Before you can perform any of Madam Kogawa’s Assignments in Hogwarts Legacy, you’ll need to purchase a broom for 600 gold from the Spintwitches Sporting Needs in Hogsmeade.
Assignment 1
The first assignment from Madam Kogawa becomes available after taking the book to the Map Chamber with Professor Fig during the main story. She will task you with popping 5 balloons over Hogsmeade Station and another 5 balloons around the Quidditch Pitch. First, travel to Hogsmeade Station, which can be found southeast of Hogsmeade. Next, you’ll need to pop the balloons by flying through them while on your broom. Afterwards, travel to the Quidditch Pitch, which can be found west of Hogwarts, and pop the balloons there. After you pop all 10 balloons, head to Madam Kogawa’s office near the Flying Class Lawn Floo Flame at the Bell Tower and speak with her to learn the Glacius spell.
Assignment 2
Madam Kogawa’s second assignment will unlock after some time passes from completing the first one. She’ll task you with popping 5 balloons near the Spires and another 5 by Keenbridge Tower. The Spires can be in the hills southwest of Hogwarts while the tower is southeast of Hogwarts near Keenbridge. After popping all 10 balloons, visit Madam Kogawa in her office near the Flying Class Floo Flame at the Bell Tower to learn Arresto Momentum.