Depth · one concept from first principles · step 2 of 4
A concert was planned to mix -minute pieces and -minute pieces so that the whole performance would take hour minutes. At the orchestra's request, the numbers of -minute pieces and -minute pieces were swapped, and the performance took hour minutes in all. There is a -minute break between each piece and the next. How many -minute pieces were originally planned?
Plan x six-minute and y eight-minute pieces. x + y pieces in a row have x + y - 1 breaks, and swapping keeps that count.
Like fence posts and gaps: n items in a line have n - 1 spaces between them.
6.EE.B.6Draw A DiagramBoth programs have breaks.
Each break sits just before one piece, and every piece except the first has exactly one break just before it, so the breaks match the pieces after the first one for one.
Swapping the counts gives pieces instead of , which is the same number, so the number of breaks does not change.
1 hour 45 minutes is 105 minutes; 1 hour 57 minutes is 117. The swapped program has y six-minute and x eight-minute pieces.
Put every time in the same unit before comparing.
8.EE.C.8Analyze The UnitsAdding gives the same coefficient on both letters.
Mirror-image equations add to a statement about the total.
8.EE.C.8Organize Information In More WaysSubtracting the first from the second leaves the difference.
...and subtract to a statement about the difference.
8.EE.C.8Organize Information In More WaysKnowing the sum and the difference gives each count.
Sum and difference together pin down two numbers.
8.EE.C.8Convert To AlgebraCount the gaps carefully, then let the mirror-image equations do the work: add for the sum, subtract for the difference.
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