Geometry and
Trigonometry
Math 5
Learning Objectives for Unit
Learning Objectives for Unit
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)
Assessment
GEOMETRY
Students 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.
• 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
Create Formal Constructions
• 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 lines and angles
• Vocabulary: regular polygon, interior angle, exterior
angle, apothem
Know properties of Polygons
• 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 triangles
Know properties of special quadrilaterals
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
Line and angle 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
Parallelogram 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.
Triangle 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
Similarity Proofs
• Prove a figure to be a special quadrilateral
• Skills to practice: Slope formula, distance formula
• Know all proving properties for parallelograms (etc.)
Use coordinates to prove
theorems and shapes
TRIGONOMETRY
Students 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.
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)
Similarity and Trigonometry
• Use trig ratios to solve real world problems
Basic trigonometry problems
• 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?
Sine - Cosine relationship
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
• Prove 𝑠𝑖𝑛2 𝑥 + 𝑐𝑜𝑠 2 𝑥 = 1
• How does this relate to the unit circle?
• How does this relate to trigonometric ratios?
• How does this relate to Pythagorean Theorem?
Sine and Cosine squared