Lauren Nowak

Capstone Project: CI 403

3 Day Unit Plan

Fall 2012

Table of Contents

Part I: Unit Introduction & Rationale Overview 2 Purpose 3 Meeting Student Needs 3 - 4 Part II: Formal Lesson Plans Day 1 5 - 10 Day 1 handout 11 - 14 Day 2 15 – 18 Day 2 handout 19 – 21 Day 3 22 – 25 Day 3 handout 26 - 29 Part III: Assessment Rationale 30 Unit Quiz 31

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Part I: Unit Introduction & Rationale

Background:

These 3 lesson plans are an expansion of two lessons I taught during field placement this

semester. I was told to teach more or less the same topics covered in a section in the textbook. These

topics included an introduction to i, powers of i, adding and subtracting complex numbers, multiplying

complex numbers, using the complex conjugate to simplify division of two complex numbers, and using

complex numbers to represent non-real solutions of quadratics.

In the book, this material was presented simply as “these are the rules, now practice them many

times,” with little explanation or sense-making offered to the students. One of the biggest challenges I

confronted with planning these lessons was balancing the very computationally-centered goals of the

text with the connections and reasoning I feel is essential in mathematics. To accomplish this, I tried to

contextualize this arithmetic through the use of the complex plane, which was not mentioned in the

text. I tried to take rules and formulas and, when possible, have the students search for the structure

and pattern themselves instead of having me immediately reveal it.

This process of balancing curricular goals with my own preferences involved compromise. There

are topics, such as seeing the complex numbers as vectors with magnitude and angle, or the graphic

explanation for multiplying two complex numbers, that I had to leave as “if time allows” extensions

instead of core parts of the discussion, due to both time constraints and the breadth of the material.

Despite these compromises, I feel these lessons present a much more coherent and intuitive

introduction to complex numbers that still covers everything I was required to teach. Moreover, I think

the process of modifying an in-place curriculum to fit your own teaching philosophy is an important part

of being an educator.

Instructional Strategies:

Overall, my instructional approach follows the pattern of, “Here is a bit of information. Now

take that bit and, either as a class, or in groups, or individually, let’s expand upon it in a way that makes

sense.” I take time to introduce an idea, and then give students time to play with this idea and construct

that concept for themselves. I chose this approach because complex numbers are not an inherently

intuitive topic. Since they aren’t something students will have encountered previously or have much

background with, I wanted to introduce some of the axioms and conventions myself (instead of defining

them as a group) so we could continue on to drawing conclusions based on those essential definitions

and prior knowledge working with the real number system.

An example of a specific action I take to make this approach work includes asking certain

students during groupwork-time if they would be willing to share their approaches at the board. I do this

because I want the students to take ownership of the mathematics (as I said to one student when

actually teaching, everyone here knows the teacher can do it; I want the class to see that someone their

own age can do it too!), as well as giving me more opportunities to evaluate where students are at with

their understanding. Though there are times when I am the one presenting new information, I make

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their participation essential to this process by always asking the class questions while we fill in the notes.

I also try to vary the difficulty of the questions I ask, for several reasons. It allows students from all

ability levels entry points into the lesson, and helps students stay more engaged since the cognitive

demands aren’t all of one intensity level.

Purpose:

One of the overarching mathematical goals of this unit is for students to develop a better idea of

what number systems are, how we’ve worked with them in the past, and how we can expand our

previous ideas and apply them to complex numbers. They will use their previous knowledge of

distributive laws, commutativity, grouping like terms, and rules with square roots / exponents to

perform arithmetic using imaginary and complex numbers. More generally, I want to promote the

mathematical idea (through the use of a number system) of axiomatic systems – we take something as a

definition, at face value, and from there use logic to construct rules and patterns from those axioms.

While we are doing this informally in this lesson, the process is mirrored (here is the definition of i; given

what we know about square roots, what does i2 mean?).

Another concept that we delve into more on the third day is the idea of functions, domains, and

ranges. They have worked with these properties with real graphs; again, I am asking them to extend it

when they plug complex points into a function. This should help them better clarify what it means to set

a domain and range for a function. Along with this, students are able to better develop 3-D spatial skills,

and can think about coordinates with 3 components to them (x, i, and f(x)).

Pedagogically, I want students to develop a mathematical disposition where they see rules as

the result of structure and our assumptions; they come about for a reason, instead of only existing

because the teacher said so. I also want them to develop a more cooperative vision of what learning is.

