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:

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 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)

Subtraction Work with your table partner to solve the subtraction problems. Remember the language of the form of the value! 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)

Division of Integers Case 1: Dividend and Divisor have the same sign— 6 ÷ 2 or -6 ÷ -2 How many times must the divisor be added to a zero charged jar to equal the dividend. So--- 6 ÷ 2 = 3 times and-6 ÷ -2 = 3 times

Case 2: Dividend and Divisor have opposite signs (-6) ÷ 2 or 6 ÷ (-2)

For (-6) ÷ 2, if we repeatedly add 2 to the jar, we will never reach (-6). We can subtract 2 from a zero charged jar three times to get (-6). We must change the form of our zero charged jar by creating 6 zero charges. Since we subtracted, our quotient is (-)(3 times) For 6 ÷ (-2), we can subtract (-2) from a zero charged jar, three times to get to (-6). Since we subtracted, our quotient is (-) (3 times).

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).