Division Using Partial Quotients (The Big 7 Model)

In 24 ÷ 8 = 3, 24 is the dividend, the amount being divided. 8 is the divisor. In different contexts, the divisor may be the number of equal groups or the number in each group. 3 is the quotient, the result of dividing.

Multiplication and division are inverse operations. A division problem is equivalent to a missing factor multiplication problem. A ÷ B = ? is equivalent to B × ? = A.

8 × 3 = 24. So, 24 ÷ 8 = 3.

I can make a rectangle and divide that rectangle into two parts. One part I can label 80 and the other part I can label 4 to show a total of 84. On one side of the rectangle I label 4 for the divisor. I can find the quotient by finding the quotient in each part of the rectangle. 84 ÷ 4 = 20. 4 ÷ 4 = 1. The quotient is 20 + 1 = 21.

120 is 1 hundred, 2 tens, and 0 ones. I cannot divide 1 hundred into 3 equal groups so I rewrite it as 12 tens. 12 tens can be divided into 3 equal groups of 4 tens. Each group has 4 tens, or 40. So, 120 ÷ 3 = 40.

Partial quotients are all the lesser quotients that I find in the process of finding the full quotient. I add the partial quotients together to find the full quotient.

I record the partial quotients on the right side of the 7. I record the subtractions showing how much is left on the left (inside) of the 7.

It is more efficient to find fewer partial quotients (using greater numbers divided in each round), but any combination is correct as long as the sum of the partial quotients is the same. So, if the quotient is 25, partial quotients could be 10 + 10 + 5, or 20 + 5, or other variations.

No. Regardless of the size of the divisor, I can subtract multiples of the divisor from the dividend and record partial quotients outside the 7.

I start by drawing the big 7 and writing 847 and 3 at the top. Then I find products of 3 and other numbers that I can subtract from the dividend, 847. 3 × 200 = 600 is a number close to 847. Write 200 as a partial quotient and record 847 – 600 = 247 inside the 7. There is 247 left to divide. 3 × 60 = 180. Write 60 as a partial quotient and record 247 – 180 = 67 inside the 7. There is 67 left to divide. 3 × 20 = 60. Write 20 as a partial quotient and record 67 – 60 = 7 inside the 7. Lastly, 3 × 2 = 6. Write 2 as a partial quotient and record 7 – 6 = 1 inside the 7. The quotient is the total of the partial quotients: 200 + 60 + 20 + 2 = 282 with 1 remaining.