Do not be fooled by low prices at departments stores and on internet auction sites: all too often, these instruments are more like toys than real musical instruments. You wouldn't let your child learn to drive with a cheap car that doesn't function properly; don't ask them to produce a musical sound with an instrument of poor quality. String instruments are ALL, at least in part, hand made, so they come with a somewhat impresive pricetag. Instruments of good quality start about $800 for a violin, more for larger instruments. Allow the rental equity you are paying to help you purchase an instrument on which your child can be successful.
If you would like to purchase a musical instrument, please see your child's teacher before making any purchases. There are many options you should investigate: rental, local music stores, reputable on-line string stores, quality used instruments. Your child's teacher will want to try the instrument and discuss its qualities with you to assist you in making your final decision. I am available outside school hours for such appointments.