# Secondary Mathematics

Welcome to the Secondary Mathematics Department

We strive to teach 7th - 12th grade Mathematics in context of Davis School District's overall mission. Below, you will find secondary Mathematics resources based upon your relationship to the school district. *The Secondary Mathematics Department feels strongly about finding strengths and best practices through collaboration, please don't hesitate to reach out using the contact information below if you have any questions, if you would like to suggest additions or changes, or if you would simply like to connect to the Secondary Mathematics Specialist!*

## Parents and Students

"at the end of the day, the most overwhelming key to a child's success is the positive involvement of parents."

Jane D. Hull

- Parent - Student Resources
- Student Opportunities
- Mathematical Practice Standards
- Secondary Mathematics Standards

## Parent - Student Resources

Carnegie Learning - My CL

*Your one-stop destination to access all Carnegie Learning Products and resources (MATHia, Resource Center, Teacher's Toolkit).*

Lance Powell's Math 7 and 8 Video Tutorials

*Short video clips that teach Middle School common core mathematics concepts.*

HippoCampus

*Get access to mathematics tutorials by concept.*

LearnZillion

*The idea for LearnZillion began at E.L. Haynes Public Charter School in Washington, D.C. where co-founder Eric Westendorf, was principal. After watching 6th grade teacher Andrea Smith teach her students what it meant to divide by fractions, Eric wondered, “Could powerful learning experiences be captured so that teachers didn’t have to re-invent the wheel every time they taught a standard?” He decided to find out. Working with Andrea and a few other E.L. Haynes teachers, he created a homemade website that featured screencasts of high quality, Common Core lessons. The website worked. Not only could teachers find examples of high quality lessons, but parents and students also benefited from the explanations.*

OnlineMath Learning

*Are you looking for free online math help, math fun and other useful resources? In this site, you will find interesting quizzes, practice, homework help and other materials to keep you occupied; or fun facts, games, puzzles and other cool stuff to make this subject something to be enjoyed rather than dreaded. Have some fun while learning some key skills to improve your grades. *

McDougal Littell - Classzone

*Access former textbooks and resources. Great resource for providing small, bite-sized tutorials.*

MobyMax

*Identify and fix your student's math gaps.*

Ask Dr. Math

*An archive of previously asked math-related questions and answers.*

If you have a resource you'd like to see represented on this page, please let us know using the contact information below.

## Student Opportunities

Salt Lake City

*A STEM Event for 7th - 10th Grade Students.*

Salt Lake City and Thanksgiving Point

*A friendly festival for all ages with maker exhibits, demos, installations, workshops, DIY, and hands-on activities.*

Parent & Daughter Engineering & Technology Night Out

Weber State University

*Parent-Daughter Engineering & Technology Night Out is designed for 7th-12th grade girls and their parents to engage in fun problem solving activities. This engaging family opportunity inspires confidence and nurtures girl's innate STEM abilities and interest. Girls are encouraged and supported by successful WSU faculty, staff, and students.*

**November 16th and 17th

Engineering Day

University of Utah

*Engineering Day is a half-day event open to high school students and parents. The event is free and designed to introduce participants to a variety of engineering disciplines.*

**October 27th

Make Salt Lake

Salt Lake City

A Makerspace where members can gather to create, invent and learn. Make Salt Lake offers access to education, tools and other makers. Interests vary from electronics to plastics and woodworking and beyond.

**Ongoing

Rube Goldberg Competition

Weber State University

*The Rube Goldberg Contest helps bring home STEM to kids, they apply what they learn in school in a fun hands-on project, using problem solving, teamwork, engineering and loads of creativity. The event is free.*

**Check link for updated dates

Online

*The Hour of Code is a global movement by Computer Science Education Week and code.org reaching tens of millions of students in 180+ countries through a one-hour introduction to computer science and computer programming.*

**December 3rd - 9th 2018

Online

*Inspired by the Educate to Innovate Campaign, President Obama's initiative to promote a renewed focus on STEM, the National STEM Video Game Challenge is a multi-year competition whose goal is to motivate interest in STEM learning among America's youth by tapping into student's natural passion for playing and making video games.*

**Check link for updated dates

Salt Lake City

*SheTech is a STEM activation and engagement platform for girls. Girls gain exposure to technology, individually create using technology, meet mentors, learn about careers in STEM fields, receive a certification and gain access to internships and scholarships.*

**Check link for updated dates

## Mathematical Practice Standards

The standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students, helping students to achieve fluency in mathematical concepts while building 21st Century Skills.

**The Eight Mathematical Practice Standards:**

- Make sense problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.

**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.

**2. Reason abstractly and quantitatively:**

Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students 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.

**3. Construct viable arguments and critique the reasoning of others:**

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

**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 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.

**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.

**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 x**2 **+ 9*x *+ 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), 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.

## Secondary Mathematics Standards

- Math 7
- Math 7 Honors
- Math 8
- Math 8 Honors
- Secondary Math 1
- Secondary Math 1 Honors
- Secondary Math 2
- Secondary Math 2 Honors
- Secondary Math 3
- Secondary Math 3 Honors
- Mathematical Decision Making for Life
- AP Statistics
- AP Caclulus AB
- AP Calculus BC

## Math 7

**Mathematical Practice Standards**

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

**Ratios and Proportional Relationships**

1. Compute unit rates using the constant of proportionality.

2. Identify and use proportional relationships to solve real world problems including tables, graphs and equations.

3. Represent proportional relationships with equations.

4. Use proportional relationships to solve multi-step ratio and percent problems.

**Number System**

1. Apply and extend previous understandings of properties of operations including negatives, fractions, and mixed numbers.

2. Apply properties of operations as strategies to solve real world problems.

3. Convert a rational number to a decimal using long division.

**Linear Expressions and Equations**

1. Use properties of operations including factoring, associative, commutative, and identity to generate equivalent expressions.

2. Solve multi-step problems and assess reasonableness of answers.

3. Use variables and symbols to represent unknown quantities and construct simple equations and inequalities to solve real-life problems. (≤ ≥ < >)

4. Solve word problems leading to equations and inequalities with rational numbers.

**Geometric Reasoning**

1. Solve problems involving scale drawings.

2. Construct geometric shapes with given conditions.

3. Describe the two-dimensional shape~~s~~ resulting from slicing a three -dimensional figure.

4. Know and use formulas to find circumference and area of a circle.

5. Recognize and use angle relationships to determine the unknown measure of an angle.

6. Solve problems involving area, surface area, and volume of triangles, quadrilaterals, polygons, cubes, right prisms, and pyramids.

**Statistical Inference and Probability**

1. Use random sampling and informal comparisons to general characteristics of a targeted group.

2. Use measures of central tendency (mean, median and mode).

3. Make informal visual and data based comparisons between two groups of data such as two: stem-and-leaf plots, line plots and box plots).

4. Understand the probability of a chance event can be expressed by a value between 0 and 1.

5. Understand experimental probability of an event will approach the expected probability as the number of trials increase.

6. Develop a uniform/non-uniform probability model using data from repeated trials of a simple event.

7. Find probabilities of compound events.

**Literacy Standards**

1. Acquire, accurately use and interpret grade-appropriate mathematical words and terms.

2. Engage in collaborative discussions with diverse partners on grade level concepts.

## Math 7 Honors

**Mathematical Practice Standards**

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

**Ratios and Proportional Relationships**

1. Compute unit rates using the constant of proportionality.

2. Identify and use proportional relationships to solve real world problems including tables, graphs and equations.

3. Represent proportional relationships with equations.

4. Use proportional relationships to solve multi-step ratio and percent problems.

**Number System**

1. Apply and extend previous understandings of properties of operations including negatives, fractions, and mixed numbers.

2. Apply properties of operations as strategies to solve real world problems.

3. Convert a rational number to a decimal using long division.

**Linear Expressions and Equations**

1. Use properties of operations including factoring, associative, commutative, and identity to generate equivalent expressions.

2. Solve multi-step problems and assess reasonableness of answers.

3. Use variables and symbols to represent unknown quantities and construct simple equations and inequalities to solve real-life problems. (≤ ≥ < >)

4. Solve word problems leading to equations and inequalities with rational numbers.

**Geometric Reasoning**

1. Solve problems involving scale drawings.

2. Construct geometric shapes with given conditions.

3. Describe the two-dimensional shape~~s~~ resulting from slicing a three -dimensional figure.

4. Know and use formulas to find circumference and area of a circle.

5. Recognize and use angle relationships to determine the unknown measure of an angle.

6. Solve problems involving area, surface area, and volume of triangles, quadrilaterals, polygons, cubes, right prisms, and pyramids.

**Statistical Inference and Probability**

1. Use random sampling and informal comparisons to general characteristics of a targeted group.

2. Use measures of central tendency (mean, median and mode).

3. Make informal visual and data based comparisons between two groups of data such as two: stem-and-leaf plots, line plots and box plots).

4. Understand the probability of a chance event can be expressed by a value between 0 and 1.

5. Understand experimental probability of an event will approach the expected probability as the number of trials increase.

