Finding Arrays In An Irregular Polygon

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Objective

SWBAT decompose an irregular polygon into arrays to find the area.

Big Idea

Context is developed for students when finding the area of a complex figure by decomposing it into separate arrays and relating it to a real world image and setting.

Warm Up

5 minutes

I begin this lesson  reviewing area by drawing rectangles marking and labeling measurements for each side. I draw the rectangles quickly, without using a ruler or straight edge, telling the students this is a rectangle or a square.  I want students to focus on multiplication facts and strategies, rather than on using a measurement tool.  It also keeps the warm up section of the lesson focused on one skill that includes multiplication fact knowledge which will support their mastery of fact fluency for the Common Core Standards.  After I model one or two rectangles using different numbers and sizes of rectangles, I ask the students to challenge their shoulder partner with an area problem and shape.

Transitioning from the warm up to the mini lesson, I draw a shape with alternating heights similar to the outline of a castle.  After a few audible sighs, one of the students whispers to another, "She is going to make work with productive struggle again."  I realize at this point, I have reached a level of rigor I know they can reach and be successful. 

Mini Lesson

5 minutes

I prepare the students by having them all draw the shape on their individual whiteboards and write the numbers to mark the lengths of the sides of different sections of the shape.  I drew my shape on a piece of blank copy paper and displayed it on the screen for the students.  I ask the students to discuss with their partner some of their ideas to solve this problem.  This discussion at this time is so important for students success because it allows them the opportunity to think about their own strategy, listen to the ideas of others, and identify any potential problems with their initial plans and ideas.  

The students share out their ideas and plans with the group which allows students to listen to the thinking of others.  I ask the students to describe what they think the shape could represent.  Several students identify it as the shape of a castle, while another student describes it as apartment buildings.  I pose the question to the students, "Do you see a pattern with this shape?"  Their response includes reasoning that it does not have a pattern because the last section is taller than all the others.  Other students defend their position with a pattern of ABAB and the last section needs to have a separate part added on.  Based on these explanations, I determine this is the best time for students to begin working with their partners to begin finding the area of this figure.  

I explain to students they will be solving the problem on their whiteboard.  When both students can provide an oral explanation of their solution, I will conference with them about their solutions.  These conferences allow me to identify their mastery of this standard, and identify where any challenges exist for individual students.  These conferences also allow me to determine if both students can explain their solution and strategy, or if one student lead the other to the solution.  The conferences provide the opportunity to differentiate my instruction and feedback specifically to their needs.  Each conference

 

Try It On Your Own

20 minutes

As students work with their partner, I circulate through the room to do a quick observational assessment and to assist students that need support.  As students begin to find a solution I begin conferencing with them about their strategy and solution.

These conferences allow me to determine the scope of mastery of this standard, and identify where any challenges exist for individual students.  These conferences also allow me to determine if both students can explain their solution and strategy, or if one student lead the other to the solution.  The conferences provide the opportunity to differentiate my instruction and feedback specifically to their needs.  Each conference varies from one group to the next based on strategy and abilities of the students.  I provide clues and supporting diagrams including dividing a few of the sections into arrays if needed, or generalized feedback such as "keep trying". 

Once students provide and explain the correct solution, I give them a piece of grid paper to prove their answer.  I chose not to use the grid paper before they came up with an answer because I did not want the students to just count the squares to find the area. I also did not provide the grid paper until they had correct answer.  During the conferences some students came to me with incorrect answers and errors in determining the length of the sides of the area, while others had only made mistakes with multiplication.

 

 

Closure

10 minutes

I close this lesson by asking students to recreate their solution in their math journals.  I also ask the students to include their description of the shape (whether they thought it was a castle, apartment buildings, or something else).  I chose this because I want to see how students view plane figures in real life. One student turned the image upside down and described it as doors.

The information recorded in their journal allows me to compare this solution with the earlier work, done with a partner, on grid paper and assess if students have mastered this concept.