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## Adding It Up: Helping Children Learn Mathematics (2001) Center for Education (CFE)

### Citation Manager

. "3 Number: What Is There to Know?." Adding It Up: Helping Children Learn Mathematics. Washington, DC: The National Academies Press, 2001.

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Adding + It Up: Helping Children Learn Mathematics
 Distributivity of multiplication over addition. When multiplying a sum of two numbers by a third number, it does not matter whether you find the sum first and then multiply or you first multiply each number to be added and then add the two products: 4×(3+2)=(4×3)+(4×2). In general, m×(n+p)=(m×n)+(m×p). Question: Is subtraction commutative? Answer: No. For example, 6–2=4, but 2–6=–4. Question: Is subtraction associative? Answer: No. For example, (7–4)–2=3–2=1, but 7–(4–2)=7–2=5.

### Subtraction and Division

So far we have talked only about addition and multiplication. It is traditional, however, to list four basic operations: addition and subtraction, multiplication and division. As implied by the usual juxtapositions, subtraction is related to addition, and division is related to multiplication. The relation is in some sense an inverse one. By this, we mean that subtraction undoes addition, and division undoes multiplication. This statement needs more explanation.

Just as people sometimes want to join sets, they sometimes want to break them apart. If Eileen has eight apples and eats three, how many does she have left? The answer can be pictured by thinking of eight apples as composed of two groups, a group of five apples and a group of three apples. When the three are taken away, the five are left. In this solution, you think of eight as 5+3, and then when you subtract the three, you are again left with five. Thus subtracting three undoes the implicit addition of three and leaves you with the original amount. It is the same no matter what amount you start

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 Front Matter (R1-R20) Executive Summary (1-14) 1 Looking at Mathematics and Learning (15-30) 2 The State of School Mathematics in the United States (31-70) 3 Number: What Is There to Know? (71-114) 4 The Strands of Mathematical Proficiency (115-156) 5 The Mathematical Knowledge Children Bring to School (157-180) 6 Developing Proficiency with Whole Numbers (181-230) 7 Developing Proficiency with Other Numbers (231-254) 8 Developing Mathematical Proficiency Beyond Number (255-312) 9 Teaching for Mathematical Proficiency (313-368) 10 Developing Proficiency in Teaching Mathematics (369-406) 11 Conclusions and Recommendations (407-432) Biographical Sketches (433-440) Index (441-454)