Learning Math With Manipulatives - Base Ten Blocks (Part II)

In part one of this article, you read about representing and adding numbers using base ten blocks. Once these two skills are mastered, it is time to move onto many a child's nightmare: subtraction. Subtraction, as you may have heard, is essentially addition in reverse. It can be an arduous task on paper, but it can be quite easy with base ten blocks.
Recall that there are four different base ten blocks: cubes (ones), rods (tens), flats (hundreds), and blocks (thousands). Groups of ten base ten blocks can be regrouped or traded for equivalent amounts of other base ten blocks; for instance, ten cubes can be traded for one rod because both are worth ten. For subtraction, it is useful to know how to trade down rods, flats, and blocks. Trading down means converting larger place value blocks into smaller place value blocks. For instance, one flat can be traded for ten rods since they are both worth 100.
Before describing the subtraction procedure, let's go over some vocabulary . . .
Minuend - The amount from which you are subtracting.
Subtrahend - The amount that you are subtracting.
Difference - The answer.
In the equation, 234 - 187 = 47, the minuend is 234, the subtrahend is 187, and the difference is 47. Most people don't bother with the terms minuend and subtrahend, but they are useful in describing the subtraction procedure using base ten blocks.
To begin, represent the minuend with base ten blocks. Try to keep the blocks in order from largest to smallest as this will help to transfer knowledge and skills to paper and pencil methods later on. Remove from the minuend piles, enough blocks to represent the subtrahend. If there aren't enough blocks available, trade some of the larger place value blocks until there are enough smaller place value blocks to remove. The resulting piles after the subtrahend is removed represents the difference.
In the example, begin by representing 234 with 2 flats, 3 rods, and 4 cubes. The goal is to remove 187 or 1 flat, 8 rods, and 7 cubes from these piles. Removing one flat is simple enough, but 8 rods and 7 cubes are difficult to remove if there are only 3 rods and 4 cubes! To solve this problem, trade in one flat for 10 rods, and one rod for 10 cubes. The result would be 1 flat, 12 rods, and 14 cubes. Removing the subtrahend - 1 flat, 8 rods, and 7 cubes - at this point would leave no flats, 4 rods, and 7 cubes. The difference is whatever is left after removing the subtrahend, so the difference is 47.
For beginners, it would be wise to start with subtraction that does not require trading. For example 1954 - 1831 would require no trading because there are enough blocks in the minuend to remove the subtrahend. For more advanced students, questions that include zeros can present a bit of a challenge. For example, 4000 - 3657 would require several trading steps all starting with four blocks. http://www.math-drills.com has several thousand free math worksheets including subtraction questions with no regrouping (trading). One of the nice features of this website is that answer keys are provided, so students can get feedback on their results.
With enough experience, students learn subtraction on a conceptual level and are better equipped to apply it to pencil and paper methods later on. Students who only learn the paper and pencil method don't always develop a conceptual understanding of subtraction and are less able to identify errors in their work.
Base ten blocks are not limited to just addition and subtraction of whole numbers. In part III of this series, several other uses of base ten blocks will be explored.