Quotients and remainders

First, students will learn how to divide multiples of powers of ten by other multiples of powers of ten, where the divisor is some one-digit number multiplied by a power of ten. The first method that students should learn is as follows:

$$\begin{align} 240{,}000 \div 300 &= \dfrac{24 \cdot 10{,}000}{3 \cdot 100} \\[1em] &= \dfrac{24}{3} \cdot \dfrac{10{,}000}{100} \\[1em] &= 8 \cdot 100 \\[1em] &= 800 \end{align}$$

The second method is to simply cross out zeros. For example, to find \(240{,}000 \div 300,\) I have four zeros on the left and two on the right, so I will finish with two zeros. Now I find the quotient of the numbers, disregarding trailing zeros. I find \(24 \div 3 = 8,\) and tacking on my two zeros gives me \(800.\) This method is less formal, but it's a method I can apply mentally. Here's a video on both methods.

Next, students will learn how to divide up to four-digit dividends by one-digit divisors, using concrete, visual, and abstract methods. For example, divide \(3{,}225\) by \(3.\) Students should also be given such problems with trailing zeros tacked on. For example, divide \(32{,}250{,}000\) by \(300.\) When teaching the abstract methods, start with the box method, as seen here. I think the box method is a bit easier than the usual abstract method, as the box method emphasizes place value. Students should also learn that the dividend is a multiple of the divisor if and only if the remainder is 0. For example, 105 / 7 = 15 R 0, so 105 is a multiple of 7.

Here's another interesting idea to discuss: Find the quotient and remainder of 4, 11, 18, and 25, when each is divided by 7. What do you notice? Can you make a similar conjecture about 6, 13, 20, 27? What about 2, 5, 8, 11?

Next, give your students this challenge. Its only prereq is being able to easily divide small numbers, mentally or otherwise.