Reasoning · Grade 4-1 Clocks and Angles
Problem
One hour mark is 30 degrees.
Cutting a full turn into 12 equal pieces is exactly the fraction-of-a-circle idea used to measure any angle: each piece is one twelfth of 360 degrees.
4.MD.C.5Analyze The UnitsThe hour hand creeps one mark per hour.
Taking a fraction of 30 degrees is the same multiply-a-fraction-by-a-whole-number move used for "two thirds of 30 apples"; here the whole is one hour-mark gap.
4.NF.B.4Analyze The UnitsMeasure both hands from 12.
Because angles around the same centre simply add up, measuring both hands from one fixed starting ray turns "the angle between them" into an ordinary subtraction.
4.MD.C.7Draw A DiagramMeasuring both hands from the 12 turns the angle between them into an ordinary subtraction.
Angles sharing the same centre and the same starting ray add and subtract just like lengths along a line.
The two angles at the centre must fill one full turn together, so whenever the subtraction passes 180 the smaller angle is what remains of 360.
8:00 gives 120 degrees.
At an exact o'clock nothing has crept, so the angle really is just a count of hour marks — the easiest case on the list.
4.G.A.1Make A Systematic List3:30 gives 75 degrees, acute.
Adding the creep to the hour mark and then subtracting is angle addition and subtraction on a single centre point — the same skill as splitting one angle into two smaller ones.
4.MD.C.7Make A Systematic List5:10 gives 95 degrees — just over.
A 5-degree creep is small, but when the answer sits right on the 90-degree boundary a small amount is exactly what decides it — which is why every time gets calculated, not eyeballed.
4.MD.C.7Make A Systematic List7:50 gives 65 degrees, acute.
By 50 minutes past the hour the hour hand is nearly at the next number, so it has almost caught up with the minute hand — the picture and the arithmetic agree.
4.MD.C.7Make A Systematic List2:55 gives 117.5 degrees.
The two angles at the centre have to add to a full turn, so whenever the first difference passes 180 degrees the smaller angle is what is left of the 360.
4.MD.C.7Make A Systematic List11:15 gives 112.5 degrees.
Even a quarter-hour of creep only moves the hour hand 7.5 degrees, nowhere near enough to pull this angle under 90.
4.MD.C.7Make A Systematic ListThe acute ones are 3:30 and 7:50.
Sorting angles into acute, right and obtuse is just comparing each measure with 90 — the table makes the comparison a single glance.
4.G.A.1Make A Systematic ListThe short hand never waits on the number — it creeps forward all hour, and that little creep is what decides the close calls!