Complex numbers are amazingly fun to teach in Algebra 2 because it is the first time students have ever seen them. It turns many of their previous “no solutions” into answers, how exciting is that? It also enables them to understand the majority of math jokes and memes!
I start my students off by letting them read about who uses complex numbers (because they always ask and I find this saves time), and then I let them read the Math Forum’s John and Betty’s Journey Into Complex Numbers.
Then, I show them this…
We have been studying patterns since the first day of school. Patterns are part of the beauty of mathematics. And the math they currently know does not let this pattern continue.
Enter complex numbers.
Some of them don’t buy it at first, but then I ask them, “Can you SEE zero cookies?” “Can you SEE -5 dollars?” Hmmmm…..
After the fun stuff comes the real work. And that is where the foldable comes in. I saw this awesome circular diagram idea for powers of i on Bonnie’s blog, Teaching On The East Side. I also used her great answer scramble idea (see below)!
I use graphing a + bi to motivate why the heck we can’t leave i the denominator and must use the complex conjugate to simplify. Fun times!
Here is the link to the word doc for the foldable.
After the foldable we needed a break before getting more practice so we played this Kahoot. I liked it because every slide has a fun math joke or meme involving complex numbers!
Finally, for the practice worksheet I made a Holiday Scramble! Where the answers are all on the board, but mixed up so you have to find them. This is great for the students that do NOT like to even see the hint of an answer before starting a problem, but do want to check their answers after they are finished.
Julie – I always love your enthusiasm. And your Holiday Scramble gave me a great idea which I want to share. I just received a task back from students on which there was an inordinate amount of sharing. The only way to get them to do their own work is to have them create something, and something that they can’t just find online. So what if I give them a Holiday Scramble – maybe 10 answers or so – and they have to create problems for each one, covering specific topics we have covered this term? Your cheery tree helped me, and I thank you!
Oh! That is an awesome idea! Please blog about it after you are finished! 🙂
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Do you have a link for the practice worksheet?
Wondering about your comment re: motivating the need to rationalize with the complex conjugate. What’s the purpose of putting it on the complex plane? That seems to require a decent answer to make it a motivator to use the complex conjugate.
We graph everything we do in my class. Everything. So when I introduced complex numbers it was just natural for us to graph them.
Groovy. I struggle with asking students to memorize skills like using a complex conjugate when the result offers them little in terms of application or understanding than the prior form. With all respect for the awesome work you share and way its prepared for students, I still don’t feel like graphing a complex number in itself is sufficient reason to learn how to use a complex conjugate, but.. it’s the best I’ve heard thus far. So point for that! 🙂
They have used the conjugate before with square roots. And for my class it’s perfect timing because quadratics are next. I don’t have them memorize the complex conjugate or the procedure for writing it in a + bi form. The conjugate is just the difference of squares. They have done that in Alg 1 so they usually recognize the pattern when given a couple of basic difference of squares problems. They sometimes do struggle with the procedure because they struggle with fractions in general, especially large equivalent fractions. But it’s great pre practice for rationals coming up and a great reminder that they are not changing the value of the number, just its appearance. There is so much great math going on in there and I love for them to be exposed to it all! 🙂
Hmm. Lost me in there a bit but loved where it was going. Changing forms is an awesome idea and valuable skill set. Still wonder – as CCSS authors do – if it’s valuable devoid of student interest. Simplifying root 24 is great if your trying to add root 24 and root 54 but also masks some things more easily seen like the approximate value of root 24.
No critique of you or this work but I’m still in a spot where graphing complex numbers needs an impetus before I can use that as the reason why we need to rationalize (is that even the proper term for simplifying a rational equation with a complex number in denominator?) a complex expression.
Hahaha. I sometimes call it rationalizing too for lack of a better word. I definitely see your point as it is a bear of a job for students, with all of the distributing and simplifying complex numbers. I think it is useful to learn about but don’t spend a ton of time on it. What other ways have you justified “rationalizing”? I would love to have more reasons to share with my students in order to motivate them. Or if you are still searching and find some more authentic ones please be sure to let me know. Thank you! 🙂
I’ve got nothing as solid justification, frankly. Being able to graph the point is the strongest reason I’ve heard. Somewhere I read a piece of writing from Bill McCallum that when authoring the CCSS they intentionally avoided any standard that mandate simplification based on the fact that algebraically manipulating expressions simplifies them in some ways only to muddy and convolute them in another. I’m fairly certain I recall his example being a simplification of the square root of 20 which I tried to allude to in that earlier post…. reducing root 20 is great if you’re adding it to the square root of 5, but likely inefficient if you’re asked to approximate it’s value.
My biggest frustration with a high school math curriculum is how often we stretch beyond the realm of what is clear and relevant to our students. A requirement to graph complex plane points is great, but then I would be struggling to justify THAT need…. I suspect I need to investigate how close the physics applications are to being relevant and accessible to my students. 🙂 Thanks for the good dialogue… keep up the blogging!
Thanks for sharing this foldable and your lesson and materials! I also enjoy teaching this in Algebra 2, and I find that the students seem to get a kick out of learning about complex numbers as well. They groan about how it stretches their brains, but most of my students seem to like this topic and how it feels like it is just at the edge of what they can wrap their heads around. I think that the puzzle itself is appealing for them, and as long as it’s in our standards, we can teach the complex conjugate just as an extension of this “puzzle.” However, the applications of complex numbers are mostly beyond what students can do at this point. We can say that there are applications in electrical engineering, but is an example of this meaningful for 10th graders?
When you say, “I start my students off by letting them read about who uses complex numbers…” I am curious about what resources you give them. Is there an article you keep handy for this, or do you release them to Google?
I release them to Google! I also have a short excerpt from a book about the origins of complex numbers. I find that even though they don’t understand how people are using them, just knowing someone does helps tremendously. 🙂