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  • Core Connections Integrated III (2015)

Core Connections Integrated III (2015)

1: Investigations and Functions

1.1: Section 1.1

1.1.2: Using a Graphing Calculator to Explore a Function

Radical Functions

Compare the graph of a radical function to its equation. Vary the terms of the equation. Explore how the graph is translated and stretched by the changes to the equation. 5 Minute Preview


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1.1.3: Function Investigation

General Form of a Rational Function

Compare the equation of a rational function to its graph. Multiply or divide the numerator and denominator by linear factors and explore how the graph changes in response. 5 Minute Preview


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Rational Functions

Compare the graph of a rational function to its equation. Vary the terms of the equation and explore how the graph is translated and stretched as a result. Examine the domain on a number line and compare it to the graph of the equation. 5 Minute Preview


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1.1.4: Combining Linear Functions

Addition and Subtraction of Functions

Explore the graphs of two polynomials and the graph of their sum or difference. Vary the coefficients in the polynomials and investigate how the graphs change in response. 5 Minute Preview


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1.2: Section 1.2

1.2.3: Describing Data

Box-and-Whisker Plots

Construct a box-and-whisker plot to match a line plots, and construct a line plot to match a box-and-whisker plots. Manipulate the line plot and examine how the box-and-whisker plot changes. Then manipulate the box-and-whisker plot and examine how the line plot changes. 5 Minute Preview


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Real-Time Histogram

Try to click your mouse once every 2 seconds. The time interval between each click is recorded, as well as the error and percent error. Data can be displayed in a table, histogram, or scatter plot. Observe and measure the characteristics of the resulting distribution when large amounts of data are collected. 5 Minute Preview


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2: Transformations of Parent Graphs

2.1: Section 2.1

2.1.1: Transformig Quadratric Functions

Quadratics in Polynomial Form

Compare the graph of a quadratic to its equation in polynomial form. Vary the coefficients of the equation and explore how the graph changes in response. 5 Minute Preview


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Quadratics in Vertex Form

Compare the graph of a quadratic to its equation in vertex form. Vary the terms of the equation and explore how the graph changes in response. 5 Minute Preview


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2.2: Section 2.2

2.2.1: Transforming Other Parent Graphs

Exponential Functions

Explore the graph of an exponential function. Vary the coefficient and base of the function and investigate the changes to the graph of the function. 5 Minute Preview


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Radical Functions

Compare the graph of a radical function to its equation. Vary the terms of the equation. Explore how the graph is translated and stretched by the changes to the equation. 5 Minute Preview


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Rational Functions

Compare the graph of a rational function to its equation. Vary the terms of the equation and explore how the graph is translated and stretched as a result. Examine the domain on a number line and compare it to the graph of the equation. 5 Minute Preview


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Translating and Scaling Functions

Vary the coefficients in the equation of a function and examine how the graph of the function is translated or scaled. Select different functions to translate and scale, and determine what they have in common. 5 Minute Preview


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2.2.2: Describing (h, k) for Each Family of Functions

Quadratics in Vertex Form

Compare the graph of a quadratic to its equation in vertex form. Vary the terms of the equation and explore how the graph changes in response. 5 Minute Preview


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Radical Functions

Compare the graph of a radical function to its equation. Vary the terms of the equation. Explore how the graph is translated and stretched by the changes to the equation. 5 Minute Preview


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Translating and Scaling Functions

Vary the coefficients in the equation of a function and examine how the graph of the function is translated or scaled. Select different functions to translate and scale, and determine what they have in common. 5 Minute Preview


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2.2.3: Transformations of Functions

Graphs of Polynomial Functions

Study the graphs of polynomials up to the fourth degree. Vary the coefficients of the equation and investigate how the graph changes in response. Explore things like intercepts, end behavior, and even near-zero behavior. 5 Minute Preview


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2.2.4: Transforming Non-Functions

Circles

Compare the graph of a circle with its equation. Vary the terms in the equation and explore how the circle is translated and scaled in response. 5 Minute Preview


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3: Solving and Inequalities

3.1: Section 3.1

3.1.1: Strategies for Solving Equations

Absolute Value Equations and Inequalities

Solve an inequality involving absolute values using a graph of the absolute-value function. Vary the terms of the absolute-value function and vary the value that you are comparing it to. Then explore how the graph and solution set change in response. 5 Minute Preview


