AMC 8 · 2008 · #3
Easy mode Grade 4Problem
Picture a calendar for February.
In this February, the 13th falls on a Friday.
What day of the week is February 1?
Pick an answer.
AMC 8 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: February has a Friday the $13^{\text{th}}$. What day of the week is February $1$?
Givens: February $13$ is a Friday; Days of the week repeat every $7$ days; Answer choices: (A) Sunday, (B) Monday, (C) Wednesday, (D) Thursday, (E) Saturday
Unknowns: The day of the week of February $1$
Understand
Restated: February has a Friday the $13^{\text{th}}$. What day of the week is February $1$?
Givens: February $13$ is a Friday; Days of the week repeat every $7$ days; Answer choices: (A) Sunday, (B) Monday, (C) Wednesday, (D) Thursday, (E) Saturday
Plan
Primary tool: #12 Use Parity or Modular Arithmetic
Secondary: #8 Work Backwards
Weekdays cycle every $7$ days, so the only thing that matters when stepping back from February $13$ to February $1$ is the remainder of $12$ divided by $7$. Tool #12 (Use Parity or Modular Arithmetic) names that idea: $12 \equiv 5 \pmod 7$. Tool #8 (Work Backwards) tells us where to start — we know February $13$ is a Friday, and we want February $1$, so we walk backwards in time. Together, the moves are: go back $7$ days (one full week, same weekday), then go back the remaining $5$ days.
Execute — Answer: A
4.OA.A.3 Step 1 - Find how many days February $1$ is before February $13$.
- Subtract the dates.
💡 Working backwards from a known day means measuring the gap first, just like reading a number line from right to left.
4.OA.B.4 Step 2 - Break $12$ days into full weeks plus leftover days.
- One full week ($7$ days) lands on the same weekday, so it can be skipped.
💡 Dividing by $7$ and keeping the remainder is the modular arithmetic move — the weekly cycle erases the $7$, only the $5$ leftover days change the weekday.
3.OA.D.9 Step 3 - Apply the $7$-day skip.
- Go back exactly $7$ days from Friday February $13$ — that lands on Friday February $6$, still a Friday.
💡 The Grade 3 "identify a pattern" move: every $7$ steps, the weekday repeats.
4.OA.A.3 Step 4 - Now walk back the $5$ leftover days from Friday February $6$ to February $1$.
- Count weekdays backwards: Friday, Thursday, Wednesday, Tuesday, Monday, Sunday.
💡 Five backward steps from Friday lands on Sunday — the last leftover days do the actual weekday shift.
4.OA.A.3 Step 5 - Read off the answer.
- February $1$ is a Sunday.
💡 The mod-$7$ shortcut plus a short backward count gives the weekday directly.
4.OA.A.3 Find how many days February $1$ is before February $13$. Subtract the dates. 4.OA.B.4 Break $12$ days into full weeks plus leftover days. One full week ($7$ days) lan 3.OA.D.9 Apply the $7$-day skip. Go back exactly $7$ days from Friday February $13$ — tha 4.OA.A.3 Now walk back the $5$ leftover days from Friday February $6$ to February $1$. Co 4.OA.A.3 Read off the answer. February $1$ is a Sunday. Review
Reasonableness: Check forward instead of backward. If February $1$ is Sunday, then Sunday $+ 7$ days $=$ Sunday February $8$, and Sunday $+ 12$ days $=$ five weekdays after Sunday February $8$, which lands on Friday February $13$. That matches the problem, so February $1$ being Sunday is consistent.
Alternative: Tool #2 (Make a Systematic List): write the Fridays of February in order — $13, 6,$ and (going back another week) $-1$. So one week before February $6$ would be "February $-1$," meaning January $30$ is also a Friday. Counting forward from a Friday on "day $-1$": day $0$ is Saturday, day $1$ is Sunday. So February $1$ is a Sunday, giving (A).
CCSS standards used (min grade 4)
3.OA.D.9Identify arithmetic patterns and explain them using properties of operations (Recognizing that the weekday repeats every $7$ days, so adding or subtracting a multiple of $7$ does not change the weekday.)4.OA.A.3Solve multistep word problems using the four operations, including problems with remainders (Setting up the gap of $12$ days from February $1$ to February $13$ and stepping backwards from a known Friday to find the unknown weekday.)4.OA.B.4Find factors and multiples of a whole number (used here as division-with-remainder) (Splitting $12 = 7 + 5$ — one full week plus $5$ leftover days — which is the same as computing $12 \bmod 7 = 5$.)
⭐ Whole weeks are free moves — only the leftover days after dividing by $7$ change the weekday. That mod-$7$ shortcut turns a calendar puzzle into a one-line count.
⭐ Whole weeks are free moves — only the leftover days after dividing by $7$ change the weekday. That mod-$7$ shortcut turns a calendar puzzle into a one-line count.
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