Patterns have fascinated mathematicians for years. You will find that many of your students have already caught on to some number patterns, using them to make their work either easier or quicker. Students may also realize that the basic division facts are at the very root of division patterns.

On their own, some of your students may have learned that a pattern of zeros can be used to divide by multiples of 10. When dividing by multiples of ten, they can just cross off the zeros. You will, however, want to fine-tune these discoveries for them.

When crossing off zeros in the dividend, the same number of zeros must be crossed off in the divisor. For example:

If the number of zeros is not the same, you can only cross off the same number of zeros in both the divisor and dividend. For example:

In each example shown above, the basic division fact 6 ÷ 3 = 2, was used to find the quotient.

When dividing, students can make comparisons to help them decide where to place the first digit.

Find 197 ÷ 36.

Step 1 Decide where to place the first digit in the quotient.

36 > 1 There are not enough hundreds.
36 > 19 There are not enough tens.
36 < 197 There are not enough ones.

Step 2 Divide the ones.

In a division problem, each time there is an amount left over, the amount must be compared to the divisor. The number left over must be less than the divisor.

Point out to students that they can also use an estimate to decide where to place the first digit in the quotient. All division problems can be estimated using multiples of ten and basic division facts.

To estimate 197 ÷ 36, help students use basic facts and multiples of 10 to find a new dividend and a new divisor.

Knowing that 197 ÷ 36 is about 5 helps students to decide where to place the first digit in the actual quotient.

Make a point of showing students that the number left over is less than the divisor: 17 < 36. If the number left over is greater than the divisor, the quotient is less than it should be.

The numbers students choose to estimate the quotient influences the ease with which they perform the estimate.

For 6,643 ÷ 81:

Notice how the above estimate is nearly as difficult to divide as the original problem. Point out that 60 ÷ 8 is not a basic fact. Emphasize to students the importance of estimating with basic facts.

Knowing that 6,643 ÷ 81 is about 80 or 8 tens, helps students to decide where to place the first digit in the actual quotient.

Sometimes, the first estimate of the quotient is either too large or too small. You may find that adjusting the quotient is a trouble spot for students.

When the number being subtracted is greater than the number being subtracted from, adjust the quotient to a lesser number.

For 719 ÷ 81:

When the answer to the subtraction is greater than the divisor, adjust the quotient to a greater number.

For 1,157 ÷ 18:

It is important for students to know that division is not a one-approach-only operation. As students increase their knowledge in analyzing quotients in division, you will notice that their ease in analyzing approaches to other situations will also improve. Knowing that more than one approach to a division problem results in the correct answer should help students when they are solving a word problem. Often, more than one operation is needed, and they need to decide which ones are appropriate and then to apply them.

The Little League baseball coaches opened 5 boxes of baseball cards. Each box contained 24 packs. The coaches gave out the packs as gifts to the players. Each player got the same number of packs. There were 40 players. How many packs did each player get?

To solve, you need to understand what to find (how many packs each player got), plan how to find the answer, (multiply to find the total number of packs; divide the total number of packs by the number of players), and then to look back to check.

Each player got 3 packs.