In third grade math, teachers mainly emphasize compatible numbers in addition and subtraction. Compatible numbers are numbers that are easy to work with mentally, such as parts of 10. Students who have memorized 8 + 2 = 10 can more easily reason that 10 - 2 = 8. By third grade, they should thus be able to quickly answer 80 + 20 or 100 - 20 by recognizing compatible numbers.
Compatible numbers are "friendly numbers" that make it quicker to solve problems. By fifth grade, students are asked what friendly numbers to use in estimating the answer to questions such as 2,012 / 98. Those who understand estimating will use 2,000 / 100 to approximate an answer. When a student understands parts of each number from 1 to 20, that knowledge later becomes a friendly helper when the student is confronted with solving questions such as 33 + 16.
Compatible Number Hiding Game
The skill of identifying compatible numbers begins in kindergarten or earlier, as children learn parts of numbers ranging from 3 (1 + 1+ 1 or 1 + 2) to 10. A common way to learn compatible parts of small numbers in kindergarten and first grade is to play the "hiding game." After displaying six cubes, a player holds them behind her back, brings out two and asks the other player how many are "hiding."
Benchmark Compatible Numbers
Benchmark numbers are another form of compatible numbers that third graders are expected to know. They end either in 0 or 5 and they make the process of estimating much easier; for example, students can use 25 + 75 to approximate the sum of 27 + 73. Mentally calculating a reasonable answer to "about how big" a sum or difference will be demonstrates development of the same skill adults use in situations such as estimating whether savings are sufficient to pay bills.
Parts of 10 and 20
Third graders are usually able to quickly answer questions related to benchmark numbers, such as the difference when subtracting 20 from 40. However, they may stumble when calculating answers related to parts of 10 that they haven't memorized, such as 40 - 26. Even if they understand that it is necessary to trade a ten so that the ones column becomes 10 - 6, their thinking will slow if they haven't memorized that 4 completes 6 to make 10. Similarly, if they don't automatically remember that 6 + 4 = 10, they will be slower to calculate 16 + 4, a parts-of-20 fact.
Becoming Independent Problem Solvers
The understanding of compatible numbers is a tool that helps students become quick, independent problem solvers who don't need to ask friends for help. It is also a major step toward students becoming abstract rather than concrete thinkers. Instead of depending on concrete objects called manipulatives (counters, linking cubes and base-10 blocks) for modeling answers, students rely on automatic knowledge about how the number system works.
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