By having them work in pairs and groups, and consistently directing them toward relying on their

groupmates to work through questions instead of always asking me, I want them to see each other as

valuable sources of knowledge.

Meeting student needs:

- For ELL students, I would ask that they work in pairs with each other for the first two days, with

one of the more fluent students paired with a less fluent student. In addition to this, I would

provide a “key words” half-sheet of paper (included below) with key mathematical terms, since

these are words the students may not be familiar with in their native language and I want to

reinforce that, whether they are speaking Spanish or English, using precise math vocabulary is

important. In addition to these supports, boardwork will be detailed, so verbal explanations are

also written down (making it more accessible to ELL students). Finally, since I have some

intermediate Spanish skills, I would also take time during group activities to speak with the ELL-

student pairs and ask them questions, see what problems they are having, and encourage them

to share their work with the rest of the class, knowing I can help with some of the translating.

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- For students with IEP’s due to reading ability, I have structured most of the materials so there is

minimal reading needed to continue with the notes. All of the questions and instructions will

also be relayed verbally. For the “Project Quiz” formal assessment, these students will have the

option to have the quiz read to them, to take time during the beginning of the 4th day of class

(which will consist of finishing up group-work) and instead begin the quiz, or to come in during a

study hall or lunch to finish up the quiz.

- For students with varying academic levels and motivation, the above supports allow information

to be presented in multiple ways, and for processes to be explained in a clear, detailed fashion.

The note keys provided below represent how the work would actually look at the board –

everything labeled, organized, and thus easier to follow. In particular, one way I try to make the

notes more organized is by color-coding, especially by consistently using blue for imaginary units

when first introducing the concept. Allowing group work means students of different abilities

can rely on each other to help make sense of the content, and providing tasks at varying levels

of difficulty mean all students can become more engaged in the process. Since I can walk

around and talk with students during group work, I can get a better sense of where each student

is at and provide individual attention to help meet their needs. This works for students of both

high and low ability levels – for students who are struggling, I can offer a more personal

approach to helping them make sense of the problem; for students of high ability, there are

many conceptually difficult tasks (such as determining how multiplication of complex numbers

relates to the plane; how to graph a quadratic function when the domain includes complex

numbers) that they can work through at a pace appropriate for them. Finally, while students

may not find this topic to be “fun”, the fact that we’re exploring something new, and that this

viewpoint is emphasized (we don’t know how these weird numbers work! Let’s find out!) means

that students don’t have to be afraid of being wrong, since we don’t even know just yet what a

right answer might look like! Exploration and social learning also is more motivating than

lecturing on many new facts for students to memorize.

Important Words / Palabras importantes:

- Square root / raíz cuadrada

o √(4) = 2, “La raíz cuadrada de 4 es 2”

- Imaginary number / número imaginario

- Complex number / número complejo

- Complex conjugate / el conjugado de un número complejo

- Reflection / la reflección

- Rotation / la rotación

- Power = exponent / la potencia de un numero = la exponente

o Ex: x3, “la potencia de x es 3”

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Part II: Formal Lesson Plans

Note: Areas where I describe “anticipated student responses” are highlighted.

Day 1

Grade: 11th-12th

Class: Precalculus

Time allotted: 50 mins

Number of students: 26

Complex Numbers: Introduction

I. Goal

To introduce students to basic topics regarding complex numbers: the imaginary unit i, the complex plane, the complex conjugate, and adding complex numbers

II. Objectives

Students will identify i as the square root of -1

Students will simplify powers of i

Students will graph complex numbers on the complex plane

Students will add complex numbers

Students will identify the complex conjugate, and explain its graphic interpretation

III. Materials

Notes sheet for each student

Calculators – optional; some students may want for adding / evaluating square roots

Enoboard with notes sheet available as OneNote file

IV. Motivation (5 mins) – Before class officially begins, make sure:

to hand out notes sheets to each student

that Enoboard works and has notes up on the screen

that today’s agenda is written on the board (2.4 notes part 1 – lecture #1, pairwork #1, pairwork #2, lecture #2)

Ask students to choose their partners now & make eye-contact so that when they work in pairs later, they know who they’re working with

Begin class by asking students to brainstorm whatever they remember or think might be connected to complex numbers. Give them 30 seconds. Most students will not remember anything about complex numbers from their Algebra II course; however, a few of them might. When asking them to share their results, start by asking, “Who has absolutely no idea what a complex number is?” and take a tally. This way, students who didn’t remember

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anything still feel like they’ve contributed to the activity, and the instructor knows roughly where the class stands. After that, have the rest of the students share what they remember.

