When you strum you don't necessarily have to strum the entire chord every pick. You can break up the strum by picking the bass note or lowest note of the chord followed by strumming the rest of the chord in a follow up strum. This technique makes you sound like you are combing the bass part with the guitar chord part. Bass note strumming is very common in folk, country and other acoustic guitar based music.
The bass note of your chord is usually the root note of your chord. The root note of your chord names the key of the chord. So a G chord would have a G bass/root note, a D chord would have a D bass/root note and so on. When you play a chord on the guitar the lowest pitched note of your chord is generally the bass/root note.
To play the most basic base note strum in a 4/4 measure, you'd pick the single bass note on beats 1 and 3 and strum the rest of the chord on beats 2 and 4. The following example shows a basic bass note strum switching between the open G and C string chords. Note how the bass note of G is on the low 6th string while the bass note of C is on the 5th string. This means you need to be more precise when picking a bass note strum because the bass note is a single specific string.
When playing a base note strum in 3/4 time signature you only play the bass note on beat 1 and strum the rest of the chord on beats 2 and 3. This example is switching between G and C again but the time signature is now 3/4 and the bass note strum is adjusted accordingly.
Now that you know what a base note strum is, we can apply our strumming rules for eighth notes. That means any note on the beat is a downstroke and any note on the offbeat is an upstroke.
Same thing as the previous example but you are in 3/4 time now.
This little example shows a base note strum with eighth note and sixteenth note strums mixed in. The chords are also switching faster since you have 2 per measure now. If you can play this little song at a quicker tempo and repeat several times without mistakes you are well on your way to mastering strumming on guitar.