integers as charges michael t. battista “a complete model for operations on integers” arithmetic...

Integers as Charges

Michael T. Battista“A Complete Model for Operations on

Integers”Arithmetic Teacher, May 1983

• Every integer can be represented by a jar of charges in a variety of ways.

Yellow represents positive charges.

Red represents negative charges.

Creating a Zero Charge (Zero Pairs)A positive charge and a negative charge have a net

value of zero charge.

This concept is foundational to understanding the addition and subtraction of integers.

How do we make a zero charge?

Make five different representations of zero

The charge in the jar represents a given integer.

Integers are represented by a collection of charges.

Integers have multiple representations.

What are some ways that we could represent: - 3 + 5

Addition

• We can ground them in what they already know about quantity.

• Addition is an extension of the cardinal number model of whole number addition. It is a joining action.

• 3 + 2 • - 3 + - 2 • 3 + - 2• - 3 + 2• Commutative Property

Subtraction

• Just as addition is a joining action, subtraction is a “take away” action.

• Represent the first integer (minuend) in a jar.• Remove from this jar the second integer

(subtrahend) • The new charge on the first jar is the

difference in the two integers.• 3 – 2 - 3 – (-2) 3 – (- 2) -

3 - 2

Subtraction

Work with your table partner to solve the subtraction problems. Remember the language of the form of the value!• 4 – 3

• - 4 – (-3)

• 4 – (- 3) • - 4 - 3

Multiplication

• The multiplication structure is based on our defined representations for addition and subtraction operations.

• If the first factor in a multiplication problem is positive, we interpret the multiplication as repeated addition of the second factor.

• If the first factor in a multiplication problem is negative, we interpret the multiplication as repeated subtraction of the second factor.

Examples of Multiplication of Integers

Begin with a zero charged jar.

(+ 3 ) ● (+2)

(+ 3 ) ● (-2)

(- 3 ) ● (+2)

(- 3 ) ● (-2)

Connections!Multiplication is repeated addition.Division is repeated subtraction.Division and Multiplication are opposite

operations.

More Connections

Each division question can be rephrased into a multiplication problem by asking:

What number must the divisor be multiplied by in order to get the dividend?

The sign of the quotient automatically is tied to our multiplication model.

---and again

If repeated addition is involved, the first factor (the quotient) is positive.

If repeated subtraction is involved, the first factor (the quotient) is negative.

6 ÷ 2 can be rewritten as ( ? ● 2 = 6)

(-6) ÷ (-2) can be rewritten as (? ● (-2) = -6). 6 ÷ (-2) can be rewritten as ( ? ● (-2) = 6)

(-6) ÷ 2 can be rewritten as (? ● 2 = (-6).