Decimal Decoder

Jordan's Claim:

"Since \(0.08\bar{3}\) has 3 digits past the decimal point, it equals \(\frac{83}{1000}\)."

Is Jordan correct or incorrect?

Explain why Jordan is wrong using what you know about repeating decimals:

EXIT TICKET

Mastery Check: \(0.08\bar{3}\) in 30 Seconds

Hand in to teacher

A. Name the Repeating Part:

In \(0.08\bar{3}\), which digit repeats forever?

Just 8 Just 3 Both 8 and 3

B. The Subtraction Result:

What does \(83.333... - 8.333...\) equal?

C. Simplest Form:

\(0.08\bar{3}\) simplified as a fraction is:

How confident do you feel cancelling repeating tails?

🧗 Still practicing 👍 Got it! 🚀 Can teach a friend

Extension Challenge (Quick Finishers):

• Challenge: Can you decode \(0.00\bar{6}\)? (Answer: \(\frac{6}{900} = \frac{1}{150}\)).
• The 0.999... Paradox: What happens when you apply the decoder to \(x = 0.\bar{9}\)? (\(10x - x = 9x = 9 \rightarrow x = 1\))!

Decimal Decoder Teacher Facilitation Guide • Tier 2/3 Math Intervention See Page 2 for Complete Answer Key & Rubric →

Answer Key & Worked Exemplars

Decimal Decoder: Student Worksheet Key

FOR TEACHER USE ONLY

Problem 1 (Model): Convert \(0.08\bar{3}\) to a Fraction Key Answer: \(\frac{1}{12}\)

1. Shifts:
\(x = 0.08333...\)
\(100x = 8.3333...\)
\(1000x = 83.3333...\)

2. Subtract:
\(1000x - 100x = 900x\)
\(83.333... - 8.333... = 75\)
\(x = \frac{75}{900}\)

3. Simplify:
\(\frac{75 \div 25}{900 \div 25} = \frac{3}{36}\)
\(\frac{3 \div 3}{36 \div 3} = \frac{\mathbf{1}}{\mathbf{12}}\)
Shade 1 out of 12 boxes.

Problem 2 (Guided Practice): Convert \(0.1\bar{6}\) to a Fraction Key Answer: \(\frac{1}{6}\)

1. Shifts:
\(x = 0.1666...\)
\(10x = 1.666...\)
\(100x = 16.666...\)

2. Subtract:
\(100x - 10x = 90x\)
\(16.666... - 1.666... = 15\)
\(x = \frac{15}{90}\)

3. Simplify:
\(\frac{15 \div 15}{90 \div 15} = \frac{\mathbf{1}}{\mathbf{6}}\)
(Or divide by 5 then 3)
Check: \(1 \div 6 = 0.1666...\) ✓

Problem 3 (Independent Practice): Convert \(0.41\bar{6}\) to a Fraction Key Answer: \(\frac{5}{12}\)

1. Shifts:
\(x = 0.41666...\)
\(100x = 41.666...\)
\(1000x = 416.666...\)

2. Subtract:
\(1000x - 100x = 900x\)
\(416.666... - 41.666... = 375\)
\(x = \frac{375}{900}\)

3. Simplify:
\(\frac{375 \div 75}{900 \div 75} = \frac{\mathbf{5}}{\mathbf{12}}\)
(Or: \(\div 25 \rightarrow \frac{15}{36} \div 3 \rightarrow \frac{5}{12}\))
Check: \(5 \div 12 = 0.41666...\) ✓

Problem 4 (Error Detective): Jordan's Blunder Key

Is Jordan correct or incorrect? Incorrect.

Exemplar Student Response: Jordan is incorrect because \(0.08\bar{3}\) does NOT have 3 digits past the decimal point; the bar over the 3 means it repeats infinitely (\(0.0833333...\)). The fraction \(\frac{83}{1000}\) equals the terminating decimal \(0.083\), which stops. To represent the repeating tail, you must subtract equations to get \(\frac{75}{900}\), which simplifies to \(\frac{1}{12}\).

Exit Ticket Scoring Key

Question A:
Correct: "Just 3"
The bar is only drawn directly over the 3.

Question B:
Correct: 75
\(83 - 8 = 75\); repeating tails cancel to zero.

Question C:
Correct: \(\frac{1}{12}\)
\(\frac{75}{900}\) reduced to simplest terms.

Quick Rubric: 3/3 Exit Check: Mastered • Ready for independent multi-digit repeaters 2/3 Exit Check: Minor calculation error • Check reduction 0-1/3: Re-teach with highlighter shifting protocol