# MA.912.GR.4.4

Solve mathematical and real-world problems involving the area of two-dimensional figures.

### Examples

Example: A town has 23 city blocks, each of which has dimensions of 1 quarter mile by 1 quarter mile, and there are 4500 people in the town. What is the population density of the town?

### Clarifications

Clarification 1: Instruction includes concepts of population density based on area.
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Geometric Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

• Area

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In elementary grades, students are introduced to the concepts of perimeter and area, with a focus on rectangles. In middle grades, students expand their knowledge of areas of quadrilaterals and triangles. In Geometry, students solve mathematical and real-world problems involving areas of two-dimensional figures, including population density. In Calculus, students will use integrals to connect the concept of area to many other real-world and mathematical contexts.
• Instruction includes reviewing units and conversions within and across different measurement systems (as this was done in middle grades).
• Instruction includes discussing the convenience of answering with exact values (e.g., the simplest radical form or in terms of pi) or with approximations (e.g., rounding to the 22 nearest tenth or hundredth or using 3.14, $\frac{\text{22}}{\text{7}}$ or other approximations for pi). It is also 7 important to explore the consequences of rounding partial answers on the accuracy or precision of the final answer, especially when working in real-world contexts.
• Instruction includes exploring the area of regular polygons and the formula based on the perimeter and the apothem ($A$ = $\frac{\text{1}}{\text{2}}$$a$$p$, where $a$ is the length of the apothem and $p$ is the perimeter). The apothem is the line segment from the center to the midpoint of on the sides of a regular polygon. In many cases, finding the length of the apothem will require the use of trigonometric ratios.
• The population density based on area is calculated by the quotient of the total population and the total area. Have students practice finding the population density or the total population, given the dimensions of a two-dimensional figure. That is, part of their work includes finding the area based on the dimensions. (MTR.7.1
• Instruction includes exploring a variety of real-world situations where finding the area is relevant for different purposes. Problem types include components like percentages, cost and budget, constraints, comparisons and others.
• Problem types include finding missing dimensions given the area of a two-dimensional figure or finding the area of composite figures.

### Common Misconceptions or Errors

• Students may not be careful with units of measurement involving area, particularly when converting from one unit to another.
•  For example, since there are 100 centimeters in a meter, a student may incorrectly conclude that there are 100 square centimeters in a square meter.

• In 2019, the population of Leon County was 293,582 and the population of Sarasota County was 433,742. The area of Sarasota County is 752 square miles, while the area of Leon County is 702 square miles.
•  Part A. Which county has a higher population density?
•  Part B. If the physical shape of the county identified in Part A was a rectangle, what are possible dimensions of the county if the length is greater than the width?
•  Part C. If the county identified in Part A was the physical shape of a right triangle, what are possible dimensions of the base and height of the county?
•  Part D. Does changing the shape of the tract of land change the population density of the county?

• The area of a regular decagon is 24.3 square meters. Determine the side length, in meters, of the regular decagon.

### Instructional Items

Instructional Item 1
• In 2019 the population for Siesta Key, FL, was 5,573 while Destin, FL, had a population of 13,702. Siesta Key is 3.475 square miles and Destin is 8.46 square miles. Which location has a smaller population density?

*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.
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206310: Geometry (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206320: Geometry Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912070: Access Mathematics for Liberal Arts (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 - 2023, 2023 and beyond (current))
1206315: Geometry for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912065: Access Geometry (Specifically in versions: 2015 - 2022, 2022 and beyond (current))
1207350: Mathematics for College Liberal Arts (Specifically in versions: 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.912.GR.4.AP.4: Solve mathematical and/or real-world problems involving the area of triangles, squares, circles or rectangles.

## Related Resources

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

## Formative Assessments

Area and Circumference – 1:

This task is the first in a series of three tasks that assess the students’ understanding of informal derivations of the formulas for the area and circumference of a circle. In this task, students are shown a regular n-gon inscribed in a circle. They are asked to use the formula for the area of the n-gon to derive an equation that describes the relationship between the area and circumference of the circle.

Type: Formative Assessment

Softball Complex:

Students are asked to solve a design problem in which a softball complex is to be located on a given tract of land subject to a set of specifications.

