In 2018 I’ve started creating little exercises that target some focused aspect of playing. I think of these things all the time and often invent them during lessons.
I’ll post new ones in my blog and try to keep a running summary on this page. They appear in reverse chronological order for the year (so the newest addition is always at the top).
This is related to the “Organ Symphony” of Camille Saint-Saëns (1835-1921). Some players argue that excerpt is best performed with no alternate positions or valved notes. Others argue in favor of these things. Ultimately, it needs to be smooth, centered and in-tune.
Maybe this little exercise can help.
Here’s a link to the pdf of Alternate Organic.
This one is mostly for younger players. Because of those initial ‘band keys’ of B-flat, F and E-flat, younger players start to fuse 2nd and 3rd positions into a one-size-fits-all position somewhere between 2nd and 3rd position. During band scale tests, many band directors won’t call this out.
Over time this leads to a chronically flat 2nd position and high 3rd position.
This exercise helps to clarify the difference.
Here’s a pdf of Accu-Slide
This slur exercise is not just useful by itself but can demonstrate an important point. The intervals are derived from the opening of the Hindemith trombone sonata. Whenever we confront awkward articulated intervals, it can be very helpful to slur them as a practice technique. If the embouchure can cleanly navigate these leaps while slurring, chances are the tongued version will sound much better!
Here’s a pdf of these slurs.
This one is useful for soft control. One level of challenging is fading out to nothing while maintaining a good centered tone (and pitch!).
Another level of challenge is fading IN from nothing without a sudden bump in the sound. This one will take some time a patience to master but yield nice benefits. By sure to stay relaxed on ‘re-entry.’
Here’s a pdf of Faders 
Yellow Brick Road
In legato, I like to keep half steps in adjacent positions. In this octave of the A-flat major scale, that creates a conundrum. We would normally play B-flat in first followed by trigger C. However, that leads to an awkward leap to the D-flat. By playing the B-flat near fourth position (probably raised) we can allow the slide to glide out to 6th, facilitating the half step move to 5th.
Alternate positions are always a trade-off. You gain facility but risk intonation problems and uncentered notes.
If only there were a practice technique [like slowing it down] that could help us master our technique [you know, like slowing it down]. I’ll have to give this some thought [while practicing licks slowly]. I know it will come to me….
Here’s a pdf of Yellow Brick Road 
Achieving a solid sound and attack in the lower register takes work. You can’t force it. Personally, I (and many people) pivot the mouthpiece a bit in search of the best placement for lower notes. I believe all of this pivoting serves the goal of allowing the lower lip to set in the right place for best vibration.
Also, lower notes call for a more air stream to allow good resonance. Some people like to think of ‘warmer’ air such as that you would use to fog up a mirror.
Here’s a pdf of Octave Leaps 
These mini exercises also have a
dedicated page on TromboneZone.org.