In-Depth Notes on Division of Whole Numbers
Division of Whole Numbers
Inverse Operation
- Division is the inverse operation of multiplication.
- Division problems can be rewritten as multiplication problems and vice versa.
- These equations are part of "fact families" since they express the same relationship:
- π₯ β
π¦ = π§
- π¦ β
π₯ = π§
- π§ Γ· π₯ = π¦
- π§ Γ· π¦ = π₯
Importance of Fact Families
- Understanding different forms of these equations aids in problem-solving.
- Example: Rewriting π₯ Γ· π¦ = π§ in different formats:
- Fraction format: π§ = ( \frac{x}{y} )
- Long division format: ( x \div y = z )
- Example word problem: If there are 6 cookies and 3 friends, how many cookies does each friend get?
- Solution: 6 Γ· 3 = 2 cookies each, no leftover.
Models of Division
- Set (Partition) Model: Answers "how many per group"
- Repeated Subtraction (Measurement): Answers "how many groups"
- Missing Factor: For any whole numbers π and π (π β 0), ( a \div b = c ) if and only if ( b \cdot c = a )
Properties of Division
- Division does not share all properties with multiplication:
- Commutative Property: Does not hold (e.g., π Γ· π β π Γ· π)
- Associative Property: Does not hold (e.g., (π Γ· π) Γ· π β π Γ· (π Γ· π))
- Identity Property: Does not hold (no identity element for division in whole numbers)
- Closure Property: Not satisfied for division of whole numbers (e.g., 3 Γ· 2 = 1.5, not a whole number)
Division Algorithms
Partial Quotients
- Example problem: 739 Γ· 6
- Each step involves estimating how many groups of the divisor fit into parts of the dividend, maintaining accuracy through subtraction.
Standard Algorithm
- Base ten blocks or money can be used to illustrate the standard algorithm for division.
- Example process for 739 Γ· 6:
- Determine how many times 6 fits into 739 in organized steps, leading to quotient and remainder.
Division Algorithm
- The Division Algorithm states that for any whole numbers π and π (with π β 0), there exist unique whole numbers π (quotient) and π (remainder) such that:
[ a = bq + r \text{ with } 0 \leq r < b ] - Example checking: If 21 Γ· 4 yields 5 with a remainder of 2, check if 21 = 4*5 + 2 holds (true).
- Important to have the condition ( 0 \leq r < b ) for the validity of the remainder.
Order of Operations
- To correctly evaluate expressions, follow PEMDAS:
- P: Parentheses
- E: Exponents
- MD: Multiplication and Division (left to right)
- AS: Addition and Subtraction (left to right)
- Example Evaluation: 20 β 8 Γ· 2(2 + 2) + 5 must adhere to the order to yield a correct answer.
Division with Zero
- Discuss the results of dividing by zero under various scenarios:
- a) 0 Γ· 4 = 0 (any number of groups of nothing is zero)
- b) 5 Γ· 0 is undefined (no groups can be formed)
- c) 0 Γ· 0 is indeterminate (since any number multiplied by zero yields zero)
- Understanding the implications of dividing by zero reinforces the rules of arithmetic.
Practical Application
- Try modeling division problems using chosen methods. For base six, visualize or use tangible items like blocks to solve. Example: 143six Γ· 4six in base six using models or manipulation methods.