Remember, young children need a snack, and play time (this is their down time). Then they will be ready to face the homework.. Make a schedule so that every day when she gets home, you have snack, playtime and homework time. Let her decide in what order. Let her know the quicker she finishes homework, the more time to play. Have a special place that she always works on her homework.
We were told at the beginning of the school year that every week there was a spelling test, reading every night and a math work sheet for our daughter all through 1st grade. Then special projects would also be expected. We knew our daughter always had homework. Instead of asking "have you finished your homework", we always said, "bring your completed home work over here so we can see what you are learning" For the practice of the spelling words, some days , I would have her call them out to me.. I would make some mistakes so she could grade me. Sometimes, I would have a pan with rice for her to "write with her finger in the rice". In the car we would try to remember the word list "by heart""and try to remember how to spell the words..
She is testing you guys. Make her responsible for her homework, but remind her, it is becoming hard to believe her since she has started lying.
I am not suggesting this is the problem but, when I was in elementary school, I would also lie about homework. The reason? My father was scary about home work.. He would push and push, make me redo the things that did not look right and get frustrated when I did not remember the correct spelling of the words on the spelling list.
Also both of my parents worked, so by the time we all got home, everybody was tired, hungry and stressed. Homework was always "why do they have to give you homework? Why didn't you finish this earlier? What do you mean you do not remember what the teacher said about how to do this? " I was just trying to avoid the whole thing. I would rather lie to my parents than have to be stressed by all of this.