Partial Quotients Division By Jennifer Adams

Why not the Traditional Method? The traditional approach (or algorithm) for large number division is the most abstract and difficult approach to division. Yet many adults think it is the only approach. Take a simple problem like 7,248 divided by 3 and try to explain the traditional method: three goes into seven twice, write down the two over the seven, multiply two times three and write the answer, six, under the seven, then subtract six from seven. That leaves 1,248. Since three won't go into one you have to move over a column and divide three into 12... You get the idea. Students often have no idea why they do what they do in this process. They do it because the teacher said to, if they can do it at all...

Why do they learn partial quotients? The partial quotient method of solving large division problems has two main advantages. It allows elementary school students to see the problem in a less abstract form. They actually ask concrete questions like "So, how many nines are there in 2,079?" **Division becomes an idea instead of a long division template. It allows the students to work the solution flexibly, using numbers they're comfortable with, instead of requiring a rigid mathematical process that the student may not find comfortable. (How that works will become more obvious as the method is presented.)

STEP 1: Setting up the problem The first step in the partial quotient approach is to set up the problem. This looks much like a traditional long division problem except that a vertical line gets drawn along the right side of the problem to create space for the student to track "partial quotients.” We call the division symbol the “table” and the vertical line down the side the “curtain” 6 72

STEP 2: Picking easy multiples The question in the mind of the student is simple: "How many groups of six are there in 72?" The partial quotient method tries to get the student to the answer through basic logic. So the next question the student asks is, "Well, are there at least blank number of groups of six in 72?" Fill in the blank with a number the student is comfortable with - let's say 5. So the question becomes "Well, are there at least 5 groups of nine in 72?" The student does the math and figures that 5 groups of six (5 x 6) is 30. The student writes 5 in the partial quotient column and writes 30 under 72 in the problem template; then he does the subtraction to see how much is left. 6 72 30 5 (5*6=30) 42

STEP 3: Can I do that again? With 42 left in his dividend, the student should ask this basic question: "Can I take that many out again?" If the answer is "yes" (like in this case), the student should do that. If the answer is "no," the student has to find a smaller easy multiple to take out. In this case the student writes 5 in the partial quotient column again and writes 30 under 42 in the problem template; then he does the subtraction again to see how much is left. 6 72 30 5 (5*6=30) 42 30 5 (5*6=30) 12

STEP 4: Can I do it one more time? With 12 left in his dividend, the student should ask that basic, logical question again: "Can I take that many out one more time?" If the answer is "yes," the student should do that. But in this case the answer now is "no," so the student has to find a smaller easy multiple to take out. Most of our students know their 2 facts, so they should all be able to see that there are two 6’s in 12 and repeat the steps we have been doing. 6 72 30 5 (5*6=30) 42 30 5 (5*6=30) 12 12 2 (2*6=12) 00

STEP 5: Add up the Partial Quotients When the student has taken this process as far as possible (the dividend left is less than the divisor) the final step is easy: add up the partial quotients to get the whole quotient. We like to say that you add up the numbers in the “curtain.” The student in this particular case will add 5+5+2 and get 12. There are 12 groups of six in 72. Or, phrased more traditionally, 72 divided by 6 is 12. 6 72 30 5 (5*6=30) 42 30 5 (5*6=30) 12 12 2 (2*6=12) 00 12

12 Step 6: You’re DONE! 6 72 30 5 (5*6=30) 42 30 5 (5*6=30) 12 We like to say that you put the answer on the “table” because it’s DONE! Just like you put the food on the table when it’s done! 6 72 30 5 (5*6=30) 42 30 5 (5*6=30) 12 12 2 (2*6=12) 00 12

Hope this helps! I hope this really helps you understand what your students are doing in math. We will also send home the parent packet that goes with this unit. If you have any questions, feel free to call or send me a note. Please know that this is also an introduction to division. In future years your child may use the traditional method for division. However, this is a great way for them to start learning and it’s a great transition into the traditional method because they understand why they do what they do in the traditional process.