Math Strategy: Using Partial Quotients

Math Strategy: Using Partial Quotients
Math Strand: Number Sense - Division



Overview:
A Partial quotient is a partial answer to a division question. It is a step-by-step method of division wherein at each step, a partial answer is obtained. It allows students to work their way toward the quotient by using friendly multipliers such as tens, fives and twos without having to immediately find the largest quotient. After all the steps have been completed, the partial answers are added together to get the quotient. As students chose larger multipliers the strategies become more efficient.



The Guide to Effective Instruction in Mathematics Volume 4 Division pages 18-30 has some very useful information.
http://www.eworkshop.on.ca/edu/resources/guides/NSN_vol_4_Division.pdf


How this supports student learning:
The strategy of decomposing the dividend into parts (e.g., decomposing 128 into 100+ 28) 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)+(28÷ 4). The sum of the partial quotients (25+7) provides the answer to the division expression.



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 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?
Students will use the flexible splitting of numbers and transition into flexible division algorithms, like the standard algorithm. With flexible algorithms, however, 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”.

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Math Strategy: Repeated Subtraction

Math strategy - Repeated Subtraction
Math strand - Number Sense




Overview:
Repeated subtraction is the process of subtracting the same number from another number two or more times until 0 is reached. This strategy specifically supports division. To use this strategy, you take the dividend and repeatedly take the divisor away from it.



On the Lawson continuum, it is located in the Working With Numbers section.



Students may begin using repeated subtraction with manipulatives and then transfer this strategy to working with numbers. See video below:





With smaller numbers, this strategy is fairly efficient and effective. However, as students progress to larger numbers this can become challenging and inefficient.


How this supports student learning:
Repeated subtraction helps prepare students for the strategy partial quotients where they are taking off chunks of the dividend. When students are comfortable subtracting from larger numbers down to 0 with a constant amount, they will better understand how to subtract from larger numbers in different sized chunks. Below is an example of partial quotients to help you better understand.




Where to next?
Once students are comfortable with the repeated subtraction strategy, they should move towards practising partial quotients, which involves students solving a division problem by subtracting multiples until they get down to 0, or as close to 0 as possible. The subtraction of friendly multiples is key, such as using 5, 10, 25, 100 etc. Students will then add the multiples up to find the answer.

An activity that might help to support repeated subtraction would be simple subtraction activities from the Guides to Effective Instruction, such as any of the Learning Connections from the Counting unit, starting on page 143.

http://oame.on.ca/eduproject/ontariomathedresources/files/Number%20Sense%20and%20Numeration%201-3%20Revised.pdf


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