**Title:** What you know about the Bermuda Triangle?

**Description:**
The basis of this lesson is for students to form a connection between the importance of geometry as it relates to the Bermuda Triangle. Students should think about the geographical locations and how they form a particular type of triangle. Within the triangle exists a relationship between the angles and the length of the sides. As the students reflect prior concepts and skills should come to mind to help them solve the problem. The ideas presented here can be used as the final lesson in a unit on triangles to summarize all that students have learned. Alternatively, the main problem from this lesson could be used as a final assessment activity for a unit on triangles.
This is a College- and Career-Ready Standards showcase lesson plan.
**Standard(s): **

[MA2015] GEO (9-12) 10: Prove theorems about triangles. *Theorems include measures of interior angles of a triangle sum to 180 ^{o}, 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, and the medians of a triangle meet at a point.* [G-CO10]

**Subject:**Mathematics (9 - 12)

**Title:**What you know about the Bermuda Triangle?

**Description:**The basis of this lesson is for students to form a connection between the importance of geometry as it relates to the Bermuda Triangle. Students should think about the geographical locations and how they form a particular type of triangle. Within the triangle exists a relationship between the angles and the length of the sides. As the students reflect prior concepts and skills should come to mind to help them solve the problem. The ideas presented here can be used as the final lesson in a unit on triangles to summarize all that students have learned. Alternatively, the main problem from this lesson could be used as a final assessment activity for a unit on triangles. This is a College- and Career-Ready Standards showcase lesson plan.

**Title:** Investigation of Special Segments of Triangles

**Description:**
This lesson will enable students to investigate three special segments of triangles in a very concrete way. The students will fold paper triangles to create the segments. This lesson would be a great way for students to explore the properties of the segments and their intersections.
**Standard(s): **

[MA2015] GEO (9-12) 26: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. [G-C3]

**Subject:**Mathematics (9 - 12)

**Title:**Investigation of Special Segments of Triangles

**Description:**This lesson will enable students to investigate three special segments of triangles in a very concrete way. The students will fold paper triangles to create the segments. This lesson would be a great way for students to explore the properties of the segments and their intersections.

**Title:** The Triangle Inequality

**Description:**
In this reproducible activity, from an Illuminations lesson, students investigate how three sides can be arranged to make both triangles and non-triangles.
**Standard(s): **

[MA2015] GEO (9-12) 17: Prove theorems about triangles. *Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle similarity.* [G-SRT4]

**Subject:**Mathematics

**Title:**The Triangle Inequality

**Description:**In this reproducible activity, from an Illuminations lesson, students investigate how three sides can be arranged to make both triangles and non-triangles.

**Thinkfinity Partner:**Illuminations

**Grade Span:**9,10,11,12

**Title:** Inequalities for Sides and Angles

**Description:**
This reproducible activity sheet, from an Illuminations lesson, guides students as they examine the relationship between the locations of sides and angles in a triangle.
**Standard(s): **

[MA2015] GEO (9-12) 10: Prove theorems about triangles. *Theorems include measures of interior angles of a triangle sum to 180 ^{o}, 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, and the medians of a triangle meet at a point.* [G-CO10]

**Subject:**Mathematics

**Title:**Inequalities for Sides and Angles

**Description:**This reproducible activity sheet, from an Illuminations lesson, guides students as they examine the relationship between the locations of sides and angles in a triangle.

**Thinkfinity Partner:**Illuminations

**Grade Span:**9,10,11,12

**Title:** Beyond Triangula Island

**Description:**
Using this reproducible activity sheet, from an Illuminations lesson, students record their findings as they measure distances from interior points to sides of an equilateral triangle, a square, an octagon and a hexagon. Students answer questions based on their findings.
**Standard(s): **

[MA2015] GEO (9-12) 10: Prove theorems about triangles. *Theorems include measures of interior angles of a triangle sum to 180 ^{o}, 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, and the medians of a triangle meet at a point.* [G-CO10]

**Subject:**Mathematics

**Title:**Beyond Triangula Island

**Description:**Using this reproducible activity sheet, from an Illuminations lesson, students record their findings as they measure distances from interior points to sides of an equilateral triangle, a square, an octagon and a hexagon. Students answer questions based on their findings.