Next, introduce the topic by acknowledging that imaginary and complex numbers are strange.

“They sound fake, right? Why would we be working with fake numbers? Well, we do it all the time. When you all were in Kindergarten, the only numbers you knew about were 0, 1, 2, 3…maybe some of you had an idea of what infinity meant (and infinity itself is kind of made-up and weird too, right?), but beyond that, those were the only numbers you knew. But when you were a bit older, you learned about new numbers. You learned about fractions, and they were weird. How can you have 6 things and divide them into 10 pieces? How can you multiply something by ½? So I want you to think now - what are some other weird numbers we’ve learned about? [student replies may include pi, negative numbers, variables, decimals). Exactly. Each time you learned about those types of numbers, it took practice before you felt comfortable with them, before they stopped feeling ‘fake.’ So I want you all to keep that in mind as we start working with imaginary and complex numbers”

V. Lesson Procedure (44 mins – 5 mins beginning lecture, 10 mins pairwork #1, 5 mins class discussion, 10 mins pairwork #2, 10 mins class discussion, 4 mins ending lecture) 1. (In general, you can anticipate at least one student remembering something about i or

negative square roots, so reference that student’s contribution when beginning the notes). “So [student] mentioned that imaginary numbers have something to do with square roots. That is exactly right. Just like we invented negative numbers to answer questions like ‘What’s 2 minus 10?’, imaginary numbers were invented to answer the question ‘What is the square root of -1?’” Fill in the first two blanks on the note sheet; explain how i2 = -1 by linking it to the idea of square roots and squares cancel; provide concrete example of sqrt(4) to help contextualize this. When substituting in √(-1) for i, ask students to fill in this substitution (this way they get used to the idea that they are interchangeable), and also show how i2 = i*i (so students are primed to break apart exponents this way when working through higher powers of i)

2. Next, fill in the number line with the different types of numbers we’ve learned. This should be done like a story, told visually: how our number systems have expanded. Mark zero and draw an arrow to the right, marking the counting (natural) numbers – 0, 1, 2, 3,… Then fill in fractions, ½, 16/3. Add in pi, then add in the negative numbers in the opposite direction, all the while commenting on how “We then expanded our number system to include _______ [ex: fractions] so we could answer questions like _______ [ex: how do you divide 16 into 3 parts?]”

3. Ask students, “So now we introduced this idea of i. Where would i go on this number line?” Expect silence; maybe one or two guesses. Students may not contribute, or one student might say “It doesn’t go anywhere on there.” Once the question has had time to sink in, reveal: “When mathematicians asked themselves the same question, they realized i doesn’t fit anywhere on this line. So, they drew a new one!” Draw in the imaginary axis, using blue (in general, try to use blue when working with imaginary units), label i. Ask them, “so given where i is, where might 2i go? –i?” Demonstrate that it is the same as the number line we’re used to, only now they are imaginary numbers. Also ask them where i2 goes; refer back to how we defined i2 above. Point out that even

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though this is a number that has i in it, it is actually a real number in disguise. Finally, say that this system of 2 axes creates the complex plane (fill in on notes sheet).

4. Tell students that they now are going to explore this new system in pairs by figuring out where other real and imaginary numbers would go on the complex plane. Also show them that you want them to look for a pattern from part a. Before asking them to move, tell them that when you want them to just pause for a moment and listen to an announcement, you will make a “time-out” sign. When you want everyone to come back together and conclude the pairwork, you will signal by making a “come here” motion with your hands and will stand at the front.

5. Let students get up and move into their pairs. Circulate the room, answering questions and observing student work. Expect to see some of the following:

Most students will have a hard time knowing where to plot the higher powers of i. For a first hint, ask them to look back at how we made sense of i2. Students may also need to be reminded of the rules for breaking up an exponent; if this is the case, see if they can first ask another student to explain the exponent rules. If you notice most students are struggling with this, call a time-out and ask one of them to explain how they broke up i3 into i*i2.

Once students have noticed the pattern, ask them where they think higher powers of i would go. In particular, if a group finishes very quickly, ask if they can develop a general rule for evaluating higher powers of i.

