Southern  Rhode  Island  Regional  Collaborative  with  process  support  from  The  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin  

41  

Algebra 2, Quarter 4, Unit 4.1

Trigonometry: The Unit Circle, Radian Measure, Angles of Rotation, and Basic and Pythagorean

Trigonometric Identities

Overview Number of instructional days: 19 (1 day = 45–60 minutes)

Content to be learned Mathematical practices to be integrated • Understand radian measure of an angle as the

length of the arc on the unit circle.

• Convert an angle measure from degrees to radians.

• Find the ratio of the degree measure to the radian measure.

• Graph angles on the unit circle and distinguish positive and negative rotations.

• Draw and identify an angle of rotation given a measure or representation.

• Understand the relationship between an angle and its coterminal angles.

• Prove the Pythogorean identity sin2(θ) + cos2(θ) = 1, rewriting it to find sin(θ) and cos(θ) and to express one trigonometric function in terms of the other.

• Find tan(θ) using the relationship

tan(θ ) = sin(θ )

cos(θ ).

Reason abstractly and quantitatively.

• Explore the relationship between arc length and radius for various intercepted arcs.

• Derive the conversion factor between radians and degrees and understand that radians are dimensionless quantities.

• Practice converting between degrees and radians.

Attend to precision.

• Accurately identify a given angle of rotation on the unit circle.

• Attend to all information in the problem to find the correct sign for the quantities of sine, cosine, and tangent.

• Use the angle in question, not just the reference angle.

Look for and make use of structure.

• Apply the patterns regarding signs of trigonometric functions and reference angles to reason out the values of the trigonometric functions for special angles

π3

,

2π3

,

4π3

, and

5π3

.

Algebra 2, Quarter 4, Unit 4.1 Trigonometry: The Unit Circle, Radian Measure, Angles of Rotation, and Basic and Pythagorean Trigonometric Identities (19 days)

 Southern  Rhode  Island  Regional  Collaborative  with  process  support  from  The  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin  

42  

Look for and express regularity in repeated reasoning.

• Visualize the fact that the measures of coterminal angles differ by multiples of ± 360 degrees.

• Apply knowledge to calculate measures of angles coterminal with a given angle.

Essential questions • What is radian measure?

• What is an angle of rotation? How is it measured?

• How can you check that you are using your conversion factor between degrees and radians correctly?

• How would you find the measure of any angle that is coterminal with a given angle?

• How can you use a given value of one of the trigonometric ratios to calculate the value of another?

Written Curriculum

Common Core State Standards for Mathematical Content

Trigonometric Functions F-TF

Extend the domain of trigonometric functions using the unit circle

F-TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

F-TF.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

Prove and apply trigonometric identities

F-TF.8 Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

Algebra 2, Quarter 4, Unit 4.1 Trigonometry: The Unit Circle, Radian Measure, Angles of Rotation, and Basic and Pythagorean Trigonometric Identities (19 days)

 Southern  Rhode  Island  Regional  Collaborative  with  process  support  from  The  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin  

43  

Common Core Standards for Mathematical Practice

2 Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

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.

7 Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

8 Look for and express regularity in repeated reasoning.

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Algebra 2, Quarter 4, Unit 4.1 Trigonometry: The Unit Circle, Radian Measure, Angles of Rotation, and Basic and Pythagorean Trigonometric Identities (19 days)

 Southern  Rhode  Island  Regional  Collaborative  with  process  support  from  The  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin  

44  

Clarifying the Standards

Prior Learning

In grades 7 and 8, students solved real-world and mathematical problems involving angle measure. In Geometry, they defined trigonometric ratios and solved problems involving right triangles. Students applied trigonometry to right triangles and were introduced to radians as a unit measure. They also found arc measure given a central angle measured in degrees.

Current Learning

In this unit, students explore and use radian measure. They understand radian measure of an angle as the length of the arc on the unit circle. Students convert an angle measure from degrees to radians and find the ratio of the degree measure to the radian measure. They graph angles on the unit circle and distinguish positive and negative rotations as well as draw and identify an angle of rotation given a measure or representation. Finally, students understand the relationship between an angle and its coterminal angles. They prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

Future Learning

In the next unit of Algebra 2 and future courses, students will extend the domain of the trigonometric functions using the unit circle and model periodic phenomena with trigonometric functions.

Additional Findings

As noted on the website The Math Page, “Analytic trigonometry is an extension of right triangle trigonometry. It takes place on the x-y plane. For, trigonometry as it is actually used in calculus and physics, is not about solving triangles. It becomes the mathematical description of things that rotate or vibrate, such as light, sound, the paths of planets about the sun, or satellites about the earth. It is necessary therefore to have angles of any size and to extend to them the meanings of the trigonometric functions.” (www.themathpage.com/atrig/unit-circle.htm)

Beyond Numeracy states, “A discipline’s focus shifts and definitions are generalized. In the case of trigonometry it was necessary to expand the definitions of the trigonometric functions to deal with triangles which were not right triangles and to cover the case of obtuse angles larger than 90 degrees. These more encompassing definitions are stated in terms of circles and rotation and link elementary trigonometry with its more modern incarnations. (p. 253)

 Southern  Rhode  Island  Regional  Collaborative  with  process  support  from  The  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin  

45  

Algebra 2, Quarter 4, Unit 4.2

Trigonometric Functions—Sine and Cosine

Overview Number of instructional days: 10 (1 day = 45–60 minutes)

Content to be learned Mathematical practices to be integrated • Graph the functions in the form of

g(θ ) = asinb(θ − h)+ k and

g(θ ) = acosb(θ − h)+ k and describe their transformations.

