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- Foundations of Algebra (2009)

# Foundations of Algebra (2009)

### 1: Rational Numbers

1.5: Compare and Order Rational Numbers

Modeling Fractions (Area Models)

Model and compare fractions using area models. Set the denominators with the arrow buttons, and then set the numerators with the arrow buttons or by clicking in the models. Compare fractions visually, on a number line, or numerically using the least common denominator. 5 Minute Preview

1.6: Estimate with Rational Numbers

Estimating Sums and Differences

Estimate the sum or difference of two fractions using area models. Compare estimates to exact sums and differences. 5 Minute Preview

1.7: Add Rational Numbers

Fractions with Unlike Denominators

Find the sum or difference of two fractions with unlike denominators using graphic models. Find the least common denominator graphically. 5 Minute Preview

1.8: Subtract Rational Numbers

Fractions with Unlike Denominators

Find the sum or difference of two fractions with unlike denominators using graphic models. Find the least common denominator graphically. 5 Minute Preview

1.9: Multiply Rational Numbers

Multiplying Fractions

Multiply two fractions using an area model. Vary the vertical area to change one fraction and vary the horizontal area to change the other. Then examine the intersection of the areas to find the product. 5 Minute Preview

1.10: Divide Rational Numbers

Dividing Fractions

Divide fractions using area models. Adjust the numerators and denominators of the divisor and dividend and see how the area model and calculation change. 5 Minute Preview

Dividing Mixed Numbers

Choose the correct steps to divide mixed numbers. Use step-by-step feedback to diagnose and correct incorrect steps. 5 Minute Preview

### 2: Real Numbers

2.3: Perfect Squares and Square Roots

Square Roots

Explore the meaning of square roots using an area model. Use the side length of a square to find the square root of a decimal number or a whole number. 5 Minute Preview

2.4: Estimate Square Roots

Ordering and Approximating Square Roots

Order square roots on a number line. Approximate the square roots using the side lengths of square regions in a grid. 5 Minute Preview

2.9: Pythagorean Theorem

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

### 3: Expressions and Equations

3.4: One-Step Addition and Subtraction Equations

Modeling One-Step Equations

Solve a linear equation using a tile model. Use feedback to diagnose incorrect steps. 5 Minute Preview

3.5: One-Step Multiplication and Division Equations

Modeling One-Step Equations

Solve a linear equation using a tile model. Use feedback to diagnose incorrect steps. 5 Minute Preview

3.6: Model Two-Step Equations

Modeling and Solving Two-Step Equations

Solve a two-step equation using a cup-and-counter model. Use step-by-step feedback to diagnose and correct incorrect steps. 5 Minute Preview

3.7: Two-Step Equations

Solving Two-Step Equations

Choose the correct steps to solve a two-step equation. Use the feedback to diagnose incorrect steps. 5 Minute Preview

### 5: Polynomials and Factoring

5.1: Polynomials

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

5.4: Add Polynomials

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

Addition of Polynomials

Add polynomials using an area model. Use step-by-step feedback to diagnose any mistakes. 5 Minute Preview

5.5: Subtract Polynomials

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

Addition of Polynomials

Add polynomials using an area model. Use step-by-step feedback to diagnose any mistakes. 5 Minute Preview

5.11: Factoring Trinomials: x2 + bx + c

Modeling the Factorization of *x*^{2}+*bx*+*c*

Factor a polynomial with a leading coefficient equal to 1 using an area model. Use step-by-step feedback to diagnose any mistakes. 5 Minute Preview

5.12: Factoring Special Products

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

5.13: Factoring Trinomials: ax2 + bx + c

Modeling the Factorization of *ax*^{2}+*bx*+*c*

Factor a polynomial with a leading coefficient greater than 1 using an area model. Use step-by-step feedback to diagnose any mistakes. 5 Minute Preview

### 6: Linear Functions and Inequalities

6.1: Relations and Functions

Introduction to Functions

Determine if a relation is a function using the mapping diagram, ordered pairs, or the graph of the relation. Drag arrows from the domain to the range, type in ordered pairs, or drag points to the graph to add inputs and outputs to the relation. 5 Minute Preview

Linear Functions

Determine if a relation is a function from the mapping diagram, ordered pairs, or graph. Use the graph to determine if it is linear. 5 Minute Preview

Points, Lines, and Equations

Compare the graph of a linear function to its rule and to a table of its values. Change the function by dragging two points on the line. Examine how the rule and table change. 5 Minute Preview

6.2: Graphs of Functions

Function Machines 2 (Functions, Tables, and Graphs)

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

6.3: Scatter Plots

Trends in Scatter Plots

Examine the scatter plot for a random data set with negative or positive correlation. Vary the correlation and explore how correlation is reflected in the scatter plot and the trend line. 5 Minute Preview

6.4: Slope of a Line

Slope

Explore the slope of a line, and learn how to calculate slope. Adjust the line by moving points that are on the line, and see how its slope changes. 5 Minute Preview

6.6: Linear Functions: Standard Form and Slope-Intercept Form

Slope-Intercept Form of a Line

Compare the slope-intercept form of a linear equation to its graph. Vary the coefficients and explore how the graph changes in response. 5 Minute Preview