**Thinkfinity Partner:**Illuminations

**Grade Span:**9,10,11,12

**Title:** Hospital Locator

**Description:**
In this Illuminations lesson, students begin with a problem in a real-world context to motivate the need to construct circumcenters and then incenters of triangles. Students must make sense of these constructions in terms of bisecting sides and angles. There are links to student interactives and other resources.
**Standard(s): **

[MA2015] GEO (9-12) 41: 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).* [G-MG3]

**Subject:**Mathematics

**Title:**Hospital Locator

**Description:**In this Illuminations lesson, students begin with a problem in a real-world context to motivate the need to construct circumcenters and then incenters of triangles. Students must make sense of these constructions in terms of bisecting sides and angles. There are links to student interactives and other resources.

**Thinkfinity Partner:**Illuminations

**Grade Span:**9,10,11,12

**Title:** Inequalities in Triangles

**Description:**
In this Illuminations lesson, students use pasta to create models of triangles and non-triangles in order to investigate the relationship between the longest side of the triangle and the sum of the other two sides of the triangle. In addition, students measure the sides and angles of a scalene triangle and investigate the relationship between the location of the largest angle and largest side in a triangle. There are also links to online activity sheets.
**Standard(s): **

[MA2015] GEO (9-12) 17: Prove theorems about triangles. *Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle similarity.* [G-SRT4]

**Subject:**Mathematics

**Title:**Inequalities in Triangles

**Description:**In this Illuminations lesson, students use pasta to create models of triangles and non-triangles in order to investigate the relationship between the longest side of the triangle and the sum of the other two sides of the triangle. In addition, students measure the sides and angles of a scalene triangle and investigate the relationship between the location of the largest angle and largest side in a triangle. There are also links to online activity sheets.

**Thinkfinity Partner:**Illuminations

**Grade Span:**9,10,11,12

**Title:** Pieces of Proof

**Description:**
This lesson offers an intermediate step of putting together the statements and reasons to build a formal geometry proof. There is a leap to be made from understanding postulates and theorems in geometry to writing proofs using them. Students are given the parts of the proof and have to put them in a correct order, like a puzzle, rather than starting with a blank page to generate the statements and reasons themselves.
**Standard(s): **

[MA2015] GEO (9-12) 18: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. [G-SRT5]

**Subject:**Mathematics

**Title:**Pieces of Proof

**Description:**This lesson offers an intermediate step of putting together the statements and reasons to build a formal geometry proof. There is a leap to be made from understanding postulates and theorems in geometry to writing proofs using them. Students are given the parts of the proof and have to put them in a correct order, like a puzzle, rather than starting with a blank page to generate the statements and reasons themselves.

**Thinkfinity Partner:**Illuminations

**Grade Span:**9,10,11,12

**Title:** Half Angle

**Description:**
This student interactive, from an Illuminations lesson, allows students to find the incenter of a triangle and see a visual explanation of why it occurs where it does.
**Standard(s): **

[MA2015] GEO (9-12) 17: Prove theorems about triangles. *Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle similarity.* [G-SRT4]

**Subject:**Mathematics

**Title:**Half Angle

**Description:**This student interactive, from an Illuminations lesson, allows students to find the incenter of a triangle and see a visual explanation of why it occurs where it does.

**Thinkfinity Partner:**Illuminations

**Grade Span:**9,10,11,12

**Title:** Hospital Locator

**Description:**
In this student interactive, from an Illuminations lesson, students act as community planners, trying to place a new medical center equidistant from three cities.
**Standard(s): **

[MA2015] MI1 (9-12) 11: Describe the development of mathematical tools and their applications. (Alabama)

**Subject:**Mathematics

**Title:**Hospital Locator

**Description:**In this student interactive, from an Illuminations lesson, students act as community planners, trying to place a new medical center equidistant from three cities.

**Thinkfinity Partner:**Illuminations

**Grade Span:**9,10,11,12

**Title:** IGD: Area of a Triangle

**Description:**
This student interactive, from Illuminations, helps students understand and investigate how to find the area of a triangle by finding a way to make it look like a simpler shap--one whose area formula is known. Students make a copy of the triangle, and then make a parallelogram using the triangle and its copy. Students then determine that the area of the triangle is one-half that of the parallelogram.
**Standard(s): **

[MA2015] GEO (9-12) 17: Prove theorems about triangles.

