Number and Operations (Number Sense) 1. A diagram of nested subsets of the Real numbers. Counting Numbers Whole Numbers Integers Rationals Reals 2. Definitions of “Number Sense” and “Numeracy”. What is number Sense? “Much of school mathematics depends on numbers, which are used to count, compute, measure, and estimate. Mathematical activity belongs here that centers primarily on the development of number concepts, on computation with numbers (addition, subtraction, multiplication, division, finding powers and roots, etc.), on numerations (systems for writing numbers, including base ten, fractions, negative numbers, rational numbers, percents, scientific notation, etc.), and on estimation. At higher levels this strand includes the study of prime and composite numbers, of irrational numbers and their approximation by rationals, of real numbers, and of complex numbers.” Mathematics Framework for California Public Schools, K-12, 1999 “Like common sense, number sense produces good and useful results with the least amount of effort. It is not mindlessly mechanical, but flexible and synthetic in attitude. It evolves from concrete experience and takes shape in oral, written, and symbolic expressions.” National Research Council, 1989 A person who has Number Sense has some combination of A. B. C. D. an ability to represent number in a variety of ways, an awareness of the relationships between numbers, a knowledge of the effects of operations, an ability to interpret and use numbers in real world counting and measurement. NCTM Addenda Series, 1994 3. Connections to State Standards. Number sets through the grade levels CA Standards Operations through the grade levels K: Counting Numbers through 30 K: Addition and subtraction for 2 numbers each less than 10 (Concrete) 1. Counting Numbers though 100 1. Addition facts to 20 (and corresponding differences) Addition and subtraction problems with a one-digit and a two-digit number. 2. Counting numbers through 1,000 Fractions – halves, thirds, sixths, eighths, twelfths. Decimals as related to money. 2. Sums and differences of 3-digit numbers Mental Arithmetic for sums and differences of 2digit numbers Model and solve simple multiplication and division. 3. Counting numbers through 10,000 Development of Fractions done in 2nd grade Decimals to the hundredths. 3. Multiplication tables through 10 x 10. Sums and differences of 4 digit numbers. Multiply and divide a multi-digit number by 1-digit number. Add, Subtract, Multiply, and Divide with money. Add and subtract simple fractions. 4. Counting numbers through 1,000,000 Decimals and fractions through hundredths Concepts of negative integers (Number line, counting, temperature, owing) 4. Estimate and Compute the sums or difference of whole numbers and positive decimals (2 places). Multiply or Divide a multi-digit number by a 2-digit number. Add and subtract negative numbers. 5. Decimals number from thousandths to millions Write fractions as percents and decimals Develop positive and negative integers 5. Add, subtract, multiply, and divide decimals and negative numbers. Proficient with multi-digit multiplication and division. Add, subtract, multiply, and divide simple fractions 6. Positive and negative fractions and decimals Mixed numbers Pi All of the above Differentiate between rationals and irrationals 6. Calculate (Add, subtract, multiply, and divide with all types of rational numbers. 7. All of the above Use exponents and roots when working with rational numbers. 7.