2-D Algebraic Thinking

Area model

Area models, linear models, and set models are different ways to physically represent fraction problems. In an area model, fractions are represented as part of a region. Think of a 10 by 10 grid with some of the squares shaded. The shaded squares represent a fractional part of the whole.

Linear model

In a linear model, the lengths of objects are compared to one another.

Set model

In a set model, a number of individual objects make up one whole.

[Properties of operations]

Commutative Property of Addition

Rule:

a + b = b + a

Description:

Changing the order of two numbers being added does not change the sum.

[Properties of operations]

Commutative Property of Multiplication

Rule:

a ⋅ b = b ⋅ a

Description:

Changing the order of two numbers being multiplied does not change the product.

[Properties of operations]

Associative Property of Addition

Rule:

(a + b) + c = a + (b + c)

Description:

Changing the grouping of the addends does not change the sum.

[Properties of operations]

Associative Property of Multiplication

Rule:

a ⋅(b ⋅ c) = (a ⋅ b) ⋅ c

Description:

Changing the grouping of the factors does not change the product.

[Properties of operations]

Additive Identity Property of 0

Rule:

a + 0 = 0 + a = a

Description:

Adding 0 to a number does not change the value of that number.

[Properties of operations]

Multiplicative Identity Property of 1

Rule:

a ⋅ 1 = 1 ⋅ a = a

Description:

Multiplying a number by 1 does not change the value of that number.

[Properties of operations]

Inverse Property of Addition

Rule:

For every a, there exists a number −a such that a + (−a) = (−a) + a = 0

Description:

Adding a number and its opposite results in a sum equal to 0.

[Properties of operations]

Inverse Property of Multiplication

Rule:

For every a, there exists a number 1 a such that a ⋅ 1 a = 1 a ⋅ a = a a = 1

Description:

Multiplying a number and its multiplicative inverse results in a product equal to 1.

[Properties of operations]

Distributive Property of Multiplication over Addition

Rule:

a ⋅ (b + c) = a ⋅ b + a ⋅ c

Description:

Multiplying a sum is the same as multiplying each addend by that number, then adding their products.

[Properties of operations]

Distributive Property of Multiplication over Subtraction

Rule:

a ⋅ (b − c) = a ⋅ b − a ⋅ c

Description:

Multiplying a difference is the same as multiplying the minuend and subtrahend by that number, then subtracting their products.

[Properties of operations]

Transitivity Property

Rule:

x = y and y = z, then x = z

Description:

For all real numbers x, y, and z, if x = y and y = z, then x = z. If x = y, then x may be replaced by y in any equation or expression.

PEDMAS

The acronym, PEMDAS, or the mnemonic Please Excuse My Dear Aunt Sally, is often used to remember the order of operations.

Please, or parentheses, includes all grouping symbols, which may include brackets [ ], braces { }, and absolute value bars | |. If there is math that can be computed inside grouping symbols, do that FIRST, then the grouping symbols may be removed.

Excuse, or exponents, means anything raised to a power should be simplified after there are no more parentheses.

My Dear, or multiplication or division, are essentially the same "type" of operation and are therefore done in order from left to right, just as you would read a book. All multiplication and division should be completed BEFORE any addition or subtraction that is not inside parentheses.

Aunt Sally, or addition and subtraction, are also essentially the same "type" of operation and are also done in order from left to right. These operations should always come last, unless they were inside parentheses.