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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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 Page 79

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Adding + It Up: Helping Children Learn Mathematics

with: 5+3–3=5; 9+3–3=9; 743+3–3=743. More formally, subtracting 3 is the inverse of adding 3.

It is similar with division and multiplication. Just as people sometimes want to form sets of the same size into one larger set, they sometimes want to break up a large set into equal-sized pieces. If you think of 15 as 5×3, then when you divide 15 by 3, you are again left with 5. Thus division by 3 undoes implicit multiplication by 3 and leaves you with the original amount. It is the same no matter what amount you start with: 5×3÷3=5; 9×3÷3=9; 743×3÷3=743. More formally, dividing by 3 is the inverse of multiplying by 3.

Two interpretations of division deserve particular mention here. If I have 20 cookies, and want to sort them into 5 bags, how many go in each bag? This is the so-called sharing model of division because I know in how many ways the cookies are to be shared. I can find the answer by picturing the 20 cookies arranged in 5 groups of 4 cookies, which will be the contents of 1 bag. If the cookies originally came out of 5 bags of 4 each, when I put them back into those bags, I will again have 4 in each. Thus, division by 5 undoes multiplication by 5, or division by 5 is the inverse of multiplication by 5. The picture below shows the sharing model for this situation.

To think about 20÷5, you could also use the measurement model: If I have 20 cookies that are to be packaged in bags of 5 each, how many bags will I get? In the sharing model (also called the partitioning model or partitive division), you know the number of groups and seek the number in a group. In the measurement model (also called quotative division), you know the size of the groups and seek the number of groups. The circled numbers in the figures above and below illustrate a crucial difference between the two models: the order in which the cookies are placed in bags. In the sharing

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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)