There are three basic types of yield curve trade.
1) Parallel shift in yield curve. This assumes a rise or fall in general interest rates without effecting the shape of the current yield curve.
2) Steepening yield curve. This assumes that the Long Bond futures rates will rise faster than near term Treasury Note futures rates.
3) Flattening yield curve. This assumes that the spread between short term Treasury Notes and long term 30yr T-Bondrates will compress.
Execution of Yield Curve Spreads
Typical movement of steepening yield curve:

Typical movement of flattening yield curve:

U.S. Interest Rate Futures response to parallel shift in cash yield:

The important note her is that the futures prices do not move on a one to one basis with cash yield changes. This example illustrates the point that volatility increases with time to expiration. Therefore, parallel strategies are based on a one to one spread contract ratio with the longer dated contract used to determine the direction of the parallel shift and the shorter dated contract used to help limit exposure.
Trading the shape of the yield curve requires eliminating the account value fluctuations of parallel shifts in the yield curve so that the account only experiences profit or loss based on the change of the slope of the yield curve itself. To accomplish this, adjustment in contract numbers corresponding to the volatility differential must be made. The easiest way to do this is to normalize the basis point dollar value movement.
10 basis points equalled a 15 point move in the 5 year notes which is $468.75.
10 basis points equalled a 24 point move in the 10 year notes which is $750.
$468.75 divided by 10 basis points = $46.875 per basis point.
$750 divided by 10 basis points = $75 per basis point.
$46.875 divided by $75 generates a ratio of .625.
Therefore, to trade the yield curve movement only, a spread would be created in the ratio of 1000 5 year note contracts against 625 10 year note contracts.
This position would create the following results for the following market movement:
Parallel shift - yields up:

Parallel shift - yields down:

And finally, Yield curve steepens - yields down: