012i - Recurring Decimal Notation and Patterns

Lesson 12 — Recurring Decimal Notation and Patterns

Strand: Number | Descriptor: AC9M8N03 | Duration: 45 minutes

Learning Intentions

• To write recurring decimals correctly using dot and bar (vinculum) notation.
• To identify the repeating block in a recurring decimal, including when it does not start immediately after the decimal point.
• To explore patterns in recurring decimals, including the cyclic pattern of sevenths.

Success Criteria

I can:

1. Write a recurring decimal using bar notation over the repeating block, e.g. .
2. Identify the repeating block correctly when non-repeating digits appear first, e.g. .
3. Convert a calculator’s rounded display back into correct recurring decimal notation.
4. Describe the cyclic digit pattern found in the sevenths family of fractions.

Warmup

(5 minutes — quickfire, mini whiteboards)

Display these calculator outputs. For each, write what you think the true value is, using as many repeated digits as you can see:

Discussion: Ask students to describe the pattern in words before any notation is introduced (“the 6 just keeps going”, “45 keeps repeating”, “the 3 keeps going but only after an 8”). This primes the need for a precise, compact notation — today’s focus.

Activities

Activity 1 — Explicit Instruction: bar and Dot Notation (10 min)

I do / We do / You do.

I do: Introduce the vinculum (bar) notation, drawn over the digit(s) that repeat forever.

The bar sits over exactly the digits that repeat, and only those digits. Some teachers use a single dot over a lone repeating digit () or dots over the first and last digit of a longer block () — this lesson will use the bar consistently, as it is unambiguous for blocks of any length.

I do (the tricky case): Some recurring decimals have digits before the repeating block starts.

Here the never repeats — only the does. Contrast directly with , where the repeating block starts immediately.

We do: Convert together, deciding where the bar begins and ends: , ,

You do: Write each in correct bar notation: , , , (calculator shows ), .

Activity 2 — Guided Practice: Reading the Block Length off a Calculator (10 min)

I do: Explain that a calculator’s rounded final digit can be misleading, and model a check: multiply the decimal by powers of 10 to see where the pattern realigns, or simply perform enough long division steps to see the remainder repeat (linking back to Lesson 11).

Block: , so .

We do: Complete the table, using long division to confirm the block length (number of repeating digits) for each.

Fraction	Decimal	Bar notation	Block length
			1
			1
			2
			6

You do: Find the bar notation and block length for , , .

Activity 3 — Inquiry: the Cyclic Pattern of Sevenths (14 min)

Pairs, then whole-class share.

Give students calculators and ask them to find the decimal expansions of , writing each to at least 6 decimal places.

Look closely at the six repeating blocks. What do you notice about the digits across all six results?

Students should notice that every block is a cyclic rotation of — the digits are the same, just starting at a different point in the cycle.

Socratic scaffolding for pairs who need support to generalise:

Prompt	Purpose
Understand: what pattern are you looking for?	Compare the sequence of digits in each block, not just their sum or size.
Write underneath . What do you see?	The second block starts where the first block’s third digit was — it has “rotated”.
Devise a plan	Line up all six blocks in a circle:
Carry out the plan	Confirm each of is a rotation starting at a different point on that circle.
Predict without dividing	Using the cycle, predict starts at the digit after in the rotation used by ‘s block — check it starts at : .
Looking back — why might this happen?	Each remainder in the long division of sevenths (1 through 6) appears exactly once before repeating, so every seventh “shares” the same six-digit cycle, just entered at a different remainder.

Finish with a challenge: without a calculator, use the cycle to predict the decimal expansion of , then verify.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Write using correct bar notation.
2. A calculator shows . Write this using bar notation, being careful about which digits repeat.
3. What is the block length of ?
4. Using the sevenths cycle , predict the decimal expansion of without dividing.

Answers: 1. ; 2. — only the 3 repeats, the 8 does not; 3. 2 digits; 4. (starting the cycle at 7).

Common Misconceptions

Misconception	How to pre-empt it
Putting the bar over every digit shown, including non-repeating leading digits.	Explicitly contrast with side by side, asking “does this digit come back?” for each position.
Assuming the calculator’s last displayed digit is part of the repeating block.	Reinforce that calculators round the final digit; use long division to find the true block.
Believing all recurring decimals have a one-digit block.	Show and ‘s six-digit block early, before students over-generalise from and .
Writing as “0.3 repeated, so it equals about 0.3”.	Emphasise that the bar means forever, not “roughly” — is exactly , an exact value, not an approximation.
Thinking the cyclic pattern in sevenths is a coincidence specific to the number 7.	Briefly mention (without full proof) that any prime denominator other than 2 or 5 produces a repeating block whose length is related to the remainders possible — sevenths simply produce a particularly clean, complete cycle.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). What is the 100th digit after the decimal point in the expansion of ?

Answer

Every digit is , so the 100th digit is 3.

E2 (AMC Junior style). The decimal equals which fraction with denominator 7?

Answer

Using the sevenths cycle , the block starts at , matching .

E3 (Challenge). What is the 100th digit after the decimal point in the expansion of ?

Answer

The block has length 6. remainder , so the 100th digit is the 4th digit of the block , which is 8.

E4 (Investigation). , , . Predict as a decimal (hint: simplify first, or think about what comes after ).

Answer

. (A famous related curiosity: exactly — well beyond today’s scope, but worth a mention for curious students.)

Homework

1. Write each in correct bar notation: (a) (b) (c) (d) .
2. State the block length of each: (a) (b) (c) .
3. Using long division, find the bar notation for and .
4. Using the sevenths cycle , write the decimal expansions of and without a calculator.
5. Reasoning. Explain why and are different numbers, even though they use the same digits. (Convert each to a fraction-style estimate or compare their first six decimal places if unsure.)
6. Challenge. The decimal expansion of has a repeating block of length 16 (it uses every digit’s worth of remainders). Without finding the whole block, explain how you know the block length must be less than or equal to 16 for any fraction .

Answers: Q1 — (a) (b) (c) (d) . Q2 — (a) 1 (b) 2 (c) 6. Q3 — ; . Q4 — ; . Q5 — (only the 5 repeats) while (the block “25” repeats) — comparing decimal places from the third digit onward shows they diverge. Q6 — dividing by 17, every remainder must be one of (it cannot be 0, since does not terminate); with only 16 possible non-zero remainders, a repeat must occur within 16 steps, so the block length cannot exceed 16.