### 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.

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**912

**Strand:**Geometric Reasoning

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- 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.

### Instructional Tasks

*Instructional Task 1 (*

*MTR.7.1*)- 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?

Instructional Task 2 (MTR.3.1)

Instructional Task 2 (MTR.3.1)

- 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

## Related Access Points

## Related Resources

## Formative Assessments

## Lesson Plans

## Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

## Perspectives Video: Teaching Ideas

## Problem-Solving Tasks

## STEM Lessons - Model Eliciting Activity

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.

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.

"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.

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.

## MFAS Formative Assessments

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.

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*) = *r*^{2} · *A*(1)].

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.

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.

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

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.

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

## Perspectives Video: Expert

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

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiast

<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

## Problem-Solving Tasks

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

## Parent Resources

## Perspectives Video: Professional/Enthusiast

<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

## Problem-Solving Tasks

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.

Type: Problem-Solving Task

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task