# Using Area Models for Multiplication, Perimeter, and Area

### Written by Donna Boucher

Donna has been a teacher, math instructional coach, interventionist, and curriculum coordinator. A frequent speaker at state and national conferences, she shares her love for math with a worldwide audience through her website, Math Coach’s Corner. Donna is also the co-author of Guided Math Workshop.

Yesterday, I blogged about the Number of the Day posters that one of the presenters at CAMT displayed.  The main topic of the presentation was using area models for multiplication, area, perimeter, and prime and composite numbers.  The presenter, Jana Davison from Sheldon ISD here in Houston, packed a lot of information in a short amount of time–kind of like we want to do in our classrooms!

Area models are fantastic for helping kiddos develop a deep understanding of multiplication.  Of course, we use arrays for facts, and area models are a perfect tool for multi-digit multiplication.  Let’s look at the example below.  In keeping with best practices, Jana framed it in a real world problem, but we’ll cut to the chase and simply show how to use the area model to do the multiplication.  To solve the problem, we are multiplying 12 x 13.

1. Students write the 12 and 13 along the side and top of the grid paper.  Starting in the top left hand corner, they outline a 12 x 13 rectangle.
2. Using base-10 blocks, students use the largest piece possible–a hundred–placing it in the upper left hand corner.  They continue covering the rectangle with base-10 pieces, tens and ones, until it is covered.  To determine the product, they count the base-10 pieces.  Students should have lots of concrete practice with the blocks before moving to the representational, shown below.
3. Remember, my Texas friends, the students get grid paper to use on STAAR.

The problem Jana used for perimeter and area was a great one:

Danny has a rectangular rose garden that measures 8 meters by 4 meters.  How many meters of fencing does he need to secure the garden?

Notice that the problem doesn’t mention perimeter.  Students have to know that perimeter is the distance around the outside, so it’s critical that students get lots of real world problems dealing with perimeter and area.

This time we used inch grid paper, because we were going to be using 1-inch tiles for the area portion of the problem.  To find the perimeter, we drew an 8 x 4 rectangle and numbered around it (you can see the numbers in the pictures below).

The next part of the problem:

If one bag of fertilizer can cover 16 sq meters, how many bags will he need to cover the entire garden?

Wowza.  Awesome problem.  So here we used the color tiles.  The problem told us one bag covered 12 sq meters, so we used 16 tiles to cover part of the rectangle and labeled it 1 bag.  We still had more to cover, so we used another 16 tiles to cover the rest of the rectangle.

Who doesn’t love using food as a math manipulative, right?  For the activity on prime and composite numbers, we used Cheerios.  Notice that the activity hits all the CRA bases.  The concrete part is the Cheerios, the representational is the model drawing, and finally the abstract is writing the factors.

1. The math program in our school district, Math Expressions, uses area models. Most of my students love to use it because they are able to break the problem down into smaller chunks. Though our area models look a little different. Ours use expanded notation to break the numbers and the rectangles down.
-Jenn B. NJ

• I’ll have to look those up! I think it’s great for the kids to see a variety of models.

2. Ok, I’m in heaven. THIS is why I LOVE MATH!!!I love finding new ways for students to see why math works the way it does. When I was talking with my third grade students about the grid paper that would be with the STAAR test last year, we practiced several different ways that they could make use of the paper. We talked about area and perimeter, but not about multiplication. Thanks for the wonderful visual. I have been to Math CAMT before, but was not able to go this year. Looks like I missed some good presenters. Thanks for sharing.