Easing into the A-stems:

The first group of nouns we studied were the o-stems, singular, and it all seemed straightforward:  masculines with

-os for the nominative  --> ἀγρ-ὸς

-ou for genitive    ---> ἀγρ-οῦ

-ô + i for dative    ----> ἀγρ-ῶι or ἀγρῷ

-o + n for accusative   ---> ἀγρ-ὸν

neuters differed only in using -on for both nom. & acc.-->  δένδρ-ον

Then we learned the plurals:

masc. has -o + i for nominative --> ἀγρ-οί

-ôn for genitive                       --> ἀγρ-ῶν

-o + i + s for dative                 --> ἀγρ-οῖς

-o-u-s < o + n + s                     --> ἀγρ-ούς

(So dative and accusative plurals just add -s to the singulars)

Neuters differ only in using short-a for nom. &  acc. --> δένδρα

That's how to make sense of it.

Just learn it.

Next we turn to the a-stem--these look like a mess

(3 different types, not counting the masculines)

These feminines start out oddly but the same basic algebra applies:

simple nominative singular ends in -â, then case endings are added as follows·

-â for nominative          --> ἀγορ-ά 'market(place)'

-â + s for genitive          --> ἀγορ-ᾶς

          (all genitives singular are cognate with our possessives 'his, hers'  & --'s)

-â + i  for dative   --> ἀγορ-ᾶι  or ἀγορᾷ

-a + n for accusative      --> ἀγορ-άν

And plurals follow a familiar pattern:

-a + i for nominative     --> ἀγορ-αί

-a + ôn for genitive       --> ἀγορ-ῶν (<<ἀγορ-ά-ων)

-a + i + s dative              --> ἀγορ-αῖς

-a + n + s for accusative -> ἀγορ-άς (<<ἀγορ-άνς)

Now the real trick comes with a historical change from long-α to -η.

This happens everywhere except after e, i, and r:

for a mnemonic 'try' πείρα (which means 'try' or attempt;

and notice the long a on the end does not change because of the -r-)

So to take the usual example, 'fountain' ends in eta:

κρήν-η nominative

κρήν-η-ς

κρην-η-ι

κρήν-η-ν

Notice this only affects the singulars:

the nominative plural has short-a --κρῆν-αι

genitive plural                        --> κρην-ῶν  (<< κρην-α-ων)

dative plural   a + i + s            --> κρήν-αις

acccusative plural                             --> κρήν-ας  (<<κρην -α-ν-ς

The last ending has a long-a that developed from the lost -n-, after the change to eta

 (so these long alphas don't change)

The 3rd type is peculiar because we start with a short-a, as in 'sea' θάλαττα

then borrow the long genitive and dative singulars             --> θαλάττ-ης

                                                                                                --> θαλάττ-ηι

before returning to short-a for the accusative                       --> θάλαττ-α-ν

Notice how the accent in nominative and accusative show that the -a at end is short.

The plurals for these end just like the others.

That's how to make sense of it.

Now just learn it.

Assignment:  Study grammar and vocab for  Ath 4a, esp. 40-41.

Then do exx. 4α--γ. For 4α you don't need a xerox of the verb chart: try to memorize the pattern of endings and then write out the conjugations of ἔχω and θεωρέω  from memory (without looking)