1. Excerpts of Expectations from the Number and Operations Standard Grades Pre-K-8 Principles and Standards for School Mathematics National Council of Teachers of Mathematics 2000

2. Understand numbers, ways of representing numbers, relationships among numbers, and number systems Grades Pre-K-2 Use multiple models to develop initial understandings of place value and the base-ten number system. Grades 3-5 Understand the place-value structure of the base- ten number system and be able to represent and compare whole numbers and decimals.

3. Understand numbers, ways of representing numbers, relationships among numbers, and number systems Grades 6-8 Develop an understanding of large numbers and recognize and appropriately use exponential, scientific, and calculator notation.

4. Place Value Models (for base ten) 1. Proportional-The material for 10 is ten times the size of 1; 100 is ten times the size of 10. Ex. base ten blocks, bean sticks, bundled sticks

5. Place Value Models (for base ten) Non-proportional-The material does not maintain any size relationships. Ex. money, abacus, color tiles or chips

6. Place Value Models Concrete Physical Models => Semi-concrete Organizational Models Tens | Ones Symbolic 2 | 5 Representation Models 25

7. Interviewer, "Circle in your drawing what the six means. Circle what this part (points to one) means." Abbie: Fifth Month of Second Grade

8. Interviewer, "I'm going to say a number and I want you to write it... thirteen...sixty-seven...one hundred twenty...three hundred twenty-four... four hundred eight...three thousand, five hundred twenty-three.“ Abbie: Fifth Month of Second Grade

9. Interviewer, "Circle in your drawing what the 4 means. Circle what this part (points to one) means." Clay: Fifth Month of Fourth Grade

10. Interviewer, "I'm going to say a number and I want you to write it...fifty-six...three hundred forty-eight...four hundred five... two thousand, seven hundred thirty one...thirty-five thousand, forty-eight.“ Clay: Fifth Month of Fourth Grade

11. Interviewer, "Circle in your drawing what the 4 means. Circle in your drawing what this part (points to one) means." Elsa: Fifth Month of Fourth Grade

12. Interviewer, "I'm going to say a number and I want you to write it... three hundred forty-eight...four hundred five... two thousand, seven hundred thirty-one...thirty-five thousand, forty-eight." Elsa: Fifth Month of Fourth Grade