MA.3.GR.2.4

Solve mathematical and real-world problems involving the perimeter and area of composite figures composed of non-overlapping rectangles with whole-number side lengths.

Examples

A pool is comprised of two non-overlapping rectangles in the shape of an “L”. The area for a cover of the pool can be found by adding the areas of the two non-overlapping rectangles.

Clarifications

Clarification 1: Composite figures must be composed of non-overlapping rectangles.

Clarification 2: Each rectangle within the composite figure cannot exceed 12 units by 12 units and 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

Terms from the K-12 Glossary

• Composite Figure
• Perimeter
• Rectangle

Vertical Alignment

Previous Benchmarks

Next Benchmarks

Purpose and Instructional Strategies

The purpose of this benchmark is for students to solve mathematical and real-world problems involving the perimeter and area of composite figures with whole-number side lengths. This benchmark builds on the work with perimeter done in Grade 2. The area of each rectangle in a composite figure is expected to be within the appropriate multiplication limits for Grade 3 – up to 12 units by 12 units. All side lengths of composite figures should be given, though cases can be provided where students are expected to use a ruler to measure before finding a composite figure’s perimeter and/or area.
• Students have previous experience with decomposing larger rectangles into smaller rectangles to find individual areas. When students utilized the distributive property in area models to multiply 2-digit factors by a 1-digit factors, students decomposed (broke apart) the 2-digit number as the sum of its tens and ones. Students learned that the product was the sum of smaller rectangles’ areas. Students likely used area models to build fluency within 12 $x$ 12 as well. During instruction of this benchmark, teachers should have students make connections to their previous learning as they begin decomposing the composite figures (MTR.2.1, MTR.5.1).
• Instruction of measuring area of composite figures should include opportunities for students to justify how they decompose their composite figures into 2 or more rectangles before calculating. As students share the different ways they decompose their figures, they identify that any correct decomposition will yield the correct calculation (MTR.2.1, MTR.3.1).

Common Misconceptions or Errors

• Students can confuse the side lengths when determining area calculations once a composite figure has been decomposed. For example, a potential way to decompose the figure from the task below is seen on the right. A student may not yet understand that once decomposed in this way, the length of 14 cm on the right side of the figure is now the sum of 5 cm + 9 cm. The student may continue to multiply by 14 cm to find the areas of each rectangle instead. Likewise, they may multiply 5 cm by 6 cm (instead of 9 cm) to find the area of the upper rectangle. During instruction, encourage students to label how side lengths change once a composite figure has been decomposed.

• Students may add all sides of the parts of the figure once it is decomposed to determine the perimeter of the composite figure. For example in the figure shown, the student may find the perimeter to be (9 cm + 9 cm + 5 cm + 5 cm) + (9 cm + 9 cm + 3 cm + 3 cm) = 52 cm, when the correct answer is 9 cm + 14 cm + 3 cm + 9cm + 6cm + 5cm = 46cm.

Strategies to Support Tiered Instruction

• Instruction includes decomposing figures in multiple ways, finding the area of each individual rectangle and then finding the sum of the two rectangles.
• For example, the teacher provides students with a composite figure and asks them to decompose the figure into two rectangles. Students draw and label the two rectangles as separate parts to show their understanding of how the side lengths have changed once the figure was decomposed. Then, students find the area of each individual rectangle. By drawing the two separate rectangles, students identify which measurements to use due to the decomposing of the figures. One example of how the figure can be decomposed is shown. Students may come up with other ways.

• Teacher provides composite figures created with unit squares. Students cut the figures to decompose them into two separate rectangles and label the dimensions for each figure. Students then find the sum of the area of the two rectangles.
• For example, teacher provides students with figures similar to the one below. Students determine how they can decompose them into two rectangles (there could be more than one way). Students then cut the figure apart to show the two rectangles and writes multiplication equations to represent the area of each part. Finally, students find the sum of the two areas and determines if the area is the same as the whole figure.
• Students can find the perimeters of the two separate rectangles and then determine if the sum of the two perimeters is the same as the perimeter of the composite figure.

What is the perimeter? ___________ What is the area? __________

Instructional Items

Instructional Item 1

A drawing of the top of a desk is shown.

What is the area of the top of the desk?
• a. 14 square feet
• b. 16 square feet
• c. 20 square feet
• d. 25 square feet

*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.
5012050: Grade Three Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7712040: Access Mathematics Grade 3 (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.3.GR.2.AP.4: Explore the perimeter and area of composite figures composed of two non-overlapping rectangles with whole-number side lengths.

Related Resources

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

Formative Assessments

Perimeters of Polygons with All Sides Known:

Students are asked to find the perimeters of three different polygons.

Type: Formative Assessment

Decompose Shapes to Find Area:

Students are given a rectilinear shape and asked to find its area.

Type: Formative Assessment

Lesson Plans

Area: We Need to Know:

In this lesson students build on their knowledge of area by finding the area of a variety of composite figures and create a composite shape when given an area.

Type: Lesson Plan

Area Architects, Lesson 5:

In this unit on area, students explore geometric measurement by becoming "Area Architects" in order to learn the concepts of area and relate area to multiplication and addition. This lesson is the fifth and final lesson of the unit. In this lesson, students will recognize area as additive. Students will find areas of rectangular figures by decomposing them into non-overlapping parts in order to solve a real-world problem. This lesson is focused on single-digit x single-digit dimensions using proper units for dimensions (e.g. ft, yd, m) and square units for the area (e.g. sq. ft, sq. yd, sq. m).

Type: Lesson Plan

Perspectives Video: Teaching Idea

Double Bubble Diagram to Compare Math Concepts:

Unlock an effective teaching strategy for helping students make connections in mathematics using the Double Bubble Diagram in this Teacher Perspectives video for educators.

Type: Perspectives Video: Teaching Idea

The Square Counting Shortcut:

This is a rectangle subdivision task; ideally instead of counting each square. students should break the letters into rectangles, multiply to find the areas, and add up the areas. However, students should not be discouraged from using individual counting to start if they are stuck. Often students will get tired of counting and devise the shortcut method themselves.

MFAS Formative Assessments

Decompose Shapes to Find Area:

Students are given a rectilinear shape and asked to find its area.

Perimeters of Polygons with All Sides Known:

Students are asked to find the perimeters of three different polygons.

Student Resources

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

The Square Counting Shortcut:

This is a rectangle subdivision task; ideally instead of counting each square. students should break the letters into rectangles, multiply to find the areas, and add up the areas. However, students should not be discouraged from using individual counting to start if they are stuck. Often students will get tired of counting and devise the shortcut method themselves.