Mathematics

Proportional Drawings and Relationships  

Unit A 

In this unit, students study scaled copies of pictures and plane figures, then apply what they have learned to scale drawings (maps and floor plans). Students begin by looking at copies of pictures and use their own words to describe what differentiates scaled and non-scaled copies of a picture. As the unit progresses, students learn that all lengths in a scaled copy are multiplied by a scale factor and all angles stay the same. They draw scaled copies of figures. They learn that if the scale factor is greater than 1, the copy will be larger, and if the scale factor is less than 1, the copy will be smaller. Next, students see that the principles and strategies that they used to reason about scaled copies of figures can be used with scale drawings. They work with scales that involve units (e.g., “1 cm represents 10 km”), and scales that do not include units (e.g., “the scale is 1 to 100”). They understand that actual lengths are products of a scale factor and corresponding lengths in the scale drawing, thus lengths in the drawing are the product of the actual lengths and the reciprocal of that scale factor.

The second portion of this unit focuses on understanding what a proportional relationship is, how it is represented, and what types of contexts give rise to proportional relationships. In a table of equivalent ratios, a multiplicative relationship between the pair of rows is given by a scale factor. By contrast, the multiplicative relationship between the columns is given by a unit rate. The relationship between pairs of values in the two columns is called a proportional relationship, the unit rate that describes this relationship is called a constant of proportionality, and the quantity represented by the right column is said to be proportional to the quantity represented by the left. Students learn that any proportional relationship can be represented by an equation of the form y=kx where k is the constant of proportionality, that its graph lies on a line through the origin and that the constant of proportionality indicates the steepness of the line. By the end of the unit, students should be able to easily work with common contexts associated with proportional relationships (such as constant speed, unit pricing, and measurement conversions) and be able to determine whether a relationship is proportional or not.

Throughout the unit, they discuss their mathematical ideas and respond to the ideas of others. In the culminating PBA of this unit, students make a floor plan of a room of their choice. This is an opportunity for them to apply what they have learned in the unit to everyday life.

Profile of a Graduate Capacities: Collective Intelligence, Product Creation

Measuring Circles 

Unit B 

In this brief unit students apply their knowledge of proportional relationships to the study of circles. The unit begins with activities designed to help students come to a more precise understanding of the characteristics of a circle. Students measure circular objects, investigating the relationship between measurements of circumference and diameter by making tables and graphs. Next, students encounter at least one informal derivation of the fact that the area of a circle is equal to times the square of its radius.

Students select and use formulas for the area and circumference of a circle to solve abstract and real-world problems that involve calculating lengths and areas. They express measurements in terms of pi or using appropriate approximations of pi to express them numerically. Finally, students work with volume, using abilities developed in earlier work with geometry and geometric measurement to calculate the volume of a sphere, a cylinder, and a cone. 

The PBA about stained glass windows gives students an opportunity to once again use their collective intelligence to inform their solution. They are asked to create a design with specific criteria and support their choices through mathematics.

Profile of a Graduate Capacities: Collective Intelligence, Product Creation

Proportional Relationships and Percents 

Unit C 

In this unit, students deepen their understanding of ratios, scale factors, unit rates (constants of proportionality), and proportional relationships, using them to solve multi-step problems that are set in a wide variety of contexts that involve fractions, decimals and percentages. Throughout the unit students work together to make sense of problems and create solutions.

The unit begins by revisiting scale factors and proportional relationships, each of which has been the focus of a previous unit. Both of these concepts can be used to solve problems that involve equivalent ratios. However, it is often more efficient to view equivalent ratios as pairs that are in the same proportional relationship rather than seeing one pair as obtained by multiplying both entries of the other by a scale factor. From the proportional relationship perspective, all that is needed is the constant of proportionality—which is the same for every ratio in the proportional relationship.

Next students learn about percent increase and decrease. Students consider situations for which percentages can be used to describe a change relative to an initial amount, e.g., prices before and after a 25% increase. They begin by considering situations with unspecified amounts. They next consider situations with a specified amount and percent change, or with initial and final amounts, using double number line diagrams to find the unknown amount or percent change. Next, they use equations to represent such situations, using the distributive property to show that different expressions for the same amount are equivalent.

Students work with long division to make connections between fraction and decimal representation. 

Students then use their abilities to find percentages and percent rates to solve problems that involve sales tax, tip, discount, markup, markdown, and commission and percent error.

The PBA, In the News, gives students an opportunity to examine current news using new skills involving percents especially those using percent increase/decrease. They will work with a partner to pose and solve questions creating a visual display of their solution as they have throughout the unit.

Profile of a Graduate Capacities: Collective Intelligence, Product Creation

Rational Number Arithmetic 

Unit D 

This unit begins by revisiting ideas familiar from grade 6: how signed numbers are used to represent quantities such as measurements of temperature and elevation, opposites (pairs of numbers on the number line that are the same distance from zero), and absolute value.

 Next, students extend addition and subtraction from fractions to all rational numbers. They begin by considering how changes in temperature and elevation can be represented—first with tables and number line diagrams, then with addition and subtraction expressions and equations. Initially, physical contexts provide meanings for sums and differences that include negative numbers. Students work with numerical addition and subtraction expressions and equations, becoming more fluent in computing sums and differences of signed numbers. Using the meanings that they have developed for addition and subtraction of signed numbers, they write equivalent numerical addition and subtraction expressions, e.g., -8 + (-3) and -8 - 3; and they write different equations that express the same relationship.

Next students study multiplication and division. They build understanding of these operations through repeated addition of signed numbers and by examining patterns. Later, in the process of solving problems set in contexts, they write and solve multiplication and division equations.

By this point students will need practice using all four operations on rational numbers, making use of structure, e.g., to see without calculating that the product of two factors is positive because the values of the factors are both negative. They solve problems that involve interpreting negative numbers in context, for instance, when a negative number represents a rate at which water is flowing. They begin working with linear equations in one variable that have rational number coefficients. At first the focus is representing situations with equations and what it means for a number to be a solution for an equation, rather than methods for solving equations. Such methods are the focus of a later unit.

During the Stock Market PBA students select a group of stocks, track their value and then compare the percent increase/decrease in value to that of their peers to determine a winner.

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Expressions, Equations and Inequalities 

Unit E 

Students begin by representing relationships of two quantities with tape diagrams and with equations noticing that one tape diagram can be described by different (but related) equations. The two main types of situations examined can be modeled with the equations of the form px +q = r and p(x +q) = r, where p, q, and r are rational numbers.

Next, students solve equations of the forms px +q = r and p(x +q) = r, then solve problems that can be represented by such equations. They begin by considering balanced and unbalanced “hanger diagrams,” matching hanger diagrams with equations, and using the diagrams to understand two algebraic steps in solving equations of the form px +q = r: subtract the same number from both sides, then divide both sides by the same number. Like a tape diagram, the same balanced hanger diagram can be described by different (but related) equations, e.g. 2(x + 3) = 18, and 2x + 6 = 18. They use the distributive property to transform an equation of one form into the other and note how such transformations can be used strategically in solving an equation.

Students also work with inequalities. They begin by examining values that make an inequality true or false, and using the number line to represent values that make an inequality true. They solve equations, examine values to the left and right of a solution, and use those values in considering the solution of a related inequality. Finally, students solve inequalities that represent real-world situations.