• Determine domain, range, intervals, maximum, minimum, zeros, and periodicity given a sine or cosine graph.

• Identify amplitude, frequency, and midline of trigonometric functions to model real-world applications.

• Use function values for special angles to graph the sine and cosine functions over one revolution around the unit circle and within the domain [0, 2π].

• Model and analyze motion of periodic phenomena using a sine and/or cosine function.

Use appropriate tools strategically.

• Sketch a sine and cosine graph of a parent function, and then transform that function using a pencil-and-paper method.

• Analyze graphs of functions and solutions generated using a graphing calculator.

• Use technological tools to explore and deepen understanding of the sine and cosine graph.

Look for and make use of structure.

• Compare points on the graph to points on the unit circle and understand how the periodicity is related to revolutions about the unit circle.

• Identify repeated patterns of the sine and cosine functions to find general methods of graphing.

Essential questions • How can sine and cosine functions be defined

using the unit circle?

• What are the key features of the graphs of the sine and cosine functions?

• How do the constants a and b in the functions

g(θ ) = asinbθ , g(θ ) = acosbθ , affect their graphs?

• How do the constants h and k in the functions

g(θ ) = asinb(θ − h)+ k and

g(θ ) = acosb(θ − h)+ k affect their graphs?

• How can the graph of a cosine function be described as a type of transformation as compared to the sine function?

• How can a sine function be used to model and analyze motion of periodic phenomena?

Algebra 2, Quarter 4, Unit 4.2 Trigonometric Functions—Sine and Cosine (10 days)

 Southern  Rhode  Island  Regional  Collaborative  with  process  support  from  The  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin  

46  

Written Curriculum

Common Core State Standards for Mathematical Content

Building Functions F-BF

Build new functions from existing functions [Include simple radical, rational, and exponential functions; emphasize common effect of each transformation across function types]

F-BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Trigonometric Functions F-TF

Model periodic phenomena with trigonometric functions

F-TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.★

Common Core Standards for Mathematical Practice

5 Use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

7 Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several

Algebra 2, Quarter 4, Unit 4.2 Trigonometric Functions—Sine and Cosine (10 days)

 Southern  Rhode  Island  Regional  Collaborative  with  process  support  from  The  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin  

47  

objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

Clarifying the Standards

Prior Learning

In Geometry, students used right angle trigonometry. Previously in Algebra 2, students graphed angles on the unit circle and identified rotations.

Current Learning

Students build new sine and cosine graphs from existing functions and model periodic phenomena with those trigonometric functions.

Future Learning

Students will apply their knowledge of graphing sine and cosine functions to future learning of tangent functions and all inverse trigonometric functions.

Additional Findings

PARCC Model Content Frameworks states, “The habit of seeing subexpressions as single entities will serve students well in areas such as trigonometry, where, for example, the factorization of x4 – y4 can be used to show that the functions cos4x – sin4x and cos2x – sin2x are, in fact, equal.” (p. 57)

Beyond Numeracy states that modern trigonometry is more concerned with periodicity and other properties of the trigonometric functions. (p. 254)

Teacher Notes

Using the Modeling with Trig function activity (section 8-9) in Algebra 2 On Core Mathematics, teachers can focus on drawing students’ attention to how they used the information given in the problem to “customize” the sine function. The constants in the model can be related to some physical characteristic in the problem situation.

Be sure to include phase shifts even though it is not addressed in the standards (e.g., y = asinb(x − h)+ k as the general form of the function).

Algebra 2, Quarter 4, Unit 4.2 Trigonometric Functions—Sine and Cosine (10 days)

 Southern  Rhode  Island  Regional  Collaborative  with  process  support  from  The  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin  

48  

 Southern  Rhode  Island  Regional  Collaborative  with  process  support  from  The  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin  

49  

Algebra 2, Quarter 4, Unit 4.3

Statistics—Experiments, Surveys, and Studies Including the Normal

Distribution and Standard Deviation

Overview Number of instructional days: 11 (1 day = 45–60 minutes)

Content to be learned Mathematical practices to be integrated • Use statistics as a process to make inferences

about a population based on a random sample.

• Decide if a specified model is consistent with results from a given data-generating process.

• Determine and explain how randomization relates to sample surveys, experiments, and observations.

• Use data to estimate a population mean and develop a margin of error.

