| Code |
Description |
| MA.912.GR.1.1: | Prove relationships and theorems about lines and angles. Solve mathematical and real-world problems involving postulates, relationships and theorems of lines and angles.Clarifications:
Clarification 1: Postulates, relationships and theorems include vertical angles are congruent; when a transversal crosses parallel lines, the consecutive angles are supplementary and alternate (interior and exterior) angles and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.Clarification 2: Instruction includes constructing two-column proofs, pictorial proofs, paragraph and narrative proofs, flow chart proofs or informal proofs. Clarification 3: Instruction focuses on helping a student choose a method they can use reliably.
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| MA.912.GR.1.2: | Prove triangle congruence or similarity using Side-Side-Side, Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, Angle-Angle and Hypotenuse-Leg.Clarifications:
Clarification 1: Instruction includes constructing two-column proofs, pictorial proofs, paragraph and narrative proofs, flow chart proofs or informal proofs.Clarification 2: Instruction focuses on helping a student choose a method they can use reliably.
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| MA.912.GR.1.3: | Prove relationships and theorems about triangles. Solve mathematical and real-world problems involving postulates, relationships and theorems of triangles.Clarifications:
Clarification 1: Postulates, relationships and theorems include measures of interior angles of a triangle sum to 180°; measures of a set of exterior angles of a triangle sum to 360°; triangle inequality theorem; 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.
Clarification 2: Instruction includes constructing two-column proofs, pictorial proofs, paragraph and narrative proofs, flow chart proofs or informal proofs. Clarification 3: Instruction focuses on helping a student choose a method they can use reliably.
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| MA.912.GR.1.4: | Prove relationships and theorems about parallelograms. Solve mathematical and real-world problems involving postulates, relationships and theorems of parallelograms.Clarifications:
Clarification 1: Postulates, relationships and theorems include opposite sides are congruent, consecutive angles are supplementary, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and rectangles are parallelograms with congruent diagonals.Clarification 2: Instruction includes constructing two-column proofs, pictorial proofs, paragraph and narrative proofs, flow chart proofs or informal proofs. Clarification 3: Instruction focuses on helping a student choose a method they can use reliably.
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| MA.912.GR.1.5: | Prove relationships and theorems about trapezoids. Solve mathematical and real-world problems involving postulates, relationships and theorems of trapezoids.Clarifications:
Clarification 1: Postulates, relationships and theorems include the Trapezoid Midsegment Theorem and for isosceles trapezoids: base angles are congruent, opposite angles are supplementary and diagonals are congruent.Clarification 2: Instruction includes constructing two-column proofs, pictorial proofs, paragraph and narrative proofs, flow chart proofs or informal proofs. Clarification 3: Instruction focuses on helping a student choose a method they can use reliably.
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| MA.912.GR.1.6: | Solve mathematical and real-world problems involving congruence or similarity in two-dimensional figures.Clarifications:
Clarification 1: Instruction includes demonstrating that two-dimensional figures are congruent or similar based on given information. |
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This cluster includes the following access points.
Vetted resources educators can use to teach the concepts and skills in this topic.
| Name |
Description |
| The Measure of an Angle of a Triangle: | Students are given the measure of one interior angle of an isosceles triangle and are asked to find the measure of another interior angle. |
| Comparing Lengths in a Parallelogram: | Students are given parallelogram ABCD along with midpoint E of diagonal AC and are asked to determine the relationship between the lengths AE + ED and BE + EC. |
| Finding Angle C: | Students are given expressions that represent the measures of two angles of a parallelogram and are asked to find the measure of an angle opposite one of the given angles. |
| Proving the Triangle Inequality Theorem: | Students are asked to prove the Triangle Inequality Theorem. |
| An Isosceles Trapezoid Problem: | Students are asked to explain why the sum of the lengths of the diagonals of an isosceles trapezoid is less than its perimeter. |
| Frame It Up: | Students are asked to explain how to determine whether a four-sided frame is a rectangle using only a tape measure. |
| Two Congruent Triangles: | Students are asked to explain why a pair of triangles formed by the sides and diagonals of a parallelogram are congruent. |
| Angles of a Parallelogram: | Students are given expressions that represent the measures of two angles of a parallelogram and are asked to find the measures of all four angles describing any theorems used. |
| Triangles and Midpoints: | Students are asked to explain why a quadrilateral formed by drawing the midsegments of a triangle is a parallelogram and to find the perimeter of the triangle formed by the midsegments. |
| Interior Angles of a Polygon : | Students are asked to explain why the sum of the measures of the interior angles of a convex n-gon is given by the formula (n – 2)180°. |