**Subject:**Mathematics

**Title:**IGD: Area of a Triangle

**Description:**This student interactive, from Illuminations, helps students understand and investigate how to find the area of a triangle by finding a way to make it look like a simpler shap--one whose area formula is known. Students make a copy of the triangle, and then make a parallelogram using the triangle and its copy. Students then determine that the area of the triangle is one-half that of the parallelogram.

**Thinkfinity Partner:**Illuminations

**Grade Span:**3,4,5,6,7,8

**Title:** IGD: Euler Line

**Description:**
This student interactive, from Illuminations, helps students understand and investigate Euler lines. Students read a definition and then construct the Euler line of a triangle using the interactive tool. They learn that the Euler line involves three points associated with triangles: the circumcenter, orthocenter, and centroid.
**Standard(s): **

[MA2015] MI1 (9-12) 11: Describe the development of mathematical tools and their applications. (Alabama)

**Subject:**Mathematics

**Title:**IGD: Euler Line

**Description:**This student interactive, from Illuminations, helps students understand and investigate Euler lines. Students read a definition and then construct the Euler line of a triangle using the interactive tool. They learn that the Euler line involves three points associated with triangles: the circumcenter, orthocenter, and centroid.

**Thinkfinity Partner:**Illuminations

**Grade Span:**9,10,11,12

**Title:** IGD: Median

**Description:**
This student interactive, from Illuminations, helps students understand and investigate the median of a triangle. Students read a definition and then use the interactive tool to manipulate and explore the properties of triangle medians by changing the shape of a triangle.
**Standard(s): **

[MA2015] GEO (9-12) 26: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. [G-C3]

**Subject:**Mathematics

**Title:**IGD: Median

**Description:**This student interactive, from Illuminations, helps students understand and investigate the median of a triangle. Students read a definition and then use the interactive tool to manipulate and explore the properties of triangle medians by changing the shape of a triangle.

**Thinkfinity Partner:**Illuminations

**Grade Span:**9,10,11,12

**Title:** IGD: Simson Line

**Description:**
This student interactive, from Illuminations, helps students understand and investigate a triangle s Simson line, which is the line passing through the feet of the perpendiculars dropped from P to the three sides of a triangle. Students construct the Simson line for a given triangle using the interactive applet.
**Standard(s): **

[MA2015] MI1 (9-12) 11: Describe the development of mathematical tools and their applications. (Alabama)

**Subject:**Mathematics

**Title:**IGD: Simson Line

**Description:**This student interactive, from Illuminations, helps students understand and investigate a triangle s Simson line, which is the line passing through the feet of the perpendiculars dropped from P to the three sides of a triangle. Students construct the Simson line for a given triangle using the interactive applet.

**Thinkfinity Partner:**Illuminations

**Grade Span:**9,10,11,12

**Title:** Isosceles Triangle Investigation

**Description:**
This student interactive, from an Illuminations lesson, allows students to investigate the relationship between the area of the triangle and the length of its base.
**Standard(s): **

[MA2015] ALT (9-12) 33: Write a function that describes a relationship between two quantities.* [F-BF1]

**Subject:**Mathematics

**Title:**Isosceles Triangle Investigation

**Description:**This student interactive, from an Illuminations lesson, allows students to investigate the relationship between the area of the triangle and the length of its base.

**Thinkfinity Partner:**Illuminations

**Grade Span:**9,10,11,12

**Title:** Triangle Classification

**Description:**
This student interactive, from an Illuminations lesson, allows students to turn a segment into a right triangle, an isosceles triangle, and an obtuse triangle.
**Standard(s): **

[MA2015] GEO (9-12) 17: Prove theorems about triangles.

**Subject:**Mathematics

**Title:**Triangle Classification

**Description:**This student interactive, from an Illuminations lesson, allows students to turn a segment into a right triangle, an isosceles triangle, and an obtuse triangle.

**Thinkfinity Partner:**Illuminations

**Grade Span:**9,10,11,12

**Title:** Triangle Island

**Description:**
This student interactive, from an Illuminations lesson, allows students to explore the distances from an interior point of a polygon to its sides and determine the sum of these distances.
**Standard(s): **

[MA2015] GEO (9-12) 17: Prove theorems about triangles.

**Subject:**Mathematics

**Title:**Triangle Island

**Description:**This student interactive, from an Illuminations lesson, allows students to explore the distances from an interior point of a polygon to its sides and determine the sum of these distances.

**Thinkfinity Partner:**Illuminations

**Grade Span:**9,10,11,12