Lesson 3: Connecting Partial Products to the Standard Algorithm

Property

The final product of a multiplication problem is the sum of all its partial products.

Examples

Property

The standard algorithm for multiplication is a condensed way of calculating and adding partial products. Each row in the standard algorithm corresponds to the sum of partial products involving one of the digits of the bottom multiplier.
For $24 \times 35$:

• Standard Algorithm Row 1: $5 \times 24 = 120$
• This equals the sum of two partial products: $(5 \times 4) + (5 \times 20) = 20 + 100 = 120$
• Standard Algorithm Row 2: $30 \times 24 = 720$
• This equals the sum of the other two partial products: $(30 \times 4) + (30 \times 20) = 120 + 600 = 720$

Examples

For $24 \times 35$:

• Standard Algorithm Row 1: $5 \times 24 = 120$
• This equals the sum of two partial products: $(5 \times 4) + (5 \times 20) = 20 + 100 = 120$
• Standard Algorithm Row 2: $30 \times 24 = 720$
• This equals the sum of the other two partial products: $(30 \times 4) + (30 \times 20) = 120 + 600 = 720$
• Final Product: $120 + 720 = 840$

For $123 \times 45$:

• Standard Algorithm Row 1: $5 \times 123 = 615$
• This equals the sum of three partial products: $(5 \times 3) + (5 \times 20) + (5 \times 100) = 15 + 100 + 500 = 615$
• Standard Algorithm Row 2: $40 \times 123 = 4920$
• This equals the sum of the other three partial products: $(40 \times 3) + (40 \times 20) + (40 \times 100) = 120 + 800 + 4000 = 4920$
• Final Product: $615 + 4920 = 5535$

Explanation

This skill connects the expanded partial products method to the compact standard algorithm. Understanding this connection shows why the standard algorithm works. Each line you write in the standard algorithm is a shortcut for adding a set of partial products together. This helps to build a deeper understanding of the multiplication process beyond just memorizing steps.

Property

To multiply multi-digit numbers, multiply the top number by each digit of the bottom number, one at a time, from right to left. These intermediate results are called partial products. The final product is the sum of the partial products.

Examples

• To calculate $86 \times 42$:

$$

\begin{array}{r r r r}
& & & 8 & 6 \\
\times & & & 2 & 4 \\
\hline
&& 3 & 4& 4 \\

• & 1 & 7 & 2 &0 \\

\hline
& 2 & 0 & 6& 4 \\
\end{array}

\begin{array}{r r r r r}
& & & 5 & 2 & 7 \\
\times & & && 3 & 5 \\
\hline
&& 2 & 6 & 3& 5 \\

• & 1 & 5 & 8 & 1& 0\\

\hline
& 1 & 8 & 4 & 4& 5 \\
\end{array}