Lesson and Assessments - Finding Area of Irregular Shapes

Progression of Skills (outline)

Before the lesson:

• Prior to the lesson the instructor should provide the pre-assessment to determine the appropriateness of this toolkit for the student. Students should have an understanding of 1:1 correspondence. It will be helpful for students to be able to understand the concept of repeated addition but not required. Students will also find the toolkit more accessible if they are able to understand the connection between repeated addition and multiplication; however, this is not a requirement to access this toolkit.

During this toolkit lesson:

• While using this toolkit, students will demonstrate an understanding of what area measures, learn how to measure area using square units and multiplication, understand that the area of a shape changes depending on the size and shape and be able to calculate the area of shapes composed of rectangles.

After the lesson:

• After using this toolkit, students should practice finding the perimeter and area of irregular shapes composed of rectangles and justify their answer based on their understanding of the difference between the two concepts of measurement

Directions for Intervention Activity:

• After there has been discussion about the shapes presented on slide 2, advise the student(s) that today they are going to work to answer the questions: “What is the area of a shape?” and “How do you find the area of a shape?” Ask the student(s) if they think they may have an answer to one of the questions. The instructor should affirm the responses by repeating anything that is mathematically accurate. After listening to the answer(s) given, click to slide 3 and read the definition of area to the students. Using the image, explain that area is the number of square units that cover a flat figure, without any gaps or overlaps. Remind students that the square units should cover the inside of the shape. Point out that the squares that are covering the shape, not only do they not overlap or have gaps, but they also touch the sides of the shape, without going outside of it. 
• Let students know that they are going to have the chance to determine the area of shapes using squares. Model with the example on the work mats with the square manipulatives (you should use 1 square for each square unit of the shape). When modeling how to place the squares on the irregular shape, also model how to determine the area by counting the squares used (your example should have 8 squares on it and count each square with 1:1 correspondence). Once counted, share with the students: “The area of this shape is 8 square units.”  

Progress Monitoring:

• Place some square manipulatives in front of the student on the table and pass the irregular shape pages. Students should be given the opportunity to find the area for shapes 1-3. The instructor should inquire with the student(s) to check for understanding. The instructor should ensure that the student(s) is/ are using multiple squares, not overlapping the squares, not leaving gaps and that all squares are touching all of the boundary lines of the shape, without going outside. The teacher should also check for understanding by asking the student(s) to complete the sentence frame when complete, “The area of this shape is __ square units.” (Sentence frame on slide 4 for reference). 
• If the student(s) are able to determine the area of the shapes in the above activity, continue onto lesson 2. If student(s) are unable to complete the activity, give corrective feedback. Possible errors to consider are: overlapping of the squares, leaving gaps between squares, the squares not touching the boundary lines or the student is not using 1 square manipulative to represent each square unit. 

Lesson 2:

Materials for Lesson: 

• Slide Presentation (slides 3-6)
• Grid paper
• Dice (digital dice tool if needed) 
• Crayons, colored pencils or markers
• Pencils
• “The Hershey’s Milk Chocolate Multiplication Book” by Jerry Pallotta Illustrated by Rob Bolster ISBN: 0-439-25412-4 (optional)

Directions for Intervention Activity: 

• The instructor should start the lesson by reviewing the definition of area and the concept that was taught yesterday using slide 3. 
• The instructor should explain that the area of a shape can be found in a variety of ways. On slide 5, the instructor should point out the given shape and ask how they see the image. 
  • Possible responses may be 20 square units (which is a great answer that connects lesson 1 learning to today’s lesson). Other possible answers may be 4 rows of 5. Or 5 columns of 4. Students may not use this vocabulary but express 4 lines of 5 or 5 lines of 4, which are also acceptable responses. 
  • The instructor should affirm the responses and ask how someone may write a math equation to model the answer when finding the area of a shape. Some responses may include 4+4+4+4+4=20, 5+5+5+5=20, counting by 5’s (5, 10, 15, 20), counting by 4’s (4, 8, 12, 16, 20), 45=20 or 54=20. If possible, I encourage the instructor to allow the student to model their thinking on the board for others to see (including the instructor as this is a great way to informally assess the students understanding and thought process). Again, affirm the students’ thinking and use this time to clear up any misconceptions about the 1:1 correspondence of the square units in the shape. 
  • Ask students to make connections between the different equations. (Can they all be correct? Why? How?) 
  • Optional: A great book to use as a model is “The Hershey’s Milk Chocolate Multiplication Book” by Jerry Pallotta. I would not suggest reading the whole book but turn to pages 21-25 and talk about how they use multiplication to determine how many sections of chocolate are in the bar. (This is not required). 
  • Using the shape on slide 5, model multiplying the base and height by saying, “A more efficient way, or faster way, to determine the area of a rectangle would be to multiply the base times the height. If the base of the rectangle is 5 square units and the height is 4 square units, I can multiply those numbers. 54=20. 54 means that there are 5 groups of 4 and is the same as repeated addition of 4+4+4+4+4.” Let the student(s) know that they will be playing a game with a partner (or the instructor if needed). 
• Students will be playing in pairs (either with another student or the instructor, if needed). Each pair will get a piece of grid paper and a pair of dice. There will be a partner A and a partner B. Each partner will choose a color to use on the grid paper to keep track of turns. Partner A will roll the die and use the numbers rolled to draw a rectangle on the grid paper. One die will represent the square units of the base and the second die will represent the square units of the height. Partner A will draw their rectangle on the grid paper and write the multiplication problem inside the shape. Partner B will have a turn to do the same. Each rectangle cannot overlap any other rectangles. An example of this activity can be found on slide 6. If the instructor is not a partner, the instructor should be doing informal checks to ensure that students are using base times height and representing the shapes with multiplication equations. 