| The Third Side of a Triangle: | Students are given the lengths of two sides of a triangle and asked to describe all possible lengths of the remaining side. |
| Name That Triangle: | Students are asked to describe a triangle whose vertices are the endpoints of a segment and a point on the perpendicular bisector of a segment. |
| Locating the Missing Midpoint: | Students are given a triangle in which the midpoints of two sides are shown and are asked to describe a method for locating the midpoint of the remaining side using only a straight edge and pencil. |
| Finding Angle Measures - 1: | Students are asked to find the measures of angles formed by three concurrent lines and to justify their answers. |
| Finding Angle Measures - 3: | Students are asked to find the measures of angles formed by two parallel lines and two transversals. |
| Finding Angle Measures - 2: | Students are asked to find the measures of angles formed by two parallel lines and a transversal. |
| Pythagorean Theorem Proof: | Students are asked to prove the Pythagorean Theorem using similar triangles. |
| Camping Calculations: | Students are asked to find the measure of an angle formed by the support poles of a tent using the properties of geometric shapes. |
| Geometric Mean Proof: | Students are asked to prove that the length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments of the hypotenuse. |
| Converse of the Triangle Proportionality Theorem: | Students are asked to prove that if a line intersecting two sides of a triangle divides those two sides proportionally, then that line is parallel to the third side. |
| Justifying a Proof of the AA Similarity Theorem: | Students are asked to justify statements of a proof of the AA Similarity Theorem. |
| Describe the AA Similarity Theorem: | Students are asked to describe the AA Similarity Theorem. |
| Triangle Proportionality Theorem: | Students are asked to prove that a line parallel to one side of a triangle divides the other two sides of the triangle proportionally. |
| What Is the Triangle Relationship?: | Students are asked to write an informal justification of the AA Similarity Theorem. |
| Same Side Interior Angles: | Students are asked to describe and justify the relationship between same side interior angles. |
| Justifying the Triangle Sum Theorem: | Students are asked to provide an informal justification of the Triangle Sum Theorem. |
| Justifying Angle Relationships: | Students are asked to describe and justify the relationship between corresponding angles and alternate interior angles. |
| Proving Congruent Diagonals: | Students are asked to prove that the diagonals of a rectangle are congruent. |
| Proving a Rectangle Is a Parallelogram: | Students are asked to prove that a rectangle is a parallelogram. |
| Proving Parallelogram Angle Congruence: | Students are asked to prove that opposite angles of a parallelogram are congruent. |
| Proving Parallelogram Diagonals Bisect: | Students are asked to prove that the diagonals of a parallelogram bisect each other. |
| Proving Parallelogram Side Congruence: | Students are asked to prove that opposite sides of a parallelogram are congruent. |
| Drawing Triangles SSA: | Students are asked to draw a triangle given the lengths of two of its sides and the measure of a nonincluded angle and to decide if these conditions determine a unique triangle. |
| Drawing Triangles SAS: | Students are asked to draw a triangle given the measures of two sides and their included angle and to explain if these conditions determine a unique triangle. |
| Drawing Triangles ASA: | Students are asked to draw a triangle given the measures of two angles and their included side and to explain if these conditions determine a unique triangle. |
| Drawing Triangles AAS: | Students are asked to draw a triangle given the measures of two angles and a non-included side and to explain if these conditions determine a unique triangle. |
| Drawing Triangles AAA: | Students are asked to draw a triangle with given angle measures, and explain if these conditions determine a unique triangle. |
| Proving the Alternate Interior Angles Theorem: | In a diagram involving two parallel lines and a transversal, students are asked to use rigid motion to prove that alternate interior angles are congruent. |
| Proving Congruence: | Students are asked to explain congruence in terms of rigid motions. |
| County Fair: | Students are given a diagram of a county fair and are asked to use similar triangles to determine distances from one location of the fair to another. |
| Basketball Goal: | Students are asked to decide if a basketball goal is regulation height and are given enough information to determine this using similar triangles. |
| Prove Rhombus Diagonals Bisect Angles: | Students are asked to prove a specific diagonal of a rhombus bisects a pair of angles. |
| Similar Triangles - 2: | Students are asked to locate a pair of similar triangles in a diagram, explain why they are similar, and use the similarity to find an unknown length in the diagram. |
| Similar Triangles - 1: | Students are asked locate a pair of similar triangles in a diagram, explain why they are similar, and use the similarity to find two unknown lengths in the diagram. |
| Constructions for Parallel Lines: | Students are asked to construct a line parallel to a given line through a given point. |
| Median Concurrence Proof: | Students are asked to prove that the medians of a triangle are concurrent. |
| Triangle Sum Proof: | Students are asked prove that the measures of the interior angles of a triangle sum to 180°. |
| Equidistant Points: | Students are asked to prove that a point on the perpendicular bisector of a line segment is equidistant from the endpoints of the segment. |
