Download Indirect Measurement - Providence Public Schools

Mathematical Models with Applications, Quarter 1, Unit 1.1
Indirect Measurement
Overview
Number of instruction days:
8 -10
Content to Be Learned
(1 day = 53 minutes)
Mathematical Practices to Be
Integrated

Understand relationships in special right
triangles.

Understand definitions of trigonometric ratios
for acute angles in right triangles.


Understand that side ratios in right triangles are
properties of the angles in the triangle because
of triangle similarity.
4 Model with mathematics.

Solve right triangles in real world problems
using trigonometric ratios, special right
triangles and the Pythagorean Theorem.
1 Make sense of problems and persevere in
solving them.

Identify what parts of the triangle need to be
used in order to find a solution.
Model with mathematics to solve real world
problems involving trigonometric ratios.
6 Attend to precision.

Use precise language to communicate accurate
solutions to trigonometric problems.

Why are trigonometric ratios important?
Essential Questions

Where would you use each of the following to
solve a problem: trigonometry, special right
triangles or the Pythagorean Theorem?

How are angles of elevation or depression
similar and different?
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Math Models, Quarter 1, Unit 1.1
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Indirect Measurement (8 -10 Days)
Standards
Common Core State Standards for Mathematical Content
Geometry
Similarity, Right Triangles, and Trigonometry
G-SRT
Define trigonometric ratios and solve problems involving right triangles
G-SRT.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.
G-SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied
problems.★
Common Core State Standards for Mathematical Practice
1
Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and
looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They
make conjectures about the form and meaning of the solution and plan a solution pathway rather than
simply jumping into a solution attempt. They consider analogous problems, and try special cases and
simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate
their progress and change course if necessary. Older students might, depending on the context of the
problem, transform algebraic expressions or change the viewing window on their graphing calculator to
get the information they need. Mathematically proficient students can explain correspondences between
equations, verbal descriptions, tables, and graphs or draw diagrams of important features and
relationships, graph data, and search for regularity or trends. Younger students might rely on using
concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students
check their answers to problems using a different method, and they continually ask themselves, “Does
this make sense?” They can understand the approaches of others to solving complex problems and
identify correspondences between different approaches.
4
Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in
everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition
equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a
school event or analyze a problem in the community. By high school, a student might use geometry to
solve a design problem or use a function to describe how one quantity of interest depends on another.
Mathematically proficient students who can apply what they know are comfortable making assumptions
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Indirect Measurement (8 -10 Days)
Math Models, Quarter 1, Unit 1.1
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and approximations to simplify a complicated situation, realizing that these may need revision later. They
are able to identify important quantities in a practical situation and map their relationships using such
tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships
mathematically to draw conclusions. They routinely interpret their mathematical results in the context of
the situation and reflect on whether the results make sense, possibly improving the model if it has not
served its purpose.
6
Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear
definitions in discussion with others and in their own reasoning. They state the meaning of the symbols
they choose, including using the equal sign consistently and appropriately. They are careful about
specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem.
They calculate accurately and efficiently, express numerical answers with a degree of precision