6. Develop a uniform/non-uniform probability model using data from repeated trials of a simple event.

7. Find probabilities of compound events.

** ** **Honors**

1. Use math in creating codes

2. Recognize and appreciate patterns in nature, art, and mathematics

3. Understand and use number systems in different bases

4. Research and analyze ancient number systems

**Literacy Standards**

1. Acquire, accurately use and interpret grade-appropriate mathematical words and terms.

2. Engage in collaborative discussions with diverse partners on grade level concepts.

## Math 8

**Mathematical Practice Standards**

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

** **1. Demonstrate every rational number can be written as a decimal.

2. Use, compare, estimate and locate irrational numbers on a number line.

3. Simplify, add, subtract, multiply and divide radicals, including square roots.

**Expressions and Equations**

** **1. Understand and apply integer exponent rules.

2. Use square root and cube root operations and symbols.

3. Compare and perform operations with numbers written in scientific notation.

4. Graph proportional relationships, interpreting the unit rate as the slope.

5. Use similar triangles to understand slope in order to write an equation.

6. Solve linear equations in one variable.

7. Analyze and solve systems (pairs) of linear equations using graphs.

**Functions**

** **1. Understand that a function assigns each input with exactly one output.

2. Compare functions represented in different ways. (ie: graphs, equations, tables, etc..)

3. Understand that not all functions are linear.

4. Determine the rate of change and initial value of a function.

5. Be able to understand and graph functions.

**Geometry **

** **1. Verify the properties of rotations, reflections and translations.

2. Show that transformations can create congruent figures.

3. Describe the effect of transformations on two-dimensional figures using coordinates.

4. Identify the similarity between two-dimensional figures created by the dilation transformation.

5. Explore and explain the relationship between angles created by parallel lines and transversals.

6. Explain and apply the Pythagorean Theorem.

7. Know and use the formulas for the volumes of cones, cylinders, and spheres.

** **1. Construct and interpret scatter plots.

2. Estimate and write an equation for a line of best fit in a scatter plot.

3. From two sets of data interpret the slope and y-intercept of a line.

4. Make comparisons using scatter plots, lines and two-way frequency tables.

**Literacy Standards**

** **1. Acquire, accurately use and interpret grade-appropriate mathematical words and terms

2. Engage in collaborative discussions with diverse partners on grade-level concepts.

## Math 8 Honors

**Mathematical Practice Standards**

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

** **1. Demonstrate every rational number can be written as a decimal.

2. Use, compare, estimate and locate irrational numbers on a number line.

3. Simplify, add, subtract, multiply and divide radicals, including square roots.

**Expressions and Equations**

1. Understand and apply integer exponent rules.

2. Use square root and cube root operations and symbols.

3. Compare and perform operations with numbers written in scientific notation.

4. Graph proportional relationships, interpreting the unit rate as the slope.

5. Use similar triangles to understand slope in order to write an equation.

6. Solve linear equations in one variable.

7. Analyze and solve systems (pairs) of linear equations using graphs.

**Functions**

** **1. Understand that a function assigns each input with exactly one output.

2. Compare functions represented in different ways. (ie: graphs, equations, tables, etc..)

3. Understand that not all functions are linear.

4. Determine the rate of change and initial value of a function.

5. Be able to understand and graph functions.

**Geometry **

** **1. Verify the properties of rotations, reflections and translations.

2. Show that transformations can create congruent figures.

3. Describe the effect of transformations on two-dimensional figures using coordinates.

4. Identify the similarity between two-dimensional figures created by the dilation transformation.

5. Explore and explain the relationship between angles created by parallel lines and transversals.

6. Explain and apply the Pythagorean Theorem.

7. Know and use the formulas for the volumes of cones, cylinders, and spheres.

**Statistics and Probability**

** **1. Construct and interpret scatter plots.

2. Estimate and write an equation for a line of best fit in a scatter plot.

3. From two sets of data interpret the slope and y-intercept of a line.

4. Make comparisons using scatter plots, lines and two-way frequency tables.

**Honors**

1. Explore 3-D graphing and graph theory.

2. Understand the concepts and applications of fair division and apportionment.

3. Use sets and set notation to communicate mathematics.

4. Examine different methods of voting.

**Literacy Standards**

** **1. Acquire, accurately use and interpret grade-appropriate mathematical words and terms

2. Engage in collaborative discussions with diverse partners on grade-level concepts.

## Secondary Math 1

**Mathematical Practices**

The Standards for Mathematical Practice in Secondary Mathematics I describe mathematical habits of mind that teachers should seek to develop in their students. Students become mathematically proficient in engaging with mathematical content and concepts as they learn, experience, and apply these skills and attitudes

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others

4. Model with mathematics

5. Use appropriate tools strategically

6. Attend to precision

7. Look for and make use of structure

8. Look for and express regularity in repeated reasoning

**Number and Quantity**

Reason quantitatively and use units to solve problems. Working with quantities and the relationships between them provides grounding for work with expressions, equations, and functions.

1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

2. Define appropriate quantities for the purpose of descriptive modeling.

3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

**Algebra-Seeing Structure in Expressions**

Interpret the structure of expressions.

1. Interpret linear expressions and exponential expressions with integer exponents that represent a quantity in terms of its context

a. Interpret parts of an expression, such as terms, factors, and coefficients.

b. Interpret complicated expressions by viewing one or more of their parts as a single entity.

**Algebra-Creating Equations**

Create equations that describe numbers or relationships. Limit these to linear equations and inequalities, and exponential equations. In the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs.

1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and simple exponential functions.

2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

3. Represent constraints by equations or inequalities and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.

4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

**Algebra-Reasoning with Equations and Inequalities**

Understand solving equations as a process of reasoning and explain the reasoning. Solve equations and inequalities in one variable (Standard A.REI.3). Solve systems of equations. Build on student experiences graphing and solving systems of linear equations from middle school. Include cases where the two equations describe the same line—yielding infinitely many solutions—and cases where two equations describe parallel lines—yielding no solution; connect to GPE.5, which requires students to prove the slope criteria for parallel lines. Represent and solve equations and inequalities graphically.

1. Explain each step in solving a linear equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Students will solve exponential equations with logarithms in Secondary Mathematics III.

- Solve equations and inequalities in one variable.

a. Solve one-variable equations and literal equations to highlight a variable of interest.

b. Solve compound inequalities in one variable, including absolute value inequalities.

c. Solve simple exponential equations that rely only on application of the laws of exponents (limit solving exponential equations to those that can be solved without logarithms). *For example, 5*^{x} *= 125 or 2*^{x} *= 1/16.*

- Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
- Solve systems of linear equations exactly and approximately (numerically, algebraically, graphically), focusing on pairs of linear equations in two variables.
- Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
- Explain why the
*x*-coordinates of the points where the graphs of the equations*y*=*f(x)*and*y*=*g(x)*intersect are the solutions of the equation*f(x)*=*g(x)*; find the solutions approximately; e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where*f(x)*and/or*g(x)*are linear and exponential functions. - Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

**Functions-Interpreting Linear and Exponential Functions**

Understand the concept of a linear or exponential function and use function notation. Recognize arithmetic and geometric sequences as examples of linear and exponential functions. Interpret linear or exponential functions that arise in applications in terms of a context. Analyze linear or exponential functions using different representations.

1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and *x* is an element of its domain, then *f(x)* denotes the output of *f* corresponding to the input *x*. The graph of *f* is the graph of the equation *y = f(x).*

2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Emphasize arithmetic and geometric sequences as examples of linear and exponential functions. *For example, the Fibonacci sequence is defined recursively by *f*(0) = *f*(1) = 1, *f*(*n*+1) = *f(n) *+ *f*(*n*-1) for *n *≥** 1*.

4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. *Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; and end behavior.*

5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. *For example, if the function *h(n) *gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*

6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

a. Graph linear functions and show intercepts.

b. Graph exponential functions, showing intercepts and end behavior.

8. Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). *For example, compare the growth of two linear functions, or two exponential functions such as *y*=3 ^{n} and *y

*=100•2*.

^{n}**Functions-Building Linear or Exponential Functions**

Build a linear or exponential function that models a relationship between two quantities. Build new functions from existing functions.

1. Write a function that describes a relationship between two quantities.

a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

b. Combine standard function types using arithmetic operations. *For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.*

2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Limit to linear and exponential functions. Connect arithmetic sequences to linear functions and geometric sequences to exponential functions.

3. Identify the effect on the graph of replacing *f(x) by f(x) + k, *for specific values of *k *(both positive and negative); find the value of *k *given the graphs. Relate the vertical translation of a linear function to its *y*-intercept. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Create equations that describe numbers or relationships. Limit these to linear equations and inequalities, and exponential equations. In the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs.

**Functions-Linear and Exponential**

Construct and compare linear and exponential models and solve problems. Interpret expressions for functions in terms of the situation they model.

1. Distinguish between situations that can be modeled with linear functions

2. and with exponential functions.

a. Prove that linear functions grow by equal differences over equal intervals; exponential functions grow by equal factors over equal intervals.

b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

3. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

4. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly.

5. Interpret the parameters in a linear or exponential function in terms of a context. Limit exponential functions to those of the form *f(x)* = *b ^{x}* +

*k*.

**Geometry-Congruence**

Experiment with transformations in the plane. Build on student experience with rigid motions from earlier grades Understand congruence in terms of rigid motions. Rigid motions are at the foundation of the definition of congruence. Reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Make geometric constructions.

1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

2. Represent transformations in the plane using, for example, transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, for example, graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Point out the basis of rigid motions in geometric concepts, for example, translations move points a specified distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle.

6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide whether they are congruent.

7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

9. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Emphasize the ability to formalize and defend how these constructions result in the desired objects. *For example, copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.*

10. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Emphasize the ability to formalize and defend how these constructions result in the desired objects.

**Geometry-Expressing Geometric Properties With Equations**

Use coordinates to prove simple geometric theorems algebraically.

1. Use coordinates to prove simple geometric theorems algebraically. *For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2)*.

2. Prove the slope criteria for parallel and perpendicular lines; use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

3. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles; connect with The Pythagorean Theorem and the distance formula.

**Statistics and Probability-Interpreting Categorical and Quantitative Data**

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

(Standards S.ID.1–3). Summarize, represent, and interpret data on two categorical and quantitative

variables (Standard S.ID.6). Interpret linear models building on students’ work with

linear relationships, and introduce the correlation coefficient (Standards S.ID.7–9).

