From her oral history as archived by the National Visionary Leadership Project:

[I]n June 1953, Katherine was contracted as a research mathematician at the Langley Research Center. At first she worked in a pool of women performing math calculations. Katherine has referred to the women in the pool as virtual “computers who wore skirts”.

Their main job was to read the data from the black boxes of planes and carry out other precise mathematical tasks.

Then one day, Katherine (and a colleague) were temporarily assigned to help the all-male flight research team. Katherine’s knowledge of analytic geometry helped make quick allies of male bosses and colleagues to the extent that, “they forgot to return me to the pool”.

While the racial and gender barriers were always there, Katherine says she ignored them. She simply told people she had done the work and that she belonged.

At NASA, Johnson started work in the all-male Flight Mechanics Branch and later moved to the Spacecraft Controls Branch.

She calculated the trajectory for the space flight of Alan Shepard, the first American in space, in 1959 and the launch window for his 1961 Mercury mission. She plotted backup navigational charts for astronauts in case of electronic failures.

In 1962, when NASA used computers for the first time to calculate John Glenn’s orbit around Earth, officials called on her to verify the computer’s numbers. Ms. Johnson later worked directly with real computers.

…and a ton of other projects, including the Apollo 11 lunar trajectory, plans for a mission to Mars, and the Space Shuttle program.

Top photo credit NASA/David C. Bowman.
Center photo by Brian Koberlein.
Bottom photo via the 1962 NASA documentary Friendship 7, captured and enhanced by Colin Mackellar.
Bottom quoted text via Wikipedia.