| Proving the Vertical Angles Theorem: | Students are asked to identify a pair of vertical angles in a diagram and then prove that they are congruent. |
| Isosceles Triangle Proof: | Students are asked to prove that the base angles of an isosceles triangle are congruent. |
| Name |
Description |
| Transformation and Similarity: | Using non-rigid motions (dilations), students learn how to show that two polygons are similar. Students will write coordinate proofs confirming that two figures are similar. |
| Triangle Mid-Segment Theorem: | The Triangle Mid-Segment Theorem is used to show the writing of a coordinate proof clearly and concisely. |
| Proof of Quadrilaterals in Coordinate Plane: | This lesson is designed to instruct students on how to identify special quadrilaterals in the coordinate plane using their knowledge of distance formula and the definitions and properties of parallelograms, rectangles, rhombuses, and squares. Task cards, with and without solution-encoded QR codes, are provided for cooperative group practice. The students will need to download a free "QR Code Reader" app onto their SmartPhones if you choose to use the cards with QR codes. |
| To Be or Not to Be a Parallelogram: | Students apply parallelogram properties and theorems to solve real world problems. The acronym, P.I.E.S. is introduced to support a problem solving strategy, which involves drawing a Picture, highlighting important Information, Estimating and/or writing equation, and Solving problem. |
| Parallel Thinking Debate: | Students prove theorems related to parallel lines using vertical, corresponding, and alternate interior angles. |
| Match That!: | Students will prove that two figures are congruent based on a rigid motion(s) and then identify the corresponding parts using paragraph proof and vice versa, prove that two figures are congruent based on corresponding parts and then identify which rigid motion(s) map the images. |
| Airplanes in Radar's Range: | For a given circle m and point A in a coordinate plane, students will be able to show whether this given point A lies on the circumference of the given circle m using the Pythagorean Theorem. Subsequently, this can be used to prove that an airplane lies within or outside the radar's range with a given radius of detection. |
| Vertical Angles: Proof and Problem-Solving: | Students will explore the relationship between vertical angles and prove the Vertical Angle Theorem. They will use vertical angle relationships to calculate other angle measurements. |
| What's the Problem: | Students solve problems using triangle congruence postulates and theorems. |
| Diagonally Half of Me!: | This lesson is an exploration activity assisting students prove that diagonals of parallelograms bisect each other. It allows them to compare other quadrilaterals with parallelograms in order to make conjectures about the diagonals of parallelograms. |
| Who Am I?: Quadrilaterals: | Students will use formulas they know (distance, midpoint, and slope) to classify quadrilaterals. |
| Triangle Medians: | This lesson will have students exploring different types of triangles and their medians. Students will construct mid-points and medians to determine that the medians meet at a point. |
| Turning to Congruence: | This lesson uses rigid motions to prove the ASA and HL triangle congruence theorems. |
| Proving and Using Congruence with Corresponding Angles: | Students, will prove that corresponding angles are congruent. Directions for using GeoGebra software to discover this relationship is provided. |
| Slip, Slide, Tip, and Turn: Corresponding Angles and Corresponding Sides: | Using the definition of congruence in terms of rigid motion, students will show that two triangles are congruent. |
| Proving Parallelograms Algebraically: | This lesson reviews the definition of a parallelogram and related theorems. Students use these conditions to algebraically prove or disprove a given quadrilateral is a parallelogram. |
| How Do You Measure the Immeasurable?: | Students will use similar triangles to determine inaccessible measurements. Examples include exploring dangerous caves and discovering craters on Mars. |
| How Much Proof Do We Need?: | Students determine the minimum amount of information needed to prove that two triangles are similar. |
| Proving quadrilaterals algebrically using slope and distance formula: | Working in groups, students will prove the shape of various quadrilaterals using slope, distance formula, and polygon properties. They will then justify their proofs to their classmates. |
| Quadrilaterals and Coordinates: | In this lesson, students will use coordinates to algebraically prove that quadrilaterals are rectangles, parallelograms, and trapezoids. A through introduction to writing coordinate proofs is provided as well as plenty of practice. |
| Observing the Centroid: | Students will construct the medians of a triangle then investigate the intersections of the medians. |
| Triangles on a Lattice: | In this activity, students will use a 3x3 square lattice to study transformations of triangles whose vertices are part of the lattice. The tasks include determining whether two triangles are congruent, which transformations connect two congruent triangles, and the number of non-congruent triangles (with vertices on the lattice) that are possible. |
| Determination of the Optimal Point: | Students will use dynamic geometry software to determine the optimal location for a facility under a variety of scenarios. The experiments will suggest a relation between the optimal point and a common concept in geometry; in some cases, there will be a connection to a statistical concept. Algebra can be used to verify some of the conjectures. |
