Mi hija barrió el suelo y yo recogí el polvo con el recogedor.

Barrió is the preterite (simple past) of barrer and means swept.

• barrer = to sweep
• barrió = he/she swept

It is used because the sentence describes a completed action in the past. In this context, your daughter finished sweeping the floor.

Form:

• barrer → barrió
• 3rd person singular preterite of an -er verb

Recogí comes from the verb recoger, which means to pick up, to gather, or to collect.

Here it is:

• recoger = infinitive
• recogí = I picked up / I collected

It is the 1st person singular preterite, so it matches yo.

A useful breakdown:

• yo recogí
• tú recogiste
• él/ella recogió

So the sentence is contrasting what your daughter did and what you did.

Yes, Spanish often omits subject pronouns because the verb ending already shows who is doing the action.

So recogí by itself already means I picked up.

But yo is included here for contrast or emphasis:

• Mi hija barrió el suelo y yo recogí el polvo...

This feels like:

• My daughter swept the floor, and I picked up the dust...

So yo is not required, but it sounds natural because it highlights the change of subject:

• my daughter did one action
• I did the other

In standard modern Spanish, possessive adjectives like mi, tu, su, nuestro usually replace the article, not go with it.

So:

• mi hija = my daughter

Not:

• la mi hija ❌

This is one of the basic patterns in Spanish:

• mi casa
• tu libro
• su coche

English speakers sometimes expect both words because English uses my differently from the, but in Spanish mi already does that possessive job.

Here suelo means floor as the physical surface you walk on.

In Spain, suelo is the most natural word in this context:

• barrer el suelo = to sweep the floor

Why not piso? Because in Spain, piso very often means apartment/flat. It can mean floor in some contexts and in some varieties of Spanish, but suelo is clearer and more typical here.

So for a learner of Spanish from Spain, barrió el suelo is a very good model.

Because polvo here works like an uncountable noun, just like dust in English.

So:

• el polvo = the dust

Spanish normally uses the singular for this idea.
Los polvos would usually sound different and could mean powders or refer to specific kinds of powder, not ordinary dust on the floor.

So in a cleaning sentence, el polvo is exactly what you would expect.

They are related words, but they are different parts of speech:

• recoger = to pick up / collect → a verb
• recogedor = dustpan → a noun

So:

• recogí el polvo = I picked up the dust
• con el recogedor = with the dustpan

This is a nice example of Spanish using related forms from the same root:

• recoger = the action
• recogedor = the tool used for that action

Because con shows the instrument used to do the action.

• con el recogedor = with the dustpan

That means the dustpan is the tool you used.

If you said en el recogedor, the focus would be more on the dust ending up inside the dustpan, not on the dustpan as the instrument. That can work in other sentence types, for example:

• Puse el polvo en el recogedor = I put the dust in the dustpan

But with recoger, con el recogedor is the natural choice.

Because the sentence tells a sequence of completed actions:

• your daughter swept
• you picked up the dust

That is exactly what the preterite is often used for in Spanish: finished events in a story or sequence.

Compare:

• Mi hija barrió el suelo y yo recogí el polvo...
  = completed events

Versus:

• Mi hija barría el suelo y yo recogía el polvo...
  = more like background, repeated action, or an action in progress

So the preterite fits this sentence better.

Yes, they are necessary in standard Spanish writing.

They show where the stress falls:

• barrió = stress on the last syllable
• recogí = stress on the last syllable

Without the accents, they would either be incorrect or could look like different words/forms.

For example:

• barrió is a past-tense verb form
• barrio is a different word meaning neighborhood

So the written accents are important both for pronunciation and meaning.

The given word order is the most natural and neutral one:

• Mi hija barrió el suelo y yo recogí el polvo con el recogedor.

Spanish does allow some flexibility, but changes in order usually add emphasis or change what sounds most important.

For example:

• Yo recogí el polvo con el recogedor y mi hija barrió el suelo.

That is still grammatical, but it shifts the flow and emphasis.

In your original sentence, the order is very natural because it presents:

1. one person and her action
2. then the other person and the next action

So for a learner, this is a good standard model to follow.