1. Chapter 8 To accompany Helping Children Learn Math Cdn Ed, Reys et al. ©2010 John Wiley & Sons Canada Ltd.

2. Guiding Questions 1.What are some instructional models that can help develop children’s understanding of our number system? 2.Why is composing and decomposing numbers important in developing place value? 3.How can you help children bridge the gap between the concrete and symbolic representations of numbers? 4.How can calculators help children develop place- value concepts?

3. Our number system Our number system is based on the “Hindu- Arabic” number system and has the following four principles: – Place value – Base of ten – Use of zero – Additive property

4. Development of Place-Value Development of place value promotes number sense and rests on two key ideas: Explicit grouping and trading rules are defined and consistently followed. The position of the digit determines the number being represented.

5. Models to Help Young Children Figure 8-3 Ungrouped and pregrouped models are used to help young children develop a beginning understanding of place value.

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 (e.g., base ten blocks, bean sticks, bundled sticks)

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

8. Proportional and Nonproportional Models

9. Place Value Models The idea of place value can help children cross from concrete models of numbers to their symbolic representations.

10. Beginning Place Value (Early Childhood) A Place to Start: Common Errors

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 below.

12. Ten-Frame Model This provides a visual reminder of the physical differences between 25 and 52.

13. Extending Place Value Using the same digits to represent different numbers helps children appreciate the importance of representing the place values accurately.

14. Extending Place Value The front-end approach can be naturally extended and applied to larger numbers Which number is larger when the front-end digits are the same? What about when the back-end digits are the same?

15. Extending Place Value: Counting and Patterns Calculators are useful in counting and pattern recognition Hundreds charts can also help children notice similarities between numbers

16. Extending Place Value: Regrouping and Renaming Nonproportional model illustrating relation between regrouping and place value

17. Reading and Writing Numbers Reading and writing numbers are symbolic activities and should follow much modelling and talking about numbers.

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 100203. If a child made this mistake, the teacher could use the place value mat as a model to demonstrate the correct number formation.

19. The precision of the rounded numbers reflects the problem context. For example, a metre stick could serve as a number line. Consider this train of rods, with 7 decimetre rods and 4 centimetre rods. Is the train closer to 7 or 8 decimetres? [7 decimetres] Is it closer to 0 or 1 metre? [1 metre] Rounding

20. Rounding (cont.) A roller coaster model could be used to develop rounding skills.

21. For class discussion… The following slides will show you some samples of student thinking around place value. Please take some time to discuss each sample with others.

22. Interviewer: “Circle in your drawing what the six means. Circle what this part (points to one) means.” Abbie: Fifth month of grade two Student Interviews

23. Interviewer: “I'm going to say some numbers and I want you to write them... thirteen...sixty- seven...one hundred twenty...three hundred twenty- four... four hundred eight...three thousand, five hundred twenty-three.” Abbie: Fifth month of grade two Student Interviews

24. Interviewer: “Circle in your drawing what the 4 means. Circle what this part (points to one) means.” Clay: Fifth Month of Grade Four Student Interviews

25. Interviewer: “I'm going to say some numbers and I want you to write them...fifty-six...three hundred forty-eight...four hundred five... two thousand, seven hundred thirty one...thirty-five thousand, forty-eight.” Clay: Fifth Month of Grade Four Student Interviews

26. Interviewer: “Circle in your drawing what the 4 means. Circle in your drawing what this part (points to one) means.” Elsa: Fifth Month of Grade Four

27. Student Interviews Interviewer: “I'm going to say some numbers and I want you to write them... three hundred forty-eight... four hundred five... two thousand, seven hundred thirty-one...thirty-five thousand, forty-eight.” Elsa: Fifth Month of Grade Four

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