Standard Form of a Line

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

6.7: Linear Functions: Point-Slope Form

Point-Slope Form of a Line

Compare the point-slope form of a linear equation to its graph. Vary the coefficients and explore how the graph changes in response. 5 Minute Preview

6.9: Direct Variation

Direct and Inverse Variation

Adjust the constant of variation and explore how the graph of the direct or inverse variation function changes in response. Compare direct variation functions to inverse variation functions. 5 Minute Preview

6.10: Solve Systems of Equations by Graphing

Cat and Mouse (Modeling with Linear Systems)

Experiment with a system of two lines representing a cat-and-mouse chase. Adjust the speeds of the cat and mouse and the head start of the mouse, and immediately see the effects on the graph and on the chase. Connect real-world meaning to slope, y-intercept, and the intersection of lines. 5 Minute Preview

Solving Linear Systems (Slope-Intercept Form)

Solve systems of linear equations, given in slope-intercept form, both graphically and algebraically. Use a draggable green point to examine what it means for an *x*, *y*)

6.11: Solve Systems of Equations by Substitution and Elimination

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

6.12: Linear Inequalities in Two Variables

Linear Inequalities in Two Variables

Find the solution set to a linear inequality in two variables using the graph of the linear inequality. Vary the terms of the inequality and vary the inequality symbol. Examine how the boundary line and shaded region change in response. 5 Minute Preview

### 7: Ratio and Proportion

7.1: Ratios, Rates, and Unit Rates

Beam to Moon (Ratios and Proportions)

Apply ratios and proportions to find the weight of a person on the moon (or on another planet). Weigh an object on Earth and on the moon and weigh the person on Earth. Then set up and solve the proportion of the Earth weights to the moon weights. 5 Minute Preview

Measuring Motion

Go on an African safari and observe a variety of animals walking and running across the savanna. Videotape the animals, and then play back the videotape to estimate animal speeds. Which animals run fastest? 5 Minute Preview

7.2: Proportions

Beam to Moon (Ratios and Proportions)

Estimating Population Size

Adjust the number of fish in a lake to be tagged and the number of fish to be recaptured. Use the number of tagged fish in the catch to estimate the number of fish in the lake. 5 Minute Preview

7.9: Similarity

Perimeters and Areas of Similar Figures

7.10: Trigonometric Ratios

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

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

### 8: Percent Applications

8.1: Fractions, Decimals, and Percents

Percents, Fractions, and Decimals

Compare a quantity represented by an area with its percent, fraction, and decimal forms. 5 Minute Preview

8.3: Percentage and Applications

Percents and Proportions

Find a part from the percent and whole, a percent from the part and whole, or a whole from the part and percent using a graphic model. 5 Minute Preview

8.6: Percent of Change

Percent of Change

Apply markups and discounts using interactive "percent rulers." Improve number sense for percents with this dynamic, visual tool. Reinforce the original cost (or original price) as the baseline for percent calculations. 5 Minute Preview

8.7: Simple and Compound Interests

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

### 9: Two-Dimensional Geometry

9.1: Angle Pairs

Investigating Angle Theorems

Explore the properties of complementary, supplementary, vertical, and adjacent angles using a dynamic figure. 5 Minute Preview

9.2: Angles of Parallel Lines

Investigating Angle Theorems

Explore the properties of complementary, supplementary, vertical, and adjacent angles using a dynamic figure. 5 Minute Preview

9.3: Classifying Polygons

Classifying Quadrilaterals

Apply constraints to a quadrilateral, and then reshape and resize it. Classify the figure by its constraints. Explore the differences between the different kinds of quadrilaterals. 5 Minute Preview

Parallelogram Conditions

Apply constraints to a dynamic quadrilateral. Then drag its vertices around. Determine which constraints guarantee that the quadrilateral is always a parallelogram. 5 Minute Preview

Special Parallelograms

Apply constraints to a parallelogram and experiment with the resulting figure. What type of shape can you be sure that you have under each condition? 5 Minute Preview

9.4: Angles of Polygons

Polygon Angle Sum

Derive the sum of the angles of a polygon by dividing the polygon into triangles and summing their angles. Vary the number of sides and determine how the sum of the angles changes. Dilate the polygon to see that the sum is unchanged. 5 Minute Preview

9.7: Make Circle Graphs

Mascot Election (Pictographs and Bar Graphs)

A brand new school is opening and it is time to elect the school mascot! Students can choose the Eagle, Lion, Bear, or Wolf. Voting results can be displayed in a table, tally chart, pictograph, bar graph, circle graph, or dot plot. You can change student votes by selecting a group of students and clicking a mascot. 5 Minute Preview

9.8: Angle Constructions

Constructing Congruent Segments and Angles

Construct congruent segments and angles using a straightedge and compass. Use step-by-step explanations and feedback to develop understanding of the construction. 5 Minute Preview

9.9: Line Constructions

Constructing Parallel and Perpendicular Lines

Construct parallel and perpendicular lines using a straightedge and compass. Use step-by-step explanations and feedback to develop understanding of the construction. 5 Minute Preview

