# Geometry Q and A Lesson Plan: Student-Generated Questions

### Submitted by: Angela Watson

Grade Levels: 3-5, 6-8, 9-12

In this lesson plan, which is adaptable for grades 3-12, students use BrainPOP resources to explore the unifying principles of geometry. Students will identify pertinent questions related to geometry, answer student-generated geometry questions using BrainPOP resources, and share research findings.

### Lesson Plan Common Core State Standards Alignments

Grade: 03

CCSS.Math.Content.3.G.A.1

Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.

Grade: 03

CCSS.Math.Content.3.G.A.2

Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.

Grade: 04

CCSS.Math.Content.4.G.A.1

Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.

Grade: 04

CCSS.Math.Content.4.G.A.2

Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.

Grade: 05

CCSS.Math.Content.5.G.A.1

Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).

Grade: 05

CCSS.Math.Content.5.G.A.2

Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.

Grade: 06

CCSS.Math.Content.6.G.A.1

Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

Grade: 06

CCSS.Math.Content.6.G.A.2

Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

Grade: 07

CCSS.Math.Content.7.G.A.1

Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

Grade: 07

CCSS.Math.Content.7.G.A.2

Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

Grade: 08

CCSS.Math.Content.8.G.A.1

Verify experimentally the properties of rotations, reflections, and translations:

Grade: 08

CCSS.Math.Content.8.G.A.1a

Lines are taken to lines, and line segments to line segments of the same length.

Grade: 08

CCSS.Math.Content.8.G.A.1b

Angles are taken to angles of the same measure.

Grade: 08

CCSS.Math.Content.8.G.A.1c

Parallel lines are taken to parallel lines.

Grade: 08

CCSS.Math.Content.8.G.A.2

Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

Grade: 08

CCSS.Math.Content.8.G.A.3

Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Grade: 08

CCSS.Math.Content.8.G.A.4

Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

Grade: 08

CCSS.Math.Content.8.G.A.5

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

Grade: 08

CCSS.Math.Content.8.G.B.6

Explain a proof of the Pythagorean Theorem and its converse.

Grade: 08

CCSS.Math.Content.8.G.B.7

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

Grade: 08

CCSS.Math.Content.8.G.B.8

Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-C.A.1

Prove that all circles are similar.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-C.A.2

Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-C.A.3

Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-C.A.4

(+) Construct a tangent line from a point outside a given circle to the circle.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-C.B.5

Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-CO.A.1

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-CO.A.2

Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-CO.A.3

Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-CO.A.4

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

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-CO.A.5

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-CO.B.6

Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-CO.B.7

Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-CO.B.8

Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-CO.C.10

Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-CO.C.11

Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-CO.C.9

Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-CO.D.12

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-CO.D.13

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-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. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-GMD.A.2

(+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-GMD.A.3

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-GMD.B.4

Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-GPE.A.1

Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-GPE.A.2

Derive the equation of a parabola given a focus and directrix.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-GPE.A.3

(+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-GPE.B.4

Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-GPE.B.5

Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-GPE.B.6

Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-GPE.B.7

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-MG.A.1

Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).★

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-MG.A.2

Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).★

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-MG.A.3

Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).★

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-SRT.A.1

Verify experimentally the properties of dilations given by a center and a scale factor:

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-SRT.A.1a

A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-SRT.A.1b

The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-SRT.A.2

Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-SRT.A.3

Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-SRT.B.4

Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-SRT.B.5

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

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-SRT.C.6

Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-SRT.C.7

Explain and use the relationship between the sine and cosine of complementary angles.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-SRT.C.8

Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-SRT.D.10

(+) Prove the Laws of Sines and Cosines and use them to solve problems.

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-SRT.D.11

(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

Grade: 09, 10, 11, 12

CCSS.Math.Content.HSG-SRT.D.9

(+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

### Students will:

- Explore the unifying principles of geometry.
- Identify pertinent questions related to geometry.
- Answer student-generated geometry questions using BrainPOP resources and share research findings.

### Materials:

- Internet and projector to show BrainPOP
- Computers for students to use BrainPOP independently
- Photocopies of the Activity and Graphic Organizer (can be copied back to back)

### Vocabulary:

geometry; euclidian geometry; point; dimensions; line; plane; compass; protractor; sextant

### Preparation:

Use this lesson to introduce or culminate a geometry unit. Prepare copies of the activity and graphic organizer, and determine how students will compile their Q&A (see last step of the procedure.)### Lesson Procedure:

- As a warm-up, have students complete the Activity page. Discuss the key terms in the 'Matching' section as a class.
- Show students the Geometry movie. Instruct students to listen for different types of geometric figures as they view: no dimensions, one-, two-, and three-dimensional.
- Play the movie through again, and this time have students complete the Graphic Organizer. Pause throughout the movie to give them time to take notes on the different types of figures.
- Project the Q&A for students to see. Tell students they will be composing their own Q&A questions related to geometry. After showing a few examples, give students time to brainstorm independently or with a partner and determine which questions they'd like to answer.
- Give students time to explore the geometry movies available on BrainPOP (listed at the top of this page.) Students who are reading below grade level may complete this activity using the BrainPOP Jr. movies listed above. Each student should watch the movie(s) related to his or her Q&A and compose the answers for the questions chosen.
- Compile the class' geometry Q&A. These may be shared on a class blog, Voicethread, podcast, Glogster, or other web 2.0 tool. A low-tech alternative is to create an interactive bulletin board in which the questions are written on slips of paper which can be lifted up to reveal the answers underneath.

