Using words to describe key features of functions can be tricky. What do we really mean when we say "increasing"? And what's the difference between that and "positive"?
Students will disagree about what these terms mean. They might try to lure you in to settle their disputes by asking, "So, what's the answer?" Don’t take the bait! Instead, let students lead you through their thought process. Encourage your students to focus on the terms that have similar connotations and notice what’s different about their graphs.
Keep an ear out for students who have head tilt moments—those who thought one thing originally but can now make sense of why that term means something else. Ask them this follow-up question: “Why do you think this term can be misleading?” When we let students lead us through their understanding instead of us leading them through our understanding, students begin to see themselves as capable mathematical thinkers.
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