1. Represent data with plots on the real number line (dot plots, histograms, and box plots).

2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). Calculate the weighted average of a distribution and interpret it as a measure of center.

4. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

a. Fit a linear function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions, or choose a function suggested by the context. Emphasize linear and exponential models.

b. Informally assess the fit of a function by plotting and analyzing residuals. Focus on situations for which linear models are appropriate.

c. Fit a linear function for scatter plots that suggest a linear association.

5. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

6. Compute (using technology) and interpret the correlation coefficient of a linear fit.

7. Distinguish between correlation and causation.

**Literacy Standards**

1. Acquire, accurately use and interpret grade-appropriate mathematical words and terms.

2. Engage in collaborative discussions with diverse partners on grade level concepts.

## Secondary Math 1 Honors

**Mathematical Practices**

The Standards for Mathematical Practice in Secondary Mathematics I describe mathematical habits of mind that teachers should seek to develop in their students. Students become mathematically proficient in engaging with mathematical content and concepts as they learn, experience, and apply these skills and attitudes

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others

4. Model with mathematics

5. Use appropriate tools strategically

6. Attend to precision

7. Look for and make use of structure

8. Look for and express regularity in repeated reasoning

**Number and Quantity**

Reason quantitatively and use units to solve problems. Working with quantities and the relationships between them provides grounding for work with expressions, equations, and functions.

1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

2. Define appropriate quantities for the purpose of descriptive modeling.

3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

**Algebra-Seeing Structure in Expressions**

Interpret the structure of expressions.

1. Interpret linear expressions and exponential expressions with integer exponents that represent a quantity in terms of its context

a. Interpret parts of an expression, such as terms, factors, and coefficients.

b. Interpret complicated expressions by viewing one or more of their parts as a single entity.

**Algebra-Creating Equations**

Create equations that describe numbers or relationships. Limit these to linear equations and inequalities, and exponential equations. In the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs.

1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and simple exponential functions.

2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

3. Represent constraints by equations or inequalities and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.

4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

**Algebra-Reasoning with Equations and Inequalities**

Understand solving equations as a process of reasoning and explain the reasoning. Solve equations and inequalities in one variable (Standard A.REI.3). Solve systems of equations. Build on student experiences graphing and solving systems of linear equations from middle school. Include cases where the two equations describe the same line—yielding infinitely many solutions—and cases where two equations describe parallel lines—yielding no solution; connect to GPE.5, which requires students to prove the slope criteria for parallel lines. Represent and solve equations and inequalities graphically.

1. Explain each step in solving a linear equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Students will solve exponential equations with logarithms in Secondary Mathematics III.

- Solve equations and inequalities in one variable.

a. Solve one-variable equations and literal equations to highlight a variable of interest.

b. Solve compound inequalities in one variable, including absolute value inequalities.

c. Solve simple exponential equations that rely only on application of the laws of exponents (limit solving exponential equations to those that can be solved without logarithms). *For example, 5*^{x} *= 125 or 2*^{x} *= 1/16.*

- Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
- Solve systems of linear equations exactly and approximately (numerically, algebraically, graphically), focusing on pairs of linear equations in two variables.
- Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
- Explain why the
*x*-coordinates of the points where the graphs of the equations*y*=*f(x)*and*y*=*g(x)*intersect are the solutions of the equation*f(x)*=*g(x)*; find the solutions approximately; e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where*f(x)*and/or*g(x)*are linear and exponential functions. - Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

**Functions-Interpreting Linear and Exponential Functions**

Understand the concept of a linear or exponential function and use function notation. Recognize arithmetic and geometric sequences as examples of linear and exponential functions. Interpret linear or exponential functions that arise in applications in terms of a context. Analyze linear or exponential functions using different representations.

1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and *x* is an element of its domain, then *f(x)* denotes the output of *f* corresponding to the input *x*. The graph of *f* is the graph of the equation *y = f(x).*

2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Emphasize arithmetic and geometric sequences as examples of linear and exponential functions. *For example, the Fibonacci sequence is defined recursively by *f*(0) = *f*(1) = 1, *f*(*n*+1) = *f(n) *+ *f*(*n*-1) for *n *≥** 1*.

4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. *Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; and end behavior.*

5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. *For example, if the function *h(n) *gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*

6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

a. Graph linear functions and show intercepts.

b. Graph exponential functions, showing intercepts and end behavior.

8. Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). *For example, compare the growth of two linear functions, or two exponential functions such as *y*=3 ^{n} and *y

*=100•2*.

^{n}**Functions-Building Linear or Exponential Functions**

Build a linear or exponential function that models a relationship between two quantities. Build new functions from existing functions.

1. Write a function that describes a relationship between two quantities.

a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

b. Combine standard function types using arithmetic operations. *For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.*

2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Limit to linear and exponential functions. Connect arithmetic sequences to linear functions and geometric sequences to exponential functions.

3. Identify the effect on the graph of replacing *f(x) by f(x) + k, *for specific values of *k *(both positive and negative); find the value of *k *given the graphs. Relate the vertical translation of a linear function to its *y*-intercept. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Create equations that describe numbers or relationships. Limit these to linear equations and inequalities, and exponential equations. In the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs.

**Functions-Linear and Exponential**

Construct and compare linear and exponential models and solve problems. Interpret expressions for functions in terms of the situation they model.

1. Distinguish between situations that can be modeled with linear functions

2. and with exponential functions.

a. Prove that linear functions grow by equal differences over equal intervals; exponential functions grow by equal factors over equal intervals.

b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

3. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

4. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly.

5. Interpret the parameters in a linear or exponential function in terms of a context. Limit exponential functions to those of the form *f(x)* = *b ^{x}* +

*k*.

**Geometry-Congruence**

Experiment with transformations in the plane. Build on student experience with rigid motions from earlier grades Understand congruence in terms of rigid motions. Rigid motions are at the foundation of the definition of congruence. Reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Make geometric constructions.

1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

2. Represent transformations in the plane using, for example, transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, for example, graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Point out the basis of rigid motions in geometric concepts, for example, translations move points a specified distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle.

6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide whether they are congruent.

7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

9. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Emphasize the ability to formalize and defend how these constructions result in the desired objects. *For example, copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.*

10. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Emphasize the ability to formalize and defend how these constructions result in the desired objects.

**Geometry-Expressing Geometric Properties With Equations**

Use coordinates to prove simple geometric theorems algebraically.

1. Use coordinates to prove simple geometric theorems algebraically. *For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2)*.

2. Prove the slope criteria for parallel and perpendicular lines; use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

3. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles; connect with The Pythagorean Theorem and the distance formula.

**Statistics and Probability-Interpreting Categorical and Quantitative Data**

Summarize, represent, and interpret data on a single count or measurement variable (Standards S.ID.1–3). Summarize, represent, and interpret data on two categorical and quantitative

1. Represent data with plots on the real number line (dot plots, histograms, and box plots).

2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). Calculate the weighted average of a distribution and interpret it as a measure of center.

4. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

a. Fit a linear function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions, or choose a function suggested by the context. Emphasize linear and exponential models.

b. Informally assess the fit of a function by plotting and analyzing residuals. Focus on situations for which linear models are appropriate.

c. Fit a linear function for scatter plots that suggest a linear association.

5. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

6. Compute (using technology) and interpret the correlation coefficient of a linear fit.

7. Distinguish between correlation and causation.

**Number and Quantity: Vector and Matrix Quantities---HONORS TOPICS**

Represent and model with vector quantities. Perform operations on vectors. Perform operations on matrices and use matrices in applications.

1. Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., *v, |v|, ||v||, v*)*.*

2. Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

3. Solve problems involving velocity and other quantities that can be represented by vectors.

4. Add and subtract vectors.

a. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

c. Understand vector subtraction *v – w *as *v *+ *(–w)*, where *–w *is the additive inverse of *w*, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.

5. Multiply a vector by a scalar.

a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as *c(v _{x} , v_{y} ) = (cv_{x} ,cv_{y})*.

b. Compute the magnitude of a scalar multiple *cv *using ||*cv*|| = |*c*|*v*. Compute the direction of *cv *knowing that when |*c*|*v *≠ 0, the direction of *cv *is either along *v (*for *c > 0) *or against *vs *(for *c < 0*).

6. Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

7. Multiply matrices by scalars to produce new matrices, e.g., as when all of the pay-offs in a game are doubled.

8. Add, subtract, and multiply matrices of appropriate dimensions.

9. Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

10. Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

11. Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

12. Work with 2 . 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

13. Solve systems of linear equations up to three variables using matrix row reduction.

**Literacy Standards**

1. Acquire, accurately use and interpret grade-appropriate mathematical words and terms.

2. Engage in collaborative discussions with diverse partners on grade level concepts.

## Secondary Math 2

**Mathematical Practices**

The Standards for Mathematical Practice in Secondary Mathematics I describe mathematical habits of mind that teachers should seek to develop in their students. Students become mathematically proficient in engaging with mathematical content and concepts as they learn, experience, and apply these skills and attitudes

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others

4. Model with mathematics

5. Use appropriate tools strategically

6. Attend to precision

7. Look for and make use of structure

8. Look for and express regularity in repeated reasoning

**Number and Quantity-The Real Number System**

Extend the properties of exponents to rational exponents. Use properties of rational and irrational numbers.

1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. *For example, we define 5*^{1/3} *to be the cube root of 5 because we want (5*^{1/3}*)*^{3} *= 5*^{(1/3)3} *to hold, so (5 *^{1/3}*)*^{3} *must equal 5.*

2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.

