# MA.8.GR.1.6

Develop and use formulas for the sums of the interior angles of regular polygons by decomposing them into triangles.

### Clarifications

Clarification 1: Problems include representing angle measures as algebraic expressions.
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Geometric Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Regular Polygon

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grades 6 and 7, students found areas of quadrilaterals and other polygons by decomposing them into triangles and trapezoids. In grade 8, students develop and use formulas for the sums of the interior angles of regular polygons by decomposing them into triangles. In Geometry, students will use this knowledge to prove relationships and theorems about triangles, parallelograms, trapezoids and other polygons.
• Once students understand the conceptual understanding associated with this benchmark, students should progress from numerical expressions to algebraic expressions.
• When beginning the exploration with polygons with four or more sides, students should be able to use one vertex to draw diagonals to non-adjacent vertices. Once students have drawn the diagonals, have them cut along the diagonals to showcase triangles.

• Once the triangles are cut, then students can lay them out to see the number of triangles and relate the work to prior work with the sum of the angles of a triangle. Students can label their angles and show their equations that help provide information on the sum of the interior angles as shown below.

• Students should record this information in a chart like the one shown below to help them create a rule to use instead of counting the triangles each time.

• Once students understand the sum of the interior angles, connections should be made to regular polygons. Students can add a column to indicate the regular polygon measurements of each angle.
• Encourage students to use proper vocabulary terms for polygons and regular polygons.

### Common Misconceptions or Errors

• Students may incorrectly draw additional lines from the vertex to create additional triangles.

### Strategies to Support Tiered Instruction

• Teacher encourages students to begin at an identified vertex and move around the polygon from that vertex when decomposing the polygons into triangles.

• Part A. Draw a pentagon, hexagon, heptagon and an octagon.
• Part B. Determine the number of triangles that can be drawn from one vertex to each of the others in each figure.
• Part C. Develop a conjecture to determine if there is a pattern or formula that can be determined to find the sums of the interior angles for any polygon.

### Instructional Items

Instructional Item 1
Find the number of degrees for the sum of the interior angles of a regular 12-sided figure.

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
1205050: M/J Accelerated Mathematics Grade 7 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current))
1205070: M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7812030: Access M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.8.GR.1.AP.6: Use tools to calculate the sum of the interior angles of regular polygons when given the formula.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Lesson Plans

Discovering Triangle Sum:

This lesson is designed to address all levels and types of learners to improve understanding of the triangle sum theorem from the simplest perspective and progress steadily by teacher led activities to a more complex level. It is intended to create a solid foundation in geometric reasoning to help students advance to higher levels in confidence.

Type: Lesson Plan

Geometer Sherlock: Triangle Investigations:

The students will investigate and discover relationships within triangles; such as, the triangle angle sum theorem, and the triangle inequality theorem.

Type: Lesson Plan

Shape It Up:

Students will derive the formula for the sum of the interior angles of a polygon by drawing diagonals and applying the Triangle Sum Theorem. The measure of each interior angle of a regular polygon is also determined.

Type: Lesson Plan

The Ins and Outs of Polygons:

In this lesson, students will explore how to find the sum of the measures of the angles of a triangle, then use this knowledge to find the sum of the measures of angles of other polygons. They will also be able to find the sum of the exterior angles of triangles and other polygons. Using both concepts, students will be able to find missing measurements.

Type: Lesson Plan

How Many Degrees?:

This lesson facilitates the discovery of a formula for the sum of the interior angles of a regular polygon. Students will draw all the diagonals from one vertex of various polygons to find how many triangles are formed. They will use this and their prior knowledge of triangles to figure out the sum of the interior angles. This will lead to the development of a formula for finding the sum of interior angles and the measure of one interior angle.

Type: Lesson Plan

Tile Patterns II: hexagons:

This task is ideally suited for instruction purposes where students can take their time and develop several of the standards, as the mathematical content is directly related to, but somewhat exceeds, the content of the standard on sums of angles in triangles. Careful analysis of the angles requires students to construct valid arguments using abstract and quantitative reasoning. Producing the picture in part (c) helps students identify a common mathematical argument repeated multiple times. Students may use pattern blocks to develop the intuition for decomposing the hexagon into triangles.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

Tile Patterns II: hexagons:

This task is ideally suited for instruction purposes where students can take their time and develop several of the standards, as the mathematical content is directly related to, but somewhat exceeds, the content of the standard on sums of angles in triangles. Careful analysis of the angles requires students to construct valid arguments using abstract and quantitative reasoning. Producing the picture in part (c) helps students identify a common mathematical argument repeated multiple times. Students may use pattern blocks to develop the intuition for decomposing the hexagon into triangles.