Partial Quotient Algorithm

[Oliverio Gallman]

Thepartial quotient method is used in long division to break down a division problem into simpler, more manageable steps. It involves finding partial quotients for each part of the dividend. These partial quotients are then added together to obtain the final quotient.

The division operation is defined as the process of repeated subtraction. It is exactly the opposite of multiplication. In the standard form of division, the divisor is used to determine how many times it can be subtracted from the dividend.

In the partial quotient division, we break the dividend into smaller parts by subtracting multiples of the divisor until the remainder is 0 or less than the divisor. The multipliers (numbers used to multiply the divisor to find these multiples) are the partial quotients, which are then summed to find the final quotient.

Thus, 5 can be subtracted 57 times from 285. To reach the final quotient of 57, we preferred chunking the division process into smaller steps. The outcome is the same, no matter how many steps you choose. The numbers with which the divisor is multiplied during this process are called the partial quotients.

The multiplier of the divisor at every step of this division is the partial quotient. This multiple of the divisor should be as close as possible to the dividend, that is, less than or equal to the dividend.

The area model division method visually represents division using rectangles, where the partial quotients and the divisor determine the length and width of these rectangles. Each partial quotient corresponds to the length of one rectangle. The combined area of these rectangles equals the dividend of the division equation.

Partial quotients provide the simple transition from a visual approach to division to a more abstract one, addressing a gap in understanding. It helps students to simplify the long division process into smaller and easy-to-understand steps.

Partial quotients in division are not unique; they can vary depending on the way you choose to approach a particular division problem. The flexibility of the partial quotients method is one of its advantages, as it allows for different approaches that suit the problem and the student's understanding.

The Partial Quotients method is one of these strategies. It is a mental math based approach that will enhance number sense understanding. Students solve the equation by subtracting multiples until they get down to 0, or as close to 0 as possible.

If you would like to try this strategy in your classroom, you may want to start with the Box Method/Area Model, which you can read more about HERE. The Box Method uses the same approach as Partial Quotients, but is organized differently and works well as an introduction.

These task cards give students the opportunity to practice the partial quotients strategy for long division in a variety of different ways. Students will calculate quotients, solve division problems, figure out missing dividends and divisors, think about how to efficiently solve an equation using the partial quotients strategy, and more. See the Partial Quotients Task Cards HERE or the Big Bundle of Long Division Task Cards HERE.

The Long Division Station is a self-paced, student-centered math station for long division. Students gradually learn a variety of strategies for long division, the partial quotients strategy being one of them. One of the greatest advantages to this Math Station is that is allows you to target every student and their unique abilities so that everyone is appropriately challenged. See The Long Division Station HERE.

Division with partial quotients is a deviation from the standard method. The divisor is multiplied with a number and the multiple obtained is deducted from the dividend. This multiple of the divisor is as close as possible to the dividend, that is, less than or equal to the dividend. The factor or the number with which the divisor is multiplied is the partial quotient.

So, we keep multiplying the divisor with a number and subtract that product from the dividend until the difference is less than the divisor or equal to 0. If the difference is less than the divisor then that difference is the remainder. To achieve the final quotient, all the partial quotients are combined together.

An area model is a method of expressing what division looks like using rectangles. The partial quotients and the divisor are taken as the length and width of this rectangle. Hence, we will have as many rectangles as the number of partial quotients.

Here we have two partial quotients 10 and 2. So, there are two rectangles with both their widths being the divisor, that is, 14 units and a length of 10 units and 2 units respectively; the two partial quotients. The sum of the areas of the two rectangles is the dividend, which is 168 square units.

Example 3: There are 75 students in a room and they were asked to distribute chocolates to the rest of the students in the school. There are a total of 6255 chocolates. Find out how many chocolates will each student get if the total number is divided equally. Also, take note of the extra chocolates that are remaining.

Example 5: 1335 square meters of cloth was totally available to a workshop for garment making. If each participant is given 15 square meters of cloth, how many participants were there in the workshop? Also, draw the area model for the dimensions provided.

When dividing a larger number by a smaller number using the partial quotient method, the divisor is multiplied by a number and the product obtained is subtracted from the dividend. This product should be equal to or less than the dividend. This process is continued until the difference between the dividend and the multiple of the divisor is zero or less than the divisor. The factors or the numbers with which the divisor is multiplied are known as the partial quotients. These partial quotients are added together to get the quotient.

The area model denotes the process of partial quotient division in the form of a rectangle. The area of the whole rectangle denotes the dividend. The rectangle may be further divided into smaller parts depending on the number of partial quotients. The breadth of the rectangle is the divisor and the length of the rectangle is the dividend. The sum of the partial quotients is also the length of the rectangle.

When dividing a larger number by a smaller number using the partial quotient method, the divisor is multiplied by a number and the product obtained is subtracted from the dividend. This product should be equal to or less than the dividend. This process is continued until the difference between the dividend and the multiple of the divisor is zero or less than the divisor. The factors or the numbers with which the divisor is multiplied are known as the partial quotients. These partial quotients are added together to get the quotient."}},"@type":"Question","name":"How can you use an area model to find a partial quotient?","acceptedAnswer":"@type":"Answer","text":"The area model denotes the process of partial quotient division in the form of a rectangle. The area of the whole rectangle denotes the dividend. The rectangle may be further divided into smaller parts depending on the number of partial quotients. The breadth of the rectangle is the divisor and the length of the rectangle is the dividend. The sum of the partial quotients is also the length of the rectangle."]} Check out our other courses CodingGrades 1 - 12Explore MusicAll agesExplore CodingGrades 1 - 12

In mathematics education at the primary school level, chunking (sometimes also called the partial quotients method) is an elementary approach for solving simple division questions by repeated subtraction. It is also known as the hangman method with the addition of a line separating the divisor, dividend, and partial quotients.[1] It has a counterpart in the grid method for multiplication as well.

In the UK, this approach for elementary division sums has come into widespread classroom use in primary schools since the late 1990s, when the National Numeracy Strategy in its "numeracy hour" brought in a new emphasis on more free-form oral and mental strategies for calculations, rather than the rote learning of standard methods.[2]

My fourth grade group just finished up a unit on division. They spent a great deal of time exploring division and what that means in the context of a variety of situations. One of the more interesting parts of the unit delved into the use of the partial-quotients method.

For the past ten years I have been introducing the partial-quotients method to students. This method brings more meaning to why the division process works. Students also seems to grasp a better understanding of how partial-quotients can add up to a quotient with a remainder.

This year I introduced something different to the students. Students were asked to use the partial-quotients method to divide numbers and then create an area model of the process. I had quite a few confused looks as most students think of multiplication when using area models. After modeling this a bit I noticed that students continued to have issues with appropriately spacing out their area models. This was a great opportunity for students to use trial-and-error.

After the class had a discussion about long division we explored the partial quotients algorithm. I explained to the students that this was another method to divide larger numbers. As we progressed through and explained the steps, students became more aware of how partial quotients seemed to make sense to them. For many, this was the first time exploring the partial quotients algorithm. The students were able to explain why each step was taken in the process.

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