Learning Mathematics for Teaching Presented by Donna McLeish, IMI Staff To IMI Cohort III Kindergarten and Grade One Teachers September 29-30, 2004

Numeration and Computation Standards I.Order of Operations II.Alternate Algorithms

Order of Operations Please Excuse Me, Dear Aunt Sally P - Parentheses E - Exponents M - Multiplication D - Division A - Addition S - Subtraction

Please Excuse Me, Dear Aunt Sally P >> remove all parentheses by performing operations within parentheses E >> raise any quantities with an exponent to the given power M and D >> perform all multiplication and division from left to right A and S >> perform all addition and subtraction from left to right

3 + (4 – 1) 2 * 5 First >> (4 – 1) is * 5 Second >> 3 2 is * 5 Third >> 9 * 5 is Last >> is 48 Answer is 48

Order of Operations Show all steps in the solution of the problem * (7-1) 2 ÷ 4 – 9 = 2.6 ÷ 3 * (7-1) = 3.(5 + 2) ÷ 7 = 4.(9 + 7) – 4 * 3 ÷ 6 =

Order of Operations - Solutions * (7-1) 2 ÷ 4 – 9 = * (6) 2 ÷ 4 – 9 = * 36 ÷ 4 – 9 = ÷ 4 – 9 = – 9 = 31 – 9 = ÷ 3 * (7-1) = 6 ÷ 3 * (6) = 6 ÷ 3 * = 2 * = = = 77 3.(5 + 2) ÷ 7 = (7) ÷ 7 = ÷ 7 = 49 – 1 = 48 4.(9 + 7) – 4 * 3 ÷ 6 = 16 – 4 * 3 ÷ 6 = 16 – 12 ÷ 6 = 16 – 2 = 14

Fact Power Knowing the basic number facts is ESSENTIAL for success in math In EM, good fact habits are called Fact Power By end of grade two, most children using EM will have mastered the basic addition and subtraction facts By end of grade four, most children using EM will have mastered the basic facts of multiplication and division

Algorithms An algorithm is a well-defined, step-by- step procedure guaranteed to achieve a certain objective, often with several steps that “loop” as many times as necessary. A good algorithm is efficient, unambiguous, and reliable.

Research shows that teaching the standard algorithms in the standard ways does fail with a large number of children. In one study, only 60 percent of US ten-year-olds achieved mastery of subtraction using the standard regrouping (borrowing) algorithm.

Focus Algorithms In EM, children are encouraged to invent their own procedures during the early phases of learning an operation. Later, when children fully understand the concept of the operation, standard algorithms are introduced. Children are urged to experiment with various algorithms in order to become proficient at using at least one alternative. Everyday Mathematics introduces a focus algorithm for each operation. All children are expected to master the focus algorithms, though they are not required to use them if they have alternatives they prefer. Focus algorithms provide a common ground for further work and offer reliable alternatives for children who have not developed effective procedures of their own.

Addition Algorithms – Partial Sums – Column Addition – Opposite Change Subtraction Algorithms – Trade-First – Counting Up – Left to Right – Same Change

Multiplication Algorithms Partial Products Modified Repeated Addition Lattice Division Algorithms Partial Quotients Column

UCSMP Website Everyday Mathematics Center