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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, :15 – 11:30 a.m.

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3, ,246 Simplify the following expression:

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

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2 nd Grader Simplifying 3, ,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, )He then associated (3, ) + 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 x Is this how you simplified it? 4 48 x

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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

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

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

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Example of 2 nd Grade String Big Idea: Keeping One Number Whole and Taking Leaps of nd 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 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, 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, Harel, G. & Rabin, J. (2010). Teaching Practices Associated With the Authoritative Proof Scheme – Journal for Research in Mathematics Education, Vol. 41, No. 1, Web Sites DreamBox Learning - To order a rekenrek:

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