Rational Operations Anchor Chart
RATIONAL OPERATIONS ANCHOR CHART
Part 1: Fraction Mastery & Decimal Multiplication
Name:
Period:
Rule 1
Multiplying Fractions: Straight Across
No common denominator needed!
1
Multiply Numerators: Top times top (\(a \times c\)).
2
Multiply Denominators: Bottom times bottom (\(b \times d\)).
3
Simplify Completely: Divide out common factors.
Worked Example: Pre-Cancel Tip
\[ \frac{3}{4} \times \frac{2}{5} = \frac{3 \times 2}{4 \times 5} = \frac{6}{20} = \frac{3}{10} \]
*Mixed Numbers: Convert to improper fractions (\(1\frac{1}{2} = \frac{3}{2}\)) before multiplying!
Rule 2
Dividing Fractions: Keep • Change • Flip
Multiply by the Reciprocal
KEEP
Leave 1st fraction alone
\( \frac{5}{8} \)
CHANGE
Division to multiplication
\( \div \;\longrightarrow\; \times \)
FLIP
Invert divisor to reciprocal
\( \frac{3}{4} \;\longrightarrow\; \frac{4}{3} \)
Full Execution: \[ \frac{5}{8} \div \frac{3}{4} = \frac{5}{8} \times \frac{4}{3} = \frac{20}{24} = \frac{5}{6} \] Result: \( \frac{5}{6} \)
Rule 3
Multiplying Decimals: Count & Shift
⚠️ DO NOT line up decimal points!
1. Line Up Rightmost Digits:
Ignore decimals at first; stack flush to the right margin.
2. Multiply as Whole Numbers:
Calculate partial products just like standard integers.
3. Count & Move Decimal Left:
Count total decimal places in factors; hop that many spots left.
2 places \(\to\) 3.42
1 place \(\to\) \(\times\) 2.6
2 0 5 2
+ 6 8 4 0
3 places \(\leftarrow\) 8.892
Estimation Check: \(3.5 \times 2.5 \approx 8.75\). Our answer (\(8.892\)) is reasonable!
Common Mistake Alert: Only flip the SECOND fraction in division. Never flip both!
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RATIONAL OPERATIONS ANCHOR CHART
Part 2: Decimal Division & Rational Classifications
Desk Reference Card
Grade 6-8 Math
Method A
Power of 10 Shortcut: Equivalent Fractions
Make the Divisor a Whole Number!
The Golden Principle:
Every division problem is a fraction: dividend ÷ divisor = \(\frac{\text{dividend}}{\text{divisor}}\). Multiply top and bottom by 10, 100, or 1000 to eliminate decimals without changing the answer.
💡 Look at the divisor (bottom) to decide how many zeros you need!
Example: \( 4.8 \div 0.12 \)
\( \frac{4.8}{0.12} \times \frac{100}{100} = \frac{480}{12} \)
\( 480 \div 12 = 40 \)
Fill empty spots with zero (\(4.8 \to 480\)).
Method B
Standard Long Division: Float Decimal Up
Never write a remainder "R"!
1
Shift Divisor Decimal:
Move decimal to the right until divisor is whole. Shift dividend by exact same amount.
2
Elevate Decimal Straight Up:
Copy the decimal point directly into the quotient above the bar.
3
Annex Zeros to Dividend:
Add trailing zeros until remainder is 0 or pattern repeats.
Solve: \( 14.7 \div 6 \)
2.4 5
6
1 4.7 0
- 1 2
2 7
- 2 4
3 0 (annex 0)
- 3 0
0 (Terminates)
Classify
Two Rational Outcomes: Terminating vs. Repeating
Every rational number \(a/b\) falls into one category!
TERMINATING
Remainder = 0
The division ends cleanly. The decimal does not continue indefinitely.
\( \frac{1}{2} = 0.5 \) Tenths
\( \frac{3}{4} = 0.75 \) Hundredths
\( \frac{5}{8} = 0.625 \) Thousandths
🔍 Denominator Rule: When simplified, prime factors of bottom number are only 2 and/or 5.
REPEATING
Infinite Cycle
A pattern loops forever. Written with a vinculum bar over repeating digits.
\( \frac{1}{3} = 0.\overline{3} \) 1 repeating
\( \frac{5}{6} = 0.8\overline{3} \) Only 3 repeats
\( \frac{4}{11} = 0.\overline{36} \) 2-digit cycle
⚠️ Bar Rule: Only put the line directly over the digits that repeat (e.g., \(0.8\overline{3}\), not \(\overline{0.83}\)).
Self-Check: 1. Mixed numbers improper? • 2. Inverted 2nd fraction? • 3. Counted decimals? • 4. Divisor whole?
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