Most everyone should be able to plot part (b). The one problem they may have is with the square root in y; if this is the case, ask the student to look back at how we defined square root of -1 at the beginning of the notes.

When a pair has made a good discovery – they’ve figured out the pattern, can explain how they evaluated one of the powers, or can plot the other points, ask if one of the two (or both of them!) would be willing to present their answer once we come back together as a class. Encourage students that they have the right idea and you’d like their classmates to see that their peers are learning to make sense of this content (instead of it only coming from you as the teacher). Also, be sure to ask students of varying ability to contribute, on both smaller/computational parts and on higher-level, conceptual parts, so everyone in the class feels invested in the activity and valued as part of the process.

6. Once students have made decent progress through the first part (either when everyone has worked through some of the powers and some of the points in part b, or once 10 minutes have passed), signal for everyone to come back together as a class.

7. During the class discussion, have the students you previously talked to present their work, highlighting the main take-away points after they give their own explanation.

8. To segue into the second round of pairwork, fill out the bottom of the sheet together. Call-and-response: What kind of numbers were we adding for w and x? Real numbers. What kind of numbers were we adding for y and z? Imaginary numbers. Then say: “So what I’m wondering now is, what if we try to add w and z together? Think-aloud: So w was 3+4, z was i-4i, so we rewrote them as z and -3i…now what? How do we smoosh those two together?” Students will, again, probably be silent – some might say you can’t. Tell them, “Well, I’m not going to give you the answer to w+z – you all will figure that out in your pairs again. What I am going to do, though, is introduce a new vocabulary term”

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9. Introduce standard form; highlight now 7-3i from above is in standard form. Point out that this is a hybrid number – it has both real and imaginary parts – and we call these hybrid numbers “complex” numbers.

10. Dismiss them into their pairs again, asking them to rewrite the numbers from part b above into standard form, and then tell them they’re going to try and figure out how those hybrid numbers look when they’re on the complex plane.

11. During pairwork:

There are several cognitive leaps they need to make at this portion – this is a more demanding task than the last one, so expect many students to have difficulties.

Problem spots include recognizing that they need to put 0 in for the real/imaginary part when rewriting; making that initial jump to plotting them as points with a real part and an imaginary part; dealing with –sqrt(17) and sqrt(-9); adding component parts

Since many groups may have trouble making that jump to plotting complex numbers as points off of the axes, if you recognize one group that made that jump, call a time-out to have them share their insight. If no group makes that jump on their own, guide one of them toward it by showing how for 3+4, we moved 3 units right and then another 4 units right on the real line; we also didn’t move up and down at all, because it’s a real number. Similar for i-4i; this will get them thinking about moving both up/down and left/right at the same time, and they should be able to get there. Once students make this connection, plotting the points is trivial, and if they have trouble adding two complex numbers, you can again refer to the idea of how many spaces they’re moving on the imaginary axis and on the real axis

Ex: “(3+4i) + (3-4i)…so we move +3 real units (right), then +4 imaginary units (up), then +3 real units again, then -4 imaginary units (down). Where did we end up?”

Students should recognize this as adding like terms, similar to when they work with variables

With –sqrt(17), students might try to put an i in there. Ask them for the difference between –sqrt(1) vs. sqrt(-1), and tell them to apply that there. [This promotes precision]

With sqrt(-9), ask them for their ideas – well it probably has an i in there somewhere – how can we find it? Review breaking apart square roots using a real example (such as sqrt(4*4)), show them how they can “find” the sqrt(-1) and simplify into a term with i in it.

Again, ask students to be ready to contribute on a specific part once we come back together as a class.

12. Run class discussion as before. If there is extra time, introduce the extension now. If there is little time left, have student presenters go through the work for a, d, e, a+b, and a+d; provide only the answers for the other problems. When going through these, be sure to highlight tricky computational parts (such as rewriting the negative square root in i form; the fact that you can’t simplify 3-sqrt(17))

13. Tell them you’re going to introduce one more new vocabulary word – complex conjugate. Define it, and provide the example that a and b from the activity above are complex conjugates. Write out 3 more complex numbers, with at least one of them

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having no imaginary part, and one of them subtracting i, and quickly ask the class to call-and-response for the complex conjugate for each of those numbers (students catch on to this easily, so call-and-response and chorus responses are acceptable here).