3. Explain why sums and products of rational numbers are rational, that the sum of a rational number and an irrational number is irrational, and that the product of a nonzero rational number and an irrational number is irrational. Connect to physical situations (e.g., finding the perimeter of a square of area 2).

**Number and Quantity-The Complex Number System**

Perform arithmetic operations with complex numbers. Use complex numbers in polynomial identities and equations.

1. Know there is a complex number *i *such that *i*** ^{2} **= -1, and every complex number has the form

*a + bi*with

*a*and

*b*real.

2. Use the relation *i*** ^{2} **= -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Limit to multiplications that involve

*i*

**as the highest power of**

^{2}*i*.

3. Solve quadratic equations with real coefficients that have complex solutions.

4. Extend polynomial identities to the complex numbers. Limit to quadratics with real coefficients. *For example, rewrite *x^{2} *+ 4 as (*x *+ 2*i*)(*x *– 2*i*)***.**

5. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

**Algebra-**** Seeing Structure in Expression**

Interpret the structure of expressions. Write expressions in equivalent forms to solve problems, balancing conceptual understanding and procedural fluency in work with equivalent expressions.

1. Interpret quadratic and exponential expressions that represent a quantity in terms of its context.

a. Interpret parts of an expression, such as terms, factors, and coefficients.

b. Interpret increasingly more complex expressions by viewing one or more of their parts as a single entity. Exponents are extended from the integer exponents to rational exponents focusing on those that represent square or cube roots.

2. Use the structure of an expression to identify ways to rewrite it. For example, see *x*** ^{4} **–

*y*

**as**

^{4}*(x*

^{2}*)*

^{2}*–*

*(y*

^{2}*)*

**, thus recognizing it as a difference of squares that can be factored as**

^{2}*(x*

^{2}*–*

*y*

^{2}*)(x*

^{2}

*+ y*

^{2}*)*.

3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. *For example, development of skill in factoring and completing the square goes hand in hand with understanding what different forms of a quadratic expression reveal.*

a. Factor a quadratic expression to reveal the zeros of the function it defines.

b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

*c. *Use the properties of exponents to transform expressions for exponential functions. *For example, the expression 1.15*^{t} *can be rewritten as (1.15 *^{1/12}*)*^{12t} *≈** 1.012*^{12t}** ***to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.*

**Algebra-Arithmetic With Polynomials and Rational Expressions**

Perform arithmetic operations on polynomials. Focus on polynomial expressions that simplify to forms that are linear or quadratic in a positive integer power of *x*.

1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

**Algebra-Creating Equations**

Create equations that describe numbers or relationships. Extend work on linear and exponential equations to quadratic equations.

1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

3. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations; extend to formulas involving squared variables. *For example, rearrange the formula for the volume of a cylinder *V *= **π** *r** ^{2} **h

*.*

**Algebra-Reasoning With Equations and Inequalities**

Solve equations and inequalities in one variable. Solve systems of equations. Extend the work of systems to include solving systems consisting of one linear and one nonlinear equation.

*1. *Solve quadratic equations in one variable.

a. Use the method of completing the square to transform any quadratic equation in *x *into an equation of the form *(x **–** p)*^{2}** ***= q *that has the same solutions. Derive the quadratic formula from this form.

b. Solve quadratic equations by inspection (e.g., for *x*** ^{2} **= 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as

*a ± bi*for real numbers

*a*and

*b*.

2. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. *For example, find the points of intersection between the line *y *= **–**3*x *and the circle *x^{2} *+ *y^{2} *= 3*.

**Functions-Building Functions**

Build a function that models a relationship between two quantities. Build new functions from existing functions.

1. Write a quadratic or exponential function that describes a relationship between two quantities.

*a. *Determine an explicit expression, a recursive process, or steps for calculation from a context.

*b. *Combine standard function types using arithmetic operations. *For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.*

2. 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. Focus on quadratic functions and consider including absolute value functions. 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.

**Functions-Linear, Quadratic and Exponential Models**

Construct and compare linear, quadratic, and exponential models and solve problems.

1. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. Compare linear and exponential growth to quadratic growth.

**Functions-Interpret Functions**

Interpret quadratic functions that arise in applications in terms of a context. Analyze functions using different representations.

1. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. *Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; and end behavior.*

2. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Focus on quadratic functions; compare with linear and exponential functions. *For example, if the function *h(n) *gives the number of person hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*

3. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

4. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

b. Graph piecewise-defined functions and absolute value functions. Compare and contrast absolute value and piecewise-defined functions with linear, quadratic, and exponential functions. Highlight issues of domain, range, and usefulness when examining piecewise-defined functions.

5. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

b. Use the properties of exponents to interpret expressions for exponential functions. *For example, identify percent rate of change in functions such as *y *= (1.02)*^{t}*, *y *= (0.97)*^{t}*, *y *= (1.01)*^{12t}*, *y *= (1.2)*^{t/10}*, and classify them as representing exponential growth or decay.*

*6. *Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Extend work with quadratics to include the relationship between coefficients and roots, and that once roots are known, a quadratic equation can be factored. *For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.*

**Functions-Trigonometric Functions**

Prove and apply trigonometric identities. Limit *θ* to angles between 0 and 90 degrees. Connect with the Pythagorean Theorem and the distance formula.

1. Prove the Pythagorean identity sin** ^{2}**(

*θ*) + cos

**(**

^{2}*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.

**Geometry-Congruence**

Prove geometric theorems. Encourage multiple ways of writing proofs, such as narrative paragraphs, flow diagrams, two-column format, and diagrams without words. Focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning.

1. Prove theorems about lines and angles. *Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.*

2. Prove theorems about triangles. *Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.*

3. Prove theorems about parallelograms. *Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.*

**Geometry-Similarity, Right Triangles and Trigonometry**

Understand similarity in terms of similarity transformations. Prove theorems involving similarity. Define trigonometric ratios and solve problems involving right triangles.

1. Verify experimentally the properties of dilations given by a center and a scale factor.

a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

2. Given two figures, use the definition of similarity in terms of similarity transformations to decide whether they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

4. Prove theorems about triangles. *Theorems include: a line parallel to one side of a triangle divides the other two proportionally and conversely; the Pythagorean Theorem (proved using triangle similarity).*

5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

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.

7. Explain and use the relationship between the sine and cosine of complementary angles.

8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

**Geometry-Circles**

Understand and apply theorems about circles. Find arc lengths and areas of sectors of circles. Use this as a basis for introducing the radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course.

1. Prove that all circles are similar.

2. Identify and describe relationships among inscribed angles, radii, and chords. *Relationships include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.*

3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

4. Construct a tangent line from a point outside a given circle to the circle.

5. Derive, using similarity, the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

**Geometry-Expressing Geometric Properties With Equations**

Translate between the geometric description and the equation for a conic section. Use coordinates to prove simple geometric theorems algebraically. Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically.

1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

2. Use coordinates to prove simple geometric theorems algebraically. *For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, **√**3) lies on the circle centered at the origin and containing the point (0, 2)*.

3. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

**Geometry-Geometric Measurement and Dimension**

Explain volume formulas and use them to solve problems.

1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Informal arguments for area formulas can make use of the way in which area scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor *k*, its area is *k*** ^{2} **times the area of the first.

*Use dissection arguments, Cavalieri’s principle, and informal limit arguments.*

2. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Informal arguments for volume formulas can make use of the way in which volume scale under similarity transformations: when one figure results from another by applying a similarity transformation, volumes of solid figures scale by *k*** ^{3} **under a similarity transformation with scale factor

*k*.

**Statistics-Interpreting Categorical and Quantitative Data**

Summarize, represent, and interpret data on two categorical or quantitative variables.

1. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and condition relative frequencies). Recognize possible associations and trends in the data.

**Statistics-Conditional Probability and the Rules of Probability**

Understand independence and conditional probability and use them to interpret data. Use the rules of probability to compute probabilities of compound events in a uniform probability model.

1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

2. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. *For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.*

3. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. *For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.*

4. Find the conditional probability of *A *given *B *as the fraction of *B*’s outcomes that also belong to *A*, and interpret the answer in terms of the model.

**Literacy Standards**

1. Acquire, accurately use and interpret grade-appropriate mathematical words and terms.

2. Engage in collaborative discussions with diverse partners on grade level concepts.

## Secondary Math 2 Honors

**Mathematical Practices**

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others

4. Model with mathematics

5. Use appropriate tools strategically

6. Attend to precision

7. Look for and make use of structure

8. Look for and express regularity in repeated reasoning

**Number and Quantity-The Real Number System**

Extend the properties of exponents to rational exponents. Use properties of rational and irrational numbers.

1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. *For example, we define 5*^{1/3} *to be the cube root of 5 because we want (5*^{1/3}*)*^{3} *= 5*^{(1/3)3} *to hold, so (5 *^{1/3}*)*^{3} *must equal 5.*

2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.

3. Explain why sums and products of rational numbers are rational, that the sum of a rational number and an irrational number is irrational, and that the product of a nonzero rational number and an irrational number is irrational. Connect to physical situations (e.g., finding the perimeter of a square of area 2).

**Number and Quantity-The Complex Number System**

Perform arithmetic operations with complex numbers. Use complex numbers in polynomial identities and equations.

1. Know there is a complex number *i *such that *i*** ^{2} **= -1, and every complex number has the form

*a + bi*with

*a*and

*b*real.

2. Use the relation *i*** ^{2} **= -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Limit to multiplications that involve

*i*

**as the highest power of**

^{2}*i*.

