Alternative or Standard Algorithm: What is that?

The alternate algorithms are partial products (multiplication) and partial quotients (division) strategies.  Students benefit from working with a partial product and partial quotient algorithms before they are introduced to the standard multiplication and division algorithms.

Overview

Alternative algorithms provide a way for students to record partial products or quotients and then add them to determine the final product.  The alternative algorithms help students to think about place value and the position of numbers in their proper place-value columns. If students are efficient with partial products and quotients, educators can take the opportunity to connect their partial products' understanding to the ‘standard’ algorithm. The thinking in both algorithms are the same and how students choose to represent their thinking will vary for every student. Some students will prefer using the standard algorithm while others may choose to use an algorithm just as efficiently.

The images below show the progression from arrays to alternative algorithms to the standard algorithms. 

Supporting Students Using Alternate or Standard Algorithms

Multiplication:

To begin using the alternative algorithm, students need to understand that numbers in a multiplication expression can be decomposed to “friendlier” numbers, and that partial products can be added to determine the product of the expression. As well, the ability to work flexibly with a range of multiplication facts is important for students to efficiently use algorithms. The introduction of any algorithm is not encouraged before students are able to efficiently work with numbers using partial products. 

Students benefit from working with a partial product algorithm before they are introduced to the standard multiplication algorithm. Then, they can connect numbers in the partial products algorithm to the numbers in the standard algorithm (as shown above).

Division: 

Both flexible division algorithms and the standard algorithm, are based on the distributive property. The standard algorithm can be very confusing for students.  They must have opportunities to solve division questions using their own strategies prior to using the standard algorithm. 

Practice with x2, x5, x10 (and multiples of 10), and x100 facts will help support students using the flexible algorithm as students use known multiplication facts to decompose the dividend into friendly “pieces”, and repeatedly subtract those parts from the whole until no multiples of the divisor are left. Students keep track of the pieces as they are “removed”.  As students become more comfortable multiplying and dividing by multiples of 10, they learn to compute using fewer partial quotients in the algorithm. 

Introducing students to algorithms before they are developmentally ready can cause significant confusion and may result in inaccurate answers. As students work with numbers using partial products or partial quotients, they will be able to gain a sense of reasonableness and recognize if the answer they have arrived upon makes sense. The ability to follow a procedure is not enough for students to build a sense of number and be flexible math thinkers.

Where to Next?

Students' next step is to use strategic and efficient strategies.  The ability to perform computations efficiently depends on an understanding of various strategies, and on the ability to select appropriate strategies in different situations. Students need opportunities to explore various strategies and to discuss how different strategies can be used efficiently in different situations.