# Division Basics Part 1: Arrays and Area Model

by OK Math and Reading Lady Division seems to be the hot topic with classes I have been visiting lately, so I thought I’d focus on that for now. Let’s look at some of the basics.  Students as young as first grade actually start thinking about division when working on fraction standards such as:  Determine fair share — equal parts. Most students have had practical experience with dividing sets of objects in their real life to share with friends, classmates, or family (cookies, pizza, crayons, money, pieces of paper). So now our job as teachers is to relate this real-life experience with the division algorithm.

Much like multiplication, there are different aspects of division children should get familiar with:

• Arrays
• Equal Groups
• Repeated Subtraction
• Number lines
• Skip counting

In this post, I will break down the benefits and uses for arrays (and the related area model) to help children understand division (and how it is related to multiplication). There’s a fun FREE game (Block-It) at the end of the post.

What is an array?  An array is a rectangular model made up of rows and columns.  When an array is constructed, the factors are represented by the number of rows and columns.  So, do your students know the difference in a row and column?  (Rows go horizontally, while columns are vertical.)  These are important math terms students should be using.

• Give students experience constructing arrays with manipulative objects (tiles, chips, cubes, etc.):
• You can be specific, such as: “Build an array using a total of 12 tiles. Put them in 3 rows.  How many columns did you create?” In this scenario, there is only 1 way to show this array. Students would be modeling 12 ÷ 3 = 4. Twelve is the dividend (the total amount you started with). The # of rows is the divisor (how it was divided).  The quotient is the result (in this case the # of columns).
• You can also be a little more open ended such as:  “Build an array using 12 tiles. Is there more than one way to do this?” If students are given the opportunity to explore, they hopefully find arrays such as 3 x 4; 4 x 3; 2 x 6; 6 x 2; 1 x 12; or 12 x 1. Students would be modeling 12 ÷ 4; 12 ÷ 2; 12 ÷1, etc. • Give students experience drawing arrays:
• You can be specific or open-ended (as above).
• Children can free-hand draw or use grid paper.  If using grid paper, then these can be cut out and displayed as “Different ways to divide 12.” • Give students experience using pre-drawn arrays:
• Students should label the sides of the array with numbers.
• Use the numbers shown to determine the fact family.  Example:  3 x 4 = 12; 4 x 3 = 12; 12 ÷ 3 = 4; and 12 ÷ 4 = 3
• After the array is made, ask questions or explore more such as:
• How many 3’s are in 12? (count the columns)
• How many 4’s are in 12? (count the rows)
• Circle the rows and / or columns to see the groups more easily.
• Help children make up story problems to match the array:  “I have 12 desks that I need to arrange in 3 rows. How many desks will be in each row?” or “I need to put 12 books equally onto 3 shelves. How many books will go on each shelf?

Relate experience with arrays to determine area of a rectangle. This mostly just adds a measurement component to the problem.

• Draw a rectangle and partition it into columns (length) and rows (width) to match the story.  Here are two sample stories:
• I am making a rectangular shaped garden which I want to be 12 square yards in size.  If the length of the garden is 4 yards, how long does the side of the garden need to be?
• I am using a rectangular piece of wood to cover a broken window that is 12 square feet.  One side of the wood is 3 feet. How long would the adjoining side be?

Here’s a great game called “Block it” which utilizes arrays. It can have variations depending on the level of your students. Here is a FREE copy of the directions: Block-It Game Directions Materials needed:

1. 1 grid paper (1/2″ is great)
2. 2 players
3. Each player needs 1 crayon or colored pencil (light colored). Different color per player.
4. 2 number cubes or dice (6 sided).

How to play:

1. Player 1 rolls the dice.  Let’s say a 3 and 4 are rolled. The player makes a 3 x 4 “block” or array. Be sure to show them how to use the lines on the grid paper to make this (as I discovered it’s not always clear to some students). Color it in with crayon.  Inside the block, write the product (12).
2. Player 2 then rolls the dice and uses their 2 numbers to create another block, colors it, labels it, etc.
3. Repeat
4. The goal is to create as many blocks / arrays as possible (more than the opponent). There is a strategy to maximize the use of the space. Repeated play helps children see they need to consider this so they don’t end up with little unusable spaces.
5. As the board gets filled up, players may have to miss a turn or roll again if not enough space is available.

Variations:

1. As the board gets filled up, students may need to start thinking of alternate ways to make their blocks to fit the available space. For example, if the player rolls a 6 and 4 but there is no room to fit a 6 by 4 array, they can think of other ways to make an array of 24 that might work (such as 8 x 3, 12 x 2).
2. Students can keep track of their score by keeping a running total of each block / array they make.
3. Use smaller size grid paper and use 9, 10, or 12 sided dice.
4. Write the fact family members for each block created.

Enjoy! Have you / your students played Block It?  Let us know if you like it!