3. Solve quadratic equations with real coefficients that have complex solutions.

4. Extend polynomial identities to the complex numbers. Limit to quadratics with real coefficients. *For example, rewrite *x^{2} *+ 4 as (*x *+ 2*i*)(*x *– 2*i*)***.**

5. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

**Algebra-**** Seeing Structure in Expression**

Interpret the structure of expressions. Write expressions in equivalent forms to solve problems, balancing conceptual understanding and procedural fluency in work with equivalent expressions.

1. Interpret quadratic and exponential expressions that represent a quantity in terms of its context.

a. Interpret parts of an expression, such as terms, factors, and coefficients.

b. Interpret increasingly more complex expressions by viewing one or more of their parts as a single entity. Exponents are extended from the integer exponents to rational exponents focusing on those that represent square or cube roots.

2. Use the structure of an expression to identify ways to rewrite it. For example, see *x*** ^{4} **–

*y*

**as**

^{4}*(x*

^{2}*)*

^{2}*–*

*(y*

^{2}*)*

**, thus recognizing it as a difference of squares that can be factored as**

^{2}*(x*

^{2}*–*

*y*

^{2}*)(x*

^{2}

*+ y*

^{2}*)*.

3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. *For example, development of skill in factoring and completing the square goes hand in hand with understanding what different forms of a quadratic expression reveal.*

a. Factor a quadratic expression to reveal the zeros of the function it defines.

b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

*c. *Use the properties of exponents to transform expressions for exponential functions. *For example, the expression 1.15*^{t} *can be rewritten as (1.15 *^{1/12}*)*^{12t} *≈** 1.012*^{12t}** ***to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.*

**Algebra-Arithmetic With Polynomials and Rational Expressions**

Perform arithmetic operations on polynomials. Focus on polynomial expressions that simplify to forms that are linear or quadratic in a positive integer power of *x*.

1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

**Algebra-Creating Equations**

Create equations that describe numbers or relationships. Extend work on linear and exponential equations to quadratic equations.

1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

3. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations; extend to formulas involving squared variables. *For example, rearrange the formula for the volume of a cylinder *V *= **π** *r** ^{2} **h

*.*

**Algebra-Reasoning With Equations and Inequalities**

Solve equations and inequalities in one variable. Solve systems of equations. Extend the work of systems to include solving systems consisting of one linear and one nonlinear equation.

*1. *Solve quadratic equations in one variable.

a. Use the method of completing the square to transform any quadratic equation in *x *into an equation of the form *(x **–** p)*^{2}** ***= q *that has the same solutions. Derive the quadratic formula from this form.

b. Solve quadratic equations by inspection (e.g., for *x*** ^{2} **= 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as

*a ± bi*for real numbers

*a*and

*b*.

2. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. *For example, find the points of intersection between the line *y *= **–**3*x *and the circle *x^{2} *+ *y^{2} *= 3*.

**Functions-Building Functions**

Build a function that models a relationship between two quantities. Build new functions from existing functions.

1. Write a quadratic or exponential function that describes a relationship between two quantities.

*a. *Determine an explicit expression, a recursive process, or steps for calculation from a context.

*b. *Combine standard function types using arithmetic operations.

2. 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. Focus on quadratic functions and consider including absolute value functions. 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.

**Functions-Linear, Quadratic and Exponential Models**

Construct and compare linear, quadratic, and exponential models and solve problems.

1. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. Compare linear and exponential growth to quadratic growth.

**Functions-Interpret Functions**

Interpret quadratic functions that arise in applications in terms of a context. Analyze functions using different representations.

1. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

2. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Focus on quadratic functions; compare with linear and exponential functions. *For example, if the function *h(n) *gives the number of person hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*

3. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

4. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

b. Graph piecewise-defined functions and absolute value functions. Compare and contrast absolute value and piecewise-defined functions with linear, quadratic, and exponential functions. Highlight issues of domain, range, and usefulness when examining piecewise-defined functions.

5. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

b. Use the properties of exponents to interpret expressions for exponential functions. *For example, identify percent rate of change in functions such as *y *= (1.02)*^{t}*, *y *= (0.97)*^{t}*, *y *= (1.01)*^{12t}*, *y *= (1.2)*^{t/10}*, and classify them as representing exponential growth or decay.*

*6. *Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Extend work with quadratics to include the relationship between coefficients and roots, and that once roots are known, a quadratic equation can be factored. *For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.*

**Functions-Trigonometric Functions**

Prove and apply trigonometric identities. Limit *θ* to angles between 0 and 90 degrees. Connect with the Pythagorean Theorem and the distance formula.

1. Prove the Pythagorean identity sin** ^{2}**(

*θ*) + cos

**(**

^{2}*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.

**Geometry-Congruence**

Prove geometric theorems. Encourage multiple ways of writing proofs, such as narrative paragraphs, flow diagrams, two-column format, and diagrams without words. Focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning.

1. Prove theorems about lines and angles. *Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.*

2. Prove theorems about triangles. *Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.*

3. Prove theorems about parallelograms. *Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.*

**Geometry-Similarity, Right Triangles and Trigonometry**

Understand similarity in terms of similarity transformations. Prove theorems involving similarity. Define trigonometric ratios and solve problems involving right triangles.

1. Verify experimentally the properties of dilations given by a center and a scale factor.

a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

2. Given two figures, use the definition of similarity in terms of similarity transformations to decide whether they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

4. Prove theorems about triangles. *Theorems include: a line parallel to one side of a triangle divides the other two proportionally and conversely; the Pythagorean Theorem (proved using triangle similarity).*

5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

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.

7. Explain and use the relationship between the sine and cosine of complementary angles.

8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

**Geometry-Circles**

Understand and apply theorems about circles. Find arc lengths and areas of sectors of circles. Use this as a basis for introducing the radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course.

1. Prove that all circles are similar.

2. Identify and describe relationships among inscribed angles, radii, and chords. *Relationships include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.*

3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

4. Construct a tangent line from a point outside a given circle to the circle.

5. Derive, using similarity, the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

**Geometry-Expressing Geometric Properties With Equations**

Translate between the geometric description and the equation for a conic section. Use coordinates to prove simple geometric theorems algebraically. Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically.

1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

2. Use coordinates to prove simple geometric theorems algebraically. *For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, **√**3) lies on the circle centered at the origin and containing the point (0, 2)*.

3. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

**Geometry-Geometric Measurement and Dimension**

Explain volume formulas and use them to solve problems.

1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Informal arguments for area formulas can make use of the way in which area scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor *k*, its area is *k*** ^{2} **times the area of the first.

*Use dissection arguments, Cavalieri’s principle, and informal limit arguments.*

2. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Informal arguments for volume formulas can make use of the way in which volume scale under similarity transformations: when one figure results from another by applying a similarity transformation, volumes of solid figures scale by *k*** ^{3} **under a similarity transformation with scale factor

*k*.

**Statistics-Interpreting Categorical and Quantitative Data**

Summarize, represent, and interpret data on two categorical or quantitative variables.

1. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and condition relative frequencies). Recognize possible associations and trends in the data.

**Statistics-Conditional Probability and the Rules of Probability**

Understand independence and conditional probability and use them to interpret data. Use the rules of probability to compute probabilities of compound events in a uniform probability model.

1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

2. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. *For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.*

3. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. *For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.*

4. Find the conditional probability of *A *given *B *as the fraction of *B*’s outcomes that also belong to *A*, and interpret the answer in terms of the model.

**Number and Quantity-Complex Number System-HONORS TOPICS**

Perform arithmetic operations with complex numbers. Represent complex numbers and their operations on the complex plane.

1. Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

2. Represent complex numbers on the complex plane in rectangular form, and explain why the rectangular form of a given complex number represents the same number.

3. Represent addition, subtraction, and multiplication geometrically on the complex plane; use properties of this representation for computation. *For example, (-1 + **√**3 *i*) ^{3} = 8 because (-1 + *

*√*

*3*i

*) has modulus 2 and argument 120°.*

**Algebra-Reasoning With Equations and Inequalities-HONORS TOPICS**

Solve systems of equations.

1. Represent a system of linear equations as a single-matrix equation in a vector variable.

2. Find the inverse of a matrix if it exists, and use it to solve systems of linear equations (using technology for matrices of dimension 3 x 3 or greater).

**Functions-Interpreting Functions-HONORS TOPICS**

Analyze functions using different representations.

1. Use sigma notation to represent the sum of a finite arithmetic or geometric series.

2. Represent series algebraically, graphically, and numerically.

**Geometry-Expressing Geometric Properties With Equations-HONORS TOPICS**

Translate between the geometric description and the equation for a conic section.

1. Derive the equation of a parabola given a focus and directrix.

2. Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

**Statistics-Conditional Probability and the Rules of Probability-HONORS TOPICS**

Understand independence and conditional probability and use them to interpret data. Use the rules of probability to compute probabilities of compound events in a uniform probability model.

1. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

2. Understand the conditional probability of *A *given *B *as *P(A and B)/P(B)*, and interpret independence of A and *B *as saying that the conditional probability of *B *given *A *is the same as the probability of *B*.

3. Apply the Addition Rule, *P(A or B) = P(A) + P(B) **–** P(A and B)*, and interpret the answer in terms of the model.

4. Apply the general Multiplication Rule in a uniform probability model, *P(A and B) = P(A)P(B|A) = P(B)P(A|B)*, and interpret the answer in terms of the model.

**Literacy Standards**

1. Acquire, accurately use and interpret grade-appropriate mathematical words and terms.

2. Engage in collaborative discussions with diverse partners on grade level concepts.

## Secondary Math 3

**Mathematical Practices**

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others

4. Model with mathematics

5. Use appropriate tools strategically

6. Attend to precision

7. Look for and make use of structure

8. Look for and express regularity in repeated reasoning

**Number and Quantity-The Complex Number System**

Use complex numbers in polynomial identities and equations. Build on work with quadratic equations in Secondary Mathematics II**.**

1. Extend polynomial identities to the complex numbers. *For example, rewrite *x^{2} *+ 4 as (*x *+ 2*i*)(*x *– 2*i*)*.

2. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Limit to polynomials with real coefficients.