Students also work with equivalent linear expressions, using properties of operations to explain equivalence. They represent expressions with area diagrams, and use the distributive property to justify factoring or expanding an expression.

Finally, students examine equations with variables on both sides of the equation. The goal is to become fluent at examining the equation before attempting to solve it to determine if there is one, none or an infinite number of solutions. As with the equations and inequalities previously worked on in this unit, students are expected to solve these equations by showing a solution pathway that demonstrates an understanding of equality in equations as well as in expressions.

The PBA is an error analysis where students identify common errors and communicate the error to the reader.

Profile of a Graduate Capacities: Analyzing

Geometry (Angles, Prisms and Transformations)

Unit F 

In this unit students briefly investigate whether sets of angle and side length measurements determine unique triangles or multiple triangles, or fail to determine triangles. Students also study and apply angle relationships, learning to understand and use the terms “complementary,” “supplementary,” “vertical angles”.  Parallel lines and the angles that result when there is a transversal are examined.  Students see a simple proof for finding the angle sum of a triangle and then use this property to solve problems including those with exterior angles. The work gives them practice working with rational numbers and equations for angle relationships. Next, moving on to three dimensional objects, students analyze and describe cross-sections of prisms, pyramids, and polyhedra. They understand and use the formula for the volume of any prism, and solve problems involving area, surface area, and volume of prims.

The students will use their geometry skills in the Poster Packaging PBA where they analyze a given box and improve upon it based on their own criteria.

The unit also includes a study of rigid transformations including dilations, rotations, reflections and translations. Students pay particular attention to similarities and differences between the image and its new image. This topic is an 8th grade standard and teachers, as a team, may decide to teach it out of sequence at another point later in the year.

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Beyond Rational Numbers 

Unit G 

This unit focuses exclusively on topics that come from grade 8 math standards. Students begin by 'discovering' the rules for working with expressions with exponents.  This work continues as we work with scientific notation. Students learn that expressing very large or small numbers in scientific notation can make computation with those numbers much more manageable.  Along with becoming fluent with converting between standard and scientific notation, students solve problems that involve any of the four operations and scientific notation. Finally, the students learn about irrational numbers, especially estimating their values, so they are prepared to use them while working with the Pythagorean Theorem in both two and three dimensional space.

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Probability and Sampling 

Unit H 

In this unit, students design and use simulations to estimate probabilities of outcomes of chance experiments and understand the probability of an outcome as its long-run relative frequency.

They represent sample spaces in tables and tree diagrams and as lists. They calculate the number of outcomes in a given sample space to find the probability of a given event. They consider the strengths and weaknesses of different methods for obtaining a representative sample from a given population. They generate samples from a given population and examine the distributions of the samples, comparing these to the distribution of the population. They compare two populations by comparing samples from each population.

The PBA, Kitten Simulation, offers students a chance to develop their own model for a simulation. They will run their simulation and then compare their results with that of their classmates.

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Pre-Algebra Topics 

Unit 1 

This unit focuses on linear relationships and the connection between situations, tables, equations, and graphs.

The unit begins with an introduction to a slope triangle and establishes that the quotient of the vertical side length and the horizontal side length does not depend on the triangle; this number is called the slope of a line.

Next, students revisit the different representations of proportional relationships (graphs, tables, and equations) and the role of the constant of proportionality in each representation and how it may be interpreted in context. Next, students are introduced to "rate of change" as a way to describe the rate per 1 in a linear relationship and note that its numerical value is the same as that of the slope that represents the relationship. Students next consider negative slopes and the slopes of vertical and horizontal lines. Students practice graphing linear functions using the slope and the vertical intercept. Students practice writing equations of lines from a graph, a table, a situation, and from two specific points. The unit concludes with students transforming equations into slope-intercept form.

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Linear Equations, Inequalities & Systems

Unit 2 

The unit begins with students learning to think of equations as a way to represent constraints or limitations on quantities. Students also see that graphs of equations can help us make sense of constraints and identify values that satisfy them. Next, students investigate different ways to express the same relationship or constraint - by analyzing and writing equivalent equations. They look at moves than can transform one equation to an equivalent equation, recognizing that these are the moves we make to solve equations. The focus here is not only on identifying acceptable moves for solving, but also on explaining why these moves keep each subsequent equation true and maintain the solutions of the original equation.

Next, students encounter situations that involve two or more constraints and realize that systems of equations are helpful for representing these constraints. Students learn to solve systems of equations by elimination and explain why the steps taken to eliminate a variable are valid. Additionally, students reinforce their awareness that a system of equations could have one solution, no solutions, or infinitely many solutions.

Finally, students explore inequalities in one and two variables. They see that inequalities are a handy way to express constraints and can be satisfied by a range of values rather than a single value. Students see that a solution to an inequality is a value or a pair of values that make the inequality true and that the solution to a system of inequalities is any pair of values that make both inequalities to the system true.

Profile of a Graduate Capacities: Product Creation

Two-Variable Statistics 

Unit 3 

The unit begins with categorical data arranged in two-way tables that students are asked to analyze. Students then examine the relative frequencies of the combinations of those categorical variables. Students find the relative frequencies for combinations relative to the whole data set, as well as row or column relative frequencies to look at subgroups within categories. The row and column relative frequency tables are ultimately used to find evidence to determine if any associations are present in the data.

The unit then transitions to bivariate numerical data, which are visualized using scatter plots and lines of best fit. Students use technology to compute the lines of best fit and observe how well the linear models match the data. Correlation coefficients are used to quantify the goodness of fit for linear models.

The unit closes with an exploration of the difference between correlation and causal relationships, as well as an opportunity to apply this learning to areas of interest, like anthropology and sports.

Profile of a Graduate Capacities: Analyzing

Functions 

Unit 4 

In this unit, students expand and deepen their understanding of functions. They develop new knowledge and skills for communicating about functions clearly and precisely, investigate different kinds of functions, and hone their ability to interpret functions. Students also use functions to model a wider variety of mathematical and real-world situations.

The unit opens with a refresher on what functions are and what they are not. Students use descriptions, tables, and graphs to reason about the idea of “exactly one output for each input.” Then, students learn that function notation is an efficient way to communicate succinctly about functions and devote some focused time to interpret this new notation and use it. They continue this work throughout the unit, employing the notation to perform increasingly sophisticated mathematical work: to analyze and compare functions, to write rules of functions (primarily linear functions), to solve for an input, to graph functions, and more.

Next, students focus their attention on graphs of functions and on how they help to tell stories about the relationships between the quantities in the functions. Students interpret features of graphs and relate them to features of situations, using terms such as “maximum,” “minimum,” and “intercepts” to describe their observations. From a graph, students can see intervals where the values of a function increase or decrease. They learn to use average rates of change to more precisely describe how quickly these values rise or fall. Students also sketch graphs to depict qualitative behavior of functions.

Students then go on to take a closer look at the input and output of a function. They think about possible and reasonable input and output values and learn to identify the domain and range of a function based on contextual and graphical information. This new awareness of input and output in turn helps students make sense of piecewise-defined functions, in which different rules apply to different intervals of the domain, producing different sets of output values.