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Quadratics in Vertex Form

Compare the graph of a quadratic to its equation in vertex form. Vary the terms of the equation and explore how the graph changes in response. 5 Minute Preview


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Roots of a Quadratic

Find the root of a quadratic using its graph or the quadratic formula. Explore the graph of the roots and the point of symmetry in the complex plane. Compare the axis of symmetry and graph of the quadratic in the real plane. 5 Minute Preview


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3.1.3: Multiple Solutions to Systems of Equations

Solving Linear Systems (Matrices and Special Solutions)

Explore systems of linear equations, and how many solutions a system can have. Express systems in matrix form. See how the determinant of the coefficient matrix reveals how many solutions a system of equations has. Also, use a draggable green point to see what it means for an (x, y) point to be a solution of an equation, or of a system of equations. 5 Minute Preview


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Solving Linear Systems (Standard Form)

Solve systems of linear equations, written in standard form. Explore what it means to solve systems algebraically (with substitution or elimination) and graphically. Also, use a draggable green point to see what it means when (x, y) values are solutions of an equation, or of a system of equations. 5 Minute Preview


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3.2: Section 3.2

3.2.1: Solving Inequalities with One or Two Variables

Absolute Value Equations and Inequalities

Solve an inequality involving absolute values using a graph of the absolute-value function. Vary the terms of the absolute-value function and vary the value that you are comparing it to. Then explore how the graph and solution set change in response. 5 Minute Preview


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Quadratic Inequalities

Find the solution set to a quadratic inequality using its graph. Vary the terms of the inequality and the inequality symbol. Examine how the boundary curve and shaded region change in response. 5 Minute Preview


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3.2.2: Using Systems to Solve a Problem

Linear Programming

Use the graph of the feasible region to find the maximum or minimum value of the objective function. Vary the coefficients of the objective function and vary the constraints. Explore how the graph of the feasible region changes in response. 5 Minute Preview


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3.2.3: Applications of Systems of Inequalities

Linear Programming

Use the graph of the feasible region to find the maximum or minimum value of the objective function. Vary the coefficients of the objective function and vary the constraints. Explore how the graph of the feasible region changes in response. 5 Minute Preview


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4: Normal Distributions and Geometric Modeling

4.1: Section 4.1

4.1.2: Samples and the Role of Randomness

Polling: City

Poll residents in a large city to determine their response to a yes-or-no question. Estimate the actual percentage of yes votes in the whole city. Examine the results of many polls to help assess how reliable the results from a single poll are. See how the normal curve approximates a binomial distribution for large enough polls. 5 Minute Preview


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Polling: Neighborhood

Conduct a phone poll of citizens in a small neighborhood to determine their response to a yes-or-no question. Use the results to estimate the sentiment of the entire population. Investigate how the error of this estimate becomes smaller as more people are polled. Compare random versus non-random sampling. 5 Minute Preview


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Populations and Samples

Compare sample distributions drawn from population distributions. Predict characteristics of a population distribution based on a sample distribution and examine how well a small sample represents a given population. 5 Minute Preview


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4.3: Section 4.3

4.3.1: Relative Frequency Histograms

Histograms

Change the values in a data set and examine how the dynamic histogram changes in response. Adjust the interval size of the histogram and see how the shape of the histogram is affected. 5 Minute Preview


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4.3.2: The Normal Probability Density Function

Polling: City

Poll residents in a large city to determine their response to a yes-or-no question. Estimate the actual percentage of yes votes in the whole city. Examine the results of many polls to help assess how reliable the results from a single poll are. See how the normal curve approximates a binomial distribution for large enough polls. 5 Minute Preview


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Populations and Samples

Compare sample distributions drawn from population distributions. Predict characteristics of a population distribution based on a sample distribution and examine how well a small sample represents a given population. 5 Minute Preview


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5: Inverses and Logarithms

5.1: Section 5.1

5.1.1: "Undo" Equations

Function Machines 1 (Functions and Tables)

Drop a number into a function machine, and see what number comes out! You can use one of the six pre-set function machines, or program your own function rule into one of the blank machines. Stack up to three function machines together. Input and output can be recorded in a table and on a graph. 5 Minute Preview


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5.2: Section 5.2

5.2.1: The Inverse of an Exponential Function

Logarithmic Functions

Compare the equation of a logarithmic function to its graph. Change the base of the logarithmic function and examine how the graph changes in response. Use the line y = x to compare the associated exponential function. 5 Minute Preview


