Problem
Use the seven-day weekday cycle
On a June calendar, the first Tuesday falls on the 1st and June has 30 days. Using the rule that the same weekday repeats every 7 days, find the dates of every Tuesday in this month.
Operations
Your answer
Watch out Just adding 7 again and again without deciding where to stop leads to mistakes. A day-gap with remainder 0 when divided by 7 means the same weekday, but June only has 30 days, so a date like the 36th — even with remainder 0 — is outside the month and must be dropped.
How to solve
Strategy Look for a Pattern — Weekdays repeat on a 7-day cycle, so I add 7 repeatedly starting from the first Tuesday and list each date until I pass the end of the month.
1STEP 1
Add 7 repeatedly from the 1st
Starting at the 1st, a Tuesday, keep adding 7 to reach each next Tuesday.
1 + 7 = 8, 8 + 7 = 15, 15 + 7 = 22, 22 + 7 = 29, 29 + 7 = 36
Tuesday comes back every 7 days because weekday names run in a fixed loop of 7 that returns to its start after 7 steps. Seven single days joined together raise the date by exactly 7.
Look For A Pattern2.NBT.A.2Adding 7 again and again, starting from the first Tuesday on the 1st, lands on the date of each following Tuesday.
Why?
Each following Tuesday sits a full 7 days after the Tuesday before it, so its date is 7 larger.
Why?
Tuesday comes back every 7 days because the weekday names run in a fixed loop of 7 that returns to where it started after 7 steps.
🧱A cycle returns after a full periodA fixed repeating cycle of length p comes back to the same position every p steps.
Why?
A gap of 7 single days raises the date by 7, since each day forward adds 1 and the seven days join with no gaps into one step of 7.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Reaching every later Tuesday means taking that same 7-day step over and over, which is just adding the same 7 repeatedly.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
2STEP 2
Check whether it passes 30
The next value, 36, passes June's 30 days, so it's dropped from the list.
29 + 7 = 36
3STEP 3
List every Tuesday in June
Keeping only values of 30 or less gives Tuesdays 1, 8, 15, 22, and 29.
{1, 8, 15, 22, 29}
Reaching every later Tuesday means taking that same 7-day step over and over, which is just adding the same 7 repeatedly.
Make A Systematic List2.OA.B.2Finding PatternsAnswer
the 1st, 8th, 15th, 22nd, and 29th
1+7=8, 8+7=15, 15+7=22, 22+7=29, 29+7=36
Check — adding 7 repeatedly from 1 gives 1, 8, 15, 22, 29, 36, but 36 exceeds June's 30 days, confirming the five Tuesdays are 1, 8, 15, 22, and 29.
Takeaway
Weekdays repeat every 7 days. Adding 7 from the 1st and keeping only values within 30 days finds every Tuesday this month.
- Add 7 repeatedly from the 1st
- Stop once it passes 30 days
- Tuesdays are 1, 8, 15, 22, and 29
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