**Algebra—Seeing Structure in Expressions**

Interpret the structure of expressions. Extend to polynomial and rational expressions**. **Write expressions in equivalent forms to solve problems**.**

1. Interpret polynomial and rational expressions that represent a quantity in terms of its context.

a. Interpret parts of an expression, such as terms, factors, and coefficients.

*b. *Interpret complex expressions by viewing one or more of their parts as a single entity. *For example, examine the behavior of *P*(1*+r/n*)*^{nt} *as *n *becomes large.*

2. Use the structure of an expression to identify ways to rewrite it. *For example, see *x^{4} *– *y^{4} *as (*x^{2}*)*^{2} *– (*y^{2}*)*^{2}*, thus recognizing it as a difference of squares that can be factored as (*x^{2} *– *y^{2}*)(*x^{2} *+ *y^{2}*).*

3. Understand the formula for the sum of a series and use the formula to solve problems.

a. Derive the formula for the sum of an arithmetic series.

b. Derive the formula for the sum of a geometric series, and use the formula to solve problems. Extend to infinite geometric series. *For example, calculate mortgage payments.*

**Algebra-Arithmetic With Polynomials and Rational Expressions**

Perform arithmetic operations on polynomials, extending beyond the quadratic polynomials. Understand the relationship between zeros and factors of polynomials. Use polynomial identities to solve problems. Rewrite rational expressions.

1. Understand that all polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

2. Know and apply the Remainder Theorem: For a polynomial *p(x) *and a number *a*, the remainder on division by *x – a *is *p(a)*, so *p(a) = 0 *if and only if *(x – a) *is a factor of *p(x)*.

3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

4. Prove polynomial identities and use them to describe numerical relationships. *For example, the polynomial identity (*x^{2} *+ *y^{2}*)*^{2} *= (*x^{2} *– *y^{2}*)*^{2} *+ (2*xy*)*^{2} *can be used to generate Pythagorean triples.*

5. Know and apply the Binomial Theorem for the expansion of *(x + y)*** ^{n} **in powers of

*x*and

*y*for a positive integer

*n*, where

*x*and

*y*are any numbers.

*For example, with coefficients determined by Pascal’s Triangle.*

6. Rewrite simple rational expressions in different forms; write *a(x)/b(x) *in the form *q(x) + r(x)/b(x)*, where *a(x), b(x), q(x)*, and *r(x) *are polynomials with the degree of *r(x) *less than the degree of *b(x)*, using inspection, long division or, for the more complicated examples, a computer algebra system.

7. Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

**Algebra-Creating Equations**

Create equations that describe numbers or relationships, using all available types of functions to create such equations**.**

1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. *For example, maximizing the volume of a box for a given surface area while drawing attention to the practical domain.*

4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. *For example, rearrange the compound interest formula to solve for *t: A *= *P*(1*+ r/n*)*^{nt}

**Algebra-Reasoning With Equations and Inequalities**

Understand solving equations as a process of reasoning and explain the reasoning. Represent and solve equations and inequalities graphically.

1. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

2. Explain why the *x*-coordinates of the points where the graphs of the equations *y = f(x) and y = g(x) *intersect are the solutions of the equation *f(x) = g(x)*; find the solutions approximately, for example, using technology to graph the functions, make tables of values, or find successive approximations. Include cases where *f(x) *and/or *g(x) *are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

**Functions-Interpret Functions**

Interpret functions that arise in applications in terms of a context. Analyze functions using different representations.

1. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. *Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*

2. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. *For example, if the function *h(n) *gives the number of person-hours it takes to assemble *n *engines in a factory, then the positive integers would be an appropriate domain for the function.*

3. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

4. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

a. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Compare and contrast square root, cubed root, and step functions with all other functions.

b. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

c. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

d. Graph exponential and logarithmic functions, showing intercepts and end behavior; and trigonometric functions, showing period, midline, and amplitude.

**5. **Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

**6. **Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). *For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.*

**Functions-Building Functions **

Build a function that models a relationship between two quantities. Develop models for more complex or sophisticated situations. Build new functions from existing functions.

1. Write a function that describes a relationship between two quantities.

*a. *Combine standard function types using arithmetic operations.

2. 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. Note the effect of multiple transformations on a single function and the common effect of each transformation across function types. Include functions defined only by a graph. 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.

3. Find inverse functions.

*a. *Solve an equation of the form *f(x) = c *for a simple function *f *that has an inverse and write an expression for the inverse. Include linear, quadratic, exponential, logarithmic, rational, square root, and cube root functions. *For example, *f(x) *= 2*x^{3} *or *f(x) *= *(x*+1)/(*x*-1) for *x *≠ 1.*

**Functions-Trigonometric Functions**

Extend the domain of trigonometric functions using the unit circle. Model periodic phenomena with trigonometric functions.

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

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.

3. Use special triangles to determine geometrically the values of sine, cosine, tangent for *π*/3, *π*/4 and *π/6*, and use the unit circle to express the values of sine, cosine, and tangent for *π– x, π+ x, *and *2π – x *in terms of their values for *x, *where *x *is any real number.

4. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

5. Use inverse functions to solve trigonometric equations that arise in modeling context; evaluate the solutions using technology and interpret them in terms of context. Limit solutions to a given interval.

**Functions-Linear, Quadratic, and Exponential Models**

Construct and compare linear, quadratic, and exponential models and solve problems. Interpret expressions for functions in terms of the situation it models. Introduce *f(x) = e*** ^{x} **as a model for continuous growth.

1. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

2. For exponential models, express as a logarithm the solution to *ab*^{ct} *= d *where *a, c*, and *d *are numbers and the base *b *is 2, 10, or *e*; evaluate the logarithm using technology. Include the relationship between properties of logarithms and properties of exponents, such as the connection between the properties of exponents and the basic logarithm property that log *xy = log x + log y*.

3. Interpret the parameters in a linear, quadratic, or exponential function in terms of a context.

**Geometry-Similarity, Right Triangles and Trigonometry**

Apply trigonometry to general triangles. With respect to the general case of the Laws of Sines and Cosines, the definitions of sine and cosine must be extended to obtuse angles.

1. Derive the formula *A = 1/2 ab sin(C) *for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

2. Prove the Laws of Sines and Cosines and use them to solve problems.

**3. **Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

**Geometry-Geometric Measurement and Dimension**

Visualize relationships between two-dimensional and three-dimensional objects.

1. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two dimensional objects.

**Geometry-Modeling With Geometry**

Apply geometric concepts in modeling situations.

1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

**Statistics-Interpreting Categorical and Quantitative Data**

Summarize, represent, and interpret data on a single count or measurement variable. While students may have heard of the normal distribution, it is unlikely that they will have prior experience using it to make specific estimates. Build on students’ understanding of data distributions to help them see how the normal distribution uses area to make estimates of frequencies (which can be expressed as probabilities). Emphasize that only some data are well described by a normal distribution.

1. 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.

**Statistics-Making Inferences and Justifying Conclusions**

Understand and evaluate random processes underlying statistical experiments. Draw and justify conclusions from sample surveys, experiments, and observational studies. In earlier grades, students are introduced to different ways of collecting data and use graphical displays and summary statistics to make comparisons. These ideas are revisited with a focus on how the way in which data is collected determines the scope and nature of the conclusions that can be drawn from that data. The concept of statistical significance is developed informally through simulation as meaning a result that is unlikely to have occurred solely as a result of random selection in sampling or random assignment in an experiment. For Standard 3, focus on the variability of results from experiments—that is, focus on statistics as a way of dealing with, not eliminating, inherent randomness.

1. Understand that statistics allow inferences to be made about population parameters based on a random sample from that population.

2. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

3. 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.

4. Evaluate reports based on data.

**Literacy Standards**

1. Acquire, accurately use and interpret grade-appropriate mathematical words and terms.

2. Engage in collaborative discussions with diverse partners on grade level concepts.

## Secondary Math 3 Honors

**Mathematical Practices**

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others

4. Model with mathematics

5. Use appropriate tools strategically

6. Attend to precision

7. Look for and make use of structure

8. Look for and express regularity in repeated reasoning

**Number and Quantity-The Complex Number System**

Use complex numbers in polynomial identities and equations. Build on work with quadratic equations in Secondary Mathematics II**.**

1. Extend polynomial identities to the complex numbers. *For example, rewrite *x^{2} *+ 4 as (*x *+ 2*i*)(*x *– 2*i*)*.

2. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Limit to polynomials with real coefficients.

**Algebra—Seeing Structure in Expressions**

Interpret the structure of expressions. Extend to polynomial and rational expressions**. **Write expressions in equivalent forms to solve problems**.**

1. Interpret polynomial and rational expressions that represent a quantity in terms of its context.

a. Interpret parts of an expression, such as terms, factors, and coefficients.

*b. *Interpret complex expressions by viewing one or more of their parts as a single entity. *For example, examine the behavior of *P*(1*+r/n*)*^{nt} *as *n *becomes large.*

2. Use the structure of an expression to identify ways to rewrite it. *For example, see *x^{4} *– *y^{4} *as (*x^{2}*)*^{2} *– (*y^{2}*)*^{2}*, thus recognizing it as a difference of squares that can be factored as (*x^{2} *– *y^{2}*)(*x^{2} *+ *y^{2}*).*

3. Understand the formula for the sum of a series and use the formula to solve problems.

a. Derive the formula for the sum of an arithmetic series.

b. Derive the formula for the sum of a geometric series, and use the formula to solve problems. Extend to infinite geometric series. *For example, calculate mortgage payments.*

**Algebra-Arithmetic With Polynomials and Rational Expressions**

Perform arithmetic operations on polynomials, extending beyond the quadratic polynomials. Understand the relationship between zeros and factors of polynomials. Use polynomial identities to solve problems. Rewrite rational expressions.