14. Plot all of these pairs of complex conjugates, using a different color for each pair, and a different type of circle for the conjugate for each pair. Ask them to examine the graph, think individually, and when they have a hypothesis for the relationship between the pairs to raise their hand and share their hypothesis. With this many points, students should be able to recognize that it is a reflection across the real axis.

VI. Closure (1 min)

Provide a recap of the work they finished that day, and explain where we’re going next.

“You all have done a lot of great work today. We’ve started making sense of i and have figured out how to add and subtract complex numbers. Now that we know how to add and subtract, tomorrow we’re going to learn how to multiply and divide, and this is where the idea of a complex conjugate is going to come in handy.”

VII. Extension – linking addition to adding vectors, tip-to-tail

If time allows, when students have finished adding complex numbers / before proceeding to the complex conjugate, introduce the idea of complex numbers being vectors with magnitude and direction. See if anyone in the class has done vector addition in their physics class. If so, ask them to explain; otherwise, provide an explanation of tip-to-tail addition of vectors. Ask each pair to select a different two pairs of points, and have them check that this definition of addition also checks out with the addition they just did.

VIII. Assessment

Assessment occurs constantly throughout the lesson. The largest source of assessment data is the time the teacher spends circulating the room while students work in pairs. In addition to these observations, assessment occurs when pairs present their explanations to the class, and when the class is asked to volunteer information during the lecture portions of the lesson.

IX. Standards

1. Common Core Math Standards

o CCSS.Math.Content.HSN-CN.B.5 (+) Represent addition, subtraction,

multiplication, and conjugation of complex numbers geometrically on the

complex plane; use properties of this representation for computation. For

example, (-1 + √3 i)3 = 8 because (-1 + √3 i) has modulus 2 and argument

120°.

o CCSS.Math.Content.HSN-CN.A.3 (+) Find the conjugate of a complex

number; use conjugates to find moduli and quotients of complex numbers.

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2. Common Core Practice Standards: The two most prominent standards for mathematical practice promoted in this lesson are:

o CCSS.Math.Practice.MP7 Look for and make use of structure.

Students are asked to use previous structures, particularly their understanding of the real number system (addition, exponents, square roots, graphing), to come up with and use analogous practices in the complex number system. They also learn to work with given definitions and extrapolate how those definitions could lead to certain rules or behaviors (such as the cyclic nature of powers of i)

o CCSS.Math.Practice.MP6 Attend to precision.

Students must be precise when graphing and when working through multi-step arithmetic in order to accurately complete the activities and to arrive at solutions that make sense.

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2.4 - Complex Numbers

Brainstorm: What do you remember about imaginary or complex numbers?

Expanding our number system:

We've expanded our number system many times in the past

Imaginary numbers were created to help us answer the following question:

√(-1) = _______

i 2 = _______

Graphic interpretation:

We can plot imaginary and real numbers on the ____________ plane

Above, plot i and i 2

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Part I: Finding real and imaginary numbers on the complex plane

In your pairs, plot the following numbers on the complex plane:

(a) i (b) w = 3 + 4

i2 x = (-6)+2

i3 y = 2i + √(-1)

i4 z = i - 4i

i5

i6

What pattern do you notice from part (a)?

What types of numbers are being added to create w and x? ________________

What types of numbers are being added to create y and z? ________________

What if we added w + z? Where would this go on our plot?

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Part II: Plotting and adding complex numbers

Standard notation: z = ___ + ___ ___

Rewrite w, x, y, and z from part (b) using standard notation

Plot:

(c) a = 3+4i (d) a + b =

b = 3-4i

c = -3+(5/2)i c - e =

d = -√(17)-6i

e = 2+ √(-9) a + d =

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Definition: The complex conjugate of a + bi is ___________

Examples:

On the graph, this shows up as:

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Day 2

Grade: 11th-12th

Class: Precalculus

Time allotted: 50 mins

Number of students: 26

Complex Numbers: Multiplication and Division

X. Goal

To introduce students to the processes for multiplying and dividing complex numbers

XI. Objectives

Students will multiply complex numbers by powers of i

Students will explain how multiplying by i links to rotations by 90 degrees

Students will multiply complex numbers, including complex conjugates

Students will simplify fractions consisting of 2 complex numbers

XII. Materials

Notes sheet

Smartboard

Whiteboard & markers

(Extension – graph paper)

