Reasoning · Grade 6-1 Distance and Speed (1)
Problem
In (1) the gap closes by 110 a minute.
Adding the two speeds feels strange until you draw the arrows: both people are working on the same gap from opposite ends, so their efforts pile up. Once you see it as one shrinking distance there is only one rate left to think about.
6.RP.A.2Draw A DiagramWalking toward each other, the two speeds add, because both of them are eating into the same gap.
What matters is how fast the distance between them shrinks, and moving in opposite directions makes each speed count toward that shrinking.
The whole distance is what one walker covers plus what the other covers, with no gap and no overlap at the meeting point.
Times the time gives 880.
Check the units as you multiply: yards per minute times minutes leaves yards, which is the kind of thing the answer should be. If the units had come out wrong, the setup would have been wrong.
6.RP.A.3Analyze The UnitsAdding the two journeys also gives 880.
Splitting the journey into 'Emma's part' and 'Sophia's part' is the slower route but it needs no new idea at all, so it is the perfect way to check that adding the speeds really was allowed.
4.NBT.B.5Identify Subproblems(2) is the same picture with closing speed 70.
Recognising that the second question is the first question with the known and the unknown swapped means you do not have to invent anything new - the same closing-speed picture carries straight over.
6.RP.A.2Look For A PatternDividing the distance gives 5 hours.
Whenever a multiplication sentence has the box in the middle instead of at the end, division puts it right. Dividing 350 by 70 is the same as dividing 35 by 7, so the number work is easy.
6.EE.B.7Work BackwardsAdding the two journeys checks out.
Substituting the answer back into the original story is the most convincing check there is: if the two distances did not add to 350, the time would be wrong.
6.RP.A.3Identify SubproblemsWhen two people walk towards each other, add their speeds - the gap between them closes that fast, and then it is just one distance, one speed, one time.