| Proving Quadrilaterals: | This lesson provides a series of assignments for students at the Getting Started, Moving Forward, and Almost There levels of understanding for the Mathematics Formative Assessment System (MFAS) Task Describe the Quadrilateral (CPALMS Resource ID#59180). The assignments are designed to "move" students from a lower level of understanding toward a complete understanding of writing a coordinate proof involving quadrilaterals. |
| The Centroid: | Students will construct the centroid of a triangle using graph paper or GeoGebra in order to develop conjectures. Then students will prove that the medians of a triangle actually intersect using the areas of triangles. |
| Congruence vs. Similarity: | Students will learn the difference between congruence and similarity of classes of figures (such as circles, parallelograms) in terms of the number of variable lengths in the class. A third category will allow not only rigid motions and dilations, but also a single one-dimensional stretch, allowing more classes of figures to share sufficient common features to belong. |
| What's the Point?: | Students will algebraically find the missing coordinates to create a specified quadrilateral using theorems that represent them, and then algebraically prove their coordinates are correct.
Note: This is not an introductory lesson for this standard. |
| Polygon...Prove it: | While this is an introductory lesson on the standard, students will enjoy it, as they play "Speed Geo-Dating" during the Independent practice portion. Students will use algebra and coordinates to prove rectangles, rhombus, and squares. Properties of diagonals are not used in this lesson. |
| Parallel Lines: | Students will prove that alternate interior angles and corresponding angles are congruent given two parallel lines and a traversal. Students will use GeoGebra to explore real-world images to prove their line segments are parallel. |
| Let's Prove the Pythagorean Theorem: | Students will use Triangle Similarity to derive the proof of the Pythagorean Theorem and apply this method to develop the idea of the geometric mean with respect to the relationships in right triangles. |
| Altitude to the Hypotenuse: | Students will discover what happens when the altitude to the hypotenuse of a right triangle is drawn. They learn that the two triangles created are similar to each other and to the original triangle. They will learn the definition of geometric mean and write, as well as solve, proportions that contain geometric means. All discovery, guided practice, and independent practice problems are based on the powerful altitude to the hypotenuse of a right triangle. |
| Intersecting Medians and the Resulting Ratios: | This lesson leads students to discover empirically that the distance from each vertex to the intersection of the medians of a triangle is two-thirds of the total length of each median. |
| Triangles: Finding Interior Angle Measures: | The lesson begins with a hands-on activity and then an experiment with a GeoGebra-based computer model to discover the Triangle Angle Sum Theorem. The students write and solve equations to find missing angle measures in a variety of examples. |
| Right turn, Clyde!: | Students will develop their knowledge of perpendicular bisectors & point of concurrency of a triangle, as well as construct perpendicular bisectors through real world problem solving with a map. |
| Halfway to the Middle!: | Students will develop their knowledge of mid-segments of a triangle, construct and provide lengths of mid-segments. |
| Location, Location, Location, Location?: | Students will use their knowledge of graphing concurrent segments in triangles to locate and identify which points of concurrency are associated by location with cities and counties within the Texas Triangle Mega-region. |
| Help me Find my Relationship!: | In this lesson, students will investigate the relationship between angles when parallel lines are cut by a transversal. Students will identify angles, and find angle measures, and they will use the free application GeoGebra (see download link under Suggested Technology) to provide students with a visual representation of angle relationships. |
| An Investigation of Angle Relationships Formed by Parallel Lines and a Transversal Using GeoGebra: | In this lesson, students will discover angle relationships formed when two parallel lines are cut by a transversal (corresponding, alternate interior, alternate exterior, same-side interior, same-side exterior). They will establish definitions and identify whether these angle pairs are supplementary or congruent. |
| Accurately Acquired Angles: | Students will start the lesson by playing a game to review angle pairs formed by two lines cut by a transversal. Once students are comfortable with the angle pairs the teacher will review the relationships that are created once the pair of lines become parallel. The teacher will give an example of a proof using the angle pairs formed by two parallel lines cut by a transversal. The students are then challenged to prove their own theorem in groups of four. The class will then participate in a Stay and Stray to view the other group's proofs. The lesson is wrapped up through white board questions answered within groups and then as a whole class. |
| Concurrent Points Are Optimal: | Students will begin with a review of methods of construction of perpendicular bisectors and angle bisectors for the sides of triangles. Included in the review will be a careful discussion of the proofs that the constructions actually produce the lines that were intended.