1. Understand that all polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

2. Know and apply the Remainder Theorem: For a polynomial *p(x) *and a number *a*, the remainder on division by *x – a *is *p(a)*, so *p(a) = 0 *if and only if *(x – a) *is a factor of *p(x)*.

3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

4. Prove polynomial identities and use them to describe numerical relationships. *For example, the polynomial identity (*x^{2} *+ *y^{2}*)*^{2} *= (*x^{2} *– *y^{2}*)*^{2} *+ (2*xy*)*^{2} *can be used to generate Pythagorean triples.*

5. Know and apply the Binomial Theorem for the expansion of *(x + y)*** ^{n} **in powers of

*x*and

*y*for a positive integer

*n*, where

*x*and

*y*are any numbers.

*For example, with coefficients determined by Pascal’s Triangle.*

6. Rewrite simple rational expressions in different forms; write *a(x)/b(x) *in the form *q(x) + r(x)/b(x)*, where *a(x), b(x), q(x)*, and *r(x) *are polynomials with the degree of *r(x) *less than the degree of *b(x)*, using inspection, long division or, for the more complicated examples, a computer algebra system.

7. Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

**Algebra-Creating Equations**

Create equations that describe numbers or relationships, using all available types of functions to create such equations**.**

3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. *For example, maximizing the volume of a box for a given surface area while drawing attention to the practical domain.*

4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. *For example, rearrange the compound interest formula to solve for *t: A *= *P*(1*+ r/n*)*^{nt}

**Algebra-Reasoning With Equations and Inequalities**

Understand solving equations as a process of reasoning and explain the reasoning. Represent and solve equations and inequalities graphically.

1. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

2. Explain why the *x*-coordinates of the points where the graphs of the equations *y = f(x) and y = g(x) *intersect are the solutions of the equation *f(x) = g(x)*; find the solutions approximately, for example, using technology to graph the functions, make tables of values, or find successive approximations. Include cases where *f(x) *and/or *g(x) *are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

**Functions-Interpret Functions**

Interpret functions that arise in applications in terms of a context. Analyze functions using different representations.

1. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. *Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*

2. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. *For example, if the function *h(n) *gives the number of person-hours it takes to assemble *n *engines in a factory, then the positive integers would be an appropriate domain for the function.*

4. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

a. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Compare and contrast square root, cubed root, and step functions with all other functions.

b. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

c. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

d. Graph exponential and logarithmic functions, showing intercepts and end behavior; and trigonometric functions, showing period, midline, and amplitude.

**5. **Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

**6. **Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

**Functions-Building Functions **

Build a function that models a relationship between two quantities. Develop models for more complex or sophisticated situations. Build new functions from existing functions.

1. Write a function that describes a relationship between two quantities.

*a. *Combine standard function types using arithmetic operations.

2. 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. Note the effect of multiple transformations on a single function and the common effect of each transformation across function types. Include functions defined only by a graph. 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.

3. Find inverse functions.

*a. *Solve an equation of the form *f(x) = c *for a simple function *f *that has an inverse and write an expression for the inverse. Include linear, quadratic, exponential, logarithmic, rational, square root, and cube root functions. *For example, *f(x) *= 2*x^{3} *or *f(x) *= *(x*+1)/(*x*-1) for *x *≠ 1.*

**Functions-Trigonometric Functions**

Extend the domain of trigonometric functions using the unit circle. Model periodic phenomena with trigonometric functions.

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

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.

3. Use special triangles to determine geometrically the values of sine, cosine, tangent for *π*/3, *π*/4 and *π/6*, and use the unit circle to express the values of sine, cosine, and tangent for *π– x, π+ x, *and *2π – x *in terms of their values for *x, *where *x *is any real number.

4. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

5. Use inverse functions to solve trigonometric equations that arise in modeling context; evaluate the solutions using technology and interpret them in terms of context. Limit solutions to a given interval.

**Functions-Linear, Quadratic, and Exponential Models**

Construct and compare linear, quadratic, and exponential models and solve problems. Interpret expressions for functions in terms of the situation it models. Introduce *f(x) = e*** ^{x} **as a model for continuous growth.

1. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

2. For exponential models, express as a logarithm the solution to *ab*^{ct} *= d *where *a, c*, and *d *are numbers and the base *b *is 2, 10, or *e*; evaluate the logarithm using technology. Include the relationship between properties of logarithms and properties of exponents, such as the connection between the properties of exponents and the basic logarithm property that log *xy = log x + log y*.

3. Interpret the parameters in a linear, quadratic, or exponential function in terms of a context.

**Geometry-Similarity, Right Triangles and Trigonometry**

Apply trigonometry to general triangles. With respect to the general case of the Laws of Sines and Cosines, the definitions of sine and cosine must be extended to obtuse angles.

1. Derive the formula *A = 1/2 ab sin(C) *for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

2. Prove the Laws of Sines and Cosines and use them to solve problems.

**3. **Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

**Geometry-Geometric Measurement and Dimension**

Visualize relationships between two-dimensional and three-dimensional objects.

1. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two dimensional objects.

**Geometry-Modeling With Geometry**

Apply geometric concepts in modeling situations.

1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

**Statistics-Interpreting Categorical and Quantitative Data**

Summarize, represent, and interpret data on a single count or measurement variable. While students may have heard of the normal distribution, it is unlikely that they will have prior experience using it to make specific estimates. Build on students’ understanding of data distributions to help them see how the normal distribution uses area to make estimates of frequencies (which can be expressed as probabilities). Emphasize that only some data are well described by a normal distribution.

1. 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.

**Statistics-Making Inferences and Justifying Conclusions**

Understand and evaluate random processes underlying statistical experiments. Draw and justify conclusions from sample surveys, experiments, and observational studies. In earlier grades, students are introduced to different ways of collecting data and use graphical displays and summary statistics to make comparisons. These ideas are revisited with a focus on how the way in which data is collected determines the scope and nature of the conclusions that can be drawn from that data. The concept of statistical significance is developed informally through simulation as meaning a result that is unlikely to have occurred solely as a result of random selection in sampling or random assignment in an experiment. For Standard 3, focus on the variability of results from experiments—that is, focus on statistics as a way of dealing with, not eliminating, inherent randomness.

1. Understand that statistics allow inferences to be made about population parameters based on a random sample from that population.

2. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

3. 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.

4. Evaluate reports based on data.

**Number and Quantity-Complex Number System-HONORS TOPICS**

Perform arithmetic operations with complex numbers. Represent complex numbers and their operations on the complex plane. Use complex numbers in polynomial identities and equations.

1. Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

2. Represent complex numbers on the complex plane in rectangular form and polar form (including real and imaginary numbers), and explain why the rectangular form of a given complex number represents the same number.

3. Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. *For example, (-1 + √3 *i*)3 = 8 because (-1 + √3 i) has modulus 2 and argument 120°.*

4. Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

5. Multiply complex numbers in polar form and use DeMoivre’s Theorem to find roots of complex numbers.

**Functions-Interpreting Functions-HONORS TOPICS**

Analyze functions using different representations.

1. Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

a. Graph rational functions, identifying zeros, asymptotes, and point discontinuities when suitable factorizations are available, and showing end behavior.

b. Define a curve parametrically and draw its graph.

**Functions-Building Functions-HONORS TOPICS**

Build a function that models a relationship between two quantities. Build new functions from existing functions.

1. Write a function that describes a relationship between two quantities.

a. Compose functions. *For example, if *T*(*y*) is the temperature in the atmosphere as a function of height, and *h(t) *is the height of a weather balloon as a function of time, then *T(h(t)) *is the temperature at the location of the weather balloon as a function of time.*

2. Find inverse functions.

a. Verify by composition that one function is the inverse of another.

b. Read values of an inverse function from a graph or a table, given that the function has an inverse.

c. Produce an invertible function from a non-invertible function by restricting the domain.

3. Understand the inverse relationship between exponents and logarithms, and use this relationship to solve problems involving logarithms and exponents.

**Functions -Trigonometric Functions-HONORS TOPICS**

Extend the domain of trigonometric functions using the unit circle. Model periodic phenomena with trigonometric functions. Prove and apply trigonometric identities.

1. Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

2. Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

3. Use the inverse functions to solve trigonometric equations that arise in the modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

4. Prove the addition and subtraction formulas for sine, cosine, and tangent, and use them to solve problems.

**Geometry-Geometric Measurement and Dimension-HONORS TOPICS**

Explain volume formulas and use them to solve problems.

1. Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

**Statistics-Conditional Probability and the Rules of Probability-HONORS TOPICS**

Use the rules of probability to compute probabilities of compound events in a uniform probability model.

1. Use permutations and combinations to compute probabilities of compound events and solve problems.

**Literacy Standards**

1. Acquire, accurately use and interpret grade-appropriate mathematical words and terms.

2. Engage in collaborative discussions with diverse partners on grade level concepts.

## Mathematical Decision Making for Life

# Mathematical Decision Making for Life

## Prerequisite: Secondary Mathematics II

Mathematical Decision Making is a four-quarter course for seniors. The course includes mathematical decision making in finance, modeling, probability and statistics, and making choices. The four quarters of instruction are independent of each other, allowing students to enter and exit the course quarterly. Students will make sense of authentic problems and persevere in solving them. They will reason abstractly and quantitatively while communicating mathematics to others. Students will use appropriate tools, including technology, to model mathematics. Students will use structure and regularity of reasoning to describe mathematical situations and solve problems.

