Sasha and Keoni examine yet another equation of a parabola that is not in vertex form, y = x2 – 4x + 5. They start out by seeking a method to re-express the equation in vertex form.
Students’ Conceptual Challenges
Re-expressing this equation in vertex form is challenging. Many students have experiences re-expressing quadratic expressions in factored form. Keoni writes (x – 5)(x + 1). While this is true, it does not support their problem solving goal.
➤ After looking for geometric information, Keoni erases his factored representation. Sasha and Keoni keep working to find a way to re-express the equation in vertex form.
For use in a classroom, pause the video and ask these questions:
1. [Pause the video at 1:38]. Why did Keoni erase (x – 5)(x + 1)? What’s wrong?
2. [Pause the video at 3:00]. What do think of Sasha’s expression x(x – 4) + 5? Is it correct? Does it help?
Invite students to reflect on problem solving as a whole class. Elicit multiple answers from the class:
1. At 2:23, Keoni says “let’s not give up.” What do you do when you get stuck?
2. It seems that Sasha and Keoni make progress when the teacher asks how the equation y = x2 – 4x + 5 is different from y = x2 – 4x + 4. How does that question help them? How can they check that y = x2 – 4x + 5 is the same as y = (x – 2)2 + 1?
1. Rewrite the equation y = x2 – 4x + 7 in vertex form. How do you know the two equations are equivalent?
2. Rewrite the equation y = x2 – 4x + 3 in vertex form. How do you know the two equations are equivalent?