Revising Mathematical Explanations

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Objective

SWBAT explain the difference between area and perimeter.

Big Idea

After creating a class comparison chart for area and perimeter, students will revise mathematical explanations to make their paragraphs more precise.

Opening

20 minutes

Unit Explanation

During the first section of this unit, students will construct a house plan, find the area of the house plan, and calculate flooring costs. While finding the area is the focus of this unit, the first few lessons (where students explore the meaning of a polygon, construct house plans, and decompose rectangles into smaller rectangles to find the area) lay the foundation for finding the area of their home plans later on. This also provides students with a meaningful and purposeful context to find the area. 

During the second section of this unit, students will investigate dog pen designs and will primarily focus on finding the perimeter, or amount of fencing needed for different dog pens. Students will also explore odd-shaped polygons by finding the area and perimeter of odd-shaped dog pens. 

Area & Perimeter Review

To begin, I ask: What does area mean again? Some students fumbled a bit while others look for our vocabulary poster and say: Area...(making a rectangle with one hand) the number of squares (acting like they are counting squares)on the inside (pointing inside the "rectangle" hand). 

I continue: What does perimeter mean again? Again, there is some hesitation. Keeping these two measurements straight can be challenging, especially when taught together! Students remember: Perimeter (making a rectangle with one hand) the distance around the outside (running finger along the outside of their rectangles).  

Then I ask students to turn and talk: What is the difference between area and perimeter? 

At this point, we review the Area & Perimeter Song. Students have smiles on their faces as they really enjoy singing the song. 

Lesson Background 

Prior to this lesson, I notice some vague explanations and some mistakes in word choice (such as Perimeter is when you count the outside squares instead of outside units) in student journals, I decide it would be best if we start today's lesson by delving into the precise language we can use when describing area and describing perimeter. 

Precision

Today, I specifically want students to focus on Math Practice 6: Attend to precision. I tell students that I'd like to see them focusing more on precision. I explain: Precision means that you are extremely careful. When you're counting, you make sure you get the right number, when you are playing basketball, you try to have perfect shooting formation, and when you writing, you try to use exact word choices. 

Teacher Demonstration

40 minutes

Goal & Lesson Introduction

At this point, I show students the Goal at the top of the anchor chart. As we have practiced in the past, students know to write this at the top of a new page in their math journals. 

I continue: Today, we are going to make sure our words are precise when we are talking about area and when we are talking about perimeter. 

Area & Perimeter Comparison Chart

To help students compare area and perimeter using precise language, I draw a three-column chart on today's Anchor Chart. I always try to use color (such as red for area and blue for perimeter) as research shows that this makes it easier for our brains to categorize information. Students create this chart in their journals and fill in information as we discuss each category together. Here's what a student journal will look like at the end of this activity: Student Journal Example.

Also, after discussing each category (such as model), I'll ask students to turn and talk: What is the difference between the model for area and the model for perimeter? This encourages students to begin using more precise language.

Model

Starting with a model, I cut out two 2 x 4 arrays and wrote "AREA" on the inside of one (taking up all the squares) and wrote perimeter all the way around the outside of the other. 

Meaning

We then discuss the meaning of area: the number of squares on the inside and the meaning of perimeter: the number of units (not squares) around the outside.

Measurement

Next, we review how to measure area and perimeter. With area, we measure square units, and with linear units. We talk about how linear means along a straight line: Linear Measurement.

Examples

I want to provide students with several examples of measurements. For the area column, we write the following examples: square feet, square centimeters, and square inches. Then, in the perimeter column, we simply write: feet, centimeters, and inches.

Steps to Find

I ask: How do we find the area? Students remember our song and begin to sing, "To find the area, count the squares... or multiply the length times width of a polygon." We record these steps in the area column and then discuss how to find the perimeter: Add the sides!

Formula

Finally, we compare the formulas for finding area (Area = L x W) and perimeter (Perimeter = L + L + W + W or P = 2L + 2W). 

Student Practice

20 minutes

Further Comparison Practice

A couple days ago, I asked students to write to the following prompt: Area is very different from Perimeter. Having reviewed the difference between area and perimeter, I want students to now return to these journal entries to revise their explanations. 

I ask students to turn back in their math journals to this paragraph and ask: How can we use what we've learned today about area and perimeter to make these mathematical explanations more precise? Students offer great suggestions:

"We can draw a model and label it." 

"We can write the formulas for perimeter and area." 

"We can use exact definitions."

I ask students to begin revising their work for with red pens: Try to make your explanation more precise! 

ELA Connection

By integrating the writing process into this lesson, I'm also supporting students as they work toward proficiency in CCSS ELA-Literacy.W.4.5. Strengthening writing through revising and editing can take place in any subject!

Monitoring Student Understanding

Once students begin working, I conference with as many students as possible. My goal is to support students by asking guiding questions (listed below). I also want to encourage students to construct viable arguments by using evidence to support their thinking (Math Practice 3). 

  1. What did you do first? 
  2. Can you explain why you _____? 
  3. What did you just learn?
  4. How can you make your work more precise?
  5. What's the difference between area and perimeter?

This is a difficult assignment at first for several students. For some kids, going back and making changes to completed work is very difficult. I encourage students, ask questions, and celebrate students who choose to overcome this frustration. Eventually, all students were revising! Here's an example of a student's journal during this time: Example of a Student Revising.

Share Out

After about ten minutes, I invite students to share how they made their work more precise by projecting their work using a document camera! As students share out loud, I watch as others who were once confused excitedly begin making their own revisions. Also, many students work harder to make their revisions just so they can share!

With each student's journal, we discuss as a class: How did this student make his/her mathematical explanation more precise? Students point out: 

"She labeled the diagram!"

"He explained more than one way to find area."

"She used the word polygon instead of the word something."