### 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.

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

**Grade:**6

**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

- 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 units
^{2}. - 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.

### Instructional Tasks

*Instructional Task 1*

**(MTR.2.1, MTR.4.1)**Mrs. Lito asked her students to label a base $b$ and its corresponding height $h$ in the triangle shown.

- 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.

*Instructional Task 2*

**(MTR.5.1)**Look at the triangles below.

- 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

## Related Access Points

## Related Resources

## Formative Assessment

## Image/Photograph

## Lesson Plans

## Original Student Tutorial

## Problem-Solving Tasks

## Tutorial

## MFAS Formative Assessments

## Original Student Tutorials Mathematics - Grades 6-8

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

## Original Student Tutorial

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

## Problem-Solving Task

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

Type: Problem-Solving Task

## Tutorial

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

## Image/Photograph

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

Type: Image/Photograph

## Problem-Solving Task

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

Type: Problem-Solving Task