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

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

(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

(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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 100203. 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

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

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

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

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

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

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

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

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

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