Lesson Title: Similar Right Triangles and Trig Ratios
Date: _____________ Teacher(s): ____________________
Course: Common Core Geometry, Unit 2
Start/end times: _________________________
Lesson Standards/Objective(s): What mathematical skill(s) and understanding(s) will be developed? Which
Mathematical Practices do you expect students to engage in during the lesson?
G.SRT.B.6 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
MP2:
MP3:
MP5:
MP6:
MP7:
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Lesson Launch Notes: Exactly how will you use the
first ten minutes of the lesson?
Have students work in pairs. Give each pair the lesson
launch sheet or display the lesson launch sheet using a
projector. Have students find the missing side of the
similar triangles. Randomly call on a pair to show the
class how they solved for the missing side. Ask for a
pair that used a different method (ratio). (Look for and
highlight a ratio setup that uses two sides from the
same triangle in each fraction of the ratio
setup…sometimes called internal ratios. If one is not
given, it will be necessary to suggest it and have
students try it to see it also works. (Look for evidence
of MP2, MP6.)
Lesson Closure Notes: Exactly what summary activity,
questions, and discussion will close the lesson and
connect big ideas? List the questions. Provide a
foreshadowing of tomorrow.
Ask, “Now that we can find the ratio of any two sides of
any right triangle without measuring those sides, in what
mathematical or real world settings could this be useful to
us?”
Have groups write responses on the back of the graphic
organizer. Have randomly selected groups share their
reasoning and let other groups add to or edit their
responses. (Anticipate and highlight responses about
being able to solve for missing sides of right triangles.
Also highlight real world settings, (e.g., finding the height
of a flag pole) in which parts of right triangles would need
to be solved.) (Look for evidence of MP2, MP4.)
Lesson Tasks, Problems, and Activities (attach resource sheets): What specific activities, investigations,
problems, questions, or tasks will students be working on during the lesson? Be sure to indicate strategic
connections to appropriate mathematical practices.
1. Give each student a sheet of unlined paper a protractor and a ruler. Instruct students to draw, as accurately as
possible, a right triangle with one of its acute angles measuring 30o. Ask students, in pairs, to compare triangles.
Ask students to share their observations. (Students should recognize that, while some of the triangles may be
approximately congruent, all the triangles in the room are approximately similar.) (Look for evidence of MP5,
MP7.)
2. Have students form trios by finding two partners that created triangles that are “different” than theirs. Ask,
“What are the differences that made you choose your group members? What is still the same about your
triangles?” (Look for evidence of MP2, MP7.)
3. Have groups measure, as accurately as possible, the lengths of all three sides of their triangles and find the
measure of the unknown angle. (They may find the angle by measuring, but some may use the fact that the sum
of the measures of the angles of a triangle is 180o to find that the remaining angle measures 60o.) Explain to
students a system of identifying the sides of their triangles. (One side is the “hypotenuse” and will be denoted
as “H.” The hypotenuse will be a previously learned concept. It may be necessary to review it in the context of
the Pythagorean theorem. The legs can now be identified relative to one of the acute angles called the “reference
angle.” The leg across from the reference angle is called the “opposite leg,” which will be denoted as “O,” and
the leg which makes up one side of the reference angle is called the “adjacent leg” which will be denoted as
“A.”) Instruct students to use the 30o angle as the reference angle. Have each group choose one pair of sides
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: Similar Right Triangles and Trig Ratios
Course: Common Core Geometry, Unit 2
Date: _____________ Teacher(s): ____________________
Start/end times: _________________________
and create a ratio of those sides for each triangle in the group and find a decimal approximation of the ratio.
(Groups should find that the ratio they find is equal for all three triangles.) Have groups report their results.
Record the results on a board or projector (using the graphic organizer). Give each group the graphic chart to
record the results for any of the ratios found by the class. If all six ratios have not been chosen, have the class
find the remaining ratios in groups and report them to complete the table. Have groups add these labels to three
of the ratios on the chart: O/H = sine 30o, A/H = cosine 30o, O/A = tangent 30o. Point out abbreviations “sin,
cos, tan.” (Anticipate the possibility of questions as to why the other three ratios, cosecant or csc and secant or
sec and cotangent or cot, are not being named at this point. In this case, point out that these ratios are merely
reciprocals of the first three and not necessary for relating the sides at this point. Feel free to identify them by
name to satisfy curiosity.) Ask,
 In any right triangle with a 30o angle, what do we now know about the ratio of the leg opposite the 30o
reference angle to the hypotenuse (O/H)? (It is always the same and equal to .5.)
 In any right triangle with a 30o angle, what do we now know about the ratio of the leg adjacent to the 30o
reference angle to the hypotenuse (A/H)? (It is always the same and equal to approximately .8660.)
 In any right triangle with a 30o angle, what do we now know about the ratio of the leg opposite to the 30o
reference angle to the leg adjacent to the 30o reference angle (O/A)? (It is always the same and equal to
approximately .5774.)
