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Integers as Charges

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.

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Integers as Charges

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  1. Integers as Charges Michael T. Battista “A Complete Model for Operations on Integers” Arithmetic Teacher, May 1983

  2. Every integer can be represented by a jar of charges in a variety of ways. Yellow represents positive charges. Red represents negative charges.

  3. 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

  4. 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

  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

  6. 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

  7. 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

  8. 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.

  9. Examples of Multiplication of Integers Begin with a zero charged jar. (+ 3 ) ● (+2) (+ 3 ) ● (-2) (- 3 ) ● (+2) (- 3 ) ● (-2)

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

  11. 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.

  12. ---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).

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