Problem
Total Length with Gaps or Overlaps
Eight frames hang in a row across a whiteboard, evenly spaced from the very left edge to the very right. Each frame is 12 in wide and every gap is 8 in. Find the inside width of the whiteboard.
Reasoning · Grade 2-2 Bars and Length
Your answer
Watch out Eight frames suggest seven gaps, but the two ends carry one each — nine in all.
How to solve
Strategy Draw a Diagram — Sketch the row as gap–frame–gap–frame–…–frame–gap. Counting the picture shows the gaps and the frames separately; the total width is all the frame widths plus all the gap widths added together.
1STEP 1
Count the gaps
The ends carry gaps too, so there are 7 + 2 = 9.
8 - 1 = 7, 7 + 2 = 9
Drawing the row makes the extra gaps at both ends easy to see, so we don't miss them.
2.MD.B.5Draw A Diagram2STEP 2
Add up all the frame widths
The frames total 8 × 12 = 96 in.
8 × 12 = 96
All frames are the same width, so repeated addition (multiplication) gives their total quickly.
3.OA.A.3Identify Subproblems3STEP 3
Add up all the gap widths
The gaps total 9 × 8 = 72 in.
9 × 8 = 72
All gaps are equal, so multiplying the count by the gap size totals them in one step.
3.OA.A.3Identify Subproblems4STEP 4
Combine frames and gaps
Together the inside width is 168 in.
96 + 72 = 168
The board's width is just everything laid across it, so adding the two totals finishes the problem.
2.NBT.B.7Identify SubproblemsThe board's width is the frames and the gaps laid across it together, so adding the two totals finishes the job.
Why?
The frames and the gaps take turns along the board with no gap between them and no overlap, so together they are the whole width.
🧱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?
All the frames are the same width and all the gaps are the same width, so each total is a count of equal groups.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Answer
168 in
8 × 12 + 9 × 8 = 168
Build it back up piece by piece: 9 gaps of 8 in and 8 frames of 12 in alternate to fill the board, 72 + 96 = 168 in. The gap count (9) is one more than the frame count (8) because both ends are gaps — a sensible check for an alternating row that starts and ends with a gap.
Takeaway
Count the gaps carefully — both ends count, so there is one more gap than frames — then add frames and gaps; just Grade 3 times-and-add!
- Count the gaps
- Add up all the frame widths
- Add up all the gap widths
- Combine frames and gaps