Determining the Area of Irregular Figures Lesson 4
Students will estimate the area of irregular shapes and use a process of decomposition to calculate the areas of irregular polygons.
In partners or small groups, students should create an irregular shape on a blank piece of paper, such as the following.
Once all students have drawn a shape, ask them to estimate the area of their shape using any method they choose. Some students may overlay centimeter grid paper on top of the shape and then count squares. Others may draw squares, rectangles, and triangles within their shape and calculate the area of each polygon. Still others may compare their shape with other objects for which they know the exact dimensions and area, such as index cards or sticky notes.
Allow students to work and compare their shape with a partner and discuss how they estimated the area. After sharing with a partner, have several students share their process with the entire class.
Distribute copies of the Polygons or Tangram Activity Sheet (download from Materials section above) and have students cut out the shapes.
In their groups, ask students to create an irregular shape wiith their shapes. The irregular shape should have no overlaps or gaps; in other words, sides should touch to form an irregular shape. (Because students will be handling these shapes and moving them around, you might want to copy them onto more durable paper or use tangram sets.) Ask each group to determine the area of the irregular shape that they created.
Have students share out their irregular shapes and their method of determining the composite area of their figure.
What real-life applications are there to what was done in this lesson? What kinds of jobs or tasks would require someone to complete an activity like the one that you did?
[Guide them to come up with any relevant occupation, such as a flooring specialist, carpenter, painter, etc., that requires knowing the area of a surface would benefit from knowing how to decompose a shape into smaller figures to compute the area.]
Discuss possible situations when is it ok to estimate and when it is necessary to be exact.
Choose a famous room with irregular dimensions, such as the Octagon Room of the Royal Observatory in Greenwich, England.
Have students calculate the area of the floor. Some will naturally decompose the room into familiar polygons and calculate the individual areas, whereas others may use a different strategy to estimate the area. Have students justify their answer by explaining their strategy and why they chose to either estimate or find the exact area.
When creating floor plans of the room in the school building, have students draw them to scale. Students could draw their plans on centimeter grid paper, for instance. They could include desks, shelves, and other items found in the rooms in their scale drawings.
Have students determine the area of a shape that is drawn within a rectangle, such as the pentagon contained in the rectangle below.
Identify a room in your school with an irregular shape (that is, the room should not be completely rectangular; if possible, there should be at least one angle that is not 90 degrees.) If needed, provide a drawing of the layout of the room, with all lengths labeled. Using that room as a basis, pose the following problem, or a similar one, to students:
Our principal wants to know how much carpet (or how many floor tiles) would be needed to cover the entire floor of this room. Your job is to measure this room, determine the area of the floor, and write a letter to the principal telling him/ her how much carpet to buy and how you arrived at your answer. [Discuss with your students why accuracy is important.] The principal needs to know the exact area, because he/she doesn't want to order too much carpet and waste money, nor she does he/she want to order too little and not be able to cover the entire floor.
Leave your thoughts in the comments below.
Students develop the area of triangles formula using the area of rectangles and by comparing triangles with equal bases and heights.
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.
This part of the lesson was really helpful for students. It was great for them to physically manipulate the tangram pieces to develop a conceptual understanding of irregular figures. I
CCSS, Content Standards to specific grade/standard
CCSS, Standards for Mathematical Practices
PtA, highlighted Effective Teaching Practice and/or Guiding Principle CCSS