Area of Irregular Polygon

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SWBAT apply their knowledge of arrays to find the area of an irregular polygon.

Big Idea

Students look for break apart irregular polygons to make arrays to find the area of these common shapes.

Warm Up

5 minutes

Students are seated on the carpet using whiteboards to practice area problems. I purposely choose problems including multiplication facts that will be easy to computer for area. The reason I choose these familiar facts is because I want the students to feel confident in determining area.  

I begin by drawing a rectangle with side dimensions of 3 and 2.   I ask the students to draw a rectangle, write the matching multiplication sentence, and determine the product of square units.  This attention to detail with labeling is something that is necessary as students move between understanding measurements in the Common Core Standards. 

This process of drawing rectangles is repeated with three similar problems.  The challenge for the students during the warm up is to have them explain the process and their solution to a partner seated next to them on the carpet.  This could be differentiated for students by challenging them with more complex problems, or it could be modified for students needing a visual to create an array next to the rectangle for visual support. This would also be a language support for ELL students, who can refer to/create arrays to demonstrate understanding. 

Mini Lesson

10 minutes

Once students solve the basic rectangle areas, and they can confidently explain the concept of their work, I draw an L-shaped polygon on a whiteboard.  I ask the students to think about what is needed to find the area of this shape since it is not just a rectangle.  The students share with their partners, and the students are able to identify the shape is created from two rectangles with one shared line.

Following this realization, I draw two rectangles separated from each other to provide the visual model to the students.  I then mark the separated rectangles with numbers such as 7 and 5, and I ask the students to determine the area.  I again use factors that are easy for the students to multiply.

I then mark the L-shaped polygon with the same numbers that were used for the separated rectangles in the example above (7 and 5), and I ask the students if the area has stayed the same.  It is important to ask the students to make this determination so that they are using their analytical thinking skills required within the Common Core standards as this new complex shape is considered.  The opinions of my students are split and it was an important moment for my students to discuss with each other why or why not the area was the same.  Based on their discussions, one student requests centimeter grid paper to prove, perhaps understand, the area.  

The entire class uses grid paper to draw an L-shape, and create the two identical rectangles separately to see if the area remains the same.  Once it is determined the area remains the same, the students enthusiastically want to try more of these type of shapes. 

I chose at this time to mark all sides of the polygon, so they focus on the area.  I didn't want to lose the momentum by creating multiple step problems. I provide more problems, using different L-shapes, and then I introduce a U-shaped polygon.  Students work with their whiteboards and partners to solve.  

I think they're ready, so I challenge the students with a random shaped polygon similar to the teeth on a jack-o-lantern with an up - down pattern of rectangles.  

Try It On Your Own

20 minutes

Before the students begin working on the challenge of the irregular shape, I have students use their math journals with the following steps to plan out a strategy to solve.  I ask the students to include in their plan:  

1. Figure out the lengths of  each side 

2. Draw lines to create rectangles 

3. Create arrays to model multiplication, if needed

4. Multiply sides to find the area 

5. Add area measurements together

I display four complex shapes on the projector, with the length of the sides labeled. Students are given the opportunity to work with a partner and solve the area, and also determine if any of the shapes had missing numerical values.  Students use their whiteboards, and centimeter grid paper as needed for support and success.  Below is demonstration of how one student solved the area of one of the polygons. 


5 minutes

To close the lesson, I ask the students to explain to me what I would need to do to find the area of a similar shape.  I randomly draw another shape, asking the students to apply the same steps they had used when they had solved the prior problems.  Verbalizing this process requires students to dig in to and explain their thinking, and also to consider the thinking of others.