Problem
Where the hour hand sits within the hour
Only the long hand is drawn, at the 24-minute mark. The time is 7:24. Say not just which number it is near but how far past it. Find exactly where the missing short hand belongs.
Reasoning · Grade 4-1 Clocks and Angles
Your answer
Watch out Seven o'clock tempts you to the 7, but 24 minutes in it sits between 7 and 8.
How to solve
Strategy Draw a Diagram — The answer is a position on a picture, so the dial itself is the working surface: I mark what one large division and one small tick are worth in degrees and then measure the hour hand's slide in those same units. Two different units are floating around — minutes on the outer ring and degrees of turn — so I keep track of which is which and convert once, deliberately. To be sure the hour hand really does slide rather than jump, I first check the two easy times I already know, 7:00 and 8:00, and then read 7:24 as a fraction of the trip between them.
1STEP 1
Find how many degrees one large division is worth
One large division is 30 degrees.
360° ÷ 12 = 30°
A full turn measuring 360 degrees is the one angle fact fourth graders are given, and cutting it into 12 equal parts is just sharing equally.
4.MD.C.5Analyze The UnitsOne large division of the dial is worth 30 degrees, because a full turn of 360 degrees is cut into 12 equal parts.
Why?
Going all the way round the centre once is 360 degrees, whatever path the hand takes to get there.
🧱Angles around a point make 360 degreesAngles that fit side by side all the way around one point fill exactly one full turn, 360 degrees.
Why?
The twelve divisions are all the same size and fill the turn with no gap and no overlap, so each is one twelfth of it.
🧱A fraction is equal shares of one wholeThe bottom number names how many equal pieces one whole was cut into and the top counts them, which is why same-sized pieces add just by counting.
2STEP 2
Find how many degrees one small tick is worth
One small tick is 6 degrees.
30° ÷ 5 = 6°
Once you know what one tick is worth, you can read any position on the dial the way you read marks on a ruler.
4.MD.C.6Analyze The Units3STEP 3
Check the two times you already know
Check the tick values against times you know.
Solving the two easy o'clock cases first pins down the road the hand travels, so all that is left is to say how far along that road it has got.
4.MD.C.7Solve An Easier Related Problem4STEP 4
Say what part of the hour has gone by
24 minutes is 2/5 of the hour.
24/60 = 2/5
Simplifying 24/60 to 2/5 is ordinary equivalent-fraction work, and 2/5 is small enough to picture straight away as a bit less than half.
4.NF.A.1Analyze The Units5STEP 5
Turn that fraction of the hour into degrees
As an angle that is 12 degrees.
2/5 × 30° = 2 × 6° = 12°
Taking a fraction of a whole amount is done by splitting into 5 equal shares and keeping 2, which is exactly the fraction-times-whole-number skill of Grade 4.
4.NF.B.4Analyze The Units6STEP 6
Read 12 degrees off in small ticks
12 degrees is 2 small ticks.
12° ÷ 6° = 2
Converting a measurement into a count of equal marks makes it something you can actually point at on the dial rather than estimate by eye.
4.MD.C.6Analyze The Units7STEP 7
Draw the hand
So draw it two ticks past the 7.
Drawing a ray from the centre through a marked point on the rim is the same 'draw a ray from a given endpoint' move used when constructing any angle.
4.G.A.1Draw A DiagramAnswer
between 7 and 8, 12 degrees past 7
2/5 × 30 = 12
24 minutes is less than half an hour, so the short hand should be less than halfway from 7 to 8, and 12 degrees is indeed less than half of 30 degrees. Measured all the way round from the 12, seven whole hours put the hand at 7 times 30 = 210 degrees, and the extra 12 degrees brings it to 222 degrees — comfortably between the 7 at 210 degrees and the 8 at 240 degrees. The units are consistent too: the answer is an angle in degrees, and 12 degrees converts to a whole number of ticks (2), so the hand lands exactly on a printed mark rather than somewhere unmarkable.
Takeaway
The hour hand creeps all hour long, so at 7:24 it has crawled two-fifths of the way from 7 to 8 — 12 degrees, or two small ticks, past the 7.
- Find how many degrees one large division is worth
- Find how many degrees one small tick is worth
- Check the two times you already know
- Say what part of the hour has gone by
- Turn that fraction of the hour into degrees
- Read 12 degrees off in small ticks
- Draw the hand