Two variations of piecewise functions are studied here: step functions and absolute value functions. The latter are introduced with the idea of absolute errors as an entry point. Thinking about “how far away from a value” primes students to regard the absolute value function as a distance function. The graph of such a function is a distinct V shape, which is convenient for noticing the graphical effects of changing an expression that defines a function.

Students close the unit by applying their insights about functions to model real-world situations and solve problems.

Profile of a Graduate Capacities: Idea Generation

Introduction to Exponential Functions 

Unit 5 

In this unit, students are introduced to exponential relationships. Students learn that exponential relationships are characterized by a constant quotient over equal intervals, and compare it to linear relationships which are characterized by a constant difference over equal intervals. They encounter contexts that change exponentially. These contexts are presented verbally and with tables and graphs. They construct equations and use them to model situations and solve problems. Students investigate these exponential relationships without using function notation and language so that they can focus on gaining an appreciation for critical properties and characteristics of exponential relationships.

Students subsequently view these new types of relationships as functions and employ the notation and terminology of functions (for example, dependent and independent variables). They study graphs of exponential functions both in terms of contexts they represent and abstract functions that don’t represent a particular context, observing the effect of different values of a and b on the graph of the function represented by f(x)=ab^x.

In this unit, students learn that the output of an increasing exponential function is eventually greater than the output of an increasing linear function for the same input.

Profile of a Graduate Capacities: Analyzing

Quadratic Functions 

Unit 6 

This unit begins with students contrasting quadratic growth with linear and exponential growth. They further observe that eventually these quadratic patterns grow more quickly than linear patterns but more slowly than exponential patterns. Students examine the important example of free-falling objects whose height over time can be modeled with quadratic functions. They use tables, graphs, and equations to describe the movement of these objects, eventually looking at the situation where a projectile is launched upward. Through this investigation, students also begin to appreciate how the different coefficients in a quadratic function influence the shape of the graph. In addition to projectile motion, students examine other situations represented by quadratic functions including area and revenue.

Next, students examine the standard and factored forms of quadratic expressions. They investigate how each form is useful for understanding the graph of the function defined by these equivalent forms.

Finally, students investigate the vertex form of a quadratic function and understand how the parameters in the vertex form influence the graph. They learn how to determine the vertex of the graph from the vertex form of the function. They also begin to relate the different parameters in the vertex form to the general ideas of horizontal and vertical translation and vertical stretch, ideas which will be investigated further in a later course.

Profile of a Graduate Capacities: Analyzing

Quadratic Equations 

Unit 7 

In this unit, students interpret, write, and solve quadratic equations. They see that writing and solving quadratic equations enables them to find input values that produce certain output values.

Students begin solving quadratic equations by reasoning. Next, students learn that equations of the form (x - m)(x - n) = 0 can be easily solved by applying the zero product property, which says that when two factors have a product of 0, one of the factors must be 0. When the equations are not in factored form, students rearrange them so that one side is 0, and rewrite the expressions from standard form to factored form. Students soon recognize that not all quadratic expressions in standard form can be rewritten into factored form. Even when it is possible, finding the right two numbers may be tedious, so another strategy is needed.

Students encounter perfect squares and notice that solving a quadratic equation is pretty straightforward when the equation contains a perfect square on one side and a number on the other. They learn that we can put equations into this helpful format by completing the square, that is, by rewriting the equation such that one side is a perfect square. Although this method can be used to solve any quadratic equation, it is not practical for solving all equations. This challenge motivates the quadratic formula.

Once introduced to the formula, students apply it to solve contextual and abstract problems, including those that they couldn’t previously solve. After gaining some experience using the formula, students investigate how it is derived. They find that the formula essentially encapsulates all the steps of completing the square into a single expression. Just like completing the square, the quadratic formula can be used to solve any equation, but it may not always be the quickest method. Students consider how to use the different methods strategically.

Throughout the unit, students see that solutions to quadratic equations are often irrational numbers. Students reason about whether such sums and products are rational or irrational.

Toward the end of the unit, students revisit the vertex form and recall that it can be used to identify the maximum or minimum of a quadratic function. Previously students learned to rewrite expressions from vertex form to standard form. Now they can go in reverse—by completing the square. Being able to rewrite expressions in vertex form allows students to effectively solve problems about maximum and minimum values of quadratic functions.

In the final lesson, students integrate their insights and choose appropriate strategies to solve an applied problem and a mathematical problem (a system of linear and quadratic equations).

Profile of a Graduate Capacities: Analyzing

One-Variable Statistics 

Unit 1 

The unit begins with an opportunity for students to review ideas from previous courses while taking the analysis of the data displays a little deeper. The lessons build on student understanding gained in middle school grades of statistical variability, ability to describe distributions, and informally comparing distributions. They represent and interpret data using data displays such as dot plots, histograms, and box plots. They describe distributions using the appropriate terminology such as “symmetric,” “skewed,” “uniform,” “bimodal,” and “bell-shaped.” They create data displays and calculate summary statistics using technology, then interpret the values in context. They recognize a relationship between the shape of a distribution and the mean and median. They compare data sets with different measures of variability and interpret data sets with greater interquartile ranges as having greater variability.

Students explore outliers and compare data sets using measures of center and measures of variability. They recognize outliers, investigate their source, make decisions about excluding them from the data set, and understand how the presence of outliers impacts measures of center and measures of variability.

Lastly, students get a chance to practice their skills by collecting data and analyzing the values. In the culminating activity, students pose and answer a statistical question by designing an experiment, collecting data, and analyzing data.

Profile of a Graduate Capacities: Analyzing

Linear Equations, Inequalities & Systems 

Unit 2

The unit begins with students learning to think of equations as a way to represent constraints or limitations on quantities. Students also see that graphs of equations can help us make sense of constraints and identify values that satisfy them. Next, students investigate different ways to express the same relationship or constraint - by analyzing and writing equivalent equations. They look at moves than can transform one equation to an equivalent equation, recognizing that these are the moves we make to solve equations. The focus here is not only on identifying acceptable moves for solving, but also on explaining why these moves keep each subsequent equation true and maintain the solutions of the original equation.

Next, students encounter situations that involve two or more constraints and realize that systems of equations are helpful for representing these constraints. Students learn to solve systems of equations by elimination and explain why the steps taken to eliminate a variable are valid. Additionally, students reinforce their awareness that a system of equations could have one solution, no solutions, or infinitely many solutions.

Finally, students explore inequalities in one and two variables. They see that inequalities are a handy way to express constraints and can be satisfied by a range of values rather than a single value. Students see that a solution to an inequality is a value or a pair of values that make the inequality true and that the solution to a system of inequalities is any pair of values that make both inequalities to the system true.

Profile of a Graduate Capacities: Product Creation

Two-Variable Statistics 

Unit 3 

The unit begins with categorical data arranged in two-way tables that students are asked to analyze. Students then examine the relative frequencies of the combinations of those categorical variables. Students find the relative frequencies for combinations relative to the whole data set, as well as row or column relative frequencies to look at subgroups within categories. The row and column relative frequency tables are ultimately used to find evidence to determine if any associations are present in the data.

The unit then transitions to bivariate numerical data, which are visualized using scatter plots and lines of best fit. Students use technology to compute the lines of best fit and observe how well the linear models match the data. Correlation coefficients are used to quantify the goodness of fit for linear models.

