1. Strategies for Whole-Number Computation

2. Computational Strategies Direct modeling Direct modeling – –Use of base-ten models Invented strategies Invented strategies –Using written recordings – Traditional algorithms Traditional algorithms – –May influence use of invented strategies

3. More on Invented Strategies Compared to traditional algorithms Compared to traditional algorithms – –Left handed vs. right handed – Benefits Benefits –Helps with base ten concepts – –Less re-teaching is required – –Faster than traditional algorithms

4. Developing Invented Strategies Discuss student’s solutions Discuss student’s solutions Choosing numbers Choosing numbers Integrate place value development Integrate place value development Move away from direct modeling Move away from direct modeling

5. Invented Strategies Addition and subtraction Addition and subtraction –Single digits –Tens and hundreds Adding two digit numbers Adding two digit numbers –Add tens, ones, then combine –Add on tens, then ones –Move some to make tens –Use a nice # and compensate

6. Invented Strategies (cont.) Subtraction Subtraction – Add tens to get close, then ones Add tens to get close, then ones Add tens to overshoot, then come back Add tens to overshoot, then come back Add ones to make a ten, then tens and ones Add ones to make a ten, then tens and ones – Take tens from the tens, then subtract ones Take tens from the tens, then subtract ones Take away tens, then ones Take away tens, then ones Take extra tens, then add back Take extra tens, then add back Add to the whole Add to the whole

7. Traditional Algorithms Addition Addition –Begin with models –Written record Subtraction Subtraction –Begin with models –Zeros –Written record

8. Multiplication Invented strategies Invented strategies – –Single digit multiplier Complete-Number Strategies Complete-Number Strategies Compensation Strategies Compensation Strategies –Using multiples of 10 and 100

9. Multiplication (cont.) Traditional Algorithms Traditional Algorithms –Written record One digits One digits – – Two digits Two digits –

10. Division Invented strategies Invented strategies –Sharing and measurement problems –Missing factor strategies Traditional Algorithms Traditional Algorithms –Use of models –Written record

11. Early Fraction Concepts

12. Fractional Parts Sharing Sharing –Story problems Fraction language Fraction language –Number of parts –Equality of parts Models Models –Area –Length –Sets

13. Fraction Symbols Fractional-parts counting Fractional-parts counting Top and bottom numbers Top and bottom numbers –What is the bottom number? –What is the top number?

14. Fraction Symbols (cont.) Mixed numbers Mixed numbers – Parts and whole tasks Parts and whole tasks –Given the whole and the fraction, find the part –Given the part and the fraction, find the whole –Given the whole and the part, find the fraction

15. Fraction Number Sense Understanding of zero, one-half, and one Understanding of zero, one-half, and one Which is more? Which is more? – –Conceptual thought patterns for comparisons More of the same size parts More of the same size parts Same number of parts but parts of different sizes Same number of parts but parts of different sizes More and less than one-half or one whole More and less than one-half or one whole Distance from one-half or one whole Distance from one-half or one whole –Fractions tell us about _________________ between part and whole

16. Fraction Number Sense (cont.) Estimation Estimation – Equivalent-fraction concepts Equivalent-fraction concepts – –Use of models