Next, students will investigate why the perpendicular bisectors and angle bisector are concurrent, that is, all three meet at a single meet.
A more modern point of currency is the Fermat-Torricelli point (F-T). The students will construct (F-T) in GeoGebra and investigate limitations of its existence for various types of triangles.
Then a set of scenarios will be provided, including some one-dimensional and two-dimensional situations. Students will use GeoGebra to develop conjectures regarding whether a point of concurrency provides the solution for the indicated situation, and which one.
A physical model for the F-T will be indicated. The teacher may demonstrate this model but that requires three strings, three weights, and a base that has holes. A recommended base is a piece of pegboard (perhaps 2 feet by 3 feet), the weights could be fishing weights of about 3 oz., the string could be fishing line; placing flexible pieces of drinking straws in the holes will improve the performance.
The combination of geometry theorems, dynamic geometry software, a variety of contexts, and a physical analog can provide a rich experience for students. |
| What's the Point? Part 1: | This is a patty paper-folding activity where students measure and discover the properties of the point of concurrency of the perpendicular bisectors of the sides of a triangle. |
| Name |
Description |
| Tile Patterns I: octagons and squares: | Students use interior and exterior angles to to verify attributes of an octagon and square. Students are given a tile pattern involving congruent regular octagons and squares. |
| Bank Shot: | This task asks students to use similarity to solve a problem in a context that will be familiar to many, though most students are accustomed to using intuition rather than geometric reasoning to set up the shot. |
| Extensions, Bisections and Dissections in a Rectangle: | This task involves a reasonably direct application of similar triangles, coupled with a moderately challenging procedure of constructing a diagram from a verbal description. |
| Finding the area of an equilateral triangle: | This problem solving task asks students to find the area of an equilateral triangle. Various solutions are presented that include the Pythagoren theorem and trigonometric functions. |
| Solar Eclipse: | This problem solving task encourages students to explore why solar eclipses are rare by examining the radius of the sun and the furthest distance between the moon and the earth. |
| Mt. Whitney to Death Valley: | This task engages students in an open-ended modeling task that uses similarity of right triangles. |
| Joining two midpoints of sides of a triangle: | Using a triangle with line through it, students are tasked to show the congruent angles, and conclude if one triangle is similar to the other. |
| Unit Squares and Triangles: | This problem solving task asks students to find the area of a triangle by using unit squares and line segments. |
| Why Does ASA Work?: | This problem solving task ask students to show the reflection of one triangle maps to another triangle. |
| When Does SSA Work to Determine Triangle Congruence?: | In this problem, we considered SSA. The triangle congruence criteria, SSS, SAS, ASA, all require three pieces of information. It is interesting, however, that not all three pieces of information about sides and angles are sufficient to determine a triangle up to congruence. |
| Points equidistant from two points in the plane: | This task asks students to show how certain points on a plane are equidistant to points on a segment when placed on a perpendicular bisector. |
| Midpoints of the Side of a Parallelogram: | This is a reasonably direct task aimed at having students use previously-derived results to learn new facts about parallelograms, as opposed to deriving them from first principles. |
| Inscribing a square in a circle: | This task provides an opportunity for students to apply triangle congruence theorems in an explicit, interesting context. |
| Construction of perpendicular bisector: | This problem solving task challenges students to construct a perpendicular bisector of a given segment. |
| Why Does SAS Work?: | This problem solving task challenges students to explain the reason why the given triangles are congruent, and to construct reflections of the points. |
| Reflected Triangles: | This task asks students to use a straightedge and compass to construct the line across which a triangle is reflected. |
| Right triangles inscribed in circles II: | This problem solving task asks students to explain certain characteristics about a triangle. |
| Why does SSS work?: | This particular problem solving task exhibits congruency between two triangles, demonstrating translation, reflection and rotation. |
| Bisecting an angle: | This problem solving task challenges students to bisect a given angle. |