### 10: Geometric Measures and Coordinate Geometry

10.2: Perimeter and Missing Dimensions

Fido's Flower Bed (Perimeter and Area)

Construct models of gardens on a grid using squares of sod. Fence the gardens to find and compare perimeters. Work with pre-built gardens made of 36 squares each to compare perimeters of shapes with equal areas. 5 Minute Preview

Perimeter and Area of Rectangles

Discover how to find the perimeter and area of a rectangle, and of a square (which is really just a special case of a rectangle). 5 Minute Preview

10.4: Circumference and Area of Circles

Circumference and Area of Circles

Resize a circle and compare its radius, circumference, and area. 5 Minute Preview

10.8: Coordinate Plane: Reflections and Translations

Reflections

Reshape and resize a figure and examine how its reflection changes in response. Move the line of reflection and explore how the reflection is translated. 5 Minute Preview

Translations

Translate a figure horizontally and vertically in the plane and examine the matrix representation of the translation. 5 Minute Preview

10.9: Coordinate Plane: Rotations

Rotations, Reflections, and Translations

Rotate, reflect, and translate a figure in the plane. Compare the translated figure to the original figure. 5 Minute Preview

10.10: Coordinate Plane: Dilations

Dilations

Dilate a figure and investigate its resized image. See how scaling a figure affects the coordinates of its vertices, both in

### 11: Patterns and Nonlinear Functions

11.1: Extend and Create Patterns

Finding Patterns

Build a pattern to complete a sequence of patterns. Study a sequence of three patterns of squares in a grid and build the fourth pattern of the sequence in a grid. 5 Minute Preview

Pattern Finder

Observe frogs jumping around on colored lily pads. Find, test, and reason about patterns you see in their jumping. 5 Minute Preview

11.2: Arithmetic Patterns and Tables

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

11.3: Geometric Patterns and Tables

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

11.6: Nonlinear Functions: Quadratic

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

11.7: Other Nonlinear Functions

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

11.8: Inverse Variation

Direct and Inverse Variation

Adjust the constant of variation and explore how the graph of the direct or inverse variation function changes in response. Compare direct variation functions to inverse variation functions. 5 Minute Preview

11.9: Graphs of Nonlinear 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

Zap It! Game

Adjust the values in a quadratic function, in vertex form or in polynomial form, to "zap" as many data points as possible. 5 Minute Preview

### 12: Three-Dimensional Geometry

12.2: Draw Views of Three-Dimensional Figures

3D and Orthographic Views

Arrange blocks in a three-dimensional space so that the top view, front view, and side view match the target top view, front view, and side view. 5 Minute Preview

12.3: Surface Area of Prisms and Cylinders

Surface and Lateral Areas of Prisms and Cylinders

Vary the dimensions of a prism or cylinder and investigate how the surface area changes. Use the dynamic net of the solid to compute the lateral area and the surface area of the solid. 5 Minute Preview

12.4: Surface Area of Pyramids and Cones

Surface and Lateral Areas of Pyramids and Cones

Vary the dimensions of a pyramid or cone and investigate how the surface area changes. Use the dynamic net of the solid to compute the lateral area and the surface area of the solid. 5 Minute Preview

12.5: Volume of Prisms and Cylinders

Prisms and Cylinders

Vary the height and base-edge or radius length of a prism or cylinder and examine how its three-dimensional representation changes. Determine the area of the base and the volume of the solid. Compare the volume of an oblique prism or cylinder to the volume of a right prism or cylinder. 5 Minute Preview

12.6: Volume of Pyramids and Cones

Pyramids and Cones

Vary the height and base-edge or radius length of a pyramid or cone and examine how its three-dimensional representation changes. Determine the area of the base and the volume of the solid. Compare the volume of a skew pyramid or cone to the volume of a right pyramid or cone. 5 Minute Preview

### 13: Data Analysis and Statistics

13.2: Surveys and Samples

Mascot Election (Pictographs and Bar Graphs)

A brand new school is opening and it is time to elect the school mascot! Students can choose the Eagle, Lion, Bear, or Wolf. Voting results can be displayed in a table, tally chart, pictograph, bar graph, circle graph, or dot plot. You can change student votes by selecting a group of students and clicking a mascot. 5 Minute Preview

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

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

13.3: Stem-and-Leaf Plots

Stem-and-Leaf Plots

Build a data set and compare the line plot of the data set to the stem-and-leaf plot. 5 Minute Preview

13.4: Box-and-Whisker Plots

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

13.7: 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

### 14: Probability and Logic

14.1: Counting Principle and Tree Diagrams

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

14.2: Theoretical Probability

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

14.3: Experimental 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

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

14.6: Compound Events

Independent and Dependent Events

Compare the theoretical and experimental probabilities of drawing colored marbles from a bag. Record results of successive draws to find the experimental probability. Perform the drawings with replacement of the marbles to study independent events, or without replacement to explore dependent events. 5 Minute Preview

Independent and Dependent Events

14.8: Permutations

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

14.9: Combinations

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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