Area of Triangles

10 min

Teacher Prep
Setup
Display images for all to see. A minute to notice and wonder, and a minute of partner discussion. 2–3 minutes of quiet work time, followed by a whole-class discussion. Access to geometry toolkits.

Narrative

This Warm-up has two aims: to solidify what students learned about the relationship between triangles and parallelograms and to connect their new insights back to the concept of area.

Students are given a right triangle and the three parallelograms that can be composed from two copies of the triangle. Though students are not asked to find the area of the triangle, they may make some important observations along the way. They are likely to see that:

  • The triangle covers half of the region of each parallelogram. 
  • The base-height measurements for each parallelogram involve the numbers 6 and 4, which are the lengths of two sides of the triangle.
  • All parallelograms have the same area of 24 square units.

These observations enable them to reason that the area of the triangle is half of the area of a parallelogram (in this case, any of the three parallelograms can be used to find the area of the triangle). In upcoming work, students will test and extend this awareness, generalizing it to help them find the area of any triangle.

Launch

Arrange students in groups of 2. Display the image of the three parallelograms for all to see. Ask students to think of at least one thing that they notice and at least one thing that they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss with their partner the things that they notice and wonder.

Give students 2–3 minutes of quiet time to complete the activity and access to their geometry toolkits. Follow with a whole-class discussion.

Student Task

Here is Triangle M.

A triangle labeled “M”. The left side is 4 units tall and the top side is 6 units wide.

Han made a copy of Triangle M and composed three different parallelograms using the original M and the copy, as shown here.

Three different parallelograms on a grid composed of Triangle M and a copy.
Three different parallelograms on a grid composed of Triangle M and a copy. First parallelogram, Triangle M and a copy along the slanted side of the triangle. Second parallelogram, Triangle M and a copy along the horizontal side of the triangle. Third parallelogram, Triangle M and a copy along the vertical side of the triangle.

  1. For each parallelogram that Han composed, identify a base and a corresponding height, and write the measurements on the drawing.

  2. Find the area of each parallelogram that Han composed. Show your reasoning.

Sample Response

  1. First parallelogram: b=6b = 6 and h=4h = 4, second parallelogram: b=4b = 4 and h=6h = 6, third parallelogram: b=6b = 6 and h=4h = 4

    <p>3 triangles on a grid</p>

  2. The area of each parallelogram is 24 square units. Sample reasoning: The base and height measurements for the parallelograms are 4 units and 6 units, or 6 units and 4 units. 4⋅6=244 \boldcdot 6 = 24 and 6⋅4=246 \boldcdot 4 = 24.
Activity Synthesis (Teacher Notes)

Ask one student to identify the base, height, and area of each parallelogram, as well as how they reasoned about the area. If not already answered by students in their explanations, discuss the following questions:

  • “Why do all the pictured parallelograms have the same area even though they all have different shapes?” (They are composed of the same parts—two copies of the same right triangles. They have the same pair of numbers for their base and height. They all can be decomposed and rearranged into a 6-by-4 rectangle.)
  • “What do you notice about the bases and heights of the parallelograms?” (They are the same pair of numbers.)
  • “How are the base-height measurements related to the right triangle?” (They are the lengths of two sides of the right triangles.)
  • “Can we find the area of the triangle? How?” (Yes, the area of the triangle is 12 square units because it is half of the area of every parallelogram, which is 24 square units.)

Math Community
After the Warm-up, display the revised Math Community Chart created from student responses in Exercise 3. Tell students that today they are going to monitor for two things:

  • “Doing Math” actions from the chart that they see or hear happening.
  • “Doing Math” actions that they see or hear that they think should be added to the chart.

Provide sticky notes for students to record what they see and hear during the lesson.

Anticipated Misconceptions

When identifying bases and heights of the parallelograms, some students may choose a non-horizontal or non-vertical side as a base and struggle to find its length and the length of its corresponding height. Ask them to see if there's another side that could serve as a base and has a length that can be more easily determined. Clarify that we can use the grid to measure a length only if the segment is parallel to the grid lines.

Students may not immediately recall that squares and rectangles are also parallelograms. Prompt them to recall the defining characteristics of parallelograms, by asking: “What makes a figure a parallelogram? What are its characteristics?”

Standards
Addressing
  • 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.
  • 6.G.A.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.

25 min