The unit closes with an exploration of the difference between correlation and causal relationships, as well as an opportunity to apply this learning to areas of interest, like anthropology and sports.

Profile of a Graduate Capacities: Analyzing

Functions 

Unit 4 

In this unit, students expand and deepen their understanding of functions. They develop new knowledge and skills for communicating about functions clearly and precisely, investigate different kinds of functions, and hone their ability to interpret functions. Students also use functions to model a wider variety of mathematical and real-world situations.

The unit opens with a refresher on what functions are and what they are not. Students use descriptions, tables, and graphs to reason about the idea of “exactly one output for each input.” Then, students learn that function notation is an efficient way to communicate succinctly about functions and devote some focused time to interpret this new notation and use it. They continue this work throughout the unit, employing the notation to perform increasingly sophisticated mathematical work: to analyze and compare functions, to write rules of functions (primarily linear functions), to solve for an input, to graph functions, and more.

Next, students focus their attention on graphs of functions and on how they help to tell stories about the relationships between the quantities in the functions. Students interpret features of graphs and relate them to features of situations, using terms such as “maximum,” “minimum,” and “intercepts” to describe their observations. From a graph, students can see intervals where the values of a function increase or decrease. They learn to use average rates of change to more precisely describe how quickly these values rise or fall. Students also sketch graphs to depict qualitative behavior of functions.

Students then go on to take a closer look at the input and output of a function. They think about possible and reasonable input and output values and learn to identify the domain and range of a function based on contextual and graphical information. This new awareness of input and output in turn helps students make sense of piecewise-defined functions, in which different rules apply to different intervals of the domain, producing different sets of output values.

Two variations of piecewise functions are studied here: step functions and absolute value functions. The latter are introduced with the idea of absolute errors as an entry point. Thinking about “how far away from a value” primes students to regard the absolute value function as a distance function. The graph of such a function is a distinct V shape, which is convenient for noticing the graphical effects of changing an expression that defines a function.

Students close the unit by applying their insights about functions to model real-world situations and solve problems.

Profile of a Graduate Capacities: Idea Generation

Introduction to Exponential Functions 

Unit 5 

In this unit, students are introduced to exponential relationships. Students learn that exponential relationships are characterized by a constant quotient over equal intervals, and compare it to linear relationships which are characterized by a constant difference over equal intervals. They encounter contexts that change exponentially. These contexts are presented verbally and with tables and graphs. They construct equations and use them to model situations and solve problems. Students investigate these exponential relationships without using function notation and language so that they can focus on gaining an appreciation for critical properties and characteristics of exponential relationships.

Students subsequently view these new types of relationships as functions and employ the notation and terminology of functions (for example, dependent and independent variables). They study graphs of exponential functions both in terms of contexts they represent and abstract functions that don’t represent a particular context, observing the effect of different values of a and b on the graph of the function represented by f(x)=ab^x.

In this unit, students learn that the output of an increasing exponential function is eventually greater than the output of an increasing linear function for the same input.

Profile of a Graduate Capacities: Analyzing

Quadratic Functions 

Unit 6 

This unit begins with students contrasting quadratic growth with linear and exponential growth. They further observe that eventually these quadratic patterns grow more quickly than linear patterns but more slowly than exponential patterns. Students examine the important example of free-falling objects whose height over time can be modeled with quadratic functions. They use tables, graphs, and equations to describe the movement of these objects, eventually looking at the situation where a projectile is launched upward. Through this investigation, students also begin to appreciate how the different coefficients in a quadratic function influence the shape of the graph. In addition to projectile motion, students examine other situations represented by quadratic functions including area and revenue.

Next, students examine the standard and factored forms of quadratic expressions. They investigate how each form is useful for understanding the graph of the function defined by these equivalent forms.

Finally, students investigate the vertex form of a quadratic function and understand how the parameters in the vertex form influence the graph. They learn how to determine the vertex of the graph from the vertex form of the function. They also begin to relate the different parameters in the vertex form to the general ideas of horizontal and vertical translation and vertical stretch, ideas which will be investigated further in a later course.

Profile of a Graduate Capacities: Analyzing

Quadratic Equations 

Unit 7 

In this unit, students interpret, write, and solve quadratic equations. They see that writing and solving quadratic equations enables them to find input values that produce certain output values.

Students begin solving quadratic equations by reasoning. Next, students learn that equations of the form (x - m)(x - n) = 0 can be easily solved by applying the zero product property, which says that when two factors have a product of 0, one of the factors must be 0. When the equations are not in factored form, students rearrange them so that one side is 0, and rewrite the expressions from standard form to factored form. Students soon recognize that not all quadratic expressions in standard form can be rewritten into factored form. Even when it is possible, finding the right two numbers may be tedious, so another strategy is needed.

Students encounter perfect squares and notice that solving a quadratic equation is pretty straightforward when the equation contains a perfect square on one side and a number on the other. They learn that we can put equations into this helpful format by completing the square, that is, by rewriting the equation such that one side is a perfect square. Although this method can be used to solve any quadratic equation, it is not practical for solving all equations. This challenge motivates the quadratic formula.

Once introduced to the formula, students apply it to solve contextual and abstract problems, including those that they couldn’t previously solve. After gaining some experience using the formula, students investigate how it is derived. They find that the formula essentially encapsulates all the steps of completing the square into a single expression. Just like completing the square, the quadratic formula can be used to solve any equation, but it may not always be the quickest method. Students consider how to use the different methods strategically.

Throughout the unit, students see that solutions to quadratic equations are often irrational numbers. Students reason about whether such sums and products are rational or irrational.

Toward the end of the unit, students revisit the vertex form and recall that it can be used to identify the maximum or minimum of a quadratic function. Previously students learned to rewrite expressions from vertex form to standard form. Now they can go in reverse—by completing the square. Being able to rewrite expressions in vertex form allows students to effectively solve problems about maximum and minimum values of quadratic functions.

In the final lesson, students integrate their insights and choose appropriate strategies to solve an applied problem and a mathematical problem (a system of linear and quadratic equations).

Profile of a Graduate Capacities: Analyzing

Equations and Inequalities

Unit 1 

This brief unit is a quick refresher of fundamental Algebra I topics including factoring polynomials, simplifying rational expressions, solving single variable equations and inequalities. Students will use these skills throughout the entire course.

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Relations and Functions

Unit 2

In this unit we move from working with single variables to multiple variables in equations. Functions and function notation will be the focus of this unit and every unit after this unit. Students will understand the concept of function, function notation, types of functions, transformations of functions, operations on functions, inverse functions and graphing functions. Students will be able to identify the domain and range of a function. Students should be able to work with functions in multiple representations: algebraic, graph and table of values.

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Quadratics Equations and Complex Numbers

Unit 3

The goal of this unit is for students to become fluent in interpreting, solving, and graphing quadratic functions with rational and irrational solutions as well as complex roots. The connection between completing the square and equations of circles is made. Students model using quadratic functions.

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Polynomial Functions

Unit 4

We move from quadratics to a study of polynomials and the relationship between the degree, the number of terms and the zeros. Multiplicity of zeros will be investigated and students will discover the relationship between the number of zeros the graph. The Rational Roots Theorem, Remainder Theorem and Factor Theorem will also be investigated in this unit. 

