# Geometry (2008)

### 1.Tools of Geometry

1.3.Distance and Midpoints

Distance Formula

Explore the distance formula as an application of the Pythagorean theorem. Learn to see any two points as the endpoints of the hypotenuse of a right triangle. Drag those points and examine changes to the triangle and the distance calculation. 5 Minute Preview

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

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

### 2.Reasoning and Proof

2.1.Inductive Reasoning and Conjecture

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

2.2.Logic

Biconditional Statements

Make a biconditional statement from a given definition using word tiles. Use both symbolic form and standard English form. 5 Minute Preview

Conditional Statements

Make a conditional statement from a given fact using word tiles. Use both symbolic form and standard English form. 5 Minute Preview

2.3.Conditional Statements

Conditional Statements

Make a conditional statement from a given fact using word tiles. Use both symbolic form and standard English form. 5 Minute Preview

2.4.Deductive Reasoning

Biconditional Statements

Make a biconditional statement from a given definition using word tiles. Use both symbolic form and standard English form. 5 Minute Preview

Conditional Statements

Make a conditional statement from a given fact using word tiles. Use both symbolic form and standard English form. 5 Minute Preview

2.6.Algebraic Proof

Biconditional Statements

Make a biconditional statement from a given definition using word tiles. Use both symbolic form and standard English form. 5 Minute Preview

Conditional Statements

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

Solving Two-Step Equations

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

2.7.Proving Segment Relationships

Biconditional Statements

Conditional Statements

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

2.8.Proving Angle Relationships

Biconditional Statements

Conditional Statements

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

### 3.Parallel and Perpendicular Lines

3.1.Parallel Lines and Transversals

Investigating Angle Theorems

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

3.3.Slopes of Lines

Distance-Time Graphs

Create a graph of a runner's position versus time and watch the runner complete a 40-yard dash based on the graph you made. Notice the connection between the slope of the line and the speed of the runner. What will the runner do if the slope of the line is zero? What if the slope is negative? Add a second runner (a second graph) and connect real-world meaning to the intersection of two graphs. 5 Minute Preview

Distance-Time and Velocity-Time Graphs

Create a graph of a runner's position versus time and watch the runner run a 40-yard dash based on the graph you made. Notice the connection between the slope of the line and the velocity of the runner. Add a second runner (a second graph) and connect real-world meaning to the intersection of two graphs. Also experiment with a graph of velocity versus time for the runners, and also distance traveled versus time. 5 Minute Preview

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

3.5.Proving Lines Parallel

Biconditional Statements

Conditional Statements

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

Proving Triangles Congruent

3.6.Perpendiculars and Distance

Distance Formula

Explore the distance formula as an application of the Pythagorean theorem. Learn to see any two points as the endpoints of the hypotenuse of a right triangle. Drag those points and examine changes to the triangle and the distance calculation. 5 Minute Preview

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

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

### 4.Congruent Triangles

4.1.Classifying Triangles

Classifying Triangles

Place constraints on a triangle and determine what classifications must apply to the triangle. 5 Minute Preview

Isosceles and Equilateral Triangles

Investigate the graph of a triangle under constraints. Determine which constraints guarantee isosceles or equilateral triangles. 5 Minute Preview

4.2.Angles of Triangles

Classifying Triangles

Place constraints on a triangle and determine what classifications must apply to the triangle. 5 Minute Preview

4.3.Congruent Triangles

Congruence in Right Triangles

Apply constraints to two right triangles. Then drag their vertices around under those conditions. Determine under what conditions the triangles are guaranteed to be congruent. 5 Minute Preview

Proving Triangles Congruent

4.4.Proving Congruence - SSS, SAS

Biconditional Statements

Conditional Statements

Congruence in Right Triangles

Apply constraints to two right triangles. Then drag their vertices around under those conditions. Determine under what conditions the triangles are guaranteed to be congruent. 5 Minute Preview

Proving Triangles Congruent

4.5.Proving Congruence - ASA, AAS

Biconditional Statements

Conditional Statements

Congruence in Right Triangles

Apply constraints to two right triangles. Then drag their vertices around under those conditions. Determine under what conditions the triangles are guaranteed to be congruent. 5 Minute Preview

Proving Triangles Congruent

4.6.Isosceles Triangles

Classifying Triangles

Place constraints on a triangle and determine what classifications must apply to the triangle. 5 Minute Preview

Isosceles and Equilateral Triangles

Investigate the graph of a triangle under constraints. Determine which constraints guarantee isosceles or equilateral triangles. 5 Minute Preview

