Chapter 12 · Lesson 1 · Grade 5

Find the Period and Use the Remainder

找出周期，用余数定位

Learn to compress a long repeating pattern into one small block, count complete cycles, and use the leftover position to jump directly to a faraway item.

Find the block → count the cycle → read the remainder.

10 missionsInteractive cycle machinesObjective feedbackEverything needed is on this page.

Mission 1

Spot what repeats

A repeating block is one complete chunk that appears again and again. The period is the number of items in the smallest repeating block.

Not complete

Periodic ideas are all around us

A week repeats every seven days. A rotating wheel returns to earlier positions. A decimal may repeat the same digits. Chapter 12 begins by asking us to find the recurring part instead of listing every future item.

Pattern A

🔴🟡🔵🔴🟡🔵

Smallest period

Pattern B

■○■○■○

Smallest period

Pattern C

ABCDABCD

Smallest period

Important: For these questions, each pictured pattern continues indefinitely by repeating the displayed strip. A longer visible chunk may repeat, but the period uses the smallest block that generates the whole pattern.

Mission 2

Find the smallest repeating block

Some candidate blocks repeat the visible strip but are longer than necessary. Test several lengths and keep the shortest one that reproduces the sequence.

Not complete

Sequence

The sequence is built from Sun, Moon, Moon, then starts again.

Try a candidate period

Choose a length from 1 to 6.

Your selected period: —

Why length 6 can be tempting: two copies of a 3-item block make a 6-item chunk. But a period must be the shortest block that works.

Two meanings of “zero remainder”

For item positions, counting starts at 1. Item 12 in a four-item block is the last item: 12 = 3 × 4. For moves after a start, the start is Move 0. After 12 moves of a four-move cycle you are back at the starting state. Say what you are counting before calculating.

Mission 3

Separate complete cycles from the leftover

For a period of length p and a position N, divide N by p.

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N = q × p + r, where 0 ≤ r < p

Fixed challenge

The repeating block is:

ABCDE

Find the 23rd item.

Complete cycles q

Remainder r

Landing item

Cycle machine

Period p5

Position N23

4complete cycles

3remainder

3position in block

Clanding item

Mission 4

Handle remainder zero correctly

When the remainder is zero, there are no leftover items. The target lands on the last item of a complete cycle—not on a mysterious “position zero.”

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Example

In the repeating block A, B, C, D:

20 = 5 × 4 + 0

The 20th item is the 4th item of the block: D.

r = 0
→ pth item

20th item of A, B, C, D

84th item in a 7-item cycle lands at position…

Mission 5

Use the seven-day week cycle

Weekdays repeat every seven days. Treat today as offset 0, one day later as offset 1, and reduce a large offset by complete weeks.

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Weekday explorer

Starting weekday

Days later

100 = 14 × 7 + 2

Wednesday

Only the remainder changes the weekday.

Check these two fixed calendar jumps (independent of the explorer)

100 days after Monday

365 days after Friday

Mission 6

Locate a distant card without listing everything

A four-card learning routine repeats: THINK, TRY, CHECK, SHARE. The strip shows its first 12 cards. Use the position remainder instead of writing thousands of cards.

Not complete

Position 2,014

2014 = 503 × 4 + 2

The target is the 2nd card in the block.

Position 100

100 = 25 × 4 + 0

Remainder zero means the 4th card.

2,014th card

100th card

Mission 7

Use a general cycle-position laboratory

Change the period (a whole number from 1–8) and the target position (1–100,000). Item positions start at 1. The block uses letters A, B, C, … in order.

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Period p

Target position N

335q

4block position

Ditem

2014 = 335 × 6 + 4

Record both challenge results

A. p = 6, N = 2,014

Item

B. p = 5, N = 60

Item

Mission 8

Cycle error detective

Judge each statement precisely. “Unknown,” “repeats,” and “smallest period” are not interchangeable ideas.

Not complete

1. In N = qp + r, q counts complete cycles contained in the first N positions.

2. Remainder 0 means the target is the first item of the block.

3. Because the repeating sequence A, B, A, B, … shows four items, its smallest period is 4.

4. For one-based positions, use block position r when r > 0, and position p when r = 0.

Mission 9

Independent workshop

Try each new problem before opening a hint. Correct all 8 answers to complete this mission.

Not complete

1. In the repeating block A, B, C, D, what is item 31?

Hint

Divide 31 by the block length 4.

Worked explanation

31 = 7 × 4 + 3. Read position 3. The answer is C.

2. Which block position is item 96 in a 6-item cycle?

Hint

Use the remainder; if it is zero, use the last position.

Worked explanation

96 = 16 × 6 + 0, giving block position 6.

3. What weekday is 80 days after Thursday?

Hint

Today is offset zero. Remove complete seven-day weeks.

Worked explanation

80 leaves remainder 3 after division by 7. Move 3 days forward from Thursday: Sunday.

4. In the repeating block 2, 5, 8, what is item 41?

Hint

Divide 41 by the block length 3.

Worked explanation

41 = 13 × 3 + 2. Read position 2. The answer is 5.

5. Smallest period of A, B, B, A, B, B, …?

Hint

Try lengths 1, 2, then 3.

Worked explanation

ABB repeats; neither one nor two items reproduce the pattern.

6. How many full blocks are in the first 67 items of a 6-item cycle?

Hint

Find the quotient, not the remainder.

Worked explanation

67 = 11 × 6 + 1: eleven full blocks.

7. Which block position is item 109 in a 5-item cycle?

Hint

Use the remainder; if it is zero, use the last position.

Worked explanation

109 = 21 × 5 + 4, giving block position 4.

8. What weekday is 44 days after Sunday?

Hint

Today is offset zero. Remove complete seven-day weeks.

Worked explanation

44 leaves remainder 2 after division by 7. Move 2 days forward from Sunday: Tuesday.

Mission 10

Transfer exit ticket

Try each new problem before opening a hint. Correct all 5 answers to complete this mission.

Not complete

1. In the repeating block P, Q, R, S, T, what is item 87?

Hint

Divide 87 by the block length 5.

Worked explanation

87 = 17 × 5 + 2. Read position 2. The answer is Q.

2. Which block position is item 200 in an 8-item cycle?

Hint

Use the remainder; if it is zero, use the last position.

Worked explanation

200 = 25 × 8 + 0, giving block position 8.

3. What weekday is 93 days after Tuesday?

Hint

Today is offset zero. Remove complete seven-day weeks.

Worked explanation

93 leaves remainder 2 after division by 7. Move 2 days forward from Tuesday: Thursday.

4. Find the remainder when 157 is divided by 9.

Hint

Find the greatest multiple of 9 that does not exceed 157.

Worked explanation

157 = 17 × 9 + 4.

5. Smallest period of X, X, Y, X, X, Y, …?

Hint

Find the shortest block that restarts the strip.

Worked explanation

XXY is the shortest repeating block: length 3.

Optional reflection — not automatically graded

Explain in your own words why remainder zero lands on the last item of a repeating block.

Lesson checkpoints completed

Period & Remainder Navigator

You completed the checkpoints on periods, complete cycles and remainders. Revisit any steps for which you needed solution help.

Lesson 12.1 · Grade 5 Periodic Problems

Teaching notes

The bead strips, weekday explorer, four-card routine, general laboratory, workshop, and assessment are added self-contained instructional scaffolds.