Learning Mathematics Learning Mathematics for Teachingfor Teaching

Learning Mathematics Learning Mathematics for Teachingfor Teaching

Presented byPresented by Donna McLeish, Donna McLeish, IMI StaffIMI Staff

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IMI Cohort IIIIMI Cohort IIIKindergarten and Grade One TeachersKindergarten and Grade One Teachers

September 29-30, 2004September 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 33 + 32

* 5Second >> 32 is 9

3 + 9 * 5Third >> 9 * 5 is 45

3 + 45Last >> 3 + 45 is 48

Answer is 48

Order of OperationsShow all steps in the solution of the

problem.

1. 4 + 3 * (7-1)2 ÷ 4 – 9 =

2. 6 ÷ 3 * (7-1)2- 4 + 9 =

3. (5 + 2)2 - 7 ÷ 7 =

4. (9 + 7) – 4 * 3 ÷ 6 =

Order of Operations - Solutions• 1. 4 + 3 * (7-1)

2 ÷ 4 – 9 =

• 4 + 3 * (6) 2 ÷ 4 – 9 =

• 4 + 3 * 36 ÷ 4 – 9 =• 4 + 108 ÷ 4 – 9 =• 4 + 27 – 9 =• 31 – 9 = 22

• 2. 6 ÷ 3 * (7-1) 2 - 4 + 9 =

• 6 ÷ 3 * (6) 2 - 4 + 9 =

• 6 ÷ 3 * 36 - 4 + 9 =• 2 * 36 - 4 + 9 =• 72 - 4 + 9 =• 68 + 9 = 77

• 3. (5 + 2) 2 - 7 ÷ 7 =

• (7) 2 - 7 ÷ 7 =

• 49 - 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 AlgorithmsIn 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

http://everydaymath.uchicago.edu/