How To Draw An Area Model

Number and Operations

Area Model for Multiplication

And then far nosotros have focused on a linear measurement model, using the number line. But in that location'due south another common way to think about multiplication: using expanse.

For example, suppose our basic unit is 1 square:

Nosotros tin picture 4 × 3 as 4 groups, with 3 squares in each group, all lined up:

But we can also pic them stacked up instead of lined upward. Nosotros would accept four rows, with iii squares in each row, like this:

And then nosotros can think about 4 × three as a rectangle that has length 3 and width iv. The product, 12, is the total number of squares in that rectangle. (That is also the area of the rectangle, since each foursquare was one unit of measurement!)

Recollect / Pair / Share

Vera drew this motion picture every bit a model for 15 × 17. Use her picture to assist you compute xv × 17. Explain your work.

Trouble eight

Draw pictures like Vera's for each of these multiplication exercises. Use your pictures to find the products without using a estimator or the standard algorithm.

23 × 37 viii × 43 371 × 42

The Standard Algorithm for Multiplication

How were y'all taught to compute 83 × 27 in school? Were you taught to write something similar the following?

Or maybe you were taught to put in the extra zeros rather than leaving them out?

This is actually no different than drawing the rectangle and using Vera'southward picture show for calculating!

Recollect / Pair / Share

Use the instance above to explain why Vera'southward rectangle method and the standard algorithm are really the aforementioned.

Calculate the products below using both methods. Explain where you're computing the same pieces in each algorithm.

23 × fourteen 106 × 21 213 × 31

Lines and Intersections

Here's an unusual way to perform multiplication. To compute 22 × thirteen, for case, draw 2 sets of vertical lines, the left fix containing two lines and the right set two lines (for the digits in 22) and two sets of horizontal lines, the upper ready containing i line and the lower gear up three (for the digits in xiii).

There are four sets of intersection points. Count the number of intersections in each and add the results diagonally equally shown:

The respond 286 appears!

There is i possible glitch as illustrated by the ciphering 246 × 32:

Although the reply six thousands, 16 hundreds, 26 tens, and 12 ones is absolutely right, one needs to carry digits and translate this as vii,872.

Trouble 9

Compute 131 × 122 via this method. Check your answer using another method.
Compute 15 × 1332 via this method. Check your answer using another method.
Tin can you suit the method to compute 102 × 3054? (Why is some adaptation necessary?)
Why does the method piece of work in general?

Lattice Multiplication

In the 1500s in England, students were taught to compute multiplication using following galley method, now more usually known as the lattice method.

To multiply 43 and 218, for instance, draw a ii × iii grid of squares. Write the digits of the showtime number forth the right side of the filigree and the digits of the second number along the top.

Divide each prison cell of the grid diagonally and write in the product of the column digit and row digit of that prison cell, separating the tens from the units across the diagonal of that jail cell. (If the product is a ane digit respond, place a 0 in the tens place.)