Part-time isn't a right like full time home educating is - you will need the agreement of the school for this... BUT that said a few people I know do it but they end up just wishing they could full time instead!!
My DH used to be dead set against it - what are his reasons? Mine didn't realise what home-ed was about really and once he realised I wasn't going to be sitting her down at a table and TEACHING her stuff it's more going out and exploring and learning as we go along...
She does have some maths and english excercises which we do from time to time but only when she wants to really as her nature isn't to sit still and learn, it's on the move learning but she's likely on the spectrum anyway
So we just follow her lead...
We do horse riding, forest school (she LOVES), Adventure playground is a great place for her to play with other kids, we go to free sessions run at an art gallery and she does gymnastics and drama lessons too - so she's not exactly missing out and I have my daughter back instead of the miserable surly child I had when she was in school...
I'm also finding it easier to work on any emotional issues as she has consistency being with me all the time instead of her trying to get her head around the school's approach (ignore it, stop making a fuss) and mine which is helping her to understand where those feelings come from and work with them instead of ignoring them...
Anyway I'm rambling and didn't mean to hit send last time!!!
Our days vary massively and yes some days the girls play by themselves for a bit while I do laundry or cleaning and we do go shopping and to the post office and etc but those are all potential learning experiences (certinly at this age anyway) - we learnt where the money from the charity shop goes, she helped a lady doing shelf labels at tesco - etc etc...
We have found that a bit of charm on our part and most working adults provided they are not busybusy are happy to give a 5yo a few mins of their time to explain what they're doing
And she will have time on the computer too whether it's just playing cbeebies games or doing a maths game which gives me time with Grace instead of both of them, or time to do chores (or sit and have a cuppa!!
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Ah right I'm def rambling now... Will post and run!