# MA.7.GR.1.2

Solve mathematical or real-world problems involving the area of polygons or composite figures by decomposing them into triangles or quadrilaterals.

### Clarifications

Clarification 1: Within this benchmark, the expectation is not to find areas of figures on the coordinate plane or to find missing dimensions.
General Information
Subject Area: Mathematics (B.E.S.T.)
Grade: 7
Strand: Geometric Reasoning
Date Adopted or Revised: 08/20
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Area
• Composite Figure
• Polygon
• Quadrilateral
• Triangle

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 6, students solved problems involving the area of quadrilaterals and composite figures by decomposing them into triangles or rectangles. In grade 7, students solve problems involving the area of polygons or composite figures by decomposing them into triangles or quadrilaterals. In high school, students will extend this knowledge to solve mathematical and real-world problems involving the perimeter or area of any polygon using coordinate geometry and other tools.
• Instruction includes problems where multiple decompositions are possible so students understand the various pathways to a solution (MTR.5.1). Scaffolded instruction may include figures on grid paper to allow students to more easily count the total area. Select and order student solutions to be shared with the whole group (MTR.4.1), depicting various solution pathways.
• Instruction includes figures where an efficient method is to subtract a basic figure from a larger figure.
• Students should use grid paper to draw a polygon that is composed of triangles and quadrilaterals that can be exchanged with a partner or within a group to find the corresponding areas.

### Common Misconceptions or Errors

• Students may neglect to add the areas of the decomposed figures to find the total area of the composite figure. Students may also incorrectly add one (or more) of the decomposed figures more than once. To address misconceptions, have students mark or color the figures as they add them to the total to keep track of their work (MTR.3.1).
• Students may not decompose the figure into the most basic figures. To address this misconception, ask students if they can find the area of each of the pieces they have, or if they can break any of them down further to find a more familiar figure (MTR.5.1).

### Strategies to Support Tiered Instruction

• Instruction includes writing the area of each decomposed figure inside the original figure and placing a check next to each of the decomposed areas as they are added to determine the total area of the composite figure.
• Teacher provides geometric software for students to interact with composite figures to develop understanding of how to decompose two dimensional figures.
• Teacher provides paper cutouts of different composite figures for students to fold or cut into triangles or quadrilaterals to visually understand how to decompose the area.
• Instruction includes color-coding parallel bases or heights to assist in determine missing measurements of composite figures.
• For example, given the figure below (assuming the two right triangles have the same side lengths as each other), students can highlight the parallel bases of the rectangle.

• Teacher has students mark or color the figures as they add them to the total to keep track of their work (MTR.3.1).
• Teacher asks students if they can find the area of each of the pieces they have, or if they can break any of them down further to find a more familiar figure (MTR.5.1).

### Instructional Tasks

Instructional Task 1 (MTR.6.1)
After a recent storm, Evan has been offered two jobs to replace patio screens. The layouts for the screens needed at both locations are given below. The shaded part represents a stone layout that does not need to be screened.
• Job #1

• Job #2

If Evan gets paid by the square inch and would like the highest paying job, which job should he take? Justify your reasoning.

Instructional Task 2 (MTR.3.1)
Tyler and Samantha are building the set for a school play. The design shown below was cut out of wood and now needs to be covered in fabric.

• Part A. If each square in the grid has a length of one foot, estimate the total area of wood that needs to be covered. Justify your answer.
• Part B. What is the exact total area of the wood, in feet, that needs to be covered? Share your strategy with a partner.

### Instructional Items

Instructional Item 1
Find the area of the figure below. Note that the figure may not be drawn to scale.

Instructional Item 2
Bena is building a kite based on the design shown below. Determine how much ripstop nylon she will need to purchase for the sail material.

*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.
1205020: M/J Accelerated Mathematics Grade 6 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current))
1205040: M/J Grade 7 Mathematics (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))
7812020: Access M/J Grade 7 Mathematics (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.7.GR.1.AP.2: Decompose complex shapes (polygon, trapezoid, and pentagon) into simple shapes (rectangles, squares, triangles) to measure area.

