**Title:** Proofs into Practice: The Pythagorean Theorem in the Real World

**Description:**
The introduction of this lesson has students verifying the famous Pythagorean Theorem with a hands-on proof. Students will then apply the Theorem in one of two ways: by solving for the side lengths of a right triangle and by determining whether three side lengths could possibly form a right triangle. Finally, students will choose one of two real-life applicataions to explore, using the Pythagorean Theorem.
**Standard(s): **

[MA2015] GEO (9-12) 21: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.* [G-SRT8]

**Subject:**Mathematics (8 - 12)

**Title:**Proofs into Practice: The Pythagorean Theorem in the Real World

**Description:**The introduction of this lesson has students verifying the famous Pythagorean Theorem with a hands-on proof. Students will then apply the Theorem in one of two ways: by solving for the side lengths of a right triangle and by determining whether three side lengths could possibly form a right triangle. Finally, students will choose one of two real-life applicataions to explore, using the Pythagorean Theorem.

**Title:** Geometric Mean and Indirect Measurement

**Description:**
In this inquiry-based lesson, students will work in groups to gather measurements needed to approximate the height of a building. Students will discover how geometric mean can be used to find the lengths of missing sides of right triangles.
This lesson plan was created by exemplary Alabama Math Teachers through the AMSTI project.
**Standard(s): **

[MA2015] GEO (9-12) 21: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.* [G-SRT8]

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

**Title:**Geometric Mean and Indirect Measurement

**Description:**In this inquiry-based lesson, students will work in groups to gather measurements needed to approximate the height of a building. Students will discover how geometric mean can be used to find the lengths of missing sides of right triangles. This lesson plan was created by exemplary Alabama Math Teachers through the AMSTI project.

**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 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:** 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: 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