Progress Monitoring:

• The instructor should ask each student at some point, “How do you know that the area of this shape is ___?” The student(s) response should closely resemble the idea that the base is ___ and the height is ___, which means that there are ___ groups of ___ and that means there are ___ square units in the shape. If student(s) are struggling with this activity, the instructor should model the activity in a whole group setting until the concept is understood. It may also be helpful to use repeated addition as a written expression instead of multiplication, if needed. 

Lesson 3:

Materials for Lesson: 

• Slide Presentation (slides 6-8)
• Grid paper
• Dice (digital dice tool if needed) 
• Crayons, colored pencils or markers
• Pencils
• Exit Ticket

Directions for Intervention Activity:

• The instructor should start the lesson by reviewing the definition of area and the concept that was taught yesterday using slide 6. On this slide, focus on the fact that multiplying the base times the height will find the area of the rectangle. 
• Explain to students that today they will be doing the same activity, but today they will not be making rectangles but instead some irregular shapes. They will do this by connecting two rectangles together to create one shape. Present slide 7. Use this slide, model how an irregular shape can be broken into multiple rectangles to make it easier to determine area. Give the area by using the equation frame: “Area of A + Area B: ___ square units + ___ square units = ___ square units
• Present slide 8. On this slide you can see that Partner A used yellow. They rolled a 3 and a 4 on their first roll and then a 2 and 2 on their second role. Each person should draw their first rectangle in pencil so that they can outline the irregular shape and leave a pencil mark where they divided the shape to find the area. Once the area is found for each rectangle, students should then take it a step further to add the square units of each rectangle together to find the area of the entire irregular shape. 

Progress Monitoring:

• Same as in lesson 2, each student should have a partner for the game. The instructor should facilitate learning by listening to the conversations and asking questions of each student to ensure the students understand that by finding the area of the two separate shapes, they can then add those areas together to determine the area of the entire irregular shape. If the student is struggling, reteaching with manipulatives may be needed with the instructor. 

Exit Ticket: 

• This lesson has an exit ticket. This will need to be graded prior to lesson 4, as there is a scaffold option for students who were unable to complete part B. 

Lesson 4:

Materials for Lesson: 

• Slide Presentation (slides 9-10)
• Lesson 4 Work Page
• Pencils

Teacher Preparation:

• Before this lesson, the instructor should review the exit ticket from lesson 3. This should indicate if the student is still in need of the supports that grid paper provides. If the student still needs this scaffold, please use part A of the work page and then have the student continue into part B. If the student was able to complete the exit ticket correctly, they may skip part A and start with part B. 

Directions for Intervention Activity:

• If the student still needs the scaffold of the grid paper, review the first part of the lesson 3 exit ticket on slide 9 of the slide presentation. If the student does not need the scaffold of the grid paper, review the second part of the lesson 3 exit ticket on slide 10. 
• Students will work on finding the area. Part A (if needed) has grids for additional practice of finding the area of a rectangle, while part B offers for the application of finding the area without the grid paper and adds the complexity of adding a second rectangular shape. The instructor should support and reteach as needed for students who are having difficulty applying the learning from lesson 3 to the new task. 

Progress Monitoring:

• Lesson 4 Work Page allows the instructor to progress monitor the application of the learning from lesson 3. If students are having difficulty with the concept, allow the student(s) to have grid paper and draw the model to scaffold the task(s) of part B. 

Lesson 5:

Materials for Lesson: 

• Slide Presentation (slides 2-11)
• Lesson 5 Work Page
• Crayons, Colored Pencils or Markers
• Pencils

Directions for Intervention Activity:

• The instructor should review the definition of “What is area?” on slide 2 and the progression that the student(s) took throughout the lessons in this toolkit. In lesson 1, the student(s) used grid paper to determine the area of a rectangle, in lesson 3, students used the grid paper to determine the area of an irregular shape that was made up of multiple rectangles. In lesson 4, students were challenged by having to find the area of rectangles without the support of grid paper and were expected to rely on multiplication to find the area. Today, students will be asked to create their own irregular shape to match a given area. Using slide 11, the student(s) should be able to model one shape that is 7 square units and then write the math equations that demonstrate their thinking to justify their answer. There are multiple shapes that have an area of 7 square units and each student should be encouraged to model their thinking for the group. 
• For independent practice, students will complete the Lesson 5 Work Page, where they will be asked to do the same task, with different area totals. The answers will vary on the Work Page. 

Progress Monitoring:

• Lesson 5 Work Page will allow students to apply their knowledge and learning to create a shape that matches the given area and use math equations to justify their answer(s). Answers will vary. 

• Area Builder Game
• Area Extension Activity (using subtraction) 

• Materials modified from Engage NY Grade 3 Module 4-13
• Materials modified from Illustrated Mathematics Grade 3 Glossary