## Related Resources

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

## Formative Assessments

Octagon Area:

Students are asked to find the area of a composite figure.

Type: Formative Assessment

Composite Polygon Area:

Students are asked to find the area of a composite figure.

Type: Formative Assessment

## Lesson Plans

Clean It Up:

Students will help a volunteer coordinator choose cleanup projects that will have the greatest positive impact on the environment and the community.  They will apply their knowledge of how litter can impact ecosystems along with some math skills to make recommendations for cleanup zones to prioritize.  Students will explore the responsibilities of citizens to maintain a clean environment and the impact that litter can have on society in this integrated Model Eliciting Activity.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations.  Click here to learn more about MEAs and how they can transform your classroom.

Type: Lesson Plan

Breaking Up is Hard to Do:

Student will use geoboards to decompose composite figures and polygons into squares, rectangles, and triangles in order to find the total area.

Type: Lesson Plan

Building Graduation Caps:

Students will apply skills from the Geometry Domain to build graduation caps for themselves using heavyweight poster paper. They will also apply some basic mathematical skills to determine dimensions and to determine minimum cost. Some of the Geometric skills reinforced in Building Graduation Caps: Cooperative Assignment are finding area, applying the concept of similarity, and the application of the properties of parallelograms. Other skills also involved in this application are measuring, and statistical calculations, such as finding the mean and the range. In addition to the hands-on group project that takes place during the lesson, there is the Prerequisite Skills Assessment: Area that should be administered before the group activity and a home-learning activity. Building Graduation Caps: Individual Assignment is the home-learning assignment; it is designed to reinforce the skills learned in the group activity.

Type: Lesson Plan

Installing Tile Floor:

In this Model Eliciting Activity (MEA), students will analyze data related to tiling rooms in a house. Students will calculate the square footage of various rooms, convert measurements to determine the amount of tile needed, and compute both the cost of the tiles and the cost of installation. They will evaluate and compare different flooring options based on cost, quality, and installation factors, and develop a procedure to recommend the best choices.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought processes. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEAs visit: https://www.cpalms.org/cpalms/mea.aspx

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

Finding Area with Hands-On Measurement:

This lesson allows students to apply the area of triangles, quadrilaterals, and trapezoids to composite figures, and gives students a chance to work with classmates to find the area by taking measurements and making the necessary calculations. Students will also see the relationship between the area formulas for rectangles, triangles, trapezoids, and polygons.

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

## Perspectives Video: Professional/Enthusiast

Modeling with Polygons for 3D Printers:

<p>Understand 3D modeling from a new angle when you learn about surface geometry and 3D printing.</p>

Type: Perspectives Video: Professional/Enthusiast

## STEM Lessons - Model Eliciting Activity

Installing Tile Floor:

In this Model Eliciting Activity (MEA), students will analyze data related to tiling rooms in a house. Students will calculate the square footage of various rooms, convert measurements to determine the amount of tile needed, and compute both the cost of the tiles and the cost of installation. They will evaluate and compare different flooring options based on cost, quality, and installation factors, and develop a procedure to recommend the best choices.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought processes. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEAs visit: https://www.cpalms.org/cpalms/mea.aspx

## MFAS Formative Assessments

Composite Polygon Area:

Students are asked to find the area of a composite figure.

Octagon Area:

Students are asked to find the area of a composite figure.

## Student Resources

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

## Perspectives Video: Professional/Enthusiast

Modeling with Polygons for 3D Printers:

<p>Understand 3D modeling from a new angle when you learn about surface geometry and 3D printing.</p>

Type: Perspectives Video: Professional/Enthusiast

## Parent Resources

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

## Perspectives Video: Professional/Enthusiast

Modeling with Polygons for 3D Printers:

<p>Understand 3D modeling from a new angle when you learn about surface geometry and 3D printing.</p>

Type: Perspectives Video: Professional/Enthusiast