Type: Formative Assessment

How Many Trees?:

Students are asked to determine an estimate of the density of trees and the total number of trees in a forest.

Type: Formative Assessment

Population of Utah:

Students are asked to determine the population of the state of Utah given the state’s population density and a diagram of the state’s perimeter with boundary distances labeled in miles.

Type: Formative Assessment

Estimating Area:

Students are asked to select appropriate geometric shapes to model a lake and then use the model to estimate the surface area of the lake.

Type: Formative Assessment

Area and Circumference - 3:

This task is the third in a series of three tasks that assess the students’ understanding of informal derivations of the formulas for the area and circumference of a circle. In this task, students are given the definition of pi as the area of the unit circle, A(1), and are asked to use this representation of pi along with the results from the two previous tasks to generate formulas for the area and circumference of a circle.

Type: Formative Assessment

Area and Circumference - 2:

This task is the second in a series of three tasks that assesses the students’ understanding of informal derivations of the formulas for the area and circumference of a circle. In this task, students show that the area of the circle of radius r, A(r), can be found in terms of the area of the unit circle, A(1) [i.e., A(r) = r2 · A(1)].

Type: Formative Assessment

## Lesson Plans

My Geometry Classroom:

Students will learn how to find the area and perimeter of multiple polygons in the coordinate plane using the composition and decomposition methods, applying the Distance Formula and Pythagorean Theorem. Students will complete a Geometry Classroom Floor Plan group activity. Students will do a short presentation to discuss their results which leads to the realization that polygons with the same perimeter can have different areas. Students will also complete an independent practice and submit an exit ticket at the end of the lesson.

Type: Lesson Plan

Observing the Centroid:

Students will construct the medians of a triangle then investigate the intersections of the medians.

Type: Lesson Plan

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.

Type: Lesson Plan

NASA Space Shuttle Mission Patches:

Students apply geometric measures and methods, art knowledge, contextual information, and utilize clear and coherent writing to analyze NASA space shuttle mission patches from both a mathematical design and visual arts perspective.

Type: Lesson Plan

Propensity for Density:

Students apply concepts of density to situations that involve area (2-D) and volume (3-D).

Type: Lesson Plan

The Grass is Always Greener:

The lesson introduces area of sectors of circles then uses the areas of circles and sectors to approximate area of 2-D figures. The lesson culminates in using the area of circles and sectors of circles as spray patterns in the design of a sprinkler system between a house and the perimeter of the yard (2-D figure).

Type: Lesson Plan

Poly Wants a Bridge!:

"Poly Wants a Bridge" is a model-eliciting activity that allows students to assist the city of Polygon City with selecting the most appropriate bridge to build. Teams of students are required to analyze properties of bridges, such as physical composition and span length in order to solve the problem.

Type: Lesson Plan

## Perspectives Video: Expert

MicroGravity Sensors & Statistics:

Statistical analysis played an essential role in using microgravity sensors to determine location of caves in Wakulla County.

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

Design Process for a Science Museum Exhibit:

<p>Go behind the scenes and learn about science museum exhibits, design constraints, and engineering workflow! Produced with funding from the Florida Division of Cultural Affairs.</p>

Type: Perspectives Video: Professional/Enthusiast

Using Geometry and Computers to make Art with CNC Machining:

<p>See and see far into the future of arts and manufacturing as a technician explains computer numerically controlled (CNC) machining bit by bit.</p>

Type: Perspectives Video: Professional/Enthusiast

## Perspectives Video: Teaching Ideas

Ecological Sampling Methods and Population Density:

Dr. David McNutt explains how a simple do-it-yourself quadrat and a transect can be used for ecological sampling to estimate population density in a given area.

Type: Perspectives Video: Teaching Idea

KROS Pacific Ocean Kayak Journey: Kites, Geometry, and Vectors:

Set sail with this math teacher as he explains how kites were used for lessons in the classroom.

Related Resources:
KROS Pacific Ocean Kayak Journey: GPS Data Set [.XLSX]
KROS Pacific Ocean Kayak Journey: Path Visualization for Google Earth [.KML]

Type: Perspectives Video: Teaching Idea

How many leaves on a tree? (Version 2):

This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

How many leaves on a tree?:

This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

How many cells are in the human body?:

This problem solving task challenges students to apply the concepts of mass, volume, and density in the real-world context to find how many cells are in the human body.

