Today’s lesson will build upon the work that we did yesterday. We will continue to try to find roots of polynomial functions, but we will be focusing on functions that have imaginary roots. To get started, show students the second slide of the PowerPoint (Notes - Day 2 of Finding Roots) and ask students how they could quickly figure out how many roots each function has. They will likely say to check the graph, so have them use their graphing calculator to look for the number of x-intercepts. Ask them if they see a relationship between the equation of the function and the number of roots. They will likely have past knowledge from other math classes and may oversimplify to say that the exponent corresponds to the number of roots. Don’t correct them – this will make them think even deeper when we move on to the next slide! It is also fine if a student mentions the imaginary roots – it will lead nicely to the next slide.
The problem on the third slide of the (Slides 3 to 5) can be tricky if students oversimplify and say that all roots are x-intercepts (and I bet many of them will). Now this is true if the roots are all real, but now we have a function (y = x3 – 4x2 + 9x – 36) that has real and imaginary roots. Give students about 5 minutes to work on this problem in their groups. They will likely begin by graphing (Slide 3 Graph); some students may be surprised to see that there is only one x-intercept when they feel there should be three. If they get stuck, push them to still use the procedure that we learned yesterday – divide out the root we know and see if they can do anything with what is left. After they work in groups, come back together as a class and talk about the problem. If a student found the two imaginary roots, have them show their work and see if a different student can explain the work.
Next give students the problem from the fourth slide (y = x4 + 2x3 + 5x2 + 8x + 4). Start by asking students how many roots the function should have. Then ask them to graph to see if they can find any. They will know that there are four roots and they will see one from the graph (Slide 4 Graph), so they may think that three of the roots will be imaginary. Again, give students about five minutes to work on this problem with their group. Some students may remember from a previous lesson that a root is repeated an even number of times if it “bounces off” the x-axis, but it may be something you will revisit during the class discussion. Again, choose a student to show their work and have another student try to explain the work.
As a summary of the work we did today, show the last slide. This gets them thinking about the past two days and how today’s lesson felt very different than yesterday’s because of the imaginary roots. Revisit the rule about how the exponent of the function relates to the number of roots and see if students can be more specific if they didn’t get it right before the lesson.
Imaginary numbers probably seem very unnatural for students. Surprisingly, mathematicians did not arbitrarily decide that the square root of –1 is equal to i; the imaginary number i came out of necessity. Since this is our first time talking about i in this class, I want students to read an article (The Roots of Complex Numbers - Katz) that gives them some history about it. The article is taken from Math Horizons, published by the Mathematics Association of America in November 1995. I remember studying non-Euclidean geometry in college and how the history became an integral part of the curriculum and helped me to conceptualize many of the topics; I want my students to feel the same way. Give them time to read the article themselves. Although the math is somewhat complicated, they can still appreciate the information without making sense of all of the calculations. Having a quick class discussion after they read would be a good way to end the class.
Finally, an assignment is given to synthesize the last two days of class.