Reasoning · Grade 6-1 Distance and Speed (2)
Problem
Draw it as two legs.
Separating a journey into legs before touching any numbers is what keeps a multi-step distance word problem from collapsing into a single wrong division.
4.MD.A.2Draw A DiagramThe outbound leg takes 4 hours.
Miles divided by miles-per-hour leaves hours. If you can see that 4 lots of 5 miles make the 20, you have already done the division.
6.RP.A.3Analyze The UnitsThe return leg takes 2 hours.
Doubling the speed halving the time is the inverse-relationship check that tells you the second division is set up the right way round.
6.RP.A.3Analyze The UnitsTogether that is 6 hours.
Adding times is safe; averaging speeds is not. The slow leg lasts 4 hours and the fast leg only 2, so the slow speed counts for twice as much of the trip, which is exactly what a plain average would throw away.
4.MD.A.2Analyze The UnitsIn (2) one unit of trail costs half an hour.
Turning the rate upside down — hours per mile instead of miles per hour — is what makes the times addable, and one mile is a small enough case to reason about with no unknown at all. 20 minutes up plus 10 minutes down is half an hour, which you can check on a clock face.
6.RP.A.2Solve An Easier Related ProblemAsking what one mile of trail costs in time turns the whole trip into a single rate.
One mile costs the same amount of time wherever it sits on the trail, so the cost per mile is a fixed rate.
Because the cost per mile never changes, twice the distance costs twice the time and the total scales straight with the miles.
Twelve hours buys 24 miles.
Dividing by one half is the same as doubling — counting how many half-hours are in 12 hours — so the biggest-looking step in the problem is really just 12 x 2.
6.NS.A.1Analyze The UnitsChecking back gives 12 hours.
Substituting the answer back into the story is the cheapest possible check, and here it also shows why averaging the speeds fails: 12 hours split 8 and 4 is nowhere near an even split.
6.RP.A.3Guess And CheckOn a round trip at two speeds, add the time for each leg — never average the speeds, because you spend longer going slow than coming back fast.