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5.2.2: Defining the Inverse of an Exponential Function

Logarithmic Functions

Compare the equation of a logarithmic function to its graph. Change the base of the logarithmic function and examine how the graph changes in response. Use the line y = x to compare the associated exponential function. 5 Minute Preview


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5.2.3: Investigating the Family of Logarithmic Functions

Logarithmic Functions

Compare the equation of a logarithmic function to its graph. Change the base of the logarithmic function and examine how the graph changes in response. Use the line y = x to compare the associated exponential function. 5 Minute Preview


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5.2.4: Transformations of Logarithmic Functions

Logarithmic Functions: Translating and Scaling

Vary the values in the equation of a logarithmic function and examine how the graph is translated or scaled. Connect these transformations with the domain of the function, and the asymptote in the graph. 5 Minute Preview


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6: Simulating Sampling Variability

6.1: Section 6.1

6.1.1: Simulations of Probability

Probability Simulations

Experiment with spinners and compare the experimental probability of particular outcomes to the theoretical probability. Select the number of spinners, the number of sections on a spinner, and a favorable outcome of a spin. Then tally the number of favorable outcomes. 5 Minute Preview


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Theoretical and Experimental Probability

Experiment with spinners and compare the experimental probability of a particular outcome to the theoretical probability. Select the number of spinners, the number of sections on a spinner, and a favorable outcome of a spin. Then tally the number of favorable outcomes. 5 Minute Preview


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6.1.2: More Simulations of Probability

Spin the Big Wheel! (Probability)

Step right up! Spin the big wheel! Each spin can result in no prize, a small prize, or a big prize. The wheel can be spun by 1, 10, or 100 players. Results are recorded on a frequency table or a circle graph. You can also design your own wheel and a sign that describes the probabilities for your wheel. 5 Minute Preview


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6.1.3: Simulating Sampling Variability

Polling: City

Poll residents in a large city to determine their response to a yes-or-no question. Estimate the actual percentage of yes votes in the whole city. Examine the results of many polls to help assess how reliable the results from a single poll are. See how the normal curve approximates a binomial distribution for large enough polls. 5 Minute Preview


Lesson Info
Launch Gizmo

6.2: Section 6.2

6.2.1: Statistical Test Using Sampling Variability

Polling: Neighborhood

Conduct a phone poll of citizens in a small neighborhood to determine their response to a yes-or-no question. Use the results to estimate the sentiment of the entire population. Investigate how the error of this estimate becomes smaller as more people are polled. Compare random versus non-random sampling. 5 Minute Preview


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6.2.2: Variability in Experimental Results

Polling: Neighborhood

Conduct a phone poll of citizens in a small neighborhood to determine their response to a yes-or-no question. Use the results to estimate the sentiment of the entire population. Investigate how the error of this estimate becomes smaller as more people are polled. Compare random versus non-random sampling. 5 Minute Preview


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7: Logarithms and Triangles

7.1: Section 7.1

7.1.1: Using Logarithms to Solve Exponential Equations

Logarithmic Functions

Compare the equation of a logarithmic function to its graph. Change the base of the logarithmic function and examine how the graph changes in response. Use the line y = x to compare the associated exponential function. 5 Minute Preview


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7.1.3: Writing Equations of Exponential Functions

Compound Interest

Explore compound interest in-depth, from compounded annually to compounded continuously. In addition, compare the END POINTS graph, with dots that fit an exponential curve, to the ALL TIME graph, which has a more step-like appearance. 5 Minute Preview


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7.2: Section 7.2

7.2.1: Determining Missing Parts of Triangles

Pythagorean Theorem

Explore the Pythagorean Theorem using a dynamic right triangle. Examine a visual, geometric application of the Pythagorean Theorem, using the areas of squares on the sides of the triangle. 5 Minute Preview


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Pythagorean Theorem with a Geoboard

Build right triangles in an interactive geoboard and build squares on the sides of the triangles to discover the Pythagorean Theorem. 5 Minute Preview


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Sine, Cosine, and Tangent Ratios

Reshape and resize a right triangle and examine how the sine of angle A, the cosine of angle A, and the tangent of angle A change. 5 Minute Preview


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7.2.4: The Ambiguous Case

Proving Triangles Congruent

Apply constraints to two triangles. Then drag the vertices of the triangles around and determine which constraints guarantee congruence. 5 Minute Preview


