Answer: The answer to your question depends on why they are brown and dry. In some cases this happens because of physical damage -- the stem has somehow been broken. If this is the case, you will have to wait for the ivy to regrow.
In some cases this is due to lack of watering or a need for repotting or a poor repotting job which results in water running out between the soil and the side of the pot. You want to make sure the soil is kept moist but not soggy wet.
Another possible reason is an insect problem. Inspect the plant carefully under good light for evidence of pests. For instance, spider mites leave little webs under the leaves. If you think this is the cause, wash the plant with a strong spray of water and/or treat with insecticidal soap according to the label instructions. (You will also need to repeat this periodically about once a week to remove subsequent generations of the pest.)
In the meantime, remove any dead foliage or stems and keep the plant in bright to moderate light. Keep the soil moist and fertilize with a water soluble fertilizer for foliage plants such as Miracle-Gro according to the label instructions. As with most houseplants, this one will do better in cooler temperatures (eg 60's) and with higher humidity so keep it away from heat vents and if your house is dry, place it on a tray of pebbles soaked with water.
Finally, if by chance the "ball" is actually a second pot, be sure you are watering both top and bottom.
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