1. CHAPTER 8 Extending Number Sense: Place Value
Tina Rye Sloan To accompany Helping Children Learn Math9e, Reys et al.
©2009 John Wiley & Sons

2. Focus Questions
What are some instructional models that can help develop children’s understanding of our number system?
Why is composing and decomposing numbers important in developing place value?
How can calculators help children develop place value concepts?
How do number names differ from country to country?
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009

3. (NCTM, 2000). Principles and Standards for
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.
Develop a sense of whole numbers and represent and use them in flexible ways, including relating, composing, and decomposing numbers.
Connect number words and numerals to the quantities they represent, using various physical models and representations.
(NCTM, 2000). Principles and Standards for
School Mathematics.
Master 8-1: Number and Operations Excerpts (place value)
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009

4. (NCTM, 2000). Principles and Standards for
Understand numbers, ways of representing numbers, relationships among numbers, and number systems
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.
Recognize equivalent representations for the same number and generate them by decomposing and composing numbers.
(NCTM, 2000). Principles and Standards for
School Mathematics.
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009

5. Ungrouped and Pregrouped Materials
Figure 8-4 Ungrouped and pregrouped models for developing place value
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009

6. Place Value Models (for base ten)
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
Master 8-2: Proportional and Non-proportional Place Value Models
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009

7. Place Value Models (for base ten)
Non-proportional-The material does not maintain any size relationships. Ex. money, abacus, color tiles or chips
Master 8-2: Proportional and Non-proportional Place Value Models
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009

8. Proportional and Nonproportional Models
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009

9. Place Value Models
Concrete Physical Models => Semi-concrete Organizational Models Tens | Ones Symbolic 2 | 5 Representational Models 25
Master 8-3: Moving from Concrete to Symbolic Models
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009

10. Beginning Place Value (Early Childhood)
A Place to Start: Common Errors
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009

11. Early Childhood Place Value
The notion of representing a quantity with the least number of pieces for a particular model is critical in place value. Establishing its importance at an early stage can eliminate some later errors such as the one indicated.
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009

12. Ten-Frame Model
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009

13. Extending Place Value
Using the same digits to represent different numbers helps children appreciate the importance of representing the place values accurately.
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009

14. Extending Place Value
The front-end approach can be naturally extended and applied to larger numbers:
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009

15. Extending Place Value: Counting and Patterns
Calculators are useful in counting and pattern recognition.
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009

16. Extending Place Value: Regrouping and Renaming
Figure 8-14 Nonproportional model illustrating relation between regrouping and place value
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009

17. Reading and Writing Numbers
Reading and writing numbers are symbolic activities and should follow much modeling and talking about numbers.
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009

18. Reading and Writing Numbers
Although it seems logical to write number words as they sound, this procedure can lead to difficulty. If this were done, sixty one would be incorrectly written as 601 and one hundred twenty three as If a child made this mistake, the teacher could use the place value mat as a model to demonstrate:
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009

19. Rounding
The precision of the rounded numbers reflects the problem context. For example, a meter stick could serve as a number line. Consider this train of rods, with 7 decimeter rods and 4 centimeter rods.
Is the train closer to 7 or 8 decimeters? [7 decimeters]Is it closer to 0 or 1 meter? [1 meter]
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009

20. Rounding (cont’d)
A roller coaster model could be used to develop rounding skills.
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009

21. Cultural Connections
Can you write the number 751 using the Chinese numeration system?
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009

22. Student Interviews
Interviewer, "Circle in your drawing what the six means. Circle what this part (points to one) means.“  Abbie: Fifth Month of Second Grade
Master 8-4: Place Value Sample
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009

23. Student Interviews (cont’d)
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.”
Master 8-5: Place Value Sample
Abbie: Fifth Month of Second Grade
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009

24. Student Interviews (cont’d)
  Interviewer, "Circle in your drawing what the 4 means. Circle what this part (points to one) means.“ Clay: Fifth Month of Fourth Grade
Master 8-6: Place Value Sample
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009

25. Student Interviews (cont’d)
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
Master 8-7: Place Value Sample
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009

26. Student Interviews (cont’d)
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
Master 8-8: Place Value Sample
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009

27. Student Interviews (cont’d)
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
Master 8-9: Place Value Sample
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009