Overlaps are one fewer than the strips

3.MD.D.83.OA.D.8 · adapt · grade 3

Archetype: Overlap Reduces the Total · step in a 4-type progression

You want to join $5$ strips of tape, each $26 \text{ cm}$ long, by overlapping them by the same amount as shown in the figure, to make one long tape that is $1 \text{ m } 10 \text{ cm}$ long. By how many $\text{cm}$ should each overlap be?

Show solution

Understand

Five tape strips, each 26 cm long, are joined into one long row by overlapping each neighboring pair by the same amount. The finished tape is 1 m 10 cm (110 cm) long. We must find how many cm each overlap is.

Givens

5 strips, each 26 cm long
Strips overlap by the same amount between neighbors (figure shows the orange overlap regions)
Final joined length is 1 m 10 cm = 110 cm

Unknowns

The length of each overlap in cm

Constraints

All overlaps are equal
With 5 strips in a row there are 5 - 1 = 4 overlap regions

Plan

#1 Draw a Diagram · also uses: #7 Identify Subproblems#5 Look for a Pattern

The figure shows that joining strips makes them overlap, and the key pattern is that the number of overlaps is one fewer than the number of strips. The total length lost equals the combined overlaps, which we then split equally.

Execute

#7 Identify Subproblems 3.OA.A.3

Add up all 5 strips before any overlapping: 5 strips of 26 cm.

5 \times 26 = 130 \text{ cm}

If they were laid end to end with no overlap, the lengths simply add.

#1 Draw a Diagram 3.MD.D.8

The finished tape is 1 m 10 cm = 110 cm. Overlapping makes the tape shorter, so the amount lost is the separate total minus the final length.

130 - 110 = 20 \text{ cm}

Every overlap hides some tape; the missing 20 cm is all the hidden overlap together.

#5 Look for a Pattern 3.OA.D.9

In the figure the orange overlap regions sit between neighboring strips. With 5 strips there are 4 such overlaps (one fewer than the number of strips).

5 - 1 = 4

Each new strip after the first adds exactly one overlap, so overlaps are one fewer than strips.

#7 Identify Subproblems 3.OA.A.3

The 20 cm of hidden tape is shared equally among the 4 overlaps.

20 \div 4 = 5 \text{ cm}

Equal overlaps means equal sharing, so divide the total overlap by how many there are.

Answer: 5 cm

Review

With 5 cm overlaps: total 130 cm minus 4 overlaps times 5 cm = 130 - 20 = 110 cm = 1 m 10 cm, matching the target. An overlap of 5 cm is smaller than a 26 cm strip, which is sensible.

Build up by addition (tool 11, work forwards/backwards): first strip 26, then each added strip contributes 26 minus the overlap; setting 26 + 4 times (26 - overlap) = 110 also gives overlap = 5 cm.

Standards · min grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Multiplying 5 by 26 and dividing the lost length by the number of overlaps
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Reasoning about combined lengths and the length lost to overlapping
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that the number of overlaps is one fewer than the number of strips

💡 When you overlap strips in a row, the overlaps are always one fewer than the strips, so the hidden length just splits evenly among them!