Number Decoders Worksheet
Number Decoders
Rational Numbers & Decimals Guide
Name:
Date:
Step 1
Word Vault: 3 Key Ideas
Rational Number
Any number you can write as a fraction: \(\frac{a}{b}\) (bottom cannot be 0).
Examples: \(\frac{3}{4}\), \(5\), \(-2\), \(0.5\)
Terminating Decimal
A decimal that stops! The division ends with a remainder of 0.
\(\frac{1}{2} = 0.5\) • \(\frac{1}{4} = 0.25\)
Repeating Decimal
A digit pattern that never ends! Shows a bar on top (\(0.\bar{3}\)).
\(\frac{1}{3} = 0.333... = 0.\bar{3}\)
Mission 1
Rational Detective: Circle Yes or No
Hint: Can it be written as a fraction \(\frac{a}{b}\)?
A. \(\frac{4}{7}\)
Written directly as a fraction.
Rational?
YES
NO
B. \(6\)
Whole number: can be \(\frac{6}{1}\).
Rational?
YES
NO
C. \(0.75\)
Decimal stops: can be \(\frac{75}{100}\).
Rational?
YES
NO
D. \(-3\)
Negative integer: can be \(\frac{-3}{1}\).
Rational?
YES
NO
Step 2
Visual Step Guide: Fraction to Decimal
TOP number goes IN the house!
•
BOTTOM number stays OUTSIDE!
1
Set Up House
Write denominator outside and numerator inside.
\(4 \overline{) 3}\)
2
Add Point & Zeros
Put decimal point inside and float it straight up!
\(4 \overline{) 3.00}\)
3
Divide & Inspect
Does it stop with 0 remainder, or repeat forever?
Remainder \(0 \rightarrow\) Stops!
\(\frac{3}{4}\)
Terminating Model: \(3 \div 4 = \mathbf{0.75}\)
Leaves remainder 0. The decimal ends neatly!
\(\frac{2}{3}\)
Repeating Model: \(2 \div 3 = 0.666... = \mathbf{0.\bar{6}}\)
Digit 6 keeps repeating. Put a bar above it!
Number Decoders • Special Education Math Resource Page 1 of 2
PRACTICE
Convert & Classify Practice
Rule: Numerator \(\div\) Denominator
Mission 2
Guided Conversions (Follow the Template)
Problem 1: Convert \(\frac{3}{5}\) Terminating?
Divide 3 by 5. Add a decimal point and zero.
Division Work Area:
.
\(5 \ \overline{)\ 3\ .\ 0}\)
Decimal Value:
Type:
Terminating Repeating
Problem 2: Convert \(\frac{1}{3}\) Repeating?
Divide 1 by 3. Notice if the remainder repeats!
Division Work Area:
.
\(3 \ \overline{)\ 1\ .\ 0\ 0}\)
Decimal Value:
Type:
Terminating Repeating
Mission 3
Independent Practice (Show What You Know)
Problem 3: Convert \(\frac{1}{4}\)
Set up: 1 inside the house, 4 outside.
Work Area:
Decimal:
Circle One:
Terminating Repeating
Problem 4: Convert \(\frac{2}{9}\)
Set up: 2 inside the house, 9 outside.
Work Area:
Decimal:
Circle One:
Terminating Repeating
Check-In
Quick Detective Reflection
Confidence: ⭐ Learning ⭐⭐ Good ⭐⭐⭐ Expert
1. Is \(-7\) a rational number? Write your reason:
2. When a decimal repeats forever, what symbol do we draw on top of the repeating number?
Number Decoders • Special Education Math Resource Page 2 of 2
Number Decoders Answer Key
Number Decoders
Answer Key
Teacher Solutions • IEP Facilitation Guide
CCSS.MATH.CONTENT.7.NS.A.2.D
Reference
Word Vault: Verbal Prompts & Clarifications
Rational Number
Point to the root word "ratio". Remind students that any integer can sit over 1 (e.g. \(5 = \frac{5}{1}\)).
Terminating Decimal
Connect to "terminal" or "terminate" (to stop). The long division process runs out of remainders (reaches 0).
Repeating Decimal
Emphasize the bar notation. The bar only goes over the specific digits that loop infinitely.
Mission 1 Key
Rational Detective: Solutions & Explanations
A. \(\frac{4}{7}\)
YES NO
✓ Already in fraction form \(\frac{a}{b}\) where \(a=4\) and \(b=7\).
B. \(6\)
YES NO
✓ Whole number: write as \(\frac{6}{1}\). All integers are rational.
C. \(0.75\)
YES NO
✓ Terminating decimal: can be written as \(\frac{75}{100}\) or simplified to \(\frac{3}{4}\).
D. \(-3\)
YES NO
✓ Negative integer: write as \(\frac{-3}{1}\). Negative values are still rational.
Common Student Misconceptions & Fixes
- Misconception: Thinking whole numbers or negatives cannot be rational because they lack visible fraction bars.
Teacher Prompt: "Can you put a '1' under it? If yes, it is rational!"
- Misconception: Inverting the division house (\(3 \overline{) 4}\) instead of \(4 \overline{) 3}\)).
Teacher Prompt: Use the kinesthetic chant: "The top number falls forward inside the front door!"
Instructional Scaffold Tip
Have students use a colored highlighter (yellow) on the numerator before dividing, and immediately highlight the inside of the division bracket. This visual pairing reinforces position before students begin subtracting.
Number Decoders Answer Key • Special Education Math Resource Page 1 of 2
SOLUTIONS
Mission 2 & 3 Step-by-Step Solutions
Full Long Division Key
Mission 2 Key
Guided Practice Conversions
Problem 1: Convert \(\frac{3}{5}\) Stops
0.6
5 ) 3.0
- 3.0
0 → Remainder is 0!
Decimal Value: 0.6
Type: ✓ [X] Terminating
Problem 2: Convert \(\frac{1}{3}\) Repeats
