What role does place value have in whole number operations?

We often learn to add, subtract, multiply and divide using traditional algorithms. Remember, an algorithm is a set of rules. They are faster or more efficient, but it's easy to get lost or make mistakes because there are several steps to remember.

Traditional algorithms don't show true place value all the way through Traditional algorithms don't show true place value all the way through. Let’s look at each of these problems. This addition problem shows regrouping. When we add 7+9…

…we write down the 6 ones and “carry the 1” …we write down the 6 ones and “carry the 1”. The 1 we carry is actually a 10. The next problem is a subtraction problem. This subtraction problem also shows regrouping. When we regroup 1 hundred in this subtraction problem…

…it becomes 10 tens not 1 ten …it becomes 10 tens not 1 ten. The next problem is a multiplication problem. Let's look at the place values of the numbers in this multiplication problem.

We are multiplying 300, 60, and 1 by 3, not 3 and 6 We are multiplying 300, 60, and 1 by 3, not 3 and 6. Finally, let's look at the place values in this division problem.

We divide 9 into 370, not 37. In traditional algorithms, it is not clear what the place values of the digits are.

How does expanded form help us see place value?

This addition problem was solved using the traditional algorithm.

Here is the same problem solved using expanded form

The expanded form helps us clearly see the place value of each digit The expanded form helps us clearly see the place value of each digit. We are not regrouping 1.

We are regrouping 10.

In the traditional algorithm, it is not clear that we are adding 10 or 100 because we only write the 1.

This subtraction problem was solved using the traditional algorithm.

Here is the same problem solved using expanded form

We are not borrowing 1, as this algorithm suggests.

We are really borrowing 10.

This multiplication problem was solved using the traditional algorithm

Here is the same problem solved using expanded form Here is the same problem solved using expanded form. Notice how clearly we see the place value of each digit.

We are not multiplying 2, 5, and 6 by 3.

We are really multiplying 200, 50 and 6 by three We are really multiplying 200, 50 and 6 by three. The traditional method is quicker and more efficient. But it does not clearly show what number we multiply by because we can’t see the place value.