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# Virginia - Mathematics: Geometry

## VA--Standards of Learning | Adopted: 2016

### RLT: : Reasoning, Lines, and Transformations

RLT.G.1: : The student will use deductive reasoning to construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include

RLT.G.1.a: : identifying the converse, inverse, and contrapositive of a conditional statement;

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

RLT.G.1.c: : determining the validity of a logical argument.

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

RLT.G.3: : The student will solve problems involving symmetry and transformation. This will include

RLT.G.3.a: : investigating and using formulas for determining distance, midpoint, and slope;

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

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

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

RLT.G.3.c: : investigating symmetry and determining whether a figure is symmetric with respect to a line or a point; and

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

RLT.G.3.d: : determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods.

Dilations

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

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

RLT.G.4: : The student will construct and justify the constructions of

RLT.G.4.a: : a line segment congruent to a given line segment;

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

RLT.G.4.b: : the perpendicular bisector of a line segment;

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

RLT.G.4.c: : a perpendicular to a given line from a point not on the line;

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

RLT.G.4.d: : a perpendicular to a given line at a given point on the line;

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

RLT.G.4.f: : an angle congruent to a given angle;

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

RLT.G.4.g: : a line parallel to a given line through a point not on the line; and

Constructing Parallel and Perpendicular Lines

RLT.G.4.h: : an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Inscribed Angles

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

### T: : Triangles

T.G.5: : The student, given information concerning the lengths of sides and/or measures of angles in triangles, will solve problems, including practical problems. This will include

T.G.5.a: : ordering the sides by length, given angle measures;

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

T.G.5.b: : ordering the angles by degree measure, given side lengths;

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

T.G.5.c: : determining whether a triangle exists; and

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

T.G.5.d: : determining the range in which the length of the third side must lie.

Triangle Inequalities

T.G.6: : The student, given information in the form of a figure or statement, will prove two triangles are congruent.

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

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

T.G.7: : The student, given information in the form of a figure or statement, will prove two triangles are similar.

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

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

T.G.8: : The student will solve problems, including practical problems, involving right triangles. This will include applying

T.G.8.a: : the Pythagorean Theorem and its converse;

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

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

T.G.8.c: : trigonometric ratios.

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

### PC: : Polygons and Circles

PC.G.9: : The student will verify and use properties of quadrilaterals to solve problems, including practical problems.

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

PC.G.10: : The student will solve problems, including practical problems, involving angles of convex polygons. This will include determining the

PC.G.10.a: : sum of the interior and/or exterior angles;

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

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

PC.G.10.b: : measure of an interior and/or exterior angle; and

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

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

PC.G.10.c: : number of sides of a regular polygon.

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

PC.G.11: : The student will solve problems, including practical problems, by applying properties of circles. This will include determining

PC.G.11.c: : arc length; and

Inscribed Angles

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

PC.G.12: : The student will solve problems involving 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

### TDF: : Three-Dimensional Figures

TDF.G.13: : The student will use surface area and volume of three-dimensional objects to solve practical problems.

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

TDF.G.14: : The student will apply the concepts of similarity to two- or three-dimensional geometric figures. This will include

TDF.G.14.a: : comparing ratios between lengths, perimeters, areas, and volumes of similar figures;

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

TDF.G.14.b: : determining how changes in one or more dimensions of a figure affect area and/or volume of the figure;

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

TDF.G.14.d: : solving problems, including practical problems, about similar geometric figures.

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

Correlation last revised: 9/16/2020

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