the Pythogorean 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 decided 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:

  1. cooperative learning
  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 sheets

 

 

COURSE: PLANE GEOMETRY

TOPIC: Two-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 and areas of congruent and

similar polygons.

b)     Geometric Theorems and trigonometric relationships may be used to solve a variety

of problems.

 

NCTM Standards:

a)     Use trigonometric relationships to determine lengths and angle measures.

b)     Use Cartesian coordinates to analyze geometric situations.

 

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.     circles have many properties.

2.     polygons have many properties.

3.     trigonometric relationships can be used 

4.     to solve real life problems.

5.     coordinate geometry is a useful method of solving problems.

 

  1. What are the parts of a circle and what are the formulas for circumference and area?
  2. What are the different types of polygons and how do you find their areas and perimeters?
  3. How are sine, cosine and tangent defined and used to solve problems?
  4. How would slope, midpoint, and

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 sine, cosine, and tangent   

relationships with right triangles.

4.  the formulas used to solve problems using coordinate geometry.

 

 Students will be able to…

  1. find tangents, chords, radii, arc length, central and inscribed angles, angles formed by secants and tangents, circumference and area of circles.
  2. determine if a polygon is convex, find the sum of the interior and exterior angles, determine the measures of the interior and exterior angles and find the perimeter and area.
  3. find sine, cosine, and tangents as they relate to data given in real life word problems.
  4. 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 cable tv company and your job today is to help install an antenna tower.  The antenna is sitting on a tower that’s 200 feet in the air.  The ground around the tower is absolutely flat for miles.  The first set of guy wires will be attached half way up the tower and it is decided that the wires must make an angle of 45 degrees with the ground.  There are five guy wires.  You must determine the amount of wire that is needed for all five guy wires.  Write down all your formulas and show all work.

 

In addition to tests and quizzes, one or more of the following will be used:

  1. cooperative learning activities
  2. graphing