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Rational Expressions and Functions

Unit 5

In this unit students will extend their understanding of polynomials functions and their graphs to rational functions and their graphs. Students are encouraged to connect operations on rational expressions to operations on fractions learned in earlier math courses. Polynomial and rational inequalities are also explored in this unit. 

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Exponential and Log Functions

Unit 6 

This unit is the study of exponential and logarithmic functions. Understanding the inverse relationship between exponential and logarithmic functions is important. The properties and rules of logarithms will be related to exponential rules and then used in application problems including Newton's Law of Cooling, compound interest and exponential growth and decay.

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Trigonometric Functions

Unit 7

This is the first time that many students will see any trigonometry beyond SOHCAHTOA, for example, radian measure, law of sines and law of cosines, the reciprocal trigonometric functions, trigonometric graphs. We chose specifically to emphasize the Unit Circle, and to simply use "circle definitions" for problems like sin(240). Students should come away from this unit feeling like many trigonometric topics can simply be done with x, y, and r (circle definitions). There are many variations of this type of problem, but students should feel the unity among them. Students need to know the basic side patterns for special right triangles along with how to draw angles in standard position.

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Equations and Inequalities

Unit 1 

This short unit (4-6 days) focuses on prior mathematical knowledge of solving multi-step equations and inequalities. Students are expected to apply their algebraic knowledge and understanding through the application to real-world problems.

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Relations and Functions

Unit 2

Students will appreciate the importance of functions and their domains and will use input/output language throughout unit. A significant part of this unit is transformations on parent functions, having students understand how parameters affecting the inputs differ from the parameters affecting the outputs. Graph analysis is introduced but somewhat limited in scope. Students will also explore systems of linear equations, systems of inequalities, and linear programming to see real world applications.

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Quadratic Equations and Complex Numbers 

Unit 3

Students will understand what a radical is and how to simplify and combine in order to solve quadratics that are not factorable. Students will learn a variety of ways to solve quadratic equations and then will be challenged to choose the most efficient method for solving a given equation. Students can visualize the solutions of quadratic equations through graphing (e.g., min, max, transformations, complex roots). Completing the squares is used as an introduction to the equation of circles to further understanding of transformations. Students will demonstrate their efficiency through the solving of application problems.

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Polynomial Functions

Unit 4

Students will perform operations on polynomials, adding, subtracting, multiplying and dividing. They will be able to graph polynomial functions by factoring to find the zeroes and understanding end behavior and multiplicities.

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Rational Expressions and Functions

Unit 5

Students will perform operations on rational expressions: simplifying, adding, subtracting, multiplying and dividing. They will also solve rational equations. They will be able to graph rational functions by finding the vertical and horizontal asymptotes, intercepts, and testing points. 

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Exponential, Logarithmic and Additional Inverse Functions

Unit 6

Students will understand exponential functions and their graphs. Students will be introduced to logarithms as the inverse of exponential functions. They will use the properties of logarithms to solve both exponential and logarithmic equations. Real world application problems will be introduced.

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Trigonometric Functions

Unit 7 

This is the first time that many students will see any trigonometry beyond SOHCAHTOA, for example, radian measure, reciprocal trig functions, trig graphs. We introduce trig functions of all angles using a circle and reference angles and then move on to use the Unit Circle as a special case. Students learn about the basic characteristics of sine and cosine graphs and then learn about transformations of these functions.

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Rebuilding Algebra Skills

Unit A

This brief unit is a review of Pre-Algebra and Algebra 1 topics that are integral for succeeding in Algebra 2. Care should be given to teach each topic in a way that illuminates the reasons behind the methodology. For example, students should understand why, when terms are on the same side of an equation, they are combined, but when on opposite sides of the equation, we need to add the opposite to eliminate a term from one side, or why absolute value equations may have two solutions, one, or no solutions at all.

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Equations on the Coordinate Plane

Unit B

The purpose of this unit is to use math to analyze situations in which the rate of change is constant and to model those situations using linear equations. Students should make a connection between tabular, algebraic, and graphic representations of relations. In later units students will use the concepts and skills from this unit to work with quadratic and exponential functions.

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Quadratic Equations and Parabolas

Unit C

The purpose of this unit is to move beyond linear functions and to learn strategies to solve quadratic equations. Students should understand that the power of 2 creates a specific shaped graph (parabola). Students should also learn the importance of the complex number system, and should be taught about the history of complex numbers not being all that different from the history of negative numbers.

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Functions

Unit D

The focus of Unit D is for students to learn what is a mathematical function and its importance in problem solving. Students will also explore and learn to use the concept of function notation. Even though function notation is awkward to learn and seems more cumbersome, it is a great tool that allows mathematicians to communicate more clearly. Students will learn to work flexibly between all representations of a relation or function (table, list, equation, graph, and mapping diagram).

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Trigonometry

Unit E

Students will learn the basics of right triangle trigonometry, and will be able to apply trig ratios to solve word problems. Students will learn how to measure angles using radians, how to sketch angles in standard position, etc. The goal of this unit is to expose students to enough trigonometry for them to understand its value in the real world and to be successful in higher math.

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Exponential and Logarithmic Functions

Unit F

The purpose of this unit to expose students to ways of manipulating expressions using exponents. Students are expected to have a conceptual understanding of the rules around exponents and logarithms. They should explore the logic behind the development of negative exponents, zero as an exponent, and rational exponents. These should not just be taught as rules.

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Introduction to Geometry

Unit A

This unit introduces students to the majority of terminology used in Geometry. Transformations, logical thinking and proofs are all part of the unit and will be referred to throughout the course. Students will be able to complete a two-column geometric proof by the end of the unit. Geometric software, along with compass and straightedge, will be used for constructions.

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Congruent Triangles

Unit B

This unit focuses on triangle classifications and proving triangles congruent. Proof is a very important concept throughout the unit. Students should become fluent in completing proofs by the end of this unit by seeing the patterns and structure within proofs.

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Lines in a Plane

Unit C

In this unit students develop proofs to fairly complex problems. Along with two column proofs students are encouraged to give verbal and/or paragraph arguments always with the idea of a clear, logical argument with mathematical justification as a priority. 

A major focus in this unit is on quadrilaterals. Properties of quadrilaterals are introduced through various discovery activities in order to build a quadrilateral tree and to be able to classify the special quadrilaterals. Parallelograms are explored in further detail as students learn about sufficient conditions for parallelograms. Coordinate plane geometry is used to classify quadrilaterals. 

Finally, the students move beyond two dimensional shapes and study lines and planes in three dimensional space.

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Polygons

Unit D

We move from quadrilaterals to this unit which explores triangles and polygons and the measures of their interior and exterior angles, including "regular" polygons. Students are encouraged to see diagrams and shapes as compositions of smaller, often repeated, shapes. Students will learn the concept of "similar" polygons and the ratios of their corresponding sides, perimeters and areas.

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Right Triangles

Unit E

This unit is an exploration of families of right triangles, the Pythagorean theorem, and right triangle trigonometry. It includes the 30-60-90 and 45-45-90 right triangles and the relationship between the lengths of their sides. Word problems focus on angles of elevation and angles of depression.

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Circles

Unit F

During this unit students use many concepts learned throughout the course to solve problems involving circles. Segments and angles associated with circles are examined. Problems on the coordinate plane again bridge Algebra and Geometry skills and concepts.