| Are the Triangles Congruent?: | The purpose of this task is primarily assessment-oriented, asking students to demonstrate knowledge of how to determine the congruency of triangles. |
| Locating Warehouse: | This problem solving task challenges students to place a warehouse (point) an equal distance from three roads (lines). |
| Tangent Lines and the Radius of a Circle: | This problem solving task challenges students to find the perpendicular meeting point of a segment from the center of a circle and a tangent. |
| Seven Circles I: | This task is intended to help model a concrete situation with geometry. Placing the seven pennies in a circular pattern is a concrete and fun experiment which leads to a genuine mathematical question: does the physical model with pennies give insight into what happens with seven circles in the plane? |
| Angle bisection and midpoints of line segments: | This task provides a construction of the angle bisector of an angle by reducing it to the bisection of an angle to finding the midpoint of a line segment. It is worth observing the symmetry -- for both finding midpoints and bisecting angles, the goal is to cut an object into two equal parts. |
| Inscribing a circle in a triangle II: | This problem solving task focuses on a remarkable fact which comes out of the construction of the inscribed circle in a triangle: the angle bisectors of the three angles of triangle ABC all meet in a point. |
| Find the Angle: | 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. |
| Find the Missing Angle: | This task provides students the opportunity to see how the mathematical ideas embedded in the standards and clusters mature over time. The task uses facts about supplementary, complementary, vertical, adjacent, and alternate interior angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. It is a good introduction to writing paragraphs, 2-column, and/or flow chart proofs. |
| Is This a Rectangle?: | The goal of this task is to provide an opportunity for students to apply a wide range of ideas from geometry and algebra in order to show that a given quadrilateral is a rectangle. Creativity will be essential here as the only given information is the Cartesian coordinates of the quadrilateral's vertices. Using this information to show that the four angles are right angles will require some auxiliary constructions. Students will need ample time and, for some of the methods provided below, guidance. The reward of going through this task thoroughly should justify the effort because it provides students an opportunity to see multiple geometric and algebraic constructions unified to achieve a common purpose. The teacher may wish to have students first brainstorm for methods of showing that a quadrilateral is rectangle (before presenting them with the explicit coordinates of the rectangle for this problem): ideally, they can then divide into groups and get to work straightaway once presented with the coordinates of the quadrilateral for this problem. |
Vetted resources students can use to learn the concepts and skills in this topic.
| Title |
Description |
| Tile Patterns I: octagons and squares: | Students use interior and exterior angles to to verify attributes of an octagon and square. Students are given a tile pattern involving congruent regular octagons and squares. |
| Bank Shot: | This task asks students to use similarity to solve a problem in a context that will be familiar to many, though most students are accustomed to using intuition rather than geometric reasoning to set up the shot. |
| Extensions, Bisections and Dissections in a Rectangle: | This task involves a reasonably direct application of similar triangles, coupled with a moderately challenging procedure of constructing a diagram from a verbal description. |
| Finding the area of an equilateral triangle: | This problem solving task asks students to find the area of an equilateral triangle. Various solutions are presented that include the Pythagoren theorem and trigonometric functions. |
| Solar Eclipse: | This problem solving task encourages students to explore why solar eclipses are rare by examining the radius of the sun and the furthest distance between the moon and the earth. |
| Mt. Whitney to Death Valley: | This task engages students in an open-ended modeling task that uses similarity of right triangles. |
| Joining two midpoints of sides of a triangle: | Using a triangle with line through it, students are tasked to show the congruent angles, and conclude if one triangle is similar to the other. |
| Unit Squares and Triangles: | This problem solving task asks students to find the area of a triangle by using unit squares and line segments. |
| Why Does ASA Work?: | This problem solving task ask students to show the reflection of one triangle maps to another triangle. |
