Big Numbers, Small Numbers

Multiplication Algorithms

Area Model

Using an area model to solve a multiplication problem can be useful to students when computing 2 digit by 2 digit multiplication, as it is based on the place value components since each number is broken up into tens and ones. By drawing the boxes as close as possible to scale the student can see why we multiply the factors in the way we do. Van de Walle¹⁵ reiterates a benefit of the area model by stating, “The area model uses a row and column structure to automatically organize equal groups and offer a visual demonstration of the commutative and distributive properties.”  

Grid paper is a great tool to use to illustrate this concept as the students can count each box to represent the factors as each box is of equal size and it would be easy to outline an area model that represents for example, 12 × 24. Once the boxes are outlined, the student can then use Base-10 blocks to cover the respective piece. Each Base-10 piece results in the partial products that will then be added together to compute the product. When modeling this example for my students, it will be important to reiterate that each flat is composed of 10 rods, which represents 10x10 = 100.

Using the equal groups, whole unknown word problem from the chart above, I will model the steps of using an area model to solve a 2-digit by 2-digit multiplication problem. As with addition, the issues of computation are largely independent of the problem types. Students need to understand the context in which multiplication can be used as well as how to execute it computationally.

Step 1: Write the problem.

12 (columns) × 24 (rows)

Step 2: Draw the grid to scale and write each factor in expanded form.

In this step, the two digit numbers are decomposed into their place value components, and each component of one is multiplied by each component of the other, according to the Extended Distributive Rule.  Each product of place value pieces is easy to compute, following the discussion above. The multiplicand of 12 is broken down into 10 and 2 and each factor is multiplied with each piece of the 24. This step also shows a visual representation of the Extended Distributive Rule that was previously explained. Note, drawing the boxes to scale is the key feature of the Area Model.

Step 3: Write the numbers in expanded notation form.

(10 + 2) × (20 +4) = 

(10 × 20) + (10 × 4) + (2 × 20) + (2 × 4) =

It is important to note that this step illustrates the symbolic version of the Extended Distributive Rule, in that each place value piece of one factor is distributed amongst all of the place value pieces of the other factor.

Step 4: Perform all the indicated multiplications in step 3, and add the partial products.

200 + 40 + 40 + 8 = 288

Box Method

The Box Method is an abstract version of the Area Model. The students are no longer drawing proportional boxes to compute a product but equal size boxes which is more conducive to logic when multiplying with larger numbers. It would not be practical to draw the boxes to scale for three digit numbers or larger. Further, it has been my experience that the students lose valuable time trying to draw the boxes to scale when working with larger numbers as opposed to drawing equal size boxes.

Let’s look at an example of the multiplicative comparison result unknown word problem from the chart above.

Khalil was in the weight room for 55 minutes last week. Gerald was in the weight room fourteen times as long as Khalil during the same week. How long was Gerald in the weight room?

Because the Area Model and the Box Method have some similarities the steps may parallel a bit. This small progression will add to the students’ overall confidence as they are will be expanding on their previous understanding of computing 2 digit by 2 digit multiplication. This progression will also support the principles behind fostering a growth mindset by allowing the students to see that growth is a continuous process. Additionally, using the Box Method allows the students to continue to see the decomposition of the numbers into their respective place value pieces.

Step 1: Write the factors to be multiplied in expanded form.

Step 2: Draw a box with equal grid lines that correspond with the number of addends produced from writing the factors in expanded form. Place the expanded form of the multiplicand horizontally across each box and the expanded form of the multiplier vertically beside each box.

Step 3: Using the grid, multiply each number of the multiplicand on the top of the box with each number of the multiplier on the side of the box.

Step 4: Add each partial product from the boxes to obtain the total product.

500

200

50

+ 20

------

770

The Box Method example above further illustrates the Extended Distributive Rule that was previously discussed as each factor has been combined by multiplying each addend. Additionally, when looking at the Box Method, the sums along the rows of the partial products produces the addends of the Standard Algorithm as evidenced by the example below. While summing along the columns produces the addends of the Standard Algorithm when the order of the factors are reversed as 14 × 55 (illustrated below).

Standard U.S. Algorithm

The standard U.S. algorithm for solving multiplication may be the most widely used strategy and one could also suspect that it is the least understood by students in classrooms all over the world.  Many teachers tend to introduce this strategy very early when teaching students how to multiply, a time in which they are still trying to grasp why they are regrouping and inserting a “0” each time they arrive at a new row of partial products.

The example provided below is the same algorithm used for the box method above to aid as a visual to show how the pieces or the resulting partial products in each strategy are equivalent.

Step 1: Multiply the ones digit in the multiplicand (the top number) by the ones digit in the multiplier (the bottom number). Put the ones in the ones place below the equal bar and regroup the tens above the tens place in the multiplicand.

2

55

× 14

------

0

2

14

× 55

------

0

It is important to note that the regrouping is a result of: 5 ones × 4 ones = 20 ones and 20 ones = 2 tens and 0 ones. Be sure to emphasize the base-ten language as it points out the number of tens and why the regrouping is necessary.

Step 2: Multiply the ones digit of the multiplier by the tens digit of the multiplicand and add the regrouped tens from step 1. Write the total beside the “0” in the ones place.

2

55

× 14

------

220

2

14

× 55

------

70

Notice that the result, 220, is the sum of the two amounts in the lower row of the grid, while 70, is the sum of the partial products of the columns in the Box Method for this product.

It is vital that students multiply the digits before regrouping as multiplying creates a new tens number that will then need to be added to the previously created tens number from step 1. However, many students make the mistake of adding the 2 to the 5 before multiplying by 4. Meaning, 5 sets of 4 equals 20; 20 plus 2 tens equals 22 tens or 2 hundreds and 2 tens. Students who regroup first, generally do so by imitating the procedure for addition by adding the regrouped digit prior to adding the remaining digits in that place value column.

From my teaching experience, many students have a difficult time keeping the digits in line when using this method which can easily skew their final product. To limit this, I have the students use grid paper in which each number corresponds with a box on the paper or the students will turn their notebook paper vertically so that the numbers are already aligned in rows.

Step 3: Insert “0” in the second row of partial products as we will now be multiplying by the tens piece of the multiplier.

2

55

× 14

------

220

0

2

14

× 55

------

70

0

Step 4: Multiply the tens digit of the multiplier by the ones digit of the multiplicand and write the digit beside the “0”.

2

55

× 14

------

220

50

2

2

14

× 55

------

70

00

Step 5: Multiply the tens digit of the multiplier by the tens digit of the multiplicand.

2

55

× 14

------

220

+550

2

2

14

× 55

------

70

+700

At this point in the process I would show the students that the second addend, 550, is the sum of the two amounts in the upper row of the grid, and 700, is the sum of the partial products in the left column for the Box Method.

Step 6: Compute the sums of the partial products to obtain the total product.

2

55

× 14

------

220

+550

------

770

2

2

14

× 55

------

70

+700

------

770