Area and Perimeter in Real Life Day 1

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Objective

Students will be able to build models wherein area remains the same, while changing perimeters.

Big Idea

Students need time to connect area and perimeter. This two day exploration sets the stage for our short unit.

Reading of Book Spaghetti and Meatballs for Everyone

10 minutes

In order to motivate the students and have them begin to explore the real world problems of area and perimeter, I will read the book "Spaghetti and Meatballs for All" by Marilyn Burns. It is a funny story about a couple, Mr. and Mrs. Comfort, hosting a dinner party. At the party, the guests begin to move the tables around and that ruins the number of seats available.  

The story is a bit long, so I suggest you take two days to read it, with activities as you go.  On the first day, read the first half and have the students look for patterns and really try to figure out what the main problem is, which is that the perimeter, or places the guests can sit, changes as the combined areas remain the same.  

Remember, that reading the book as an engaging activity does not "teach" the mathematical concepts of area and perimeter.  However, in reading this book, you will now have a common referent as you explore area and perimeter.

As you read, allow the students to enjoy the story and the illustrations.  

Boys and Girls, Mr. and Mrs. Comfort have a real problem in the story "Spaghetti and Meatballs for All". They are having a dinner party for 32 people and when everyone comes, the company rearranges their tables!  

Have you ever seen your parents do that in a restaurant or at your home?  Sometimes they push tables together or pull them apart so different people can sit together?  That's what happens in the story. As you enjoy, watch for the problems that moving the tables causes.

Active Engagement

15 minutes

I display my book on the document camera so all students can take in the illustrations, which helps them connect to the math in this story. I think it is important to show the class the illustrations, as the story is read, so students pick up on that chairs are eliminated every time the same tables are rearranged.  

I found it helpful to pause at every page spread and have the students just observe the illustrations. They get a kick out of the antics of the cat!  Use questions to guide them to see the visual clues that the perimeter (or seating space) is getting smaller.  Have them look for stacked chairs, people standing, Mrs. Comfort acting upset.  All of this interaction will help when they work on their partner activity.

Boys and girls, can you turn and talk to your partner about what is happening to the seating?  Mrs. Comfort looks uncomfortable.  Why do you think she feels that way?

Why are there extra chairs?  At the beginning of the story there weren't enough, now there are too many?

Listen to student ideas to pull out the understanding that everywhere the tables touch, two chairs must be removed. Mathematically, they will begin to understand that the number of tables (or area) remains the same, but the perimeter changes.  In this case, it decreases as the tables are pushed together.  In focusing on this, you will really be promoting Mathematical Practice 4: Look for and express regularity in repeated reasoning.  As you read the next section, you will notice the students really grabbling with this.

Partner Work

15 minutes

I stop reading halfway through the book, at the point when I know the students understand that every time a new set of guests arrive, the tables are moved around again.  Now is the perfect time to engage them in the problem solving of this book!

To do this activity, I've created sets of 8 square inch blocks for the students to use. Their use can first be modeled on the projector to re-create the original setting of the book, which is 8 separate tables. Ask some mathematical questions:

  • How many people can sit at this arrangement?  
  • How do you know that?  What was your strategy? 
  • If we create some of the other sitting arrangements, what happens?  
  • Where are the seats lost?  Why are they lost?
  • How many are lost?

After I've established that students are recognizing the connection between the arrangement and the number of places people can sit, I send the students off with their partners to continue to figure out more seating arrangements. Give them these specific rules:

  • The tables must not overlap.  
  • When they are pushed together, the full sides must touch…no diagonal arrangements.
  • Draw your models and label how many chairs can be placed at that arrangement.

I challenged my students to look for an arrangement  that would seat the most people, one that's different than the original arrangement.  You may choose to do this, or have them explore arrangements and the seating that would go with it, for discussion later at the close. 

This group is working on how to rearrange one of their models to get more seating, and then trying to figure out a way of counting the seats. 

 

This partnership works on figuring out how to count the perimeter and realizes there is a way to increase it.

 

Wrap Up

10 minutes

For closing today, I have the students share their favorite configuration and strategy.  As the students share their area and perimeter findings, my other students are expected to use our math talking moves to respond.

I had my students explain whether they agreed or disagreed with the findings of their peers.  In doing this, both the presenter and the questioner must once again think through their strategies and often they find their own mistakes. This is much more powerful than me telling them they made an error, which they may or may not grasp. This way they internalize their strategies.  It is also a wonderful way to develop students' mathematical practices. In today's lesson I'm focusing on MP3: Construct viable arguments and critique the reasoning of others and MP7: Attend to precision, which means students advance their mathematical communication skills as they try to use the precise language to speak with each other.

Mathematicians, your thinking today was wonderful.  I am impressed with the patterns and solutions you found for our problem!  Many of you began to develop strategies that you can use every time you are working to find area or perimeter.  Tomorrow we will again work with these concepts and share out more of our thinking.