• Compare data from experiments to decide if differences are significant.

• Evaluate reports based on data.

Make sense of problems and persevere in solving them.

• Determine how to approach a problem, confirm that a distribution is normal, and plan a solution pathway.

• Use knowledge of normal distribution to gain insight into a solution and approach problems differently as needed.

Reason abstractly and quantitatively.

• Review various ways to display data and learn to recognize various data distribution shapes.

• Calculate measures of center and spread and relate them to the shapes of distributions.

Attend to precision.

• Use clear and precise vocabulary when discussing data gathering and samples.

• Use precision in communication and interpret statistical symbols correctly:

o population mean and sampling mean

o standard deviation and standard error

o sample distribution of sample means and sampling distributions of sample proportions.

Algebra 2, Quarter 4, Unit 4.3 Statistics—Experiments, Surveys, and Studies Including the Normal Distribution and Standard Deviation (11 days)

 Southern  Rhode  Island  Regional  Collaborative  with  process  support  from  The  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin  

50  

Look and make use of structure.

• Recognize the difference between a theoretically generated probability distribution and an experimentally generated probability distribution.

• Explain why theoretical and experimental probability distributions are not necessarily the same and when they may be similar.

Essential questions • What are the different methods for gathering

data about a population?

• How can you use shape, center, and spread to characterize a data distribution?

• What is a probability distribution? How is it displayed?

• How do you find percents of data and probabilities of events associated with normal distribution?

• How is the mean of a sampling distribution related to the population mean or proportion?

• How do you calculate a confidence interval and a margin of error for a population proportion or mean?

• What kinds of statistical research are there? Which ones can establish cause-and-effect relationships between variables?

Algebra 2, Quarter 4, Unit 4.3 Statistics—Experiments, Surveys, and Studies Including the Normal Distribution and Standard Deviation (11 days)

 Southern  Rhode  Island  Regional  Collaborative  with  process  support  from  The  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin  

51  

Written Curriculum

Common Core State Standards for Mathematical Content

Making Inferences and Justifying Conclusions★ S-IC

Understand and evaluate random processes underlying statistical experiments

S-IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.★

S-IC.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?★

Make inferences and justify conclusions from sample surveys, experiments, and observational studies

S-IC.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.★

S-IC.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.★

S-IC.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.★

S-IC.6 Evaluate reports based on data.★

Interpreting Categorical and Quantitative Data★ S-ID

Summarize, represent, and interpret data on a single count or measurement variable

S-ID.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.★

Common Core 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

Algebra 2, Quarter 4, Unit 4.3 Statistics—Experiments, Surveys, and Studies Including the Normal Distribution and Standard Deviation (11 days)

 Southern  Rhode  Island  Regional  Collaborative  with  process  support  from  The  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin  

52  

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.

2 Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

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.

7 Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

Clarifying the Standards

Prior Learning

In grade 6, students developed an understanding of statistical variability. They summarized and described distributions. In grade 7, students continued to investigate data in order to draw inferences about a population. They compared two data distributions and addressed questions about differences between populations. In Algebra 1, students summarized, represented, and interpreted data on a single count or measurement variable.

Algebra 2, Quarter 4, Unit 4.3 Statistics—Experiments, Surveys, and Studies Including the Normal Distribution and Standard Deviation (11 days)

 Southern  Rhode  Island  Regional  Collaborative  with  process  support  from  The  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin  

53  

Current Learning

Students calculate mean and standard deviation within a normal distribution. They use data collections to estimate a population mean to develop a margin of error through the use of simulations and experiments. Students use calculators, spreadsheets, and tables to estimate areas under the normal curve. They understand and evaluate random processes underlying statistical experiments. Students make inferences and justify conclusions from sample surveys, experiments, and observational studies.

Future Learning

Students taking a precalculus, probability and statistics, or AP statistics course will use these foundations to determine the probability distribution using the normal model. Most college majors now require a statistics course as well.

Additional Findings

PARCC Model Content Frameworks for Mathematics states that students explore and investigate the randomness underlying statistical experiments and make inferences and justify conclusions from sample surveys, experiments, and observational studies. (p. 56)

Beyond Numeracy states that statistics, more than most other areas of mathematics, is just formalized common sense, quantified straight thinking. (p. 58)

Science for All Americans states that statistics is a form of mathematics that develops useful ways for organizing and analyzing large amounts of data. Alternatively, the data set can be characterized in a summary fashion by describing where its middle is and how much variation there is around that middle.

In Principles and Standards for School Mathematics, it is mentioned that understanding how to draw inferences about a population from random samples requires understanding how those samples might be distributed. Such an understanding can be developed with the aid of simulations. (p. 329)

Teacher Notes

Center and spread measures are dependent on shape of the distribution.

Algebra 2, Quarter 4, Unit 4.3 Statistics—Experiments, Surveys, and Studies Including the Normal Distribution and Standard Deviation (11 days)

 Southern  Rhode  Island  Regional  Collaborative  with  process  support  from  The  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin  

54