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Area, Surface Area and Volume

Unit G

This short unit on area, surface area and volume gives students an opportunity to apply the Geometry they have learned throughout the year. The goal for students is to understand the formulas involved through deriving the formulas rather than simply memorize the formulas.

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Advanced Coordinate Geometry

Unit H

In this final unit, students link what they have learned in Algebra I about graphing equations to the concepts they have learned throughout this Geometry course. The work in this unit will create a smooth bridge to the work done in Algebra II. 

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Introduction to Geometry

Unit A

This unit introduces students to the majority of terminology and core concepts used in Geometry. Constructions, transformations, logical thinking and proofs are all part of the unit and will be referred to throughout the course. Students will be able to complete a two-column geometric proof by the end of the unit. Geometric software, along with compass and straightedge, may be used for constructions.

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Congruent Triangles

Unit B

This unit focuses on triangle classifications and proving triangles congruent. Proof is a fundamental concept throughout the unit. Proving and using congruent triangles will be used throughout the course.

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Lines in a Plane

Unit C

In this unit we de-emphasize two column proofs and concentrate more heavily on diagram type problems and the notion that there are other ways to make a mathematical proof including verbal and paragraph arguments but the idea of a clear, logical argument with mathematical justification for each step remains constant. 

A major focus in this unit is on quadrilaterals. Properties of quadrilaterals are introduced through various discovery activities in order to build a quadrilateral tree and to be able to classify the special quadrilaterals. Parallelograms are explored in further detail as students learn about sufficient conditions for parallelograms. Coordinate plane geometry is incorporated frequently to bridge Algebra and Geometry concepts.

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Polygons

Unit D

We move from quadrilaterals to this unit which explores triangles and polygons and the measures of their interior and exterior angles, including "regular" polygons. Students are encouraged to see diagrams and shapes as compositions of smaller, often repeated, shapes. Students will learn the concept of "similar" polygons and the ratios of their corresponding sides, perimeters and areas. Formal proofs are not done in this unit.

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Right Triangles

Unit E

This unit is an exploration of families of right triangles, the Pythagorean theorem, and right triangle trigonometry. It includes the 30-60-90 and 45-45-90 right triangles and the relationship between the lengths of their sides. Word problems focus on angles of elevation and angles of depression.

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Circles

Unit F

During this unit students use many concepts learned throughout the course to solve problems involving circles. Segments and angles associated with circles are examined. Problems on the coordinate plane again bridge Algebra and Geometry. Proofs are not done in this unit.

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Area, Surface Area, Volume

Unit G

This short unit on area, surface area and volume gives students an opportunity to apply the Geometry they have learned throughout the year. Formulas are provided to students so they focus can be on application and complex thinking rather than recall of formulas. Students solve a variety of application problems that involve surface area, area and/or volume.

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Introduction to Geometry

Unit A 

This unit introduces students to the majority of terminology used in Geometry. Constructions, transformations, logical thinking and proofs are all part of the unit and will be referred to throughout the course. Students will be able to complete a two-column geometric proof by the end of the unit. Geometric software, along with compass and straightedge, will be used for constructions and transformations.

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Triangles

Unit B 

This unit focuses on triangle classifications and proving triangles congruent. Properties of triangles are applied to proofs so that students have experienced with the proof process. Proofs can be differentiated to students as they develop skill in the process by using word banks, missing statements or reasons, or cut-up proofs where student must re-order steps to establish sequence. Students will be extended to create 5-10 step proofs without assistance by the end of the unit. Segments that can be drawn in a triangle and their properties are explored.

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Similarity

Unit C 

This unit extends students’ understanding of relationships between triangles (and shapes in general). They will learn what it means for shapes to be similar (congruent angles and proportional sides) and solve for sides of similar triangles. The PBA will have students measuring the height of unknown objects using similarity. The second half of the unit focuses on right triangles, using the idea of similarity to introduce the concept of Trigonometry.

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Polygons and Quadrilaterals

Unit D 

In this unit, students learn first about polygons and then focus on quadrilaterals. Properties of quadrilaterals are introduced through various discovery activities in order to build a quadrilateral tree and to be able to classify the special quadrilaterals. Parallelograms are explored in further detail as students learn about sufficient conditions for parallelograms. Areas of quadrilaterals are examined, and coordinate plane geometry is used to classify quadrilaterals.

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Circles

Unit E

This unit provides students a thorough study of circles. Students learn about the segments (in particular tangents) and angles (central and inscribed) associated with circles. Equations of circles on the coordinate plane are taught. The formula for circle circumference is reviewed, and students explore an informal proof of the area of a circle. Arc length and sector area are introduced.

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Area, Surface Area, Volume

Unit F

This unit explores the 3D world of surface area and volume. Students will discover the formulas for prisms, cylinders, cones, pyramids, and spheres. They will then use these concepts to determine the surface area and volume of composite figures. 2D cross-sections of 3D objects will be investigated with online applets. Finally, the effects of dilating dimensions on surface area and volume will be introduced.

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Probability

Unit G

This brief unit introduces students to probability. It begins by arranging information from sets into Venn Diagrams. Then the Venn Diagrams are used to determine probabilities. Next, some geometric probabilities related to length and area are explored. Finally, the fundamental counting principle is covered and applied to permutations and combinations.

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Rebuilding the Prerequisite Skills for Calculus

Unit A 

This course is not meant to take the place of AB Calculus or Calculus I but meant to be an introduction to the basic conceptual foundations of differentiation and integration and the algebraic applications used in formulaic differentiation and integration. Major concepts taught are differentiation (Chain rule and implicit application, curve sketching and related rates) and integration (substitution, application, simple initial value problems). 

Even though this is an introductory course, students will be required to use specific, correct notation in all written work. Limit notation, parentheses, etc, if left out, completely change the meaning of the written expression. 

The initial unit serves as a summarized review of the concepts taught in PreCalculus which are a prerequisite to the study of differentiation and integration in calculus. Students may spend up to 25% of the course time in this unit. Students are encouraged to work in groups to help each other as needed to strengthen skills and understanding.

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Limits and Continuity

Unit B 

Students are introduced to the concept of a limit. Different representations will provide students with a deep understanding of limits. Simple computations of limits are introduced. Students will understand continuity of its' implications. Formative assessments for each section can be done as pairs. The teacher can decide whether assessments are given as open note or not. 

For all written exercises, students must give a specific justification for any answer. Notation used must be correct. 

Prerequisite skills to review: 

• toolkit graphs 
• rational, absolute value functions 
• radical functions 
• domain restrictions 
• factoring polynomials

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Derivatives

Unit C 

Students are introduced to the derivative as the slope of the line tangent to a function at any point. Students will practice finding derivative first using the limit definition and then using the rules and algebraic computation and the chain rule. Student will study implicit differentiation and apply this concept to related rate problems.

By then end of this unit students should be able to determine the equation of the tangent to a graph using explicit or implicit derivatives. Students should also be able to distinguish between position, velocity and acceleration and how they relate to each other in terms of being derivatives of each other.
There are many places to be careful and apply numerous rules at the same time. If extra time is necessary, take that time. Students must know derivative before they can do integral.