appropriate for the problem context. In the elementary grades, students give carefully formulated
explanations to each other. By the time they reach high school they have learned to examine claims and
make explicit use of definitions.
Clarifying the Standards
Prior Learning
In Grade 4, students classified shapes by properties of their angles and worked with word problems
involving unknown angle measures. In Grade 5 students classified two dimensional figures based on
their properties. Students worked with ratios and proportional relationships in Grade 6. A major cluster in
Grade 7 is extending their ability to recognize, represent, and analyze proportional relationships. In Grade
8 understanding and applying the Pythagorean Theorem is a major cluster. Additionally, Grade 8 students
worked informally to establish facts about angle sum and exterior angles of triangles. Fluency with
triangle congruency and similarity was expected in Geometry. Students applied their knowledge of
similar triangles to three trigonometric ratios: sine, cosine, and tangent. Students added trigonometric
ratios to their indirect measurement toolkits. Students explored and used the relationship between sine and
cosine ratios of angles whose sum is 90. Students understood that by similarity, side ratios in right
triangles are properties of the angles in the triangle. They solved real world problems using these
relationships.
Current Learning
Using the concept of indirect measurement, students who struggled with trigonometric concepts now have
an opportunity to extend their previous knowledge through mathematical modeling of the content to be
learned in this unit. Fluency with triangle congruency and similarity is expected. Their study of special
right triangles begins their study of indirect measurement. Students apply their knowledge of similar
triangles to develop and then apply three trigonometric ratios: sine, cosine, and tangent. Students add
trigonometric ratios to their indirect measurement toolkits, adding trigonometric ratios to special right
triangles and similar triangles. Students will explore and use the relationship between sine and cosine
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ratios of angles whose sum is 90°, the acute angles in a right triangle. Students understand that by
similarity, side ratios in right triangles are properties of the angles in the triangle. They solve real world
problems using these relationships.
Future Learning
Extensions of the study of trigonometry will include the unit circle, graphing and analyzing trigonometric
functions and identities in Algebra 2. This study is extended in PreCalculus to include solving
trigonometric equations. Careers that include use of trigonometry are extensive, including surveying,
engineering, construction, physics, navigation, astronomy, etc.
Additional Findings
This material is challenging to students because they have difficulty distinguishing between the opposite
and adjacent sides in a right triangle. According to Principles and Standards of School Mathematics (p.
309), “High school students should develop facility with a broad range of representing geometric ideas . .
.” For example, desks can be arranged in a right triangle to physically represent change in opposite and
adjacent sides when the reference angle is changed.
Assessment
When constructing an end of unit assessment, be aware that the assessment should measure your students’
understanding of the “big ideas” indicated within the standards. The CCSS Content Standards and the
CCSS Practice Standards should be considered when designing assessments. Standards based
mathematics assessment items should vary in difficulty, content and type. The assessment should include
a mix of items which could include multiple choice items, short and extended response items and
performance based tasks. When creating your assessment you should be mindful when an item could be
differentiated to address the needs of students in your class.
The mathematical concepts below are not a prioritized list of assessment items and your assessment is not
limited to these concepts. However, care should be given to assess the skills the students have developed
within this unit. The assessment should provide you with credible evidence as to your students’
attainment of the mathematics within the unit.
Math Models students should be provided with multiple, alternative methods to express their
understandings of the concepts that follow:

D-4
Develop a definition of trigonometric ratios using corresponding angles of similar right
triangles to show that the relationships of the side ratios are the same.
Providence Public Schools
Indirect Measurement (8 -10 Days)
Math Models, Quarter 1, Unit 1.1
Version 2

Apply trigonometric ratios and their inverse relationships to determine missing angle
measures and side lengths of right triangles in problem situations.

Use trigonometric ratios, special right triangles, and the Pythagorean Theorem to solve real
world problems.

Solve right triangle problems using angles of elevation and angles of depression.
Instruction
Learning Objectives
Students will be able to:

Use the properties of special right triangles and the Pythagorean Theorem to solve problems.

Use similar triangles to find distances and heights indirectly.

Apply the trigonometric ratios of sine, cosine, and tangent to determine missing side lengths
of right triangles.

Apply inverse relationships and trigonometric ratios to determine missing angle measures in
right triangles from problem situations.

Apply trigonometric ratios, angles of depression, and angles of elevation to solve real-world
problems.

Review and demonstrate knowledge of important concepts and procedures related to indirect
measurement.
Resources

Modeling with Mathematics: A Bridge to Algebra II, W.H. Freeman and Company, 2006
Sections 8.1 through 8.5 (pp. 496 – 513)

Online Companion Website: http://bcs.whfreeman.com/bridgetoalgebra2/

Additional Resources located in the Supplementary Unit Materials Section of the Binder:
o
Graphic Organizer: SOHCAHTOA.
o
Building Height Activity Instructions
o
Building Height Activity Sheet
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o
Clinometer Construction
o
Water Rockets
Indirect Measurement (8 -10 Days)
Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the
Planning for Effective Instructional Design and Delivery section below for specific recommendations.
Materials
Ruler, protractor, meter sticks or tape measures (1 for each pair of students), colored pencils, TI-Nspire
graphing calculator, calculator viewscreen, document camera (ELMO), dynamic geometry software;
optional – graphic organizer, clinometers, paper clips, straws, 5 by 7 index cards, protractors (preferably
those with a center hole at 0°), string or dental floss, scissors, small weights (such as washers or coins),
drinking straws, tape.
Instructional Considerations
Key Vocabulary
angle of depression
inverse tangent ratio
angle of elevation
opposite side
adjacent side
reference angle
cosine ratio
sine ratio
indirect measurement
special right triangle
inverse cosine ratio
tangent ratio
inverse sine ratio
trigonometric ratio
Planning for Effective Instructional Design and Delivery
Reinforced vocabulary taught in previous grades or units: Pythagorean Theorem, hypotenuse,
trigonometry, ratio, and similarity.
This unit on indirect measurement was purposely placed in the beginning of the year in order to support
the Physics curriculum and provide additional support for NECAP preparation.
Students who struggle with applications of trigonometric ratios typically do so because they either select
the incorrect ratio or they perform the computations incorrectly. Help students understand how
trigonometric ratios relate to similar right triangles, students can elaborate on their knowledge by
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Indirect Measurement (8 -10 Days)
Math Models, Quarter 1, Unit 1.1
Version 2
identifying similarities and differences as they compare the ratio of the side lengths and trigonometric
values.
Create a physical model of a right triangle by arranging the desks to form a right triangle. Label the sides
and angles using poster paper to reinforce the appropriate key vocabulary in this unit. Students develop
an understanding that opposite and adjacent sides change when the reference angle changes.
Indirect measurement is a technique that uses proportions to find a measurement when direct
measurement is not possible.
A critical resource to make Math Models effective is the use of tables, handouts, and assessments
provided by the publisher at http://www.whfreeman.com/Catalog/static/whf/mma/ (or google: “Math
Models: A Bridge to Algebra 2”). Also available on this website are power point presentations, lesson
plans, assessments and activities. For initial use, you will be prompted to set an an instructors account
using an e-mail address as the UserId. You will also be prompted for the following companion website
code: BFW41INST.
To support students who select incorrect ratios, emphasize the importance of selecting a problem-solving
strategy, such as drawing a diagram to help solve the problem. Use colored pencils to color code the sides
of a right triangle. Mnemonic devices such as SOHCAHTOA help students remember the side lengths
involved in each of the three ratios. The four-step problem-solving model and nonlinguistic
representations such as graphic organizers also help struggling students, including English language
learners and students with special needs, organize their knowledge. A SOHCAHTOA graphic organizer
is available on www.sw-georgia.resa.k12.ga.us/math.html and is also provided in the supplementary
materials section of this curriculum frameworks binder.
To help increase students’ computational fluency, provide multiple opportunities for students to practice
using the graphing calculator to compute trigonometric ratios in the context of solving a problem. Using
proportional reasoning to solve equations generated using trigonometric ratios also helps increase the
accuracy of students’ computations, as it connects the idea of solving equations with trigonometric ratios
to the familiar knowledge of solving proportions.
Use real world problem situations to increase the relevance of problems involving indirect measurement.
The building height activity is provided in the supplementary materials section of this curriculum
frameworks binder and is also available on the following NCTM website:
http://illuminations.nctm.org/LessonDetail.aspx?ID=L764. In this activity, students use a clinometer (a
measuring device built from a protractor) and isosceles right triangles to find the height of a building. The
class will compare measurements, talk about the variation results. Detailed instructions for building
clinometers are also available on the Water Rockets and Clinometer Construction worksheet included in
the supplementary materials section of the binder. It will be important to organize groups and have all the
materials prepared before the activity begins in order to maintain a brisk pace for instruction.
As you formatively and summatively assess students, a cues, questions, and advance organizers
strategy can be used, since students are answering questions about content that is important.
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Notes
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Providence Public Schools