Hexagonal pattern of beehives:

The goal of this task is to use geometry to study the structure of beehives.

Archimedes and the King's Crown:

This problem solving task uses the tale of Archimedes and the King of Syracuse's crown to determine the volume and mass of gold and silver.

Inscribing a hexagon in a circle:

This problem solving task challenges students to inscribe equilateral triangles and regular hexagons on a circle with a compass and straightedge.

## STEM Lessons - Model Eliciting Activity

NASA Space Shuttle Mission Patches:

Students apply geometric measures and methods, art knowledge, contextual information, and utilize clear and coherent writing to analyze NASA space shuttle mission patches from both a mathematical design and visual arts perspective.

Poly Wants a Bridge!:

"Poly Wants a Bridge" is a model-eliciting activity that allows students to assist the city of Polygon City with selecting the most appropriate bridge to build. Teams of students are required to analyze properties of bridges, such as physical composition and span length in order to solve the problem.

## MFAS Formative Assessments

Area and Circumference – 1:

This task is the first in a series of three tasks that assess the students’ understanding of informal derivations of the formulas for the area and circumference of a circle. In this task, students are shown a regular n-gon inscribed in a circle. They are asked to use the formula for the area of the n-gon to derive an equation that describes the relationship between the area and circumference of the circle.

Area and Circumference - 2:

This task is the second in a series of three tasks that assesses the students’ understanding of informal derivations of the formulas for the area and circumference of a circle. In this task, students show that the area of the circle of radius r, A(r), can be found in terms of the area of the unit circle, A(1) [i.e., A(r) = r2 · A(1)].

Area and Circumference - 3:

This task is the third in a series of three tasks that assess the students’ understanding of informal derivations of the formulas for the area and circumference of a circle. In this task, students are given the definition of pi as the area of the unit circle, A(1), and are asked to use this representation of pi along with the results from the two previous tasks to generate formulas for the area and circumference of a circle.

Estimating Area:

Students are asked to select appropriate geometric shapes to model a lake and then use the model to estimate the surface area of the lake.

How Many Trees?:

Students are asked to determine an estimate of the density of trees and the total number of trees in a forest.

Population of Utah:

Students are asked to determine the population of the state of Utah given the state’s population density and a diagram of the state’s perimeter with boundary distances labeled in miles.

Softball Complex:

Students are asked to solve a design problem in which a softball complex is to be located on a given tract of land subject to a set of specifications.

## Student Resources

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

## Perspectives Video: Expert

MicroGravity Sensors & Statistics:

Statistical analysis played an essential role in using microgravity sensors to determine location of caves in Wakulla County.

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiast

Using Geometry and Computers to make Art with CNC Machining:

<p>See and see far into the future of arts and manufacturing as a technician explains computer numerically controlled (CNC) machining bit by bit.</p>

Type: Perspectives Video: Professional/Enthusiast

How many leaves on a tree? (Version 2):

This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

How many leaves on a tree?:

This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

How many cells are in the human body?:

This problem solving task challenges students to apply the concepts of mass, volume, and density in the real-world context to find how many cells are in the human body.

Hexagonal pattern of beehives:

The goal of this task is to use geometry to study the structure of beehives.

Archimedes and the King's Crown:

This problem solving task uses the tale of Archimedes and the King of Syracuse's crown to determine the volume and mass of gold and silver.

Inscribing a hexagon in a circle:

This problem solving task challenges students to inscribe equilateral triangles and regular hexagons on a circle with a compass and straightedge.

## Parent Resources

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

## Perspectives Video: Professional/Enthusiast

Using Geometry and Computers to make Art with CNC Machining:

<p>See and see far into the future of arts and manufacturing as a technician explains computer numerically controlled (CNC) machining bit by bit.</p>

Type: Perspectives Video: Professional/Enthusiast

How many leaves on a tree? (Version 2):

This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

How many leaves on a tree?:

This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

How many cells are in the human body?:

This problem solving task challenges students to apply the concepts of mass, volume, and density in the real-world context to find how many cells are in the human body.

Hexagonal pattern of beehives:

The goal of this task is to use geometry to study the structure of beehives.