Problem
Lengths in an Arithmetic Progression
A 20 m pole is cut once into two pieces. One piece is 8 m longer than the other. Both pieces have a positive length. Find the length of the longer piece.
Reasoning · Grade 2-2 Bars and Length
Your answer
Watch out Halving 20 to 10 and adding 8 gives 18, but then the pieces total 28. Take the 8 off first, then halve.
How to solve
Strategy Draw a Diagram — Draw the two pieces as a long bar and a short bar lined up at the left. The extra 8 m sticks out past the short bar. If we mentally trim off that 8 m, the rest of the pole splits into two equal halves, so we can find one piece and then add the extra back.
1STEP 1
Trim off the extra 8 m
Setting the extra 8 m aside leaves 20 − 8 = 12 m.
20 - 8 = 12
Drawing the bars shows the 8 m as the part that makes the pieces unequal, so removing it leaves two equal lengths to share.
2.MD.B.5Draw A DiagramTrimming the extra 8 m off the longer piece leaves two pieces of exactly the same length to share the rest.
Why?
The whole rope is the short piece and the long piece laid end to end, so removing 8 m from the total removes it from the long piece only.
🧱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?
The 8 m is exactly what made the two pieces differ, so taking it away leaves a statement that the two remaining lengths are equal.
🧱Do the same to both sides and equality holdsA balanced scale stays level when you take the same amount off both pans or add the same amount to both.
2STEP 2
Split the rest into two equal halves
That 12 m splits evenly, so the short piece is 6 m.
12 ÷ 2 = 6
After the unequal 8 m is gone, the leftover is split fairly, so halving gives the shorter piece directly.
3.OA.A.3Identify Subproblems3STEP 3
Add the extra back to get the longer piece
Adding the 8 m back makes the long piece 14 m.
6 + 8 = 14
The longer piece was just the short one with 8 m more, so adding the trimmed amount back rebuilds it.
2.MD.B.5Identify SubproblemsAnswer
14 m
(20 − 8) ÷ 2 + 8 = 14
Check the two pieces: 14 m and 6 m. Their sum is 14 + 6 = 20 m (the whole pole) ✓, and their difference is 14 − 6 = 8 m ✓. Both conditions hold, and 14 m is more than half of 20 m, which fits 'the longer piece'.
Takeaway
Trim off the 8 m that makes the pieces unequal, share the rest evenly, then add it back — just Grade 3 halving you already know!
- Trim off the extra 8 m
- Split the rest into two equal halves
- Add the extra back to get the longer piece