XIII. Motivation (5 mins) Before class officially begins, make sure:

to hand out notes sheets to each student

that Enoboard works and has notes up on the screen

that today’s agenda is written on the board (2.4 notes part 2 – lecture #1, pairwork #1, pairwork #2, lecture #2)

Tell students they have to work with a different partner today. Again, ask them to choose partners now & make eye-contact so that when they work in pairs later, they know who they’re working with

Begin class by referring back to yesterday’s results: “With your new partner, take one minute to go over one main idea or fact from yesterday’s assignment. After the minute is up, we’re going to go around the room and each pair is going to share what they chose”

XIV. Lesson Procedure (40 mins) 1. “[Student / no one] mentioned a pattern that we noticed yesterday – that when we

took powers of i, we saw a cycle. Let’s review that result again briefly.” 2. Have students supply answers for 1*i = i; i*i = -1, etc (see notes sheet). Be sure to

emphasize “So we multipled by i again and our new point was…” This way, students can start to see it as repeatedly applying a transformation to the original point 1.

3. Graph those points again, and ask students to describe, visually, what happened to 1 (Should mention that it moved around, that it stayed on those 4 points, that it rotated.

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If a student mentions a reflection, ask them to pay attention to what order the points occurred in. Also note that we did the same thing every time (multiplied by i), so if it was a reflection, we’d have to reflect across the same line every time.)

4. As a class, ask for predictions for what will happen when you muyltiply (1+i) and(3+5i) time i. Try and estimate where to plot those points. Student may have trouble with 3+5i (may want to reflect it over the imaginary axis instead of rotating), so draw in lines to help them estimate a 90 degree angle. Also point out the idea that the distance from the origin has to stay the same)

5. After making these guesses, tell students that multiplication rules for complex numbers should make sense and align with this visual idea. Ask them, for i(1+i), how they would have multiplied if they were working with regular numbers or variables (they will all be familiar with distribution, and so can supply that answer). Work through the distribution together, asking at each step, “What’s i times 1? i times i?, and then plot the point. Do this again, but a bit faster, with the second point, and they should see that their predictions and the multiplication do match up.

6. Tell students for the next part they’re going to work in their pairs. Tell them that before they pick up any pencils and start writing, though, you want them to just talk with their partner and decide on their predictions. Emphasize that this is a new system that we don’t know the right answers to yet, so an important part of learning to make sense of something new is by making guesses and checking them. Give them 2 minutes to talk in their pairs, then tell them they can pick up their pencils and check their answers by doing the multiplication.

7. While students are working in their pairs, check and see what their predictions were, and if groups didn’t make predictions, ask them what was difficult about making a guess (they might have felt intimidated by getting it wrong, or feel like they still have no idea what’s going on with multiplying these numbers, or they didn’t want to waste time guessing when they can just get the right answer by doing the arithmetic. Amongst these varied possible answers, respond in a way that reinforces that it’s okay to be wrong since we’re exploring this together; if they have no idea what’s going on, tell them it’s great that they recognize what’s giving them trouble and encourage them to pay extra attention to any patterns they might see). Again, ask certain students or pairs of students to share their ideas when we come back together.

8. If pairs finish quickly, ask them to double check that their rule works for a different point. Also ask them to look at how –i and i3 are related, and if this makes sense with the rule they came up with.

9. Have students present their findings. 10. Transition: Now we’re going to move away from the complex plane for a little bit, and

spend some more time playing with this idea of multiplication. So, keeping in mind how we used distribution to multiply by i, I want you and your partner to try the next few multiplication problems in your notes. For the second two problems, you’re multiplying complex conjugates, so for the last one, I want you to fill in the blank with the complex conjugate for 2 + sqrt(-1), and like we’ve been doing, look for an overall pattern”

11. Most pairs should be fine using the distributive property; if any seem stuck, point them in that direction. Pairs may make algebra mistakes, but they will be able to correct these when it is reviewed as a class. While they shouldn’t have a hard time finding the pattern, if it is a problem, ask them to think about what happened to the real units? What happened to the imaginary units? They should see that all the imaginary units got cancelled out

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12. Work through the algebra at the board, writing out each step explicitly as denoted in the key, and asking students to supply answers for each step (2*3 is? I2 is?). When working through the pattern-noticing question, have students supply their thoughts, and write them up on the board. Also reference the comic included

o Note: While this comic does contain a mild obscenity (blacked out), these students are old enough to have been exposed to that kind of language. To address it, as the teacher you can use humor (I don’t know what you’re talking about…all it says there is “It gets real!”) This comic is included for a purpose – the phrase it is playing with is a common phrase, and the humor will help students connect the idea of complex conjugates turning complex numbers real.

