I have to quibble with a few points on this one, though it's mainly a question of doctrination.
The way the (British) Royal Navy do it, you fix at different intervals given where you are.
In Pilotage waters - every 3 minutes
In Coastal Passages - Every 6 minutes
In Ocean Passages - Every 12 minutes (Loran/GPS)
You would then DR on from every fix, using the 6 minute rule. To avoid any of the multiplication inherent in the rules above, the RN version is simple - the number of Cables (1/10ths of a Nautical mile) travelled in 6 minutes is equal to the speed in knots you are doing - for example, at 12 knots, 2 cables per minute. This makes things very easy if you have a stopwatch - you know every 30 seconds that tick by on the stopwatch, you have gone another cable down your nav track.
Anyway, I'm getting sidetracked. Every fix, you put a DR on, be it 3 minutes or 6 depending on how frequent you are fixing. This will give you a rough estimate of where you will be, not taking into account the effects of wind or tide. The DR is marked with a vertical dash on the line and 2 figure time.
However, for accurate reckoning of your position at a given time, you use what is called an EP - Estimated Position. You have 2 fixes on while steering the same course, you simply draw a line through the two fixes and continue it in the direction you are travelling. You then take the dividers and measure the distance between the two fixes, which has been travelled in a known time period - say for our purposes, 6 minutes. You use the chart to find out that distance, and you know your speed in knots. You use the distance between the fixes when marking off on the line where you will be in the next 6, 12, 18 minutes etc. However, instead of a line through your track line, you use a triangle, with the same 2 figure time. Thus you have an EP.