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# Missouri - Mathematics: Geometry

## MO--Learning Standards | Adopted: 2016

### CO: : Congruence

CO.A: : Experiment with transformations in the plane.

CO.A.1: : Define angle, circle, perpendicular line, parallel line, line segment and ray based on the undefined notions of point, line, distance along a line and distance around a circular arc.

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

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

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

Inscribed Angles

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

Parallel, Intersecting, and Skew Lines

Explore the properties of intersecting, parallel, and skew lines as well as lines in the plane. Rotate the plane and lines in three-dimensional space to ensure a full understanding of these objects. 5 Minute Preview

CO.A.2: : Represent transformations in the plane, and describe them as functions that take points in the plane as inputs and give other points as outputs.

Dilations

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

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

Translations

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

CO.A.3: : Describe the rotational symmetry and lines of symmetry of two-dimensional figures.

Holiday Snowflake Designer

Fold paper and cut in a certain way to make symmetrical snowflakes with six sides (similar to what can be found in nature) or with eight sides (an easier folding method). This simulation allows you to cut virtual paper on the computer screen with round dot or square dot "scissors" of various sizes before using physical paper. 5 Minute Preview

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

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

Dilations

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

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

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

Translations

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

CO.A.5: : Demonstrate the ability to rotate, reflect or translate a figure, and determine a possible sequence of transformations between two congruent figures.

Dilations

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

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

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

Translations

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

CO.B: : Understand congruence in terms of rigid motions.

CO.B.1: : Develop the definition of congruence in terms of rigid motions.

Absolute Value with Linear Functions

Compare the graph of a linear function, the graph of an absolute-value function, and the graphs of their translations. Vary the coefficients and constants in the functions and investigate how the graphs change in response. 5 Minute Preview

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

Dilations

Holiday Snowflake Designer

Fold paper and cut in a certain way to make symmetrical snowflakes with six sides (similar to what can be found in nature) or with eight sides (an easier folding method). This simulation allows you to cut virtual paper on the computer screen with round dot or square dot "scissors" of various sizes before using physical paper. 5 Minute Preview

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

Reflections

Rotations, Reflections, and Translations

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

Translations

CO.B.2: : Develop the criteria for triangle congruence from the definition of congruence in terms of rigid motions.

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

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

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

CO.C: : Prove geometric theorems.

CO.C.1: : Prove theorems about lines and angles.

Investigating Angle Theorems

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

CO.C.2: : Prove theorems about 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

Triangle Angle Sum

Measure the interior angles of a triangle and find the sum. Examine whether that sum is the same for all triangles. Also, discover how the measure of an exterior angle relates to the interior angle measures. 5 Minute Preview

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

CO.C.3: : Prove theorems about polygons.

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

CO.D: : Make geometric constructions.

CO.D.1: : Construct geometric figures using various tools and methods.

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

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

Segment and Angle Bisectors

Explore the special properties of a point that lies on the perpendicular bisector of a segment, and of a point that lies on an angle bisector. Manipulate the point, the segment, and the angle to see that these properties are always true. 5 Minute Preview

### SRT: : Similarity, Right Triangles, and Trigonometry

SRT.A: : Understand similarity in terms of similarity transformations.

SRT.A.1: : Construct and analyze scale changes of geometric figures.

Dilations

Similar Figures

SRT.A.2: : Use the definition of similarity to decide if figures are similar and to solve problems involving similar figures.

Circles

Dilations

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

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

SRT.B: : Prove theorems involving similarity.

SRT.B.1: : Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Dilations

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

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

SRT.C: : Define trigonometric ratios, and solve problems involving right triangles.

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

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

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 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

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

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

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

SRT.C.1: : Understand that side ratios in right triangles define the trigonometric ratios for acute angles.

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

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

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

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

SRT.C.3: : Use trigonometric ratios and the Pythagorean Theorem to solve right triangles.

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

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

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 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

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

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

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

### C: : Circles

C.A: : Understand and apply theorems about circles.

C.A.2: : Identify and describe relationships among inscribed angles, radii and chords of circles.

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

Circumference and Area of Circles

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

Inscribed Angles

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

C.B: : Find arc lengths and areas of sectors of circles.

C.B.1: : Derive the formula for the length of an arc of a circle.

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

C.B.2: : Derive the formula for the area of a sector of a circle.

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

### GPE: : Exploring Geometric Properties with Equations

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

GPE.A.1: : Derive the equation of a circle.

Circles

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

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

GPE.A.2: : Derive the equation of a parabola given a focus and directrix.

Parabolas

Explore parabolas in a conic section context. Find the relationship among the vertex, focus, and directrix of a parabola, and how that relates to its equation. 5 Minute Preview

GPE.B: : Use coordinates to prove geometric theorems algebraically.

GPE.B.4: : Use coordinates to compute perimeters of polygons and areas of triangles and rectangles.

Distance Formula

### GMD: : Geometric Measurement and Dimension

GMD.A: : Explain volume formulas and use them to solve problems.

GMD.A.1: : Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid and cone.

Circumference and Area of Circles

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

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

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

GMD.A.2: : Use volume formulas for cylinders, pyramids, cones, spheres and composite figures to solve problems.

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

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

### CP: : Conditional Probability and Rules of Probability

CP.A: : Understand independence and conditional probability and use them to interpret data.

CP.A.1: : Describe events as subsets of a sample space using characteristics of the outcomes, or as unions, intersections or complements of other 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

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

CP.A.2: : Understand the definition of independent events and use it to solve problems.

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

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

CP.A.3: : Calculate conditional probabilities of 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

CP.A.5: : Recognize and explain the concepts of conditional probability and independence in a context.

Independent and Dependent Events

CP.A.6: : Apply and interpret the Addition Rule for calculating probabilities.

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

CP.A.7: : Apply and interpret the general Multiplication Rule in a uniform probability model.

Independent and Dependent Events

CP.A.8: : Use permutations and combinations to solve problems.

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

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

Correlation last revised: 9/16/2020

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