Geometry and Trigonometry

Math 5

Learning Objectives for Unit

Learning Objectives for Unit

Assessment

All objectives will be rated from 0 – 7

0 – 1 No data to assess or demonstrates minimal knowledge of learning objective, no mathematical practices used

2 – 3 Demonstrates some knowledge of learning objectives; concepts are missing key understanding; symbolic notation used correctly part of the time. Some mathematical practices used to communicate understanding.

4 – 5 Demonstrates knowledge of learning/practice objective, mistakes are due to algorithmic issues not missing conceptual understanding; communicates justification of mathematics, symbolic notation used correctly, formulas used correctly,

6 – 7 Demonstrates knowledge of learning/practice objective, symbolic notation used correctly, formulas used correctly, PLUS: communicates justification with proper vocabulary; demonstrates deep understanding of subject including: how to apply knowledge and extend knowledge to new circumstances.

Assessment

• Each learning objective will be measured independently • Each learning objective may be interrelated • Two-column format • left = meets expectation (no greater than 5)• Right = exceeds expectation (access to 6 – 7 band)

GEOMETRYStudents apply their earlier experience with dilations and proportional reasoning to build a formal understanding of similarity. It is in this unit that students develop facility with geometric proof. They use what they know about congruence and similarity to prove theorems involving lines, angles, triangles, and other polygons. They explore a variety of formats for writing proofs.

Create Formal Constructions

• Bisect an angle and prove that it works• Construct a perpendicular bisector and prove that it

works• Construct an isosceles triangle and prove that the base

angles are congruent

Know properties of lines and angles

• Vocabulary: angle, angle bisector, angle addition postulate, adjacent angles, linear pair, vertical angles, interior of angle

• Vocabulary: line, ray, line segment, collinear, noncollinear, midpoint, point, plane, parallel, perpendicular,

Know properties of Polygons

• Vocabulary: regular polygon, interior angle, exterior angle, apothem

Know properties of triangles

• Vocabulary: congruent, similar, bases angles, median, altitude, perpendicular bisector, opposite, legs, hypotenuse

• Combinations of sides and angles that prove triangles congruent or similar

Know properties of special quadrilaterals

Line and angle proofs

Proofs may be done with constructions, paragraph, two-column, flow charts• Vertical angles are congruent• Parallel lines and transversals/angle relationships associated• Alternate interior are congruent• Corresponding angles are congruent

• Points on a perpendicular bisector of a segment are equidistant from the segments endpoint

Parallelogram Proofs

Proofs may be done with constructions, paragraph, two-column, flow charts• Opposite sides are congruent• Opposite angles are congruent• Diagonals bisect each other• Rectangles are parallelograms with congruent diagonal

Triangle Proofs

Proofs may be done with constructions, paragraph, two-column, flow charts• Sum of a triangle is always 180 degrees, • prove two triangles congruent or similar,• base angles of an isosceles triangle are congruent • The medians of a triangle meet at one point.

Similarity Proofs

• a line passing through a triangle parallel to the base divides the triangle into similar triangles

• a segment joining the midpoints of the sides of a triangle is parallel to the base and ½ the length of the triangles base

• Prove the Pythagorean Theorem using similarity

Use coordinates to prove theorems and shapes

• Prove a figure to be a special quadrilateral• Skills to practice: Slope formula, distance formula• Know all proving properties for parallelograms (etc.)

TRIGONOMETRYStudents apply their earlier experience with dilations and proportional reasoning to build a formal understanding of similarity. They identify criteria for similarity of triangles, use similarity to solve problems, and apply similarity in right triangles to understand right triangle trigonometry, with particular attention to special right triangles and the Pythagorean Theorem.

Similarity and Trigonometry

Use similarity to justify the definitions of sine, cosine, tangent and relate these ratios to the ability to solve for an angle.

How do sin, cos, tan relate to similar right triangles?

How does the ratio for each help to find angles or sides?

(Justify with trig ratio table)

Basic trigonometry problems

• Use trig ratios to solve real world problems

Sine - Cosine relationship

• Explain the relationship between sine and cosine• How does the unit circle show the relationship?• How do the sine and cosine wave show the relationship?• How can the sine of an angle be related to the cosine of an angle?• Can the sine of an angle be rewritten in terms of the cosine?• Vice-versa?

Writing Sine in Cosine Terms and Cosine in Sine Terms (How does this relate to the waves?)

Sin(x) = cos(90 – x) and Cos(x) = sin(90 – x)

A. Write sin 52° in terms of the cosine.

sin 52° = cos(90 – 52)°

=

B. Write cos 71° in terms of the sine.

cos 71° = sin(90 – 71)°

=

Relate sine to cosine

Sine and Cosine squared

• Prove • How does this relate to the unit circle?• How does this relate to trigonometric ratios?• How does this relate to Pythagorean Theorem?