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We were to choose to do page 22 or 23 today and dd chose 22. Anyway, at the top where it has sections a,b, and c, the problem are written horizontally like 509+365 and 746+254. Are these meant to do in their head or is it okay to write them vertically and then solve?

We make the kids show their work, even when they can do it their head.

I think sometimes when they write the problems that way, they are thinking that they will be done mentally especially if that is something that they have recently worked on. Both of the problems you mentioned would be easy to do mentally (500+300=800 and 65+9=74 so 874; 700+200=900 and 46+54=100 so 900+100=1000) so I would encourage that. If your dd finds that too difficult though, I would allow her to write it out then discuss with her afterwards how quick and easy it would have been to do it mentally. I did that with my dd because she really liked to revert to writing it all down and doing it the long way before even thinking about the problem. I really wanted her to develop her mental math skills so I would let her do it the long way, see how long it took, then explain to her the short cut way of doing it mentally. It helped her see how the mental math could be much easier and quicker.