4.7.Triangles and Coordinate Proof

Biconditional Statements

Classifying Triangles

Conditional Statements

### 5.Relationships in Triangles

5.1.Bisectors, Medians, and Altitudes

Concurrent Lines, Medians, and Altitudes

Explore the relationships between perpendicular bisectors, the circumscribed circle, angle bisectors, the inscribed circle, altitudes, and medians using a triangle that can be resized and reshaped. 5 Minute Preview

5.2.Inequalities and Triangles

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

5.4.The Triangle Inequality

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

5.5.Inequalities Involving Two Triangles

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

### 6.Quadrilateral

6.2.Parallelograms

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

6.3.Tests for Parallelograms

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

6.4.Rectangles

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

6.5.Rhombi and Squares

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

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

6.7.Coordinate Proof with Quadrilaterals

Biconditional Statements

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

Conditional Statements

### 7.Proportions and Similarity

7.1.Proportion

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

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

Perimeters and Areas of Similar Figures

Manipulate two similar figures and vary the scale factor to see what changes are possible under similarity. Explore how the perimeters and areas of two similar figures compare. 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

Similar Figures

Vary the scale factor and rotation of an image and compare it to the preimage. Determine how the angle measures and side lengths of the two figures are related. 5 Minute Preview

Similar Figures

Vary the scale factor and rotation of an image and compare it to the preimage. Determine how the angle measures and side lengths of the two figures are related. 5 Minute Preview

7.2.Similar Polygons

Perimeters and Areas of Similar Figures

Manipulate two similar figures and vary the scale factor to see what changes are possible under similarity. Explore how the perimeters and areas of two similar figures compare. 5 Minute Preview

Similar Figures

Vary the scale factor and rotation of an image and compare it to the preimage. Determine how the angle measures and side lengths of the two figures are related. 5 Minute Preview

Similar Figures

7.3.Similar Triangles

Perimeters and Areas of Similar Figures

Manipulate two similar figures and vary the scale factor to see what changes are possible under similarity. Explore how the perimeters and areas of two similar figures compare. 5 Minute Preview

Similar Figures

Similar Figures

7.4.Parallel Lines and Proportional Parts

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

Perimeters and Areas of Similar Figures

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

Similar Figures

Similar Figures

7.5.Parts of Similarity Triangles

Perimeters and Areas of Similar Figures

Similar Figures

Similar Figures

Similarity in Right Triangles

Divide a right triangle at the altitude to the hypotenuse to get two similar right triangles. Explore the relationship between the two triangles. 5 Minute Preview

### 8.Right Triangles and Trigonometry

8.1.Geometric Mean

Similarity in Right Triangles

Divide a right triangle at the altitude to the hypotenuse to get two similar right triangles. Explore the relationship between the two triangles. 5 Minute Preview

8.2.The Pythagorean Theorem and Its Converse

Distance Formula

Explore the distance formula as an application of the Pythagorean theorem. Learn to see any two points as the endpoints of the hypotenuse of a right triangle. Drag those points and examine changes to the triangle and the distance calculation. 5 Minute Preview

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

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

Pythagorean Theorem

8.4.Trigonometry

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

### 9.Transformations

9.1.Reflections

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

Rotations, Reflections, and Translations

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

9.2.Translations

Rotations, Reflections, and Translations

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

Translations

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

9.3.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

9.5.Dilations

Dilations

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

9.6.Vectors

Vectors

Manipulate the magnitudes and directions of two vectors to generate a sum and learn vector addition. The x and y components can be displayed, along with the dot product of the two vectors. 5 Minute Preview

### 10.Circles

10.1.Circles and Circumference

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

10.3.Arcs and Chords

Chords and Arcs

Explore the relationship between a central angle and the arcs it intercepts. Also explore the relationship between chords and their distance from the circle's center. 5 Minute Preview

10.4.Inscribed Angles

Inscribed Angles

Resize angles inscribed in a circle. Investigate the relationship between inscribed angles and the arcs they intercept. 5 Minute Preview

10.8.Equations of Circles

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

### 11.Area of Polygons and Circles

11.5.Geometric Probability and Ares of Sectors

Geometric Probability

Randomly throw darts at a target and see what percent are "hits." Vary the size of the target and repeat the experiment. Study the relationship between the area of the target and the percent of darts that strike it 5 Minute Preview

### 13.Extending Volume

13.4.Congruent and Similar Solids

Congruence in Right Triangles

Proving Triangles Congruent

13.5.Coordinates in Space

Vectors

Manipulate the magnitudes and directions of two vectors to generate a sum and learn vector addition. The x and y components can be displayed, along with the dot product of the two vectors. 5 Minute Preview

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