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Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 72367
The Cosine Ratio
Students are asked to compare the ratio of corresponding sides of two triangles and to explain how this ratio is related to the cosine of a given
angle.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, similar triangles, cosine ratio, AA similarity
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_TheCosineRatio_Worksheet.docx
MFAS_TheCosineRatio_Worksheet.pdf
FORMATIVE ASSESSMENT TASK
Instructions for Implementing the Task
This task can be implemented individually, with small groups, or with the whole class.
1. The teacher asks the student to complete the problem on the The Cosine Ratio worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student does not demonstrate an understanding of the relationship between
and
.
Examples of Student Work at this Level
The student writes that
The triangle with side lengths
because:
and
is bigger.
page 1 of 4 The triangles are similar not congruent.
The student writes that
because:
Both triangles have a
angle.
Both triangles are special right triangles.
The sides “match up.”
The angles are congruent so the sides must be congruent.
Questions Eliciting Thinking
What do you know about these two triangles? Are they congruent? Similar?
If two triangles are similar, what do you know about the ratio of their corresponding sides?
What does
mean? Does it mean that the corresponding sides must be congruent?
Are corresponding sides of two right triangles always proportional? Why or why not?
Can you be certain that both triangles are
,
,
triangles?
Instructional Implications
Review ways to prove two triangles are similar (AA, SAS, SSS) and what must be established in order to conclude two triangles are similar using each method. Remind the
student that once two triangles are proven similar, all corresponding angles are congruent and all corresponding sides are proportional.
Explain that the two given triangles are similar by the AA Similarity Theorem. Ask the student to write ratios of corresponding sides in the form of an extended proportion.
Guide the student to use the extended proportion to deduce that
. Model explaining that
because corresponding sides of similar triangles are
proportional.
Provide the student with pairs of triangles that can be shown to be similar. Ask the student to identify the theorem that justifies the similarity and then write an extended
proportion relating the lengths of the sides and congruence statements that show how angles are related.
Review terminology related to right triangles (e.g., right angle, acute angles, legs, hypotenuse, side opposite an angle, and side adjacent to an angle) and the definitions of
the sine, cosine and tangent ratios. Provide the student with a right triangle with the lengths of the sides given. Have the student identify the sine, cosine, and tangent
ratios for both acute angles. Include a variety of right triangles in different orientations.
Have the student compare the cosine ratio for the same acute angle of several right triangles with sides of different lengths. Guide the student to understand that the
trigonometric ratios for a given angle measure are the same in all right triangles with an acute angle of that measure. Remind the student that right triangles with the same
acute angle measure are similar by the AA Similarity Theorem.
Consider implementing the MFAS task The Sine of 57 (G­SRT.3.6) to further assess the student’s understanding of trigonometric ratios.
Moving Forward
Misconception/Error
The student does not recognize the ratios as the cosine of a and the cosine of ß .
Examples of Student Work at this Level
The student recognizes that the triangles are similar and the ratio of corresponding sides is equal. However, the student does not recognize the given ratios as the cosine
ratios. The student:
May describe the lengths in the ratio as the lengths of the hypotenuse, opposite side, or adjacent side but does not identify the ratio as the cosine ratio.
page 2 of 4 Questions Eliciting Thinking
How is the cosine ratio defined?
What is the cosine of
? What is the cosine of
?
Instructional Implications
Review terminology related to right triangles (e.g., right angle, acute angles, legs, hypotenuse, side opposite an angle, and side adjacent to an angle) and the definitions of
the sine, cosine and tangent ratios. Provide the student with a right triangle with the lengths of the sides given. Have the student identify the sine, cosine, and tangent
ratios for both acute angles. Guide the student to understand that the two given ratios are cosine ratios and since
, then cos
= cos
.
Have the student compare the cosine ratio for the same acute angle of several right triangles with sides of different lengths. Guide the student to understand that the
trigonometric ratios for a given angle measure are the same in all right triangles with an acute angle of that measure. Remind the student that right triangles with the same
acute angle measure are similar by the AA Similarity Theorem.
Consider implementing the MFAS task The Sine of 57 (G­SRT.3.6) to further assess the student’s understanding of trigonometric ratios.
Almost There
Misconception/Error
The student is unable to deduce that if the cosines of two acute angles of two right triangles are congruent, then the triangles are similar.
Examples of Student Work at this Level
The student recognizes that the triangles are similar so that the ratio of corresponding sides are equal and identifies the given ratios as cosine ratios. However, the student
is unable to deduce that if the cosines of two acute angles of two right triangles are congruent, then the triangles are similar. The student:
Indicates that he or she does not understand what must be true of the two triangles.
Writes that the triangles must be congruent.
Writes that the triangles are similar but is unable to explain why.
Questions Eliciting Thinking
What must be true of
and
if cos(E) = cos(Q)?
What is the difference between similar and congruent triangles?
Instructional Implications
Remind the student of the definition of the cosine ratio. Explain that the cosine ratio is a ratio of specific sides in a class of right triangles that are related by similarity. The
class can be identified by an acute angle measure that is common to all right triangles in the class. So if cos(E) = cos(Q), then
and
are angles of similar right
triangles. Consequently, since
and
are both right triangles such that cos(E) = cos(Q), then
by definition of the cosine ratio.
Consider implementing the MFAS task The Sine of 57 (G-SRT.3.6).
Got It
Misconception/Error
The student provides complete and correct responses to all components of the task.
Examples of Student Work at this Level
The student explains that since both triangles are right triangles and
=
, the triangles are similar by the AA Similarity Theorem. The student writes that
because corresponding sides of similar triangles are proportional. The student recognizes that
deduce from cos(E) = cos(Q) that right
is similar to right
as the cosine of
and
as the cosine of
. The student is able to
.
Questions Eliciting Thinking
Could you deduce that
is similar to
if they were not both right triangles?
Why are trigonometric ratios the same in right triangles with the same acute angle measure?
Instructional Implications
Ask the student to explore and explain the relationship between the sine and cosine of complementary angles.
Consider implementing the MFAS task The Sine of 57 (G-SRT.3.6).
page 3 of 4 ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
The Cosine Ratio worksheet
SOURCE AND ACCESS INFORMATION
Contributed by: MFAS FCRSTEM
Name of Author/Source: MFAS FCRSTEM
District/Organization of Contributor(s): Okaloosa
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.G-SRT.3.6:
Description
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to
definitions of trigonometric ratios for acute angles.
page 4 of 4