# MA.6.GR.2.1

Derive a formula for the area of a right triangle using a rectangle. Apply a formula to find the area of a triangle.

### Clarifications

Clarification 1: Instruction focuses on the relationship between the area of a rectangle and the area of a right triangle.

Clarification 2: Within this benchmark, the expectation is to know from memory a formula for the area of a triangle.

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

## Benchmark Instructional Guide

• Algorithm
• Area
• Rectangle
• Triangle

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

Students will use their understanding from grade 5 of models and formulas to find the area of a rectangle to derive the formula to find the area of a triangle (MTR.1.1, MTR.2.1). Students will extend this knowledge in grade 7 to decompose composite figures into triangles and quadrilaterals in order to find area.
• Instruction includes developing the understanding that two copies of any right triangle will always form a rectangle with the same base and height. Therefore, the triangle has an area of half of the rectangle, $A$ = $\frac{\text{1}}{\text{2}}$ ($b$$a$$s$$e$ × $h$$e$$i$$g$$h$$t$). This understanding can develop from seeing how a triangle is constructed when cutting a rectangular piece of paper diagonally in half.
• Students should be flexible in their understanding of formulas to be able to use show the equivalency of $\frac{\text{1}}{\text{2}}$$b$$h$ and $\frac{\text{bh}}{\text{2}}$.
• Formulas can be a tool or strategy for geometric reasoning. Students require a solid understanding of two area concepts: (1) the area of a rectangle is $l$$e$$n$$g$$t$$h$ × $w$$i$$d$$t$$h$ or $b$$a$$s$$e$ × $h$$e$$i$$g$$h$$t$, and (2) figures of the same size and shape (congruent) have the same area.
• Instruction includes representing measurements for area as square units, units squared or units2.
• Students should understand that any side of the triangle can be a base; however, the height can only be represented as a line segment drawn from a vertex perpendicular to the base. The terms height and altitude can be used interchangeably. Students should see the right-angle symbol,  to indicate perpendicularity.
• Problem types include having students’ measure lengths using a ruler to determine the area.

### Common Misconceptions or Errors

• Students may forget that multiplying by and dividing by $\frac{\text{1}}{\text{2}}$ are the same operation.
• Students may neglect to apply the $\frac{\text{1}}{\text{2}}$ when finding the area of a triangle.
• Students may incorrectly identify a side measurement as the height of a triangle.

### Strategies to Support Tiered Instruction

• Teacher models several problems solving them both ways (using a rectangle and using a formula) and then have the students solve them step by step guiding them to the answer. This will provide students with the opportunity to see that the two operations are identical.
• Teacher reinforces that a right triangle is half of a rectangle, therefore we must cut the area in half.
• Teacher models with geometric software so students can see that a right triangle is half of a rectangle, which is why we multiply by $\frac{\text{1}}{\text{2}}$.
• Teacher models the use of manipulatives that students can measure to better understand there is a difference between a side length and the height in non-right triangles.

Mrs. Lito asked her students to label a base $b$ and its corresponding height $h$ in the triangle shown.

Three students drew the figures below.

• Part A. Which students, if any, have correctly identified a base and its corresponding height? Which ones have not? Explain what is incorrect.
• Part B. There are three possible base-height pairs for this triangle. Sketch all three.

Look at the triangles below.

Determine and explain:
• Which triangle has the greatest area?
• Which triangle has the least area?
• Do any of the triangles have the same area?
• Are some areas impossible to compare?

### Instructional Items

Instructional Item 1
Find the area of Δ$D$$E$$F$.

*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.
1205010: M/J Grade 6 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1205020: M/J Accelerated Mathematics Grade 6 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 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))
7812015: Access M/J Grade 6 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.6.GR.2.AP.1: Given the formula, find the area of a triangle.

## Related Resources

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

## Formative Assessment

Area of Triangles:

Students are asked to find the area of two different triangles.

Type: Formative Assessment

## 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

Power of a Right Triangle: Day 1 Proving Pythagoras:

In this first of three lessons on the Pythagorean Theorem students work to prove the Pythagorean theorem and verify that the theorem works.

Type: Lesson Plan

Using Nets to Find the Surface Area of Pyramids:

In this lesson, students will explore and apply the use of nets to find the surface area of pyramids.

Type: Lesson Plan

Area of a Triangle:

This lesson is primarily formative in nature and is designed to introduce students to the area of a triangle by having them derive the formula themselves using the relationship between rectangles and triangles. During the lesson the teacher will be facilitating their students as they work with their teams and shoulder partners to solve problems.

Type: Lesson Plan

Area of a Right Triangle:

Area of a Right Triangle

Type: Lesson Plan

Power of a Right Triangle: Day 1 Proving Pythagoras:

In this first of three lessons on the Pythagorean Theorem students work to prove the Pythagorean theorem and verify that the theorem works.

Type: Lesson Plan

## Original Student Tutorial

Area of Triangles:

Follow George as he explores the formula for the area of a triangle and uses it to find the area of various triangles in this interactive student tutorial.

Type: Original Student Tutorial

Base and Height:

Students are asked to determine and illustrate all possible descriptions for the base and height of a given triangle.

Same Base and Height, Variation 1:

This task is a good precursor to students developing the formula for the area of a triangle. The fact that each triangle has the same area can be used to highlight the meaning of the components of the area formula, as well as the meaning of the altitude of a triangle (an issue since the given triangles are not acute.) Students may try to determine the area of each triangle by counting the square units or using the "surround and subtract" method. Students may think that triangle ABC has the largest area because the others appear thinner.

Same Base and Height, Variation 2:

This is the second version of a task asking students to find the areas of triangles that have the same base and height. This presentation is more abstract as students are not using physical models. They still determine the area of each triangle by counting the square units or using the "surround and subtract" method, but it is a good lead-up for students to think about the formula for the area of a triangle and notice that the length of bases and altitudes are the same. Students who do not analyze the area may think that triangle ABC has the largest area because the others appear thinner.

## Tutorial

Area of Triangle on a Grid:

We will be able to find the area of a triangle in a coordinate grid. The formula for the area of a triangle is given in this tutorial.

Type: Tutorial

## MFAS Formative Assessments

Area of Triangles:

Students are asked to find the area of two different triangles.

## Original Student Tutorials Mathematics - Grades 6-8

Area of Triangles:

Follow George as he explores the formula for the area of a triangle and uses it to find the area of various triangles in this interactive student tutorial.

## Student Resources

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

## Original Student Tutorial

Area of Triangles:

Follow George as he explores the formula for the area of a triangle and uses it to find the area of various triangles in this interactive student tutorial.

Type: Original Student Tutorial

Base and Height:

Students are asked to determine and illustrate all possible descriptions for the base and height of a given triangle.

## Tutorial

Area of Triangle on a Grid:

We will be able to find the area of a triangle in a coordinate grid. The formula for the area of a triangle is given in this tutorial.

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