Below:  Why don't rocks like these appear in beginning geology texts?  The answer is that they're confusing!  This rock isn't a plain hunk of sandstone;  it isn't a plain hunk of chert.  Apparently there were several events that contributed to its formation.

The rocks came from a sandstone formation. That tells us about a significant part of their geologic history, but it sure doesn't tell us everything.
There are some portions which have fossil seashells in them, especially on the grainy "rind" that's on the outside of some pieces. This, too, is consistent with a detrital sedimentary origin.
There are some unanswered questions: why a "rind"? Are these nodules of some type? What made the patterns form the way they did? Did the rock have to deform in the earth at all to yield these patterns? Was this rock formed simply by sedimentation and compaction, or was there infiltration of silica-rich fluids? If the latter was the case, was there hot fluid that cooled rapidly, or was it the same temperature for a long time? Did every part of these rocks form at the same time?
Without studying this rock extensively (or talking to someone who has already done so), one thing we do know for certain: it can take a whole sequence of geologic events, some not obvious, to form any given rock specimen.

The photograph below shows what looks like a fault in the rock. There are very tiny quartz crystals (not shown) lining the fault. These have special significance: they tell us that this rock didn't just crack from a hammer blow or something; the crack formed at some point in the rock's geologic history-- some point before the tiny quartz crystals could have formed. Another important clue: on either side of the fault there are bands or bedding layers that don't match up! While somewhat visible in the photgraph, this mismatch is really easy to spot when you inspect the rock in person.

Note also that parts of the rock show conchoidal fracture. These particular areas do not have any visible grains; even with a magnifier, they look like a solid, amorphous mass.

A look under the microscope should answer the question: are the jasper-like areas are made of distinguishable grains... or are they a cryptocrystalline mass? We could also look at the fracture habit by examining a freshly-broken surface. Does the fracture go through the grains or around them? Is it conchoidal, like agate or chert, or is it earthy and rough?

In many cases, the collectors who live near a specific deposit can tell you about how it formed.  Franklin-Ogdensburg collectors can talk up a storm about pneumatolytic zones and contact metamorphism, but picture jasper leaves this Franklin-Ogdensburg collector simply shrugging his shoulders.

I am really curious to saw this up on the slab saw so I can see what other banding patterns might be inside it.