Area Within An Area

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SWBAT calculate area for the area surrounding an area within an area.

Big Idea

Understanding and calculating the area of a space within another defined area better represents a real world problem.

Warm Up

10 minutes

To begin this lesson, students sit on the carpet with whiteboards and work with a partner to find the area of rectangle.   The challenge at this point in the year for the class is using multiplication rather than skip counting or repeated addition to determine area.  My goal is for the students to use multiplication factors including six through nine, and for some students even extending to twelve. 

As students create rectangles, and some squares, I notice some hesitation by the students in choosing dimensions.  I ask one of the students why she is having trouble choosing a number, and she responds, "I don't know how long this side is.  I don't have anything to measure it with." I realized the previous experiences of the students included using grid paper, or numbers supplied by me to determine a length.  I explained to her and the rest of the class, their goal was not for measurement, but instead just to practice using multiplication.  

Students each gave their partner one area problem to their partner, and this gave me the opportunity to observe the students' skills with the concept of area and apply multiplication fact fluency.

Mini Lesson

15 minutes

To begin this lesson, I drew two rectangles and asked the students to find the area of each rectangle.  To keep the students engaged, I decided to have the girls in the classroom do find the area of one rectangle, and the boys were responsible for the other rectangles.  Each rectangle was in both length and width so that I could show how one rectangle would fit inside of the other rectangle.   

While the students were doing the computations, I cut out the rectangles that I had drawn on white paper so that these models could be used as a hands on manipulative.  When the students had completed the computations, I asked them to figure out which one had a larger area, and I also asked the students to think of something in real life their rectangle could represent.  

Because the students were focused on small items, I suggested the area of the rectangle could be a floor, a rug, a yard, a swimming pool, a basketball court, or a soccer field.  This prompted the students to think on a larger scale, and they provided many more sports analogies and school related areas for this lesson. 

I asked the students to consider, "What would happen if one rectangle was inside of the other rectangle?"  "What happens to the area measurements?" "Do they add on to each other, or is one subtracted out?"  "What is the area left in the bigger rectangle?"

Based on these questions, the students discussed several different possibilities including both addition and subtraction as real options.  Their math discussion which is so important to the Common Core standards, directed me to focus the students on the real world options of using a carpet on a tile floor, or a swimming pool within a yard.  I explained I wanted to know the area of the tile visible on the floor, and the amount of grass remaining within the yard.  

The students discussed this would require subtracting out the smaller area from the larger area.  I asked the students, "How could they prove this was the area remaining?"  Because we have been using graph paper with area problems, one of the students offered this immediately as an option to show their work.  

We drew solutions on graph paper, and I asked, "Does it work if it is in the middle, or just in the corner? Why?"

Try It On Your Own

20 minutes

The students create their own rectangles and settings for sets of rectangles. The students could create their own rectangular figures and dimensions to find the remaining area for an area within an area.  

Shown below is one student's solution to a diagram I drew for the students to use as a model, as they find the area of the space highlighted in pink.   


Wrap Up

10 minutes

To close the lesson I pose the the following questions to the students: 

"Is the area remaining the same if the rectangle is in any location of the rectangle, such as the corner?  

How does the remaining area change as the length of the sides of the rectangle changes?" 

Students respond on whiteboards to these questions, so that I can quickly see their responses and how they are thinking when reasoning through this concept of area within an area.