| When Does SSA Work to Determine Triangle Congruence?: | In this problem, we considered SSA. The triangle congruence criteria, SSS, SAS, ASA, all require three pieces of information. It is interesting, however, that not all three pieces of information about sides and angles are sufficient to determine a triangle up to congruence. |
| Points equidistant from two points in the plane: | This task asks students to show how certain points on a plane are equidistant to points on a segment when placed on a perpendicular bisector. |
| Midpoints of the Side of a Parallelogram: | This is a reasonably direct task aimed at having students use previously-derived results to learn new facts about parallelograms, as opposed to deriving them from first principles. |
| Inscribing a square in a circle: | This task provides an opportunity for students to apply triangle congruence theorems in an explicit, interesting context. |
| Construction of perpendicular bisector: | This problem solving task challenges students to construct a perpendicular bisector of a given segment. |
| Why Does SAS Work?: | This problem solving task challenges students to explain the reason why the given triangles are congruent, and to construct reflections of the points. |
| Reflected Triangles: | This task asks students to use a straightedge and compass to construct the line across which a triangle is reflected. |
| Right triangles inscribed in circles II: | This problem solving task asks students to explain certain characteristics about a triangle. |
| Why does SSS work?: | This particular problem solving task exhibits congruency between two triangles, demonstrating translation, reflection and rotation. |
| Bisecting an angle: | This problem solving task challenges students to bisect a given angle. |
| Are the Triangles Congruent?: | The purpose of this task is primarily assessment-oriented, asking students to demonstrate knowledge of how to determine the congruency of triangles. |
| Locating Warehouse: | This problem solving task challenges students to place a warehouse (point) an equal distance from three roads (lines). |
| Tangent Lines and the Radius of a Circle: | This problem solving task challenges students to find the perpendicular meeting point of a segment from the center of a circle and a tangent. |
| Seven Circles I: | This task is intended to help model a concrete situation with geometry. Placing the seven pennies in a circular pattern is a concrete and fun experiment which leads to a genuine mathematical question: does the physical model with pennies give insight into what happens with seven circles in the plane? |
| Angle bisection and midpoints of line segments: | This task provides a construction of the angle bisector of an angle by reducing it to the bisection of an angle to finding the midpoint of a line segment. It is worth observing the symmetry -- for both finding midpoints and bisecting angles, the goal is to cut an object into two equal parts. |
| Inscribing a circle in a triangle II: | This problem solving task focuses on a remarkable fact which comes out of the construction of the inscribed circle in a triangle: the angle bisectors of the three angles of triangle ABC all meet in a point. |
| Find the Angle: | 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. |
| Find the Missing Angle: | This task provides students the opportunity to see how the mathematical ideas embedded in the standards and clusters mature over time. The task uses facts about supplementary, complementary, vertical, adjacent, and alternate interior angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. It is a good introduction to writing paragraphs, 2-column, and/or flow chart proofs. |
| Is This a Rectangle?: | The goal of this task is to provide an opportunity for students to apply a wide range of ideas from geometry and algebra in order to show that a given quadrilateral is a rectangle. Creativity will be essential here as the only given information is the Cartesian coordinates of the quadrilateral's vertices. Using this information to show that the four angles are right angles will require some auxiliary constructions. Students will need ample time and, for some of the methods provided below, guidance. The reward of going through this task thoroughly should justify the effort because it provides students an opportunity to see multiple geometric and algebraic constructions unified to achieve a common purpose. The teacher may wish to have students first brainstorm for methods of showing that a quadrilateral is rectangle (before presenting them with the explicit coordinates of the rectangle for this problem): ideally, they can then divide into groups and get to work straightaway once presented with the coordinates of the quadrilateral for this problem. |
Vetted resources caregivers can use to help students learn the concepts and skills in this topic.