# Quarter A – Mathematical Decision Making: Finance

## Standard I: Students will use mathematical analysis to manage personal resources and make financially sound decisions.

**Objective 1: Determine, represent and analyze mathematical models for various types of income calculations.**

a. Compute and compare hourly wages, given commissions or salaries and hours worked.

b. Compute gross earnings based on commissions, salaries, hourly wages, or piece-work.

c. Compute net earnings after common payroll deductions.

d. Research and compare annual earnings for various employment opportunities.

## Objective 2: Create, represent, and justify personal budgets.

a. Create spreadsheets, tables, and charts that represent personal income and expenses.

b. Calculate the total costs of owning a car, including monthly payments, insurance, maintenance, and fuel.

c. Analyze and model periodic monthly expenditures, including those that change during the year such, as heating and cooling costs.

## Objective 3: Analyze mathematical models related to investing and borrowing money.

a. Compute and compare the anticipated earnings for investments and savings plans.

b. Interpret stock market data charts.

c. Research and predict retirement income from savings, Social Security benefits, pensions, and investments.

d. Compute the costs of loans for monthly payments.

e. Compare time and costs required to borrow money compared to saving for purchase of an item.

f. Analyze various types of loans to determine the best loan for a given situation.

## Objective 4: Analyze numerical data to make quantitative and qualitative decisions.

a. Research, compare, and contrast published ratios, rates, ratings, averages, weighted averages, and indices to make informed decisions.

b. Use spreadsheets to manage large quantities of data.

c. Understand and analyze situations involving large numbers, such as national debt or national budgets.

# Quarter B – Mathematical Decision Making: Modeling

## Standard II: Students will use mathematical models to organize, communicate, and solve problems.

**Objective 1: Use matrices to represent and analyze mathematical situations.**

a. Use matrices to represent and manipulate data.

b. Multiply matrices by scalars to produce new matrices.

c. Add, subtract, and multiply matrices of appropriate dimensions.

d. Use matrices to represent geometric transformations.

e. Use matrices to solve applied problems.

## Objective 2: Model mathematical problems with geometric tools.

a. Use geometric methods to solve design problems.

b. Calculate measures of perimeter, surface area, area, and volume, and apply those measures to relevant situations.

## Objective 3: Use mathematics to model and solve problems involving change.

a. Analyze and solve problems involving models for linear, exponential, and logistic growth and decay.

b. Identify, model, and solve problems involving cyclical change that can be represented using trigonometric functions.

c. Identify, model, and solve problems involving change that can be represented with a piecewise function.

d. Model and solve problems involving recursion or iteration.

# Quarter C – Mathematical Decision Making: Probability and Statistics

## Standard III: Students will use statistics and probability to make decisions. Objective 1: Understand and communicate statistical information.

a. Report results of statistical studies in both oral and written form, including graphical representations.

b. Describe strengths and weaknesses of sampling techniques, data and graphical displays, and interpretations of summary statistics.

c. Identify uses and misuses of statistical analyses.

## Objective 2: Develop and evaluate inferences and predictions that are based on data.

a. Understand and evaluate random processes underlying statistical experiments.

b. Determine possible sources of statistical bias and describe the potential impact of such bias on a study.

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

d. Use data from a sample survey to estimate a population mean or proportion.

## Objective 3: Apply statistical methods to design and conduct a survey or an experiment.

a. Formulate a question that can be analyzed using statistical methods.

b. Determine possible sources of variability of data, including both those that can and cannot be controlled.

c. Identify the population of interest, select an appropriate sampling technique, and collect data.

d. Create graphical displays of data.

e. Calculate and compare measures of central tendency, spread, and unusual features in data.

## Objective 4: Use the rules of probability to calculate independent and conditional probabilities in real contexts.

a. Distinguish between subjective, experimental, and theoretical probability.

b. Calculate probabilities using addition and multiplication rules, tree diagrams, and two-way tables using correct probability notation.

c. Calculate conditional probabilities of compound events using two-way tables and Venn diagrams.

d. Use permutations and combinations to find probabilities.

## Objective 5: Analyze risk and return in the context of everyday situations.

a. Construct and analyze tree diagrams, Venn diagrams, and area models to make decisions in problem situations.

b. Construct and interpret two-way frequency tables of data.

c. Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

d. Use probabilities to make fair decisions.

e. Analyze decisions and strategies using probability concepts.

Mathematical Language and Symbols Students Should Use:bias, combination, conditional, expected value, experiment, experimental probability, fair decision, frequency table, independent, measures of central tendency (mean, median, mode), measures of spread (range, standard deviation), observational study, permutation, randomization, sample, survey, theoretical probability, tree diagram, variability, Venn diagram

# Quarter D - Mathematical Decision Making: Using Models to Make Choices

## Standard IV: Students will use mathematical models to analyze situations and make choices. Objective 1: Construct viable arguments and critique the reasoning of others.

a. Use stated assumptions, definitions, and previously established results to construct an argument.

b. Make conjectures and build a logical progression of statements to explore the truth of conjectures.

c. Recognize and use counterexamples.

d. Justify and communicate conclusions, and respond to the arguments of others.

e. Compare two plausible arguments and make decisions based on correct logic.

## Objective 2: Analyze and evaluate the mathematics behind various ranking and selection methods.

a. Analyze and apply various ranking algorithms to determine an appropriate method for a given situation.

b. Evaluate various voting and selection processes to determine an appropriate method for a given situation.

c. Analyze and apply various algorithms for making fair divisions.

## Objective 3: Construct, analyze, and interpret flow charts.

a. Construct flowcharts to describe processes or problem-solving procedures.

b. Analyze flowcharts and follow procedures to solve problems.

c. Evaluate efficiency of control processes.

d. List requirements and restrictions needed for a suggested algorithm.

## Objective 4: Use a variety of graphical models to represent network and scheduling problems.

a. Solve scheduling problems using mathematical models.

b. Explore shortest route and fastest route situations.

c. Solve precedence or critical paths problems to facilitate “what if” scenarios.

Mathematical Language and Symbols Students Should Use:algorithm, counterexample, critical paths, Euler path, flow chart, logic, minimal spanning trees, truth table, vertex-edge graph

## AP Statistics

Please visit the AP Statistics page at College Board to learn more about this course.

## AP Caclulus AB

Please visit the AP Calculus AB page at College Board to learn more about this course.

## AP Calculus BC

Please visit the AP Calculus BC page at College Board to learn more about this course.

## Educators

- Content

"tell me and i forget, teach me and i remember, involve me and i learn."

Benjamin franklin

## Professional Resources

DSD Secondary Mathematics Educators - Facebook Group

*This is a private group for Davis School District secondary mathematics educators to collaborate, share ideas, and discuss math education topics! You must have a Facebook account to join.*

Utah Education Network

*UETN connects all Utah school districts, schools, and higher education institutions to a robust network and quality educational resources. UETN is one of the nation's premier education networks.*

Utah State Board of Education

*Education excellence for all students.*

If you have a resource you'd like to see represented on this page, please let us know using the contact information below.

## Content Resources

Mathematics Common Core Standards

*The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education.*

Partnership for 21st Century Learning

*P21's mission is to serve as a catalyst for 21st century learning by building collaborative partnerships among education, business, community, and government leaders so that all learners acquire the knowledge and skills they need to thrive in a world where change is constant and learning never stops.*

Illuminations

*Illuminations works to serve you by increasing access to quality standards-based resources for teaching and learning mathematics, including interactive tools for students and instructional support for teachers.*

National Council for the Teachers of Mathematics (NCTM)

*Founded in 1920, the National Council of Teachers of Mathematics (NCTM) is the world's largest mathematics education organization, with 60,000 members and more than 230 Affiliates throughout the United States and Canada. Interested in membership?*

Utah Council of Teachers of Mathematics (UCTM)

*Local chapter of NCTM*

If you have a resource you'd like to see represented on this page, please let us know using the contact information below.

## Student Engagement

Vi Hart

7th - 12th Grades

Free

*Excellent set of videos by mathematician, Vi Hart, for high student engagement that teach non-traditional mathematical topics such as hexaflexagons, Mobius strips, and a nice introduciton to imagining infinities.*

BuzzMath

6th - 9th Grades

Free Trial, subscription for ongoing use

*BuzzMath focuses on helping middle schoolers practice their math skills. It contains high-quality problems. It gives immediate and detailed feedback, letting students progress at their own pace. Randomly generated values let students to retry problems to obtain mastery. Teachers also receive detailed results that help them guide and monitor student progress.*

DreamBox

6th - 8th Grades

Free Trial, subscriptions available

*An adaptive learning platform designed to complement classroom instruction and deliver results. Includes resources for teachers, student data reports, and instructive insights. *

Padlet

All Grade Levels

Free trial, upgrades require purchases

*Create interactive message boards to promote community engagement.*

Edgenuity

K - 12th Grades

Subscription required

*Online learning digital curriculum for primary or supplementary instruction. Give students the support they need exactly when they need it. *

Desmos

K - 12th Grades

Free

*Online graphing calculator.*

GeoGebra

9th - 12th Grades

Free

*Graphing calculator for functions, geometry, algebra, calculus, statistics, and 3D math. Includes practice sheets. *

Make Salt Lake

All Ages

*A Makerspace where members can gather to create, invent and learn. Make Salt Lake offers access to education, tools and other makers. Interests vary from electronics to plastics and woodworking and beyond.*

If you have a resource you'd like to see represented on this page, please let us know using the contact information below.

Lindsey Henderson

Secondary Mathematics Supervisor

lhenderson@dsdmail.net