Cherry Picking
Also known as: "taking something out of context"
You can knowingly misrepresent someone’s argument by picking out, highlighting, or responding to fragments of a larger argument, without the additional information that might change its meaning.
Examples
In 2008, Mitt Romney wrote a NY Times op-ed entitled (perhaps unwisely) Let Detroit Go Bankrupt. Over the next approximately thousand words, Romney explained how bankruptcy would be a first, necessary step in a larger recovery plan to rescue Detroit (where he was born, incidentally), but he was nevertheless dogged throughout his presidential campaign with the phrase “let Detroit go bankrupt.”
In 2012, President Obama was campaigning in support of Hillary Clinton’s senate campaign. Several times, he made a speech pointing out that business owners depended on government expenditures like roads, police, and the education system. In one instance, Obama made the mistake of uttering this sentence: “If you’ve got a business – you didn’t build that. Somebody else made that happen.” He has claimed – and past speeches support – that the “that” in the sentence referred to his prior claims about infrastructure, but his opponents repeated that sentence to mean that Obama was claiming business owners were not responsible for any of their own success.