13. Work through the first division problem as a class. Before doing so, ask students what they would do when they had a square root in the denominator of a fraction – again, at this point, most students will recognize that they multiply by a version of 1 to cancel the square root. Ask what they might multiply the denominator by in this problem so it is easier to work with; since they just used the complex conjugate, this connection should be obvious.

14. Again, be sure to write out the algebra for this step very clearly and explicitly. It is a longer problem to work out at the board; some students might disengage a bit while wading through all that algebra, but it is good to have one example problem up at the board so students can at least see what the process of dividing two complex numbers looks like, and have a model for how to keep their own work organized.

15. After finishing the first problem, ask the class if they would like to do the second problem together as a class or individually. Either way, try to get students to complete sections of the problem (i.e. can someone write the complex conjugate up here? Can someone multiply out the numerator? The denominator?) to help keep engagement up during a more arithmetic-heavy problem

XV. Closure (5 mins)

Close class by taking a moment to again recap the types of operations (multiplication and division) that we now can do with complex numbers.

XVI. Extension: If time allows, give graph paper to each student. Have them plot a complex number on the plane, and have them multiply that number by many different real numbers, and draw in the arrows for each vector. Help them see that this is the same as stretching or shrinking the arrow, depending on what real numbers they chose. Also have them try multiplying by imaginary numbers other than i (try 2i, for example) to see that it both stretches and rotates the number.

XVII. Assessment: Again, student participation and supplying of answers is necessary and consistent throughout the lesson, through the use of groupwork, teacher observation, board work, call-and-response, and fill-in-the-blank lecturing.

XVIII. Standards

1. Common Core Math Standards

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o CCSS.Math.Content.HSN-CN.A.2 Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

o CCSS.Math.Content.HSN-CN.A.3 (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers

o CCSS.Math.Content.HSN-CN.B.5 (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.

2. Common Core Practice Standards: This lesson is very similar to the lesson from the day before, and again promotes most prominently:

o CCSS.Math.Practice.MP7 Look for and make use of structure.

Students are asked to use previous structures, particularly their understanding of the real number system (exponents, square roots, graphing, multiplying by a version of 1), to come up with and use analogous practices in the complex number system. They also learn to work with given definitions and extrapolate how those definitions could lead to certain rules or behaviors (such as multiplication by i being equivalent to a rotation by 90 degrees)

o CCSS.Math.Practice.MP6 Attend to precision.

Students must be precise when graphing and when working through multi-step arithmetic in order to accurately complete the activities and to arrive at solutions that make sense. This is especially true during distribution, as well as paying attention to negatives.

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2.4 - Complex Numbers, continued

What happened when we multiplied 1 times i? i times i ?

What does this look like on the graph?

What do we get when we multiply ( 1+i ) times i ? (3+5i) times i?

Make a guess: What happens when we multiply

◦ ( 1+i ) * i2

◦ ( 1+i ) * -i

Check your guess

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Now, we’re going to move away for a little bit from this idea of the complex plane, and delve

more into the ways we can manipulate complex numbers.

Multiplication, continued:

How do we multiply ( 2+3i )( 3+4i )?

Multiplying complex conjugates:

(7 + 3i)(7 - 3i)

(2 + √[-1]) ( )

What seems to always happen when we multiply complex conjugates?

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Division - How do you divide:

3 + 5i 2 + 6i

5 - 6i 8 + i

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Day 3

Grade: 11th-12th

Class: Precalculus

Time allotted: 50 mins

Number of students: 26

Complex Numbers: Roots of Quadratics

XIX. Goal

To introduce students to solving for the complex roots of quadratic equations, and interpreting these roots graphically

XX. Objectives

Students will solve for the complex roots of quadratic equations

Students will check that these roots do return f(x)=0 when plugged back into the equation

Students will explain where these roots exist when graphed in a 3 dimensional system

Students will construct their own 3D models of quadratic equations with complex values in the domain

XXI. Materials

Notes sheet

Enoboard

Graphing Calculators

Graph paper

Cardstock

Pipe cleaners

Scissors