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