# MA.4.GR.2.2

Solve problems involving rectangles with the same perimeter and different areas or with the same area and different perimeters.

### Examples

Possible dimensions of a rectangle with an area of 24 square feet include 6 feet by 4 feet or 8 feet by 3 feet. This can be found by cutting a rectangle into unit squares and rearranging them.

### Clarifications

Clarification 1: Instruction focuses on the conceptual understanding of the relationship between perimeter and area.

Clarification 2: Within this benchmark, rectangles are limited to having whole-number side lengths.

Clarification 3: Problems involving multiplication are limited to products of up to 3 digits by 2 digits. Problems involving division are limited to up to 4 digits divided by 1 digit.

Clarification 4: Responses include the appropriate units in word form.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Geometric Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

• Perimeter

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

The purpose of this benchmark is for students to understand the relationship between perimeter and area. Students will explore situations where the multiple shapes have the same area and different perimeters and same perimeters and different areas. This benchmark supports the perimeter and area work in MA.4.GR.2.1.
• Instruction will help students begin to generalize that when working with rectangles with the same area, squares will have the smallest perimeter and the longer one side is, the greater the perimeter is going to be.

### Common Misconceptions or Errors

• Students may believe that a rectangle with a large perimeter must also have a large area.

### Strategies to Support Tiered Instruction

• Instruction includes comparing figures with the same perimeter but different areas and the same area but different perimeters.
• For example, students find the area and perimeter for figures created using grid paper making the connection that not all figures with a large perimeter have a large area.
• Instruction includes providing several square tiles that can be arranged to make rectangular figures in many ways. Students build figures with the same area and calculate the perimeter.
• For example, students use 36 tiles to make a figure that is 2 tiles by 18 tiles. They would calculate Area = 2 × 18 = 36 square units, and then calculate Perimeter = 2 + 2 + 18 + 18 = 40 units. Students would then rearrange the tiles to create a rectangle that is 6 tiles by 6 tiles. They would calculate the Area= 6 × 6 =26 square units, and Perimeter as 6 + 6 + 6 + 6 = 24 units. Students compare the area and perimeter of both figures and make the connection that the area of a figure does not determine the perimeter.

Steve has 600 feet of fencing. He is trying to figure out how to build his fence so that he has a rectangle with the greatest square footage inside the fence.
• Part A. What are the dimensions of the fence he can build with the greatest area inside?
• Part B. What is the area inside his fence?

### Instructional Items

Instructional Item 1

Skylar built a rectangular table for her doll house. The area of the table is 105 square inches and the side lengths are whole-number inches. What are some possible perimeters of the table?
• a. 26 inches
• b. 44 inches
• c. 52 inches
• d. 76 inches
• e. 210 inches

*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.
5012060: Mathematics - Grade Four (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7712050: Access Mathematics Grade 4 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2022, 2022 and beyond (current))
5012055: Grade 3 Accelerated Mathematics (Specifically in versions: 2019 - 2022, 2022 and beyond (current))
5012015: Foundational Skills in Mathematics 3-5 (Specifically in versions: 2019 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.4.GR.2.AP.2: Explore the relationship between perimeter and area using rectangles with the same perimeter and different areas or with the same area and different perimeters.

## Related Resources

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

## Formative Assessments

Rectangles with the Same Perimeter:

Students are asked to find the whole number dimensions of every rectangle with a given perimeter and then find the area of each rectangle.

Type: Formative Assessment

Find All The Possible Rectangles:

Students are asked to find the whole number dimensions of every rectangle with a given area and then find the perimeter of each rectangle.

Type: Formative Assessment

## Lesson Plans

Zoning with Area and Perimeter Part 3:

This is Part 3 of the Zoning with Area and Perimeter lesson. Students will be asked to increase the area of each zone they created in Parts 1 and 2. They will then determine the perimeter of each zone based on its new area measurements.

Type: Lesson Plan

Zoning with Area and Perimeter Part 2:

This is Part 2 of the Zoning with Area and Perimeter unit. Students will be asked to place and zone three schools to serve the students of the community. They will be provided the perimeters of each school and will need to maximize its area in this integrated lesson plan.

Type: Lesson Plan

Zoning with Area and Perimeter Part 1:

Students will hold a town hall meeting to zone a new community. They will assign a different area measurement to each zone and then determine the zone’s perimeter. Students will explore how rectangles with the same area can have different perimeters in this integrated lesson plan.

Type: Lesson Plan

Numbers Grow Here:

The students will use prior knowledge of the area formula to design a garden with a  area. Students will compare gardens and note that rectangles with the same area could have different dimensions.

Type: Lesson Plan

Same Perimeter, Different Area:

In this lesson, students are presented with a problem that requires them to create rectangles with the same perimeter but different areas.  Students also search for relationships among the perimeters and areas of different rectangles and find which characteristics produce a rectangle with the greatest area.

Type: Lesson Plan

Cupcake Shop Creator:

In this Model Eliciting Activity, MEA, students will become architects to determine the best layout for a new cupcake shop coming to town. Students will use area and perimeter to assist in presenting the best layout of the store. The factors that the students will need to consider are: kitchen space, front counter space, a bathroom, and a wall to display and sell merchandise.

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 process. 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 MEA’s visit: https://www.cpalms.org/cpalms/mea.aspx

Type: Lesson Plan

Shoe Closet MEA:

In this open-ended problem, students will work in teams to determine a procedure for ranking shoe closet styles for a person’s dream closet. Students will need to calculate the perimeter and cost for the closet, make decisions based on a table of data, and write a letter to the client providing evidence for their decisions.

Type: Lesson Plan

Area and Perimeter of Rectangles Investigations:

Students will determine the validity of the statement, "All rectangles with the same area will have the same perimeter" through two investigations.

Type: Lesson Plan

House Building Architects:

In this lesson, students are tasked with drawing a house based on given directions. The directions include the area and perimeter of particular features of the house. This resource is recommended as a review of perimeter and area.

Type: Lesson Plan

## Tutorial

Area and Perimeter Word Problem: Table Dimensions:

This Khan Academy tutorial video presents a step-by-step solution for finding the length and width of a table when given its area and perimeter.

Type: Tutorial

## STEM Lessons - Model Eliciting Activity

Cupcake Shop Creator:

In this Model Eliciting Activity, MEA, students will become architects to determine the best layout for a new cupcake shop coming to town. Students will use area and perimeter to assist in presenting the best layout of the store. The factors that the students will need to consider are: kitchen space, front counter space, a bathroom, and a wall to display and sell merchandise.

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 process. 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 MEA’s visit: https://www.cpalms.org/cpalms/mea.aspx

Shoe Closet MEA:

In this open-ended problem, students will work in teams to determine a procedure for ranking shoe closet styles for a person’s dream closet. Students will need to calculate the perimeter and cost for the closet, make decisions based on a table of data, and write a letter to the client providing evidence for their decisions.

## MFAS Formative Assessments

Find All The Possible Rectangles:

Students are asked to find the whole number dimensions of every rectangle with a given area and then find the perimeter of each rectangle.

Rectangles with the Same Perimeter:

Students are asked to find the whole number dimensions of every rectangle with a given perimeter and then find the area of each rectangle.

## Student Resources

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

## Tutorial

Area and Perimeter Word Problem: Table Dimensions:

This Khan Academy tutorial video presents a step-by-step solution for finding the length and width of a table when given its area and perimeter.

Type: Tutorial

## Parent Resources

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