Discovering the Area Formula for Triangles Lesson 1

• Discovering the Area Formula for Triangles

Lesson 1 of 4

60–70 minutes

Description

Students develop the area of triangles formula using the area of rectangles and by comparing triangles with equal bases and heights.

Materials

Introduce

As an introductory activity, have students draw at least one square and at least one rectangle on grid paper. Have students determine the area of each shape, using multiple strategies and justifying their reasoning. Be sure to allow time for student to share strategies with the class. If necessary, reinforce the area formula for rectangles: A = b × h.

Explore

Divide students into partners or small groups. Distribute the Squares and Rectangles Activity Sheet and the Squares and Rectangles Recording Sheet (download from Materials section above). Have each group measure and record the dimensions and area of each quadrilateral.
Using rulers, have students draw a diagonal in shapes A, B, and C, and cut each shape along the diagonal. Have students estimate the area of each triangle formed by dividing shapes A, B, and C in half along the diagonal.
Students can estimate the areas using any methods they choose. Discuss strategies with the class as a whole. Teachers may want to create a chart or visual display to record student strategies. Highlight student strategies which are focused on developing precision.
Next, have students find the area of shape D. As instructed on the Activity Sheet, students should mark a point on the top edge of shape D (not choosing a vertex as a possible point). Each member of the group should pick a different point on the top of the shape. Using a straightedge, each student should draw a line from each bottom vertex to the point marked and cut out the three triangles formed. As with shapes A, B, and C, students estimate the area of the largest triangle using various strategies.
Have each student in the group share the point that he or she chose, along with their estimates. If needed, help students manipulate the two smaller triangles to cover the largest triangle. Students should also recognize that although the shape of the largest triangle may be different than other students' triangles, the base, height, and area are the same.

Synthesize

In their partners or small groups, have students create a formula for determining the area of a triangle. Have them explain their reasoning and justify that their formula works. To promote discussion, you may need to ask questions about the relationship between the area of a triangle and the area of a rectangle. Be sure to include a discussion about the base and height of the triangle and rectangle. These ideas will help students discover the formula for area of triangles on their own.

Assessment

Distribute the Unknown Triangle Activity Sheet (download from Materials section above), in which the dimensions of two triangles are known, and ask students to determine the area of both triangles. Allow them to use any method they like, but encourage them to use what they just discovered to find the area.

Students will likely realize that the first shape is a right triangle, so it is congruent to one-half a rectangle that was divided along the diagonal. However, they may have more difficulty realizing that the area of the second triangle is equal to one-half the area of a 3 × 4 rectangle. Students should be able to demonstrate their understanding by justifying their work.

Extension

Activity 1 (Technology Option)

Use the triangle tab in the Area Tool. When students share their points and estimates, they should realize that, although the shape of the triangle may change, the base, height, and area do not. To emphasize this point, ask them to drag point D so that it lies directly on top of point B; this forms a right triangle with the right angle at A, as shown above. Then, ask students to drag point D so that it lies directly above point C; this forms a right triangle with the right angle at C. Students should recognize that these triangles are congruent, so they must have the same area.

Discuss the results with the class. Students should realize that, in each case, the area of the triangle is equal to one-half the area of the rectangle. (At this point, you may be tempted to give students the formula A = ½(bh), but it will be more valuable to let them formalize the rule on their own in the ensuing discussion.)

Activity 2

The Bermuda Triangle is the triangular region defined by San Juan, Puerto Rico; Miami, Florida; and Bermuda. Using a map, students should determine the dimensions of the Bermuda Triangle, measure the distances using the scale on the map, and calculate the area of the triangle. Students can use the Bermuda Triangle Activity Sheet (download from Materials section) to record their work.

Teacher Reflection

• What strategies did students use to find the areas of the triangles? How did you sequence student’s strategies to promote conceptual understanding and student discourse?
• What were some of the ways that the students illustrated they were actively engaged in the learning process?
• How did activities build towards student understanding of determining a general rule or formula for area of triangles?
• What scaffolding and/or differentiation was needed during the lesson to ensure all students experienced success?
• What were common misconceptions that students had and how did you address them?

Related Material

• If a group of students is struggling to make the connections, then students can go to Desmos and use the sliders to observe the properties which lead to the area formula.

Other Lessons in This Activity

Students use prior knowledge of the area formula for rectangles and triangles to discover the formula for the area of parallelograms.

Students explore several strategies for calculating the area of a trapezoid while discovering the area formula for trapezoids.

Students will estimate the area of irregular shapes and use a process of decomposition to calculate the areas of irregular polygons.

• count the squares
number of columns X number of squares in each row

• How do you conceptually teach your students that the area of a rectangle is (base*height)?

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• Ratings

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• Essential Question(s)

• How can you use the area of rectangles and squares to determine the area of triangles? What is the relationship between the areas of rectangles and triangles having the same lengths of base and height?

Standards

CCSS, Content Standards to specific grade/standard

• 6.G.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

CCSS, Standards for Mathematical Practices

• SMP 5 Use appropriate tools strategically.
• SMP 8 Look for and express regularity in repeated reasoning.

PtA, highlighted Effective Teaching Practice and/or Guiding Principle CCSS

• Pose purposeful questions.