Array Model and Partial Quotients

ND State Standard
4.NO.NBT.6 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors using place value strategies. Show and justify the calculation using equations, rectangular arrays and models.

Investigate how to use the Box (Grid) Model, Rectangular Array and Partial Quotients as strategies for division. 

• Investigation 1 - Box (Grid) Model 

• Investigation 2 - Box (Grid) Model

• Investigation 3 - Rectangular Array Model

• Investigation 4 - Rectangular Array Model and Partial Quotients 

• Investigation 5 - Partial Quotients

• Investigation 6 - Partial Quotients

Record yourself explaining and showing how you would solve any 5 of the problems of your choice with the picture below using each strategy:

•  Box (Grid) Model

• Rectangular Array Model

• Partial Quotients

A description of each method.

Box Method

The Box Method is a way of organizing long division using place value. It involves creating a grid (or box) where each column represents a different place value (thousands, hundreds, tens, etc.). Here's how it works:

• Create a box with enough columns for each place value of the dividend (the number being divided).

• Start with the highest place value. Divide the part of the dividend corresponding to that place value by the divisor.

• Multiply the divisor by the quotient you got from the previous step and write it under the dividend in that column.

• Subtract this product from the dividend to find the remainder for that column.

• Move to the next place value and repeat the process until you've covered all place values.

This method helps students keep their work organized and focus on one place value at a time.

Rectangular Array Method

The Rectangular Array Method involves using a grid or array to visualize division as distributing items into equal groups. Here's how it works:

1. Represent the dividend as a total number of items or squares in a grid.

2. Divide the array into rows or columns according to the divisor. Each row or column will represent one group.

3. Count the number of rows or columns to find the quotient.

This method helps students see the relationship between division and multiplication as they can directly count the groups or items.

Partial Quotients Method

1. Start by estimating how many times the divisor can go into the dividend. It doesn't have to be exact—just a reasonable guess that is easy to work with (e.g., multiples of 10, 5, or even 1).

2. Multiply the divisor by your estimate and subtract the result from the dividend. Write down the result underneath the dividend and also record the partial quotient (your estimate) on the side.

3. Repeat the process with the new smaller number (the result from the previous subtraction). Continue subtracting multiples of the divisor until the remainder is smaller than the divisor.

4. Add up all the partial quotients to get the final quotient.

5. The leftover number, once you can no longer subtract the divisor, is the remainder.

This method focuses on estimating and subtracting multiples of the divisor from the dividend until you reach a remainder that's smaller than the divisor.