Decomposing Given Large Areas Day 2

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Students will be able to decompose large arrays in order to mentally compute the total area.

Big Idea

In this second of a three-part lesson, students extend decomposing arrays to applying a strategy for mental computation of area by finding smaller areas.


10 minutes

To warm my students up today for some mental math in the area of multiplication, I decided to play the game "I Have, Who Has?".  We usually play two or three rounds, each time trying to beat our times. If you have not played this wonderful game, visit here for complete directions and printable game sets. 

This is a wonderful game for warm-ups and sometimes closings of the math sessions.  The students must understand their task card, listen carefully to each other, and think in terms of equations and products all at once.  It only takes a few minutes to play, but is strong in its purpose. 


10 minutes

My lesson is short today, as I know the students will need a lot of exploration and work time today.  I don't want to give them too much of the "answer" or strategy, rather I want them to develop the best way for them to solve today's problems.

As the students gather in our community area, I place a paper with 3 different rectangles on it in the center of our circle.  I show the students my 1 square inch blocks and tell them that the task for today is to find the area of rectangles using at least 2 different colored tiles.  I then ask:

Can anyone show me how to fill the area of this rectangle with 2 different colored tiles?

As the volunteer works, he places the tiles in a checkerboard pattern.  Another student comments that he shouldn't do it that way because it makes it too hard to count.  He continues anyway and then proceeds to count all of the blocks when asked to report the area.  

I then ask "Can anyone think of a way to find the total area quicker using the two colors?"  Of course someone says to put all the like colors together and do the multiplication for the two small arrays. We do this and check to see if we got the same area, which we did.

Next, I show the students the counter top full of various rectangle sizes.  I give each group a large cup of colored 1 square inch tiles and ask them to choose a sheet of rectangles in which to find the areas. The rules they are to work by are written on the board:

1. No empty space

2. Put like colors together

3. Keep track of your work in your reflection journal

4. Use talking moves with your partner

Active Engagement

25 minutes

As the students work together to find the areas, I move around the room to interact and give mini-lessons to each partnership, depending on what strategies I notice and any misconceptions they may have.  Some students are trying to create equal arrays by simply trying to ballpark the half mark of the large array.  For these students, I work with them on decomposing to "easy numbers to work with", like a 5 or a 10.  Other students are already working with the easy numbers and have moved to working with creating 3 easy area configurations. With these children, I work with how to record what they have done with equations.  I also have some students that are simply working on the concept of decomposing an array.  Here, I choose to work with having them talk through where they are breaking apart the array and why they have chosen to do so.  

This student had the products written correctly and needed some prompting on writing the correct equation for each piece.  This is important, because even though he can do this work in his head, showing the work is a demonstration of his thinking, a critical skill for ongoing mathematical success.






10 minutes

As the students wrap up this lesson, I pull them to our community center and ask them to sit in a circle.  I then ask them to think silently about how today's lesson helped them solve the problem of finding large areas.  Next, I have them turn and share with a partner.  I tell them to listen very carefully to their partner, because my next step is to have them turn the opposite way and share what their partner told them.  This way, everyone hears two thoughts and must listen in very carefully!

The homework assignment for this evening is to write in their reflection journal how they would find the area of a very large array.