Array Model and Partial Products

ND State Standard
4.NO.NBT.3 Apply place value understanding to round multi-digit whole numbers to any place.

4.NO.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number and multiply two-digit numbers. Show and justify the calculation using equations, rectangular arrays, and models.

Let's investigate how to use the Area Model as a strategy for Multiplication.   The area method is a powerful method as we continue to move from the concrete to the more abstract versions and can be utilized with fractions, decimals, and even within algebra!

• Investigation 1 - Array Method (Area Method)

• Investigation 2 - Array Method (Area Method)

• Investigation 3- Array Method (Area Method) and Partial Products

• Investigation 4 - Partial Products

First, provide an estimation for each situation.  

Then, solve each scenario listed below using the Array Method (Area Model) and Partial Products.  

Record yourself explaining your steps and showing how you solved, make sure to comment on your estimation and how you chose to round and then estimate.

• Two digit number  x two digit number

• Two digit number with a nine in the ones position x two digit even number with a nine in the tens position. 

• Three digit odd number x three digit even number

• Two-digit number with a five in the tens place x two-digit number with a zero in the ones place.

• Three-digit number with a seven in the hundreds place x three-digit number with a six in the tens place and zero in the ones place.

A description of each method.

Box Method (Area Method)

The Box Method, also known as the Area Method, uses place value and a visual representation to multiply numbers. It breaks the numbers into their place values, and each part is multiplied separately. Here's how it works:

1. Draw a box and divide it into sections (or smaller boxes) based on the number of digits in the numbers being multiplied. For example, if you're multiplying a 2-digit number by another 2-digit number, you’ll need a 2x2 grid (4 boxes).

2. Decompose each number into its place values. For example, if you're multiplying 23 by 47, you decompose into:

   • 20 (for the tens place in 23)

   • 3 (for the ones place in 23)

   • 40 (for the tens place in 47)

   • 7 (for the ones place in 47)

3. Multiply the numbers in each section of the grid. For the example:

   • 20 × 40 = 800

   • 20 × 7 = 140

   • 3 × 40 = 120

   • 3 × 7 = 21

4. Add all the products together to get the final answer:

   • 800 + 140 + 120 + 21 = 1081

The grid visually represents how each part of the numbers interacts, helping students understand how multiplication works with larger numbers.

Partial Products Method

The Partial Products Method is similar to the Box Method but without the visual grid. It also breaks down multiplication into smaller parts based on place value. Here's how it works:

1. Just like in the Box Method, decompose each number into its place values.

2. Multiply each part of one number by each part of the other number. For example, if you're multiplying 23 by 47:

   • 20 × 40 = 800

   • 20 × 7 = 140

   • 3 × 40 = 120

   • 3 × 7 = 21

3. Add up all the partial products to get the final answer:

   • 800 + 140 + 120 + 21 = 1081

Unlike the Box Method, the Partial Products Method doesn't rely on a visual grid, but it still emphasizes the importance of understanding place value and breaking down the multiplication process into simpler steps.

Both methods help learners understand the concept of multiplication by decomposing the numbers into manageable parts, making it easier to multiply larger numbers and reinforcing the understanding of place value.