Problem
Log Cutting: Cuts, Pieces, and Time
A 72 cm log is cut into equal pieces. A cut takes 6 minutes and a 2-minute rest follows each cut, except after the last one. Work runs from 1:00 to 2:02. Find the length of one piece.
Reasoning · Grade 3-1 Length and Time
Your answer
Watch out Eight cuts suggest eight pieces, but on one log pieces run one ahead of cuts.
How to solve
Strategy Draw a Diagram — First I find the total time the work took. Then I picture the work as a row of cut-rest-cut-rest blocks to count how many cuts fit in that time, remembering the last cut has no rest after it. The number of cuts tells me the number of pieces, and dividing the length by the pieces gives each piece's size.
1STEP 1
Find the total time
From 1:00 to 2:02 is 62 minutes.
60 + 2 = 62 minutes
Counting up from the start time to the finish time gives the elapsed minutes.
3.MD.A.1Draw A Diagram2STEP 2
Find the time for one cut plus its rest
A cut plus its rest is 8 minutes; the last cut is just 6.
6 + 2 = 8 minutes per cut-and-rest block
Every cut except the last comes paired with a rest, so I bundle each cut with the rest that follows it.
3.NBT.A.2Identify Subproblems3STEP 3
Count how many cuts were made
Since 7 × 8 + 6 = 62, there were 8 cuts.
7 × 8 + 6 = 56 + 6 = 62 minutes
I test cut counts until the total time lands on 62 minutes; 8 cuts fits perfectly.
3.OA.D.8Guess And Check4STEP 4
Find the number of pieces
Pieces run one ahead of cuts: 9 pieces.
8 + 1 = 9 pieces
Think of cuts as the gaps between pieces in a row; a row of 9 pieces has 8 gaps.
3.OA.A.3Draw A DiagramEight cuts make nine pieces, because the cuts are the gaps between pieces standing in a row.
Why?
Each cut sits between exactly one pair of neighbouring pieces, so cuts and the gaps between pieces match up one for one.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
A row of pieces is the pieces and the gaps between them, and a row always has one more piece than it has gaps.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
5STEP 5
Find the length of each piece
Sharing 72 into 9 gives 8 cm a piece.
72 ÷ 9 = 8 cm
Sharing the whole length equally among 9 pieces means dividing 72 by 9.
3.OA.A.3Identify SubproblemsAnswer
8 cm
72 ÷ 9 = 8
Check the time with 8 cuts: 8 cuts * 6 min = 48 min of cutting, plus 7 rests * 2 min = 14 min of rest, total 48 + 14 = 62 minutes, which matches 1:00 to 2:02. And 9 pieces * 8 cm = 72 cm rebuilds the whole log, so 8 cm per piece is correct.
Takeaway
Cuts are like the gaps between pieces, and the last cut needs no rest, so count carefully!
- Find the total time
- Find the time for one cut plus its rest
- Count how many cuts were made
- Find the number of pieces
- Find the length of each piece