Project: Constructing and Measuring Triangles

Context

Area is a measure of surface size. In the real world, it is everywhere around us. From sports fields and courts of varying shapes, to pieces of fabric that are cut and sewn to make an article of clothing, to the amount of paint needed to cover a wall. Many careers, such as civil engineering, architecture, carpentry, fashion design, and interior design, require a solid understanding of how to calculate area.

For our studies, we will focus on the area of triangles. Why triangles? Well, a triangle is one of the most basic and fundamental shapes in geometry. It is a polygon with the fewest number of sides, but that does not mean that it is unimportant or simple! In fact, all polygonal shapes can be decomposed into triangles. So, understanding how to calculate the area of a triangle is a foundational step towards understanding how to calculate the area of any polygon.

The Basic Theory

The area of a triangle can be calculated using a very simple formula: one-half the product of the base and the height. (That is, Area = 1/2 * base * height) The simplicity of this formula should not fool you, as the concepts involved are fundamental to the understanding of many other more advanced topics in math, physics, engineering, and more.

For any triangle, the base can be any of the sides, and the height is a line drawn perpendicular from the chosen side to the opposite side. In a right triangle, the height is just one of the legs, but in a non-right triangle you will have to draw in the height to the triangle.

Hands-On Activity: "Constructing and Measuring Triangles"

Project Goal

The primary goal of this project is to help students develop a deep understanding of how to calculate the area of triangles. It will be completed in groups of 3-5 students to encourage collaboration, communication, and teamwork. The project should take about a week to complete, with an estimated two to four hours of work per student.

Detailed Project Description

Your group will be responsible for creating a miniature model city. In your city, you must incorporate at least 10 unique triangles (buildings, parks, etc.) of varying sizes and shapes. Each triangle should be clearly marked and distinct from all others. You will be expected to measure the base and height of each triangle and then calculate its area.

Materials List

• Cardboard or foam core (for the base of the city)
• Construction paper, popsicle sticks, clay, bottle caps, etc. (to make the buildings and decorate the city)
• Ruler or measuring tape
• Pencils, pens, or markers
• Glue, scissors
• Calculator

Step-by-Step Instructions for Completing the Activity

1. Meet with your group and discuss the layout of your city. Where will buildings be placed? How will the parks be arranged? Sketch out a plan on paper.

2. Begin constructing the triangles for your city. Remember, each triangle must be unique. For each triangle, measure the base and the height and record the measurements.

3. Calculate the area of each triangle and record your calculations.

4. Once all of the triangles have been created and their areas have been calculated, begin assembling the city on the cardboard or foam core base.

5. Once the city is complete, go back over your calculations and check for accuracy.

6. Prepare and submit the final project report, as described below.

Connecting to the Written Component

After completing the construction of the city, each group must write a report detailing their experience with the project, discussing their findings, and drawing conclusions. The sections of the report should include the following:

1. Introduction: Describe the miniature city you created, the importance of understanding area, and how you applied the calculation of triangle area to your model.

2. Body: Explain the theory you used to calculate the area of the triangles. Describe the different shapes of the triangles in your city, their measurements, and how you calculated the area of each one. Discuss which formulas and methods you used and why.

3. Conclusion: Summarize the main points you learned by completing the project. What challenges did you encounter along the way and how did your group overcome them? What conclusions did you draw about the topic of triangle area and its applications in the real world?

4. References: List all sources of information that you used to assist you with the project, including any books, videos, websites, etc.

Upon completion of this project, students should have a strong understanding of how to calculate the area of triangles as well as an improved ability to work in teams, solve problems, and manage their time.