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Using Calculus to Sketch Curves

Unit D 

Students will use methods of calculus to determine critical points, points of local and absolute extrema, intervals of increasing and/or decreasing, points of inflection, and intervals of concavity. Together with material already covered (such as x and y intercept(s), vertical and/or horizontal, and/or slant asymptotes, etc) they will draw clear sketches of graphs without using the graphing calculator. In addition, linear approximations and L’Hopital’s Rule are covered. Students are given time throughout the unit to work with peers to solve problems.

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Integration

Unit E 

Students are introduced to the concept of an antiderivative using the concepts learned in Unit C. Using Riemann sums and sigma notation, students will see the connection of an integral to the area under a curve. Students will explore definite and indefinite integrals. Students will learn the integration technique of substitution. 

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Introduction to Computer Programming with Python

Unit 1

Throughout this course students will develop algorithms and apply logic to use a computer to solve and model a real world problem.

Throughout this course students learn how to use logic and sets of instructions to have a computer accomplish a task.

In this unit, students will gain a general understanding of what a computer program is, how it works, and how to write one using a language such as Visual Basic and an Integrated Development Language such as Microsoft Visual Studio. Students will understand the flow of a program, how to respond to user-generated events, how to add user interface elements to a form, and how to save and run a program. Students will become acquainted with much of the terminology as well as the technology that is used throughout the course.

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Working with Variables, Constants and Calculations

Unit 2

In this unit, students will learn how a computer stores and manipulates various types of data including numeric and textual information. Students will learn how to perform basic arithmetic calculations such as adding, subtracting, multiplying, and dividing, as well as how to write code to count and accumulate values.

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Conditional Logic and Decision Making

Unit 3

Students will develop the ability to read, write, and use conditional statements to model the decision-making process used in the real world. Students will build on their ability to use variables and calculations by applying conditional logic to their use. Students will first learn how to create simple, single variable conditional statements, and will eventually learn how to model more complex decision making with compound conditional statements and singly nested conditional statements.

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Functions and Subroutines

Unit 4

In this unit, students will learn how we can break large tasks into smaller, reusable units of work called subroutines and functions. Students will learn the value of subroutines and functions, how to write them, and how to pass arguments to them. Students will continue exploring how to use built-in functions to efficiently code solutions to problems.

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Iteration and Computer Simulation

Unit 5

In this unit, students will learn how to solve problems that require looping, also known as iterations. Students will learn several different ways to structure loops, and how iteration can be a valuable problem solving technique. Students will also gain experience modeling real world events through computer simulations that are implemented using loops.

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Working with Strings

Unit 6

This unit marks a shift in the course from a focus on learning the building blocks of computer programming to using those building blocks to solve problems. In this unit, students are exposed to many of the built-in methods of the String class, and then use these methods to solve challenges involving strings that require many of the skills they’ve learned in prior units.

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Lists, Arrays and Problem Solving

Unit 7

This unit continues the theme of using a computer program as a problem solving tool. Students will learn how to use arrays and lists to represent real-world objects and how to manipulate those lists to arrive at solutions. A general four-step approach to problem solving will be explored, and students will have an opportunity to practice the approach on a series of challenging exercises.

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Object Oriented Concepts and Culminating Activity

Unit 8

In this final unit, students will gain an appreciation for object oriented programming concepts including inheritance, encapsulation, and polymorphism. Students will also have the opportunity to apply the knowledge they have learned throughout the course in a culminating programming activity.

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Trigonometry

Unit A

This unit covers all the basics of trigonometry, from radian measure to right triangle and unit circle definitions to graphing. These fundamentals will be built on in further units, so it is important students understand these concepts thoroughly, without relying on the calculator until the applications are taught.

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Analytic Trigonometry

Unit B

This unit extends the topics covered in unit A. It begins with simplifying expressions so that students understand how proving trigonometric identities depends on showing one side of an equation simplifies to the other without manipulating both sides simultaneously. Students also learn how to solve trigonometric equations, in particular when the angle has been multiplied by a factor within the trigonometric function. The graphing calculator is introduced here to show how the solutions may be verified. The second half of the unit covers many formulas that extend the number of angles for which exact values of the trigonometric functions may be found.

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Vectors

Unit C

This two part unit explores applications of trigonometry, most importantly vectors.

In part 1 the Laws of Sines and Cosines are derived, allowing us to solve for sides and angles in oblique (non-right) triangles. Students need to be aware that in the SSA case, there could be no, one, or two possible triangles and why this happens. Area of triangles is also covered at this time for the SAS and SSS cases (Heron’s Formula). After oblique triangles, most of the unit is spent on vectors, quantities with both magnitude and direction (velocity, force, etc). Students will learn how to express them in component form as well as a magnitude and direction angle. They will learn how to perform several operations on vectors, including the dot product. This operation is used to find angles between vectors and projections of vectors. Many applications of vectors are discussed, including plane and wind problems, force balancing, weights on ramps, and work. The unit finishes with new topic, complex numbers. By converting from a + bi form to a trigonometric form, some calculations (raising to powers and finding roots) can be done much quicker by using DeMoivre’s Theorem.

Part 2 extends the topic of vectors to 3-dimensional space. The unit begins by discussing the 3D coordinate system, including how to plot points, find distance between points, midpoints and equations of spheres. Vectors are useful in 3D to determine if points are collinear. Angles between vectors is revisited, along with the dot product. A new operation is taught, the cross product, which is only possible in 3D space. Students will learn that this operation creates a new vectors which is normal (perpendicular) to the plane containing the two original vectors and can be used to find area and volume of parallelogram-type figures. Finally, vectors are used to determine equations of lines and planes in 3D space and determining the distance between a point and a plane.

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Systems of Equations

Unit D

This short unit reviews systems of equations, with a new perspective of the graphing calculator. The traditional methods of substitution and elimination are covered, but with more complicated systems than just linear equations. The graphing calculator is used to confirm exact answers found algebraically and also used to solve equations that would be very difficult to solve by hand (ex: natural logs). No solution and infinite solutions results are discussed in the context of the intersections of the graphs of the equations. Linear systems of three variables are at first solved by hand through repeated elimination and back substitution, but these methods are quickly replaced with matrices on the graphing calculator. A brief overview of matrices is given, but most of the focus is on solving multivariable linear systems. Finally, systems are used for a new concept, partial fraction decomposition.

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Sequences, Series, and Probability

Unit E

This unit is all about recognizing patterns. Formulas for nth terms and summations of arithmetic and geometric sequences are derived. The concept of induction is introduced as a way to prove other formulas for series, divisibility, and inequalities. Then the binomial theorem is taught as an application of another pattern. Finally, combinations and permutations are covered and these counting principles are used to find probabilities.

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Topics in Analytic Geometry

Unit F

This large unit covers a variety of topics in Analytic Geometry (mostly graphing related). The concept of slope is analyzed with respect to the angles lines make with axes and other lines. A major portion of the unit focuses on the conic sections: parabolas, ellipses, and hyperbolas. These equations/graphs are introduced as loci of points and their various applications to the real world are explored, in particular their reflective properties. The rotation of their axes is not emphasized. Two new types of graphing are studied: parametric equations and polar equations which connect back to the conic sections.

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Introduction to Calculus

Unit G

This short unit introduces students to the concepts of limits and derivatives. Various techniques for determining limits are explored: graphically, numerically (table), comparing the one-sided limits, and algebraically or direct substitution, if possible. Limits are then applied to the slope of secant lines to determine the slope of a curve at a single point (slope of the tangent line) using the limit definition of a derivative. Infinite limits and limits of summations are discussed and if there is time, used to determine the area under a curve.

