Problem
Clock chimes and hand cycles
A clock chimes the hour number at each o'clock and once at every half past. Hours cycle 1 to 12, so 1 p.m. chimes once. Only chimes falling inside a span count. Total the chimes in each of the two spans.
Reasoning · Grade 2-2 Exploring Clocks
Your answer
Watch out Reading 1 p.m. as 13 suggests thirteen chimes, but the dial only counts to 12 — 1 p.m. chimes once.
How to solve
Strategy Make a Systematic List — Walk a timeline through each span and list every chime event in order. Split the count into two easier subproblems: the on-the-hour chimes (whose sizes add up like 4+5+6+7) and the once-each half-hour chimes. Listing them along a timeline guarantees none are missed or double-counted.
1STEP 1
List the chime moments in span (1)
Span (1) chimes at eight moments, from 4:00 to 7:30.
4 : 00, 4 : 30, 5 : 00, 5 : 30, 6 : 00, 6 : 30, 7 : 00, 7 : 30
Reading clock times to the nearest five minutes is a Grade 2 skill; listing the events in order is just walking the clock forward.
2.MD.C.7Make A Systematic List2STEP 2
Add the on-the-hour chimes for span (1)
The on-the-hour chimes give 4 + 5 + 6 + 7 = 22.
4+5+6+7 = 22
Each hour contributes a known count; adding within 100 is a Grade 2 skill.
2.OA.A.1Identify Subproblems3STEP 3
Add the half-hour chimes for span (1)
Adding the 4 half-hour chimes makes (1) 26.
22 + 4 = 26
Combining two simple counts is a one-step addition a Grade 2 student handles easily.
2.OA.A.1Identify Subproblems4STEP 4
List the chime moments in span (2)
Span (2) has six half-hour chimes and five on-the-hour ones.
9 : 30, 10 : 00, 10 : 30, 11 : 00, 11 : 30, 12 : 00, 12 : 30, 1 : 00, 1 : 30, 2 : 00, 2 : 30
Tracking that the hour count cycles back to 1 after 12 is the key clock idea, still within Grade 2 clock reading.
2.MD.C.7Make A Systematic ListAfter 12 the hour count starts again at 1, so the chimes of the second span repeat the same cycle of numbers.
Why?
The clock's hours form a closed ring of twelve, so stepping off the end of it simply brings you back to the beginning.
🧱A cycle returns after a full periodA fixed repeating cycle of length p comes back to the same position every p steps.
Why?
Only what is left after whole rounds of twelve decides which number the clock strikes, so full laps can be dropped.
🧱Division with remainderAny number splits into a whole count of equal groups plus a leftover, and the leftover is always smaller than the group size.
5STEP 5
Add the on-the-hour chimes for span (2)
Past 12 the count restarts: 10 + 11 + 12 + 1 + 2 = 36.
10+11+12+1+2 = 36
Adding a short list of numbers within 100 is a Grade 2 addition skill.
2.OA.A.1Identify Subproblems6STEP 6
Add the half-hour chimes for span (2)
Adding the 6 half-hour chimes makes (2) 42.
36 + 6 = 42
One more easy addition combines the two counts.
2.OA.A.1Identify SubproblemsAnswer
(1) 26, (2) 42
Each span covers about 4 to 5.5 hours, so a few dozen chimes is sensible. In span (1) the four o'clock chimes alone (4+5+6+7=22) dominate, plus 4 half-hour chimes gives 26. In span (2) the larger hour counts (10,11,12) push the o'clock total to 36, plus 6 half-hour chimes gives 42, which is reasonably larger than span (1).
Takeaway
Just walk the clock forward and tally each chime - this only needs Grade 2 clock reading and addition you already know!
- List the chime moments in span (1)
- Add the on-the-hour chimes for span (1)
- Add the half-hour chimes for span (1)
- List the chime moments in span (2)
- Add the on-the-hour chimes for span (2)
- Add the half-hour chimes for span (2)