Problem
Length of overlapping strips
Five points A, B, C, D, E sit on one line in that left-to-right order. The spans are AE = 10 cm, AC = 7 cm and BE = 8 cm. Find AB, BC, CD, DE and say which pieces cannot be known.
Reasoning · Grade 2-1 Measurable Lengths
Your answer
Watch out It is tempting to split the 3 cm evenly. No given span touches D, so CD and DE cannot be determined.
How to solve
Strategy Draw a Diagram — Drawing the points on one line and marking each known span as an arc shows how the spans overlap. Then each unknown piece is its own little subtraction subproblem: a longer known span minus another known span. Where two unknown pieces share only one known total, that total can't be split, so those pieces stay unknown.
1STEP 1
Find AB using the overlap of AE and BE
BE covers everything but AB, so 10 − 8 = 2 is AB.
AB = AE - BE = 10 - 8 = 2
BE starts at B and reaches the end, so the only part of AE it misses is from A to B - the leftover is just a subtraction.
2.MD.A.4Draw A DiagramSince BE covers everything from B to the end, the only part of AE it misses is AB, so AB is what is left after taking BE away.
Why?
The point B cuts the segment AE into two pieces that do not overlap and leave no gap, so the two pieces together are the whole of AE.
🧱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?
When the whole and one part are both known, taking the part away returns the other part, because taking away undoes putting together.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
2STEP 2
Find BC using AC and AB
AC is AB then BC, so 7 − 2 = 5 is BC.
BC = AC - AB = 7 - 2 = 5
AC is just AB and BC laid end to end, so once one piece is known the other is the leftover.
2.MD.A.4Identify Subproblems3STEP 3
Find the combined piece C to E
The rest, CE, is 10 − 7 = 3, so CD + DE = 3.
CE = AE - AC = 10 - 7 = 3, CD + DE = 3
Cutting the whole line at C, the right-hand leftover is everything past C, which is CD and DE together.
2.MD.A.4Draw A Diagram4STEP 4
See why CD and DE cannot be determined
Nothing touches D, so 1+2 or 2+1 both fit — neither piece is determined.
CD + DE = 3 → many splits (e.g. 1+2, 2+1)
If no measurement ever touches point D, there is no way to know how the 3 cm is shared between CD and DE.
2.MD.B.5Identify SubproblemsAnswer
AB = 2, BC = 5, CD + DE = 3 cm (CD and DE not separately determined)
Check the known pieces against the givens: AB + BC = 2 + 5 = 7 = AC, correct. AB + BC + CD + DE = 2 + 5 + 3 = 10 = AE, correct. BE should be BC + CD + DE = 5 + 3 = 8, correct. Every given span is reproduced, so AB = 2 and BC = 5 are right, and the 3 cm tail genuinely cannot be split.
Takeaway
Overlapping lengths just subtract: line up the spans and the leftover piece pops out - and if no mark ever touches a point, that piece has to stay a mystery!
- Find AB using the overlap of AE and BE
- Find BC using AC and AB
- Find the combined piece C to E
- See why CD and DE cannot be determined