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Graphs and Equations

Unit A

In this unit, students review graphing and properties of linear equations. Technology such as graphing calculators will be used to model the linear relationship between two variables. Students will also review equation solving techniques which will be used throughout the remainder of the course.

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Functions and Their Graphs

Unit B

In this unit, students will be re-familiarized with the definition of functions, function notation, and operations on functions. Students will learn how to express the relationship between two variables as a function, how to graph these relationships on a graphing calculator and how to determine the optimal values of a function in an appropriate domain. Function families that will be covered in this unit include polynomials, rationals and radicals. A generalized transformation rule will be introduced and will be used for the remainder of the year.

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Exponential and Logarithmic Functions

Unit C

This unit covers the basic properties of exponential and logarithmic functions. Students will be graphing and solving exponential and logarithmic functions. Students will also model “real world” situations with exponential and logarithmic functions.

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Trigonometry

Unit D

This unit covers all the basics of trigonometry, from radian measure to right triangle and unit circle definitions to graphing. These fundamentals will be built upon in further units, so it is important students understand these concepts thoroughly, without relying on the calculator.

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Applications of Trigonometric Functions

Unit E

This unit will cover solving right and oblique triangles using Trigonometry. Right triangles will be solved using right triangle trigonometry and oblique triangles will be solved using the Law of Sines or the Law of Cosines. Students will apply these concepts to “real world” problems. Area will also be covered in this unit.

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Rebuilding the Foundations

Unit 1

Many students who take this course have used memorization in previous courses. This method of learning math is not sustainable and has led to frustration and only partial understanding. In this course, students will be encouraged to reach for complete conceptual understanding of every topic, and discouraged from memorizing all but a very few small things like definitions. Learning math this way is more rewarding, enjoyable experience, and students will feel empowered to continue on in mathematics.

The goal of this mini-unit is to rewire student thinking about concepts that are fundamental to Algebra. We will look at the familiar concepts of order of operations, fractions, mathematical properties, operations on polynomials, and graphing from a new perspective that will enable students to transfer their knowledge to more abstract applications in the future. Example: fractions will be taught by prime factorization so that students can transfer their knowledge to the simplification of rational expressions.

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Equations and Lines

Unit 2

In this unit students will understand that the way we solve equations is based on an understanding of the order of operations. There are certain equations that require more than just an algebraic step (absolute value requires a leap of logic). Literal equations are a great opportunity for students to hone their equation solving skills which will be beneficial in other classes, like science.
For writing equations of lines, students should get to the point where they no longer think of every problem as a unique case, but as all being basically the same, just slight variations. Students should appreciate the usefulness of all forms of lines, not just y = mx + b, and should be allowed to leave answers in any form. Students should leave the unit displaying confidence in their understanding of how slope can be interpreted in the real world.

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Functions and Transformations

Unit 3

Students will understand the importance of functions - they give us the ability to predict because they have only one output given an input. There is an emphasis on input/output language throughout unit. Piecewise, compositions, and inverses are explored. A significant part of this unit is about transformations, having students understand how parameters affecting the inputs differ from the parameters affecting the outputs. The ABCD method of transforming functions (with point mapping) will really help for when many different parameters are used at once. Graph analysis is introduced but somewhat limited in scope.

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Polynomials

Unit 4

Students will be extremely proficient and confident at factoring in order to be able to solve polynomials later in this unit and to facilitate ease with Unit 5: simplifying rational expressions and solving rational equations. Complex numbers are briefly touched on, as is the quadratic formula, but only as tools for solving higher degree polynomial equations. Students should know by the end of this unit that the number of solutions to a polynomial equation is the same as the degree of the polynomial (The Fundamental Theorem of Algebra). Students will graph polynomials with attention to intercepts and end behavior. Quadratics are considered an optional topic.

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Rational Functions

Unit 5

The main goals of this unit include improving students’ fluency with rational expressions - understanding difference between expressions and equations and the strategies/approaches we can use to simplify each. Students will build confidence to handle more complex math problems they will encounter in future math classes. Students will make connections between equations and graphs in terms of asymptotes and domain and limits.

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Radicals, Exponents, and Logarithms

Unit 6

This unit will help students build fluency with radicals and rational exponents. They should know why negative exponents mean “divide” and rational exponents are equivalent to radical notation. We want students to know that a logarithm is used to solve for a variable in the exponent. Logarithms are a way to solve difficult (near impossible) problems in a fast, easy way. Use the language for rational exponents: “8^(⅔) means: 2 of the 3 ‘identical factors’ that multiply to 8.”

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Trigonometry

Unit 7

This is the first time that many students will see any trigonometry beyond SOHCAHTOA, for example, radian measure, law of sines and law of cosines, the reciprocal trig functions, trig graphs. We chose specifically to deemphasize Unit Circle, and to simply use “circle definitions” for problems like sin(240). Students should come away from this unit feeling like many trig topics can simply be done with x, y, and r (circle definitions). There are many variations of this type of problem, but students should feel the unity among them. Students need to know the basic side patterns for special right triangles along with how to draw angles in standard position.

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Introduction to Single Variable Statistics

Unit A

Unit A begins with an overview of statistics and how they impact our lives. Students will examine univariate and bivariate data and make sense out of it using statistical methods and displays. Graphing calculators are used throughout the course.

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Research Design

Unit B

This unit explores the process of collecting and interpreting data. Students investigate sampling as a method of understanding information about populations. It includes discussion of uncertainty in samples and how the margin of error narrows as sample size grows. Students review articles with sampling and review the validity of the statistical processes used to obtain data. Experimentation is introduced. Students learn about the basic principles of experiment design, including: explanatory versus response variables, the definition of statistical significance, adjusting for confounding variables, and double blind experiments. Students explore the ethical complexities of experimentation in a review of the movie Miss Evers’ Boys, which is a historical account of the controversial Tuskegee Syphilis Study.

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Chance

Unit C

In this unit, we discuss the basic ideas and methods of probability. Our goal is not just to help students answer questions like “What’s the probability that you get no heads if you toss a fair coin 5 times?” We aim to show students the role that probability plays in statistical inference. Contrast the previous question with this one: “Suppose you toss a coin five times and get no heads. Is the coin fair?” That’s a statistics question, but you need to understand probability to answer it.

Probability is about much more than coins, dice, and cards. It’s about making decisions in the face of uncertainty. People use probability to assess the results of drug tests, to determine the strength of certain kinds of evidence in a court case, to set insurance premiums, to choose an investment strategy, and to weigh the risks and benefits of medical treatment options. Of course, probability also plays an integral role in games of chance, from state and national lotteries to casino favorites like slot machines, craps, roulette, and Texas Hold ‘Em. In Unit C, we try to strike a balance between applications involving games of chance (which motivated the study of probability in the first place) and interesting uses of probability in everyday life.

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Inference

Unit D

Unit D deals with the reasoning of statistical inference. It presents methods for estimating and testing claims about a population proportion. Discussion about confidence intervals builds on foundations laid in previous learning about normal distributions and sampling. Extensions are made to testing for an association between two categorical variables, and estimating and testing claims about a population mean.

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