 What is the measure of the other acute angle? What would happen to the trig ratios is we used this
angle as the reference angle?
(Look for evidence of MP3, MP5, MP7.)
4. Ask, “What would happen to the trig ratios if we used a different angle? What, more specifically, do you think
would happen to the sin and cos of the angle if we reshaped the triangle so that the reference angle was changed
from 30o angle to 40o? Be prepared to explain why you think this will be true.” Make several options available
to students to explore this question, including:
 Protractor and ruler (for construction),
 Explore at a computer station using Sketchpad (see Sketchpad file),
 Explore using patty paper or other hands on tools,
 Paper and writing utensil for sketching and logical reasoning).
Have students discuss the accuracy of the answers. Elicit responses from groups who found specific answers
gleaned from ratios. Note these answers as with 30o before. Look for or make a suggestion that there are other
ways than construction of a triangle to find the ratios. (Look for evidence of MP5, MP6)
5. Make sure each group has a scientific or graphing calculator. (One calculator per student is preferred.) Ask them
to explore and find the sin30o, cos 30o, sin 40o, and cos 40o on their calculators. Tell them they will know they
have found it correctly, without using a constructed right triangle, when they get the same approximate answers
as discovered earlier. Ask them to find tan of the two reference angles as well.
6. Give each student a copy of the trigonometric table spread sheet. Ask students to find the sin, cos, and tan of 30o
and 40o on the table and compare the answer to that on the calculator. (Note that the answers are approximate
and that the decimals for the ratios non-terminating and non-repeating. Ask, “What generally happens to the sine
as the reference angle gets bigger? What happens to the cosine and tangent?” Other than, looking at answers on
the table or calculator, why do you think this is true.(Look for evidence of MP2, MP7)
7. Ask,” If one of the acute angles of any size right triangle was equal to 64o, what would be the approximate ratio
of the opposite leg to the hypotenuse?” (Use of calculator or table here is appropriate.) Have randomly selected
groups give their responses and allow for discourse on the meaning, in terms of ratios, of these results. Now ask
about the ratio of adjacent leg to hypotenuse and the ratio of opposite leg to adjacent leg and allow for similar
discourse. Point out that we could answer these questions for any right triangle. Use responses here as a
formative assessment to determine if more examples of this nature are needed for comprehension.
8. As an extension, ask, “What would have to be true about a right triangle if the tangent of one of the reference
angles was known to be equal to 1?” Students may use the same type of exploration as in section 5 above. Have
randomly selected groups share their reasoning and let other groups add to or edit their responses. Repeat this as
necessary with other answers to trig ratios.
9. As another extension, ask, “What difficulty may be presented when the angle is 0o or 90o? Why are the answers
0, 1 or undefined here? Have randomly selected groups share their reasoning and let other groups add to or edit
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: Similar Right Triangles and Trig Ratios
Date: _____________ Teacher(s): ____________________
their responses.
Course: Common Core Geometry, Unit 2
Start/end times: _________________________
Evidence of Success: What exactly do I expect students to be able to do by the end of the lesson, and how will I
measure student success? That is, deliberate consideration of what performances will convince you (and any outside
observer) that your students have developed a deepened and conceptual understanding.
Students will be able to find the sin, cos, tan, of acute angles.
Students will be able to see the sin, cos, and tan as ratios of sides of right triangles.
Students will understand that, for a given acute angle in a right triangle, the triangles are similar and the trig ratios
will be the same for all triangles of that shape.
Students will begin to be able to see the relationship between some of the values of the trig functions of various
angles.
Notes and Nuances: Vocabulary, connections, anticipated misconceptions (and how they will be addressed), etc.
Sine (sin), Cosine (cos), Tangent (tan), reference angle, opposite side, adjacent side, hypotenuse
One misconception is that these first three trig ratios are “magic” functions that give us a number. In fact, it should
be stressed that they are merely names for the ratios of sides for similar right triangles (those with the same two acute
angles).
Resources: What materials or resources are essential
for students to successfully complete the lesson tasks or
activities?
Homework: Exactly what follow-up homework tasks,
problems, and/or exercises will be assigned upon the
completion of the lesson?
Rulers, Protractors, Graphing or Scientific calculators,
Lesson Launch Resource Sheet
Graphic Chart Resource Sheet
Trigonometric Table spreadsheet
Most texts will have a set of problems asking for sin, cos,
tan of some angles. This will make a good assignment.
Lesson Reflections: How do you know that you were effective? What questions, connected to the lesson
standards/objectives and evidence of success, will you use to reflect on the effectiveness of this lesson?
How well did students connect the concept of the trig values on the table and calculator to the ratios of the sides of
the right triangle?
How did students do in comparing the values on the calculator to those on the table?
How well were students able to make conjectures about trig values when changes were made to the triangles.
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.