performance task on measurement of the sides of a triangle
Performance Task on Measurement of the Sides of a Triangle
Overview
The measurement of the sides of a triangle is a fundamental concept in geometry that can be applied to various practical and theoretical problems. This performance task is designed to help students develop their measurement skills, understand the properties of triangles, and use a variety of tools (rulers, protractors, or technology) to solve meaningful, real-world problems related to triangles.
Objectives
By completing this performance task, students will:
- Learn how to measure the sides of a triangle accurately.
- Apply the properties of triangles (e.g., the Triangle Inequality Theorem).
- Solve real-world problems involving perimeter, area, and other measurements of triangles.
- Use mathematical reasoning to classify triangles (e.g., equilateral, isosceles, scalene) based on side lengths.
- Apply geometric concepts, such as Pythagoras’ Theorem, for specific types of triangles (e.g., right triangles).
Materials
- Ruler (in cm or inches)
- Protractor (optional, for angle measurements)
- Compass (for constructing triangles)
- Graph paper (to ensure accuracy in drawings)
- Calculator (optional, for computations involving area, perimeter, or Pythagoras’ Theorem)
- Task Worksheet (if available)
Performance Task Instructions
Part 1: Measuring the Sides of Triangles
- Use a ruler to measure the sides of the given or drawn triangles. The measurements should be accurate to the nearest millimeter (mm) or tenth of an inch.
- Label each side of the triangle as a, b, and c.
Key Concepts:
- A triangle has three sides (a, b, c).
- Side lengths should always follow the Triangle Inequality Theorem: The sum of the lengths of any two sides must be greater than the length of the third side.
Example:
Suppose you have a triangle with:
- a = 5 \, \text{cm}
- b = 7 \, \text{cm}
- c = 10 \, \text{cm}
Check: a + b = 5 + 7 = 12 \, \text{cm}, which is greater than c = 10 \, \text{cm}. Thus, this is a valid triangle.
Part 2: Triangle Classification Based on Side Lengths
Using the measured side lengths, classify each triangle into one of the following categories:
- Scalene Triangle: All three sides are of different lengths.
- Isosceles Triangle: Two sides are of equal length.
- Equilateral Triangle: All three sides are of equal length.
Example:
For a triangle with side lengths a = 4 \, \text{cm}, b = 4 \, \text{cm}, and c = 6 \, \text{cm}:
- The triangle is isoceles because two of its sides are equal.
Part 3: Calculating the Perimeter
The perimeter of a triangle is the sum of its side lengths.
Formula:
\text{Perimeter} = a + b + c
Example:
For a triangle with a = 3 \, \text{cm}, b = 4 \, \text{cm}, and c = 5 \, \text{cm}:
\text{Perimeter} = 3 + 4 + 5 = 12 \, \text{cm}
Part 4: Calculating the Area
To find the area of a triangle, you can use one of the following methods depending on the given data:
1. Using the Height (Base-Height Formula):
If the height (h) corresponding to a base (b) is given:
\text{Area} = \frac{1}{2} \cdot b \cdot h
Example:
If the base of the triangle is b = 8 \, \text{cm} and the height is h = 5 \, \text{cm}:
\text{Area} = \frac{1}{2} \cdot 8 \cdot 5 = 20 \, \text{cm}^2
2. Using Heron’s Formula (When Side Lengths are Given):
When only side lengths (a, b, c) are known, use Heron’s formula:
\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}
Where s is the semi-perimeter:
s = \frac{a + b + c}{2}
Example:
For a triangle with a = 6 \, \text{cm}, b = 8 \, \text{cm}, c = 10 \, \text{cm}:
- Compute the semi-perimeter:
s = \frac{6 + 8 + 10}{2} = 12 \, \text{cm}
- Plug into Heron’s formula:
\text{Area} = \sqrt{12(12-6)(12-8)(12-10)}
\text{Area} = \sqrt{12 \cdot 6 \cdot 4 \cdot 2} = \sqrt{576} = 24 \, \text{cm}^2
Part 5: Special Right Triangles and Pythagoras’ Theorem
Pythagoras’ Theorem:
For a right triangle:
c^2 = a^2 + b^2
Where c is the hypotenuse (the triangle’s longest side).
Example:
For a triangle with sides a = 3 \, \text{cm}, b = 4 \, \text{cm}:
c^2 = 3^2 + 4^2 = 9 + 16 = 25
c = \sqrt{25} = 5 \, \text{cm}
This confirms the triangle is a right triangle.
Performance Task Activities
Activity 1: Measure and Verify
- Draw 3 triangles with random side lengths (e.g., use a ruler and compass).
- Measure the side lengths of each triangle.
- Verify whether the side lengths satisfy the Triangle Inequality Theorem.
Activity 2: Classify and Calculate
- For the triangles in Activity 1, classify them as scalene, isosceles, or equilateral.
- Calculate the perimeter of each triangle.
- Using either the base-height formula or Heron’s formula, calculate the area.
Activity 3: Real-World Application
Discuss a scenario where triangular measurements are important, such as:
- Designing a triangular garden.
- Measuring the dimensions of triangular roof trusses in construction.
- Calculating distances in navigational triangulation.
Rubric for Grading
| Criterion | Excellent (5) | Good (4) | Satisfactory (3) | Needs Improvement (2 or below) |
|---|---|---|---|---|
| Measurement Accuracy | All sides measured precisely | Most sides accurately measured | Few measurements accurate | Measurements incorrect or omitted |
| Triangle Classification | All triangles classified correctly | Most classifications correct | Some classifications correct | Incorrect or no classification |
| Mathematical Computations | All computations correct | Most computations correct | Some errors in computations | Numerous errors in computations |
| Completeness of Work | All tasks and activities done | Most tasks completed | Some tasks are incomplete | Tasks missing or incomplete |
| Logical Explanation | Clear, detailed explanations | Explanations mostly clear | Explanations somewhat unclear | Minimal/no explanations |
Reflection Questions
- How do the side lengths of a triangle affect its perimeter and area?
- Why is the Triangle Inequality Theorem important?
- In what real-world situations might you measure the sides of a triangle?
By completing this performance task, students will strengthen their understanding of triangle properties, learn to apply mathematical formulas, and appreciate the importance of geometric concepts in real-world applications. Let me know if you need additional resources or examples! @username