| Title |
Description |
| Tile Patterns I: octagons and squares: | Students use interior and exterior angles to to verify attributes of an octagon and square. Students are given a tile pattern involving congruent regular octagons and squares. |
| Bank Shot: | This task asks students to use similarity to solve a problem in a context that will be familiar to many, though most students are accustomed to using intuition rather than geometric reasoning to set up the shot. |
| Extensions, Bisections and Dissections in a Rectangle: | This task involves a reasonably direct application of similar triangles, coupled with a moderately challenging procedure of constructing a diagram from a verbal description. |
| Finding the area of an equilateral triangle: | This problem solving task asks students to find the area of an equilateral triangle. Various solutions are presented that include the Pythagoren theorem and trigonometric functions. |
| Solar Eclipse: | This problem solving task encourages students to explore why solar eclipses are rare by examining the radius of the sun and the furthest distance between the moon and the earth. |
| Mt. Whitney to Death Valley: | This task engages students in an open-ended modeling task that uses similarity of right triangles. |
| Joining two midpoints of sides of a triangle: | Using a triangle with line through it, students are tasked to show the congruent angles, and conclude if one triangle is similar to the other. |
| Unit Squares and Triangles: | This problem solving task asks students to find the area of a triangle by using unit squares and line segments. |
| Why Does ASA Work?: | This problem solving task ask students to show the reflection of one triangle maps to another triangle. |
| When Does SSA Work to Determine Triangle Congruence?: | In this problem, we considered SSA. The triangle congruence criteria, SSS, SAS, ASA, all require three pieces of information. It is interesting, however, that not all three pieces of information about sides and angles are sufficient to determine a triangle up to congruence. |
| Points equidistant from two points in the plane: | This task asks students to show how certain points on a plane are equidistant to points on a segment when placed on a perpendicular bisector. |
| Midpoints of the Side of a Parallelogram: | This is a reasonably direct task aimed at having students use previously-derived results to learn new facts about parallelograms, as opposed to deriving them from first principles. |
| Inscribing a square in a circle: | This task provides an opportunity for students to apply triangle congruence theorems in an explicit, interesting context. |
| Construction of perpendicular bisector: | This problem solving task challenges students to construct a perpendicular bisector of a given segment. |
| Why Does SAS Work?: | This problem solving task challenges students to explain the reason why the given triangles are congruent, and to construct reflections of the points. |
| Reflected Triangles: | This task asks students to use a straightedge and compass to construct the line across which a triangle is reflected. |
| Right triangles inscribed in circles II: | This problem solving task asks students to explain certain characteristics about a triangle. |
| Why does SSS work?: | This particular problem solving task exhibits congruency between two triangles, demonstrating translation, reflection and rotation. |
| Bisecting an angle: | This problem solving task challenges students to bisect a given angle. |
| Are the Triangles Congruent?: | The purpose of this task is primarily assessment-oriented, asking students to demonstrate knowledge of how to determine the congruency of triangles. |
| Locating Warehouse: | This problem solving task challenges students to place a warehouse (point) an equal distance from three roads (lines). |
| Tangent Lines and the Radius of a Circle: | This problem solving task challenges students to find the perpendicular meeting point of a segment from the center of a circle and a tangent. |
| Seven Circles I: | This task is intended to help model a concrete situation with geometry. Placing the seven pennies in a circular pattern is a concrete and fun experiment which leads to a genuine mathematical question: does the physical model with pennies give insight into what happens with seven circles in the plane? |
| Angle bisection and midpoints of line segments: | This task provides a construction of the angle bisector of an angle by reducing it to the bisection of an angle to finding the midpoint of a line segment. It is worth observing the symmetry -- for both finding midpoints and bisecting angles, the goal is to cut an object into two equal parts. |
| Inscribing a circle in a triangle II: | This problem solving task focuses on a remarkable fact which comes out of the construction of the inscribed circle in a triangle: the angle bisectors of the three angles of triangle ABC all meet in a point. |
| Find the Angle: | 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. |
| Find the Missing Angle: | This task provides students the opportunity to see how the mathematical ideas embedded in the standards and clusters mature over time. The task uses facts about supplementary, complementary, vertical, adjacent, and alternate interior angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. It is a good introduction to writing paragraphs, 2-column, and/or flow chart proofs. |
| Is This a Rectangle?: | The goal of this task is to provide an opportunity for students to apply a wide range of ideas from geometry and algebra in order to show that a given quadrilateral is a rectangle. Creativity will be essential here as the only given information is the Cartesian coordinates of the quadrilateral's vertices. Using this information to show that the four angles are right angles will require some auxiliary constructions. Students will need ample time and, for some of the methods provided below, guidance. The reward of going through this task thoroughly should justify the effort because it provides students an opportunity to see multiple geometric and algebraic constructions unified to achieve a common purpose. The teacher may wish to have students first brainstorm for methods of showing that a quadrilateral is rectangle (before presenting them with the explicit coordinates of the rectangle for this problem): ideally, they can then divide into groups and get to work straightaway once presented with the coordinates of the quadrilateral for this problem. |