### Extension Activity:

Take the Geometry POP Quiz together as a class. Then have students compose their own quiz questions. Hold a class competition and see how many questions teams of students can answer correctly!Filed as: 3-5, 6-8, 9-12, Angles, Area of Polygons, CCSS.Math.Content.3.G.A.1, CCSS.Math.Content.3.G.A.2, CCSS.Math.Content.4.G.A.1, CCSS.Math.Content.4.G.A.2, CCSS.Math.Content.5.G.A.1

CCSS.Math.Content.5.G.A.2, CCSS.Math.Content.6.G.A.1, CCSS.Math.Content.6.G.A.2, CCSS.Math.Content.7.G.A.1, CCSS.Math.Content.7.G.A.2, CCSS.Math.Content.8.G.A.1, CCSS.Math.Content.8.G.A.1a, CCSS.Math.Content.8.G.A.1b, CCSS.Math.Content.8.G.A.1c, CCSS.Math.Content.8.G.A.2, CCSS.Math.Content.8.G.A.3, CCSS.Math.Content.8.G.A.4, CCSS.Math.Content.8.G.A.5, CCSS.Math.Content.8.G.B.6, CCSS.Math.Content.8.G.B.7, CCSS.Math.Content.8.G.B.8, CCSS.Math.Content.HSG-C.A.1, CCSS.Math.Content.HSG-C.A.2, CCSS.Math.Content.HSG-C.A.3, CCSS.Math.Content.HSG-C.A.4, CCSS.Math.Content.HSG-C.B.5, CCSS.Math.Content.HSG-CO.A.1, CCSS.Math.Content.HSG-CO.A.2, CCSS.Math.Content.HSG-CO.A.3, CCSS.Math.Content.HSG-CO.A.4, CCSS.Math.Content.HSG-CO.A.5, CCSS.Math.Content.HSG-CO.B.6, CCSS.Math.Content.HSG-CO.B.7, CCSS.Math.Content.HSG-CO.B.8, CCSS.Math.Content.HSG-CO.C.10, CCSS.Math.Content.HSG-CO.C.11, CCSS.Math.Content.HSG-CO.C.9, CCSS.Math.Content.HSG-CO.D.12, CCSS.Math.Content.HSG-CO.D.13, CCSS.Math.Content.HSG-GMD.A.1, CCSS.Math.Content.HSG-GMD.A.2, CCSS.Math.Content.HSG-GMD.A.3, CCSS.Math.Content.HSG-GMD.B.4, CCSS.Math.Content.HSG-GPE.A.1, CCSS.Math.Content.HSG-GPE.A.2, CCSS.Math.Content.HSG-GPE.A.3, CCSS.Math.Content.HSG-GPE.B.4, CCSS.Math.Content.HSG-GPE.B.5, CCSS.Math.Content.HSG-GPE.B.6, CCSS.Math.Content.HSG-GPE.B.7, CCSS.Math.Content.HSG-MG.A.1, CCSS.Math.Content.HSG-MG.A.2, CCSS.Math.Content.HSG-MG.A.3, CCSS.Math.Content.HSG-SRT.A.1, CCSS.Math.Content.HSG-SRT.A.1a, CCSS.Math.Content.HSG-SRT.A.1b, CCSS.Math.Content.HSG-SRT.A.2, CCSS.Math.Content.HSG-SRT.A.3, CCSS.Math.Content.HSG-SRT.B.4, CCSS.Math.Content.HSG-SRT.B.5, CCSS.Math.Content.HSG-SRT.C.6, CCSS.Math.Content.HSG-SRT.C.7, CCSS.Math.Content.HSG-SRT.C.8, CCSS.Math.Content.HSG-SRT.D.10, CCSS.Math.Content.HSG-SRT.D.11, CCSS.Math.Content.HSG-SRT.D.9, Circles, Coordinate Plane, Geometry, Geometry and Measurement, Lesson Plans, Math, Math Lessons, Parallel and Perpendicular Lines, Polygons, Polyhedrons, Pythagorean Theorem, Similar Figures, Student Projects, Types of Triangles, Volume of Cylinders, Volume of Prisms