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Early Number Sense The “Phonics” of Mathematics Presenters: Lisa Zapalac, Head of Lower School Kevin Moore, 4 th Grade Math Brooke Carmichael, Kindergarten November 19, 2010 10:15 – 11:30 a.m. www.austintrinity.org

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3,996 + 4,246 Simplify the following expression:

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1 1 1 1 3,996 +4,246 8,242 Is this how you simplified it?

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2 nd Grader Simplifying 3,996 + 4,246 Example of 2nd Grader Using Compensation

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How did the 2 nd grader simplify the expression? He used an addition strategy called compensation, but there are many underlying concepts that are embedded in compensation 1)He noticed that 3,996 is 4 less than 4,000 2)He recognized 4,246 as being equivalent to 4,242+ 4 3)He then associated (3,996 + 4) + 2,242

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Recognizes values in their various forms Number Sense Demonstrates proficiency with estimation and evaluation of quantities Recognizes unreasonable conclusions Possesses a repertoire of mental computation strategies Evidence of Number Sense

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Let’s try another expression 50 x 48 4th Grade Video

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4 48 x50 2400 Is this how you simplified it? 4 48 x 50 00 +2400 2400

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4 th Graders Simplifying 48 (50) Example 1 Example 2 Example 3

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Solve 76 x 89 One more… 4th Grader

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Number Sense…. How do we build it? There are many effective strategies for building number sense. At Trinity, “strings” are one power practice used.

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Using “Strings” to Develop Number Sense Strings are a set of arithmetic problems in which the children are developing very specific strategies. Strings are generally done mentally. Each string begins with a known expression and moves towards the unknown, scaffolding the development of key strategies. The following slides contain examples of strings at various grade levels.

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1 st Grade String 5 + 5 5 + 6 6 + 6 6 + 7 7 + 7 7 + 8 8 + 8 9 + 7 6 + 8

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Building Number Sense through Facts Doubles plus or minus 1 – Ex. 6 + 7 = 6 + 6 + 1 (or 7 + 7 – 1) = 13 Doubles plus or minus 2 – Ex. 5 + 7 = 5 + 5 + 2 (or 7 + 7 - 2) Working with the structure of five – Ex. 6 + 7 = 5 + 1 + 5 + 2 = 10 + 3 = 13 Making tens – Ex. 8 + 4 = 8 + 2 + 2 Using tens to solve nines – Ex. 9 + 7 = 10 + 7 - 1 Using compensation – Ex. 6 + 8 = 7 + 7 (adding one to one addend, while subtracting one from the other addend) Possesses a repertoire of mental computation strategies 5 + 5 5 + 6 6 + 6 6 + 7 7 + 7 7 + 8 8 + 8 9 + 7 6 + 8

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Using Tools and Models to Develop Number Sense Recognizes values in their various forms The rekenrek, or arithmetic rack, is a tool consisting of two rows of ten beads with two sets of five in each. The rekenrek was developed by Adri Treffers, a researcher at the Freudenthal Institute in the Netherlands, and it provides a powerful model for exploring the composing and decomposing of number (Treffers 1991)

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Kindergarten String Example 5 on the top, 5 on the bottom 7 on the top, 3 on the bottom 4 on the top, 6 on the bottom 6 on the top, 4 on the bottom 8 on the top, 2 on the bottom Possesses a repertoire of mental computation strategies Recognizes values in their various forms Kindergarten String

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2 3840 80 Moving Beyond Facts Modeling 38 + 42 The open number line is a tool used to model students’ thinking. In this problem, 38 + 42, a student might solve it by moving to a landmark number first. Or, they might first make jumps of ten. 38 78 40 80 2

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Example of 2 nd Grade String Big Idea: Keeping One Number Whole and Taking Leaps of 10 75 + 20 75 + 25 75 + 24 55 + 30 55 + 39 69 + 21 69 + 29 2nd Grade String

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Building Number Sense with Multiplication Constructing facts through relationships and models 4(4) = 162[(4)2] or 2(8)

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Multiplication (3 rd & 4 th Grade Strategies) Doubling ▪6 x 6 = 2 x 3 x 6 Halving and doubling ▪4 x 3 = 2 x 6 Using the distributive property ▪7 x 8 = (5 x 8) + (2 x 8), or ▪7 x 8 = (8 x 8) – (1 x 8) Using the commutative property ▪5 x 8 = 8 x 5 Possesses a repertoire of mental computation strategies

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Example of 4 th Grade String 4 x 8 14 x 8 6 x 9 26 x 9 12 x 13 15 x 24 4th Grade Multiplication String

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4 th Grade String Revisited – Connecting to Algebra a (8) (a + b) 8 (3a)2 (2a + c) (5) (a + 3) (a + 2)

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Number sense is the bridge between arithmetic and algebra ArithmeticAlgebra Number Sense

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Resources Books Ma, L. (1999). Knowing and Teaching Elementary Mathematics. Mahwah, NJ: Lawrence Erlbaum Associates, Inc. Devlin, K. (2000). The Math Gene. Great Britain: Weidenfeld & Nicolson Stigler & Hiebert (1999). The Teaching Gap. New York, NY: The Free Press Fosnot, C., & Dolk, M. (2001). Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction. Portsmouth, NH: Heinemann Fosnot, C., & Dolk, M. (2001). Young Mathematicians at Work: Constructing Multiplication and Division. Portsmouth, NH: Heinemann Carpenter, T., Franke, M., & Levi, L. (2003). Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School. Portsmouth, NH: Heinemann Fosnot, C. & Uittenbogaard, W. (2007). Minilessons for Early Addition and Subtraction. Portsmouth, NH: Heinemann Fosnot, C. & Uittenbogaard, W. (2007). Minilessons for Extending Addition and Subtraction. Portsmouth, NH: Heinemann Fosnot, C. & Uittenbogaard, W. (2007). Minilessons for Early Multiplication and Division. Portsmouth, NH: Heinemann Fosnot, C. & Uittenbogaard, W. (2007). Minilessons for Extending Multiplication and Division. Portsmouth, NH: Heinemann Articles Faulkner, V. (2009). The Components of Number Sense – An Instructional Model for Teachers. – Teaching Exceptional Children, Vol. 41, No. 5, 24-30 Gersten, R. & Chard, D. (2010). Validating a Number Sense Screening Tool for Use in Kindergarten and First Grade: Prediction of Mathematics Proficiency in Third Grade – School Psychology Review, Vol. 39, No. 2, 181-195 Harel, G. & Rabin, J. (2010). Teaching Practices Associated With the Authoritative Proof Scheme – Journal for Research in Mathematics Education, Vol. 41, No. 1, 14-19 Web Sites DreamBox Learning - www.dreambox.comwww.dreambox.com To order a rekenrek: www.eNasco.comwww.eNasco.com

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