# MA.7.GR.1.1

Apply formulas to find the areas of trapezoids, parallelograms and rhombi.

### Clarifications

Clarification 1: Instruction focuses on the connection from the areas of trapezoids, parallelograms and rhombi to the areas of rectangles or triangles.

Clarification 2: Within this benchmark, the expectation is not to memorize area formulas for trapezoids, parallelograms and rhombi.

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

• Area
• Parallelogram
• Rectangle
• Rhombus
• Triangle
• Trapezoid

### 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 apply formulas to find the areas of trapezoids, parallelograms and rhombi. 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 using students’ prior knowledge of finding the area of a rectangle to build the area formulas for trapezoids, parallelograms and rhombi. The use of grid paper can support students in counting the squares to verify the areas are accurate.
• Investigations and explore activities for students can include:
• Draw or provide a cutout of a rhombus. Slice the rhombus vertically at a right angle. Slide the sliced off portion to form a square so students can see the base and height of the square are the same as that of the rhombus (MTR.5.1).

• Draw or provide a cutout of a parallelogram. Slice the parallelogram vertically at a right angle. Slide the sliced off portion to form a rectangle so students can see the base and height of the rectangle are the same as that of the parallelogram (MTR.5.1).

• Draw or provide a cutout of a trapezoid. Duplicate the trapezoid using patty paper or tracing paper and rotate it to form a larger parallelogram formed with both figures. The base of the parallelogram then becomes the sum of the two bases of the original trapezoid while the height is the perpendicular height of the original trapezoid. The area is one half of the area of this parallelogram since it contains two identical trapezoids (MTR.5.1).

• Instruction includes the comparison of formulas between rectangles, trapezoids, parallelograms and rhombi.

### Common Misconceptions or Errors

• Students may incorrectly identify a side length as a height rather than using the perpendicular distance between the bases. To address this misconception, use cutouts or measuring tools to show that these distances are not the same; consider using physical objects that are not square or rectangular to make sense of finding the correct height.
• Students may not properly locate the height or base(s) when using figures in various orientations. To address this misconception, provide multiple orientations of objects and figures. Note that parallelograms are like rectangles, in that any side can be considered a base, so there are two possible heights.

### Strategies to Support Tiered Instruction

• Teacher models measuring the length and height of a given shape to demonstrate the difference between the dimensions.
• Instruction includes the use of manipulatives or geometric software to demonstrate the similarity of trapezoids, parallelograms and rhombi to square, rectangles, and triangles.
• Teacher co-creates a graphic organizer with images and formulas for trapezoids, parallelograms and rhombi and uses different colors to connect the dimensions of the figures to the variables within the formulas.
• Teacher uses cutouts or measuring tools to show that the length of the side of a figure is not necessarily the same as its height. Consider using physical objects that are not square or rectangular to make sense of finding the correct height.

Trace a parallelogram or a rhombus on a sheet of graph paper. Highlight (or color) each of the sides a different color. Slice your two-dimensional figure vertically from a vertex at a right angle to an opposite side, to create a right triangle. Move the sliced-off portion to form a rectangle.

• Part A. What are the length and width of the created rectangle?
• Part B. Determine the formula for finding the area of your two-dimensional figure.
• Part C. Compare your two-dimensional figure and formula with a partner. What do you notice?

Duplicate the trapezoid given below using patty paper or tracing paper and color the corresponding bases the same color. Cut out both trapezoids. Rotate one trapezoid 180o and line it up next to the other.

• Part A. What figure has formed?
• Part B. What is the formula for figure’s area? Use this information to determine the formula for finding the area of a trapezoid.

### Instructional Items

Instructional Item 1
A new park is being built in the shape of a trapezoid, as shown in the diagram below. The builders will cover the ground with a solid rubber surface to ensure the children playing have a safe and soft place to land when they jump or fall. How many square yards of rubber will be needed for this park?

Instructional Item 2

Find the area of the figure below.

*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.1: Given the formulas, find the area of parallelograms and rhombi.

## Related Resources

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

## Image/Photograph

Clipart: Geometric Shapes:

In this lesson, you will find clip art and various illustrations of polygons, circles, ellipses, star polygons, and inscribed shapes.

Type: Image/Photograph

## 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.

Type: Lesson Plan

Guiding Grids: Math inspired self-portraits:

Students will create a proportional self portrait from a photo using a gridded drawing method and learn how a grid system can help accurately enlarge an image in a work of art. Students will use the mathematical concepts of scale, proportion and ratio, to complete their artwork.

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

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

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

## Perspectives Video: Teaching Idea

Surface Area Misconception:

Unlock an effective teaching strategy for identifying the base and height of figures in this Teacher Perspectives video for educators.

Type: Perspectives Video: Teaching Idea

Surface Area and Volume:

In this activity, students adjust the dimensions of either a rectangular or triangular prism and the surface area and volume are calculated for those dimensions. Students can also switch into compute mode where they are given a prism with certain dimensions and they must compute the surface area and volume. The application keeps score so students can track their progress. This application allows students to explore the surface area and volume of rectangular and triangular prisms and how changing dimensions affect these measurements. This activity also includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

## Tutorials

Area of a Parallelogram:

This video portrays a proof of the formula for area of a parallelogram.

Type: Tutorial

Area of a Trapezoid:

A trapezoid is a type of quadrilateral with one set of parallel sides. Here we explain how to find its area.

Type: Tutorial

Perimeter and Area:

Students will learn the basics of finding the perimeter and area of squares and rectangles.

Type: Tutorial

## Student Resources

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

## Tutorials

Area of a Parallelogram:

This video portrays a proof of the formula for area of a parallelogram.

Type: Tutorial

Area of a Trapezoid:

A trapezoid is a type of quadrilateral with one set of parallel sides. Here we explain how to find its area.

Type: Tutorial

Perimeter and Area:

Students will learn the basics of finding the perimeter and area of squares and rectangles.

Type: Tutorial

## Parent Resources

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

## Image/Photograph

Clipart: Geometric Shapes:

In this lesson, you will find clip art and various illustrations of polygons, circles, ellipses, star polygons, and inscribed shapes.

Type: Image/Photograph