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Triangle Inequalities

Discover the inequalities related to the side lengths and angle measures of a triangle. Reshape and resize the triangle to confirm that these properties are true for all triangles. 5 Minute Preview


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8: Polynomials

8.1: Section 8.1

8.1.1: Sketching Graphs of Polynomial Functions

Graphs of Polynomial Functions

Study the graphs of polynomials up to the fourth degree. Vary the coefficients of the equation and investigate how the graph changes in response. Explore things like intercepts, end behavior, and even near-zero behavior. 5 Minute Preview


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Polynomials and Linear Factors

Create a polynomial as a product of linear factors. Vary the values in the linear factors to see how their connection to the roots of the function. 5 Minute Preview


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8.1.2: More Graphs of Polynomial Functions

Graphs of Polynomial Functions

Study the graphs of polynomials up to the fourth degree. Vary the coefficients of the equation and investigate how the graph changes in response. Explore things like intercepts, end behavior, and even near-zero behavior. 5 Minute Preview


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Polynomials and Linear Factors

Create a polynomial as a product of linear factors. Vary the values in the linear factors to see how their connection to the roots of the function. 5 Minute Preview


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8.2: Section 8.2

8.2.1: Writing Equations Using Complex Roots

Points in the Complex Plane

Identify the imaginary and real coordinates of a point in the complex plane. Drag the point in the plane and investigate how the coordinates change in response. 5 Minute Preview


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Quadratics in Vertex Form

Compare the graph of a quadratic to its equation in vertex form. Vary the terms of the equation and explore how the graph changes in response. 5 Minute Preview


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Roots of a Quadratic

Find the root of a quadratic using its graph or the quadratic formula. Explore the graph of the roots and the point of symmetry in the complex plane. Compare the axis of symmetry and graph of the quadratic in the real plane. 5 Minute Preview


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8.2.2: More Real and Complex Roots

Graphs of Polynomial Functions

Study the graphs of polynomials up to the fourth degree. Vary the coefficients of the equation and investigate how the graph changes in response. Explore things like intercepts, end behavior, and even near-zero behavior. 5 Minute Preview


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8.3: Section 8.3

8.3.1: Polynomial Division

Dividing Polynomials Using Synthetic Division

Divide a polynomial by dragging the correct numbers into the correct positions for synthetic division. Compare the interpreted polynomial division to the synthetic division. 5 Minute Preview


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8.3.2: Factors and Rational Zeros

Graphs of Polynomial Functions

Study the graphs of polynomials up to the fourth degree. Vary the coefficients of the equation and investigate how the graph changes in response. Explore things like intercepts, end behavior, and even near-zero behavior. 5 Minute Preview


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8.3.4: Special Cases of Factoring

Factoring Special Products

Choose the correct steps to factor a polynomial involving perfect-square binomials, differences of squares, or constant factors. Use the feedback to diagnose incorrect steps. 5 Minute Preview


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9: Trigonometric Functions

9.1: Section 9.1

9.1.2: Graphing the Sine Function

Sine Function

Compare the graph of the sine function with the graph of the angle on the unit circle. Drag a point along the sine curve and see the corresponding angle on the unit circle. 5 Minute Preview


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9.1.3: Unit Circle

Sine Function

Compare the graph of the sine function with the graph of the angle on the unit circle. Drag a point along the sine curve and see the corresponding angle on the unit circle. 5 Minute Preview


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9.1.4: Graphing and Interpreting the Cosine Function

Cosine Function

Compare the graph of the cosine function with the graph of the angle on the unit circle. Drag a point along the cosine curve and see the corresponding angle on the unit circle. 5 Minute Preview


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9.1.5: Defining a Radian

Radians

As factory belt operator, your job is to move boxes just the right distance on the belt, so they can be stamped for delivery. Your only controls are the radius and rotation of the belt’s wheel. How do you set these to get the distance right? See how this relates to arc length, and discover how radians help make this task easier. 5 Minute Preview


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9.1.6: Building a Unit Circle

Cosine Function

Compare the graph of the cosine function with the graph of the angle on the unit circle. Drag a point along the cosine curve and see the corresponding angle on the unit circle. 5 Minute Preview


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Sine Function

Compare the graph of the sine function with the graph of the angle on the unit circle. Drag a point along the sine curve and see the corresponding angle on the unit circle. 5 Minute Preview


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9.1.7: The Tangent Function

Tangent Function

Compare the graph of the tangent function with the graph of the angle on the unit circle. Drag a point along the tangent curve and see the corresponding angle on the unit circle. 5 Minute Preview


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9.2: Section 9.2

9.2.1: Transformations of y = sin(x)

Translating and Scaling Sine and Cosine Functions

Experiment with the graph of a sine or cosine function. Explore how changing the values in the equation can translate or scale the graph of the function. 5 Minute Preview


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9.2.2: One More Parameter for a Periodic Function

Translating and Scaling Sine and Cosine Functions

Experiment with the graph of a sine or cosine function. Explore how changing the values in the equation can translate or scale the graph of the function. 5 Minute Preview


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9.2.3: Period of a Trigonometric Function

Translating and Scaling Sine and Cosine Functions

Experiment with the graph of a sine or cosine function. Explore how changing the values in the equation can translate or scale the graph of the function. 5 Minute Preview


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9.2.4: Graph

Translating and Scaling Sine and Cosine Functions

Experiment with the graph of a sine or cosine function. Explore how changing the values in the equation can translate or scale the graph of the function. 5 Minute Preview


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10: Series

10.1: Section 10.1

10.1.1: Introduction to Arithmetic Series

Arithmetic Sequences

Find the value of individual terms in arithmetic sequences using graphs of the sequences and direct computation. Vary the common difference and examine how the sequences change in response. 5 Minute Preview


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Arithmetic and Geometric Sequences

Find the value of individual terms in an arithmetic or geometric sequence using graphs of the sequence and direct computation. Vary the common difference and common ratio and examine how the sequence changes in response. 5 Minute Preview


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10.2: Section 10.2

10.2.1: Geometric Series

Arithmetic and Geometric Sequences

Find the value of individual terms in an arithmetic or geometric sequence using graphs of the sequence and direct computation. Vary the common difference and common ratio and examine how the sequence changes in response. 5 Minute Preview


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Geometric Sequences

Explore geometric sequences by varying the initial term and the common ratio and examining the graph. Compute specific terms in the sequence using the explicit and recursive formulas. 5 Minute Preview


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10.3: Section 10.3

10.3.1: Using a Binomial Probability Model

Binomial Probabilities

Find the probability of a number of successes or failures in a binomial experiment using a tree diagram, a bar graph, and direct calculation. 5 Minute Preview


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Permutations and Combinations

Experiment with permutations and combinations of a number of letters represented by letter tiles selected at random from a box. Count the permutations and combinations using a dynamic tree diagram, a dynamic list of permutations, and a dynamic computation by the counting principle. 5 Minute Preview


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10.3.2: Pascal's Triangle and the Binomial Theorem

Binomial Probabilities

Find the probability of a number of successes or failures in a binomial experiment using a tree diagram, a bar graph, and direct calculation. 5 Minute Preview


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10.3.3: The Number e

Compound Interest

Explore compound interest in-depth, from compounded annually to compounded continuously. In addition, compare the END POINTS graph, with dots that fit an exponential curve, to the ALL TIME graph, which has a more step-like appearance. 5 Minute Preview


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Logarithmic Functions

Compare the equation of a logarithmic function to its graph. Change the base of the logarithmic function and examine how the graph changes in response. Use the line y = x to compare the associated exponential function. 5 Minute Preview


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12: Analytic Trigonometry

12.1: Section 12.1

12.1.2: Solutions to Trigonometric Equations

Cosine Function

Compare the graph of the cosine function with the graph of the angle on the unit circle. Drag a point along the cosine curve and see the corresponding angle on the unit circle. 5 Minute Preview


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Tangent Function

Compare the graph of the tangent function with the graph of the angle on the unit circle. Drag a point along the tangent curve and see the corresponding angle on the unit circle. 5 Minute Preview


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12.2: Section 12.2

12.2.1: Trigonometric Identities

Simplifying Trigonometric Expressions

Choose the correct steps to simplify a trigonometric function. Use step-by-step feedback to diagnose incorrect steps. 5 Minute Preview


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12.2.2: Proving Trigonometric Identities

Simplifying Trigonometric Expressions

Choose the correct steps to simplify a trigonometric function. Use step-by-step feedback to diagnose incorrect steps. 5 Minute Preview


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12.2.3: Angle Sum and Difference Identities

Sum and Difference Identities for Sine and Cosine

Choose the correct steps to evaluate a trigonometric expression using sum and difference identities. Use step-by-step feedback to diagnose incorrect steps. 5 Minute Preview


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