Using Partial Quotients

Using Partial Quotients: What is it?

 Partial quotients is a division strategy in which the numbers are decomposed to arrive at an answer. 

The strategy of decomposing the dividend into parts (e.g., decomposing 128 into 100+ 20+8) and then dividing each part by the divisor is an application of the distributive property. According to the distributive property, division expressions, such as 128÷ 4, can be split into smaller parts, for example, (100÷ 4)+(20÷ 4)+(8÷4).  The sum of the partial quotients (25+5+2) provides the answer to the division expression.

Supporting Students Using Partial Quotients

The array is often used to model partial quotients. The dividend has been decomposed into numbers that are easier to work with. Consider the division expression 195 ÷ 15. Students can rework the problem into friendly numbers: 195 can be decomposed into 150 + 45, and each part can be divided by 15.

The same problem is modeled in an open array to link the operations of multiplication and division. Students might decide to ‘multiply up’ to reach the dividend in order to find the quotient.

Students will need to use their factual knowledge in order to decide how to decompose the number. Students learn that facts involving 10 × and 100 × are helpful when using the distributive property. To solve 888 ÷ 24, for example, students might take a “stepped” approach to decomposing 888 into groups of 24.

Where to Next?

Once students are flexibly splitting of numbers they can be encouraged to transition into flexible division algorithms, such as alternate or standard algorithm.