Division
Grade 5 Math – Patterns, Properties & Strategies
Division Patterns over Increasing Place Values
- Dividing by 10, 100, 1,000, etc. means the digits shift LEFT as many times as zeros.
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Move the decimal LEFT by the number of zeros in the divisor.
E.g., \(4200 \div 100 = 42\)
Formula:
\(a \div 10^n\) → move decimal \(n\) places left (or remove \(n\) zeros)
E.g., \(85000 \div 1000 = 85\)
\(a \div 10^n\) → move decimal \(n\) places left (or remove \(n\) zeros)
E.g., \(85000 \div 1000 = 85\)
Divide Numbers Ending in Zeros
- Cancel out: When both numbers end in zeros, "cancel" the same number of zeros from each factor.
- E.g., \(2400 \div 80 = (24 \div 8 = 3)\) and remove two zeros from both: \(24\div8=3\).
Formula:
\(\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}\)
(as long as \(m \ge n\))
(as long as \(m \ge n\))
Estimate Quotients (with 2-digit divisors)
- Round both dividend and divisor to friendly numbers (multiples of 10).
- Divide using basic facts, ignore remainder for rough estimate.
Formula:
Round dividend: \(453 \rightarrow 450\)
Round divisor: \(54 \rightarrow 50\)
\(450 \div 50 = 9\)
Round divisor: \(54 \rightarrow 50\)
\(450 \div 50 = 9\)
Divide Multi-digit Numbers by 1-digit & 2-digit Numbers
- Standard algorithm: Divide, Multiply, Subtract, Bring Down (Repeat!)
- Long division: Line up digits, work left to right, remainder if needed.
- Partial quotients: Subtract chunks repeatedly, sum all partial quotients.
Example:
\(452 \div 7 = ?\) (step by step)
Find how many 7s in 45: 6
\(6\times7=42\), subtract, bring down 2.
\(1\times7=7\), remainder 4.
\(452 \div 7 = ?\) (step by step)
Find how many 7s in 45: 6
\(6\times7=42\), subtract, bring down 2.
\(1\times7=7\), remainder 4.
Interpret Remainders
- If remainder in word problems, decide if you round up, leave as a remainder, or use only quotient.
- Example: You have 23 candies for 6 children. Each gets 3, 5 left.
Does everyone get equal? What to do with leftover?
Divide by 2-digit Numbers: Models & Partial Quotients
- Models: Use base-ten blocks/area diagrams to split numbers into groups for visual division.
- Partial quotients: Subtract multiples of the divisor (friendly chunks: 20×, 10×, 5×...) and keep a running total.
Example: \(276 \div 24\)
Subtract \(24 \times 10 = 240\) from 276, get 36. \(36 \div 24 = 1\), remainder 12. Total quotient: 11 remainder 12.
Subtract \(24 \times 10 = 240\) from 276, get 36. \(36 \div 24 = 1\), remainder 12. Total quotient: 11 remainder 12.
Adjust & Complete Quotients
- If your estimate is too high or too low, adjust by recalculating with closer numbers or checking the remainder.
- Fill in missing values in division sentences: \(a \div b = c\), so \(c \times b = a\).
Relate Multiplication and Division
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Inverse Operation: Division "undoes" multiplication.
If \(a \times b = c\), then \(c \div b = a\), \(c \div a = b\). - Check your division by multiplying quotient × divisor, add remainder.
Quick Reference Summary
- Dividing by 10, 100, etc.: Move decimal left, remove zeros.
- Estimate quotients: Round, use friendly numbers.
- Partial quotients: Subtract in chunks, add up.
- Division Models: Use diagrams or blocks for visualization.
- Multiplication is the inverse of division!
- Always check your work by multiplying quotient and divisor (+ remainder if any).
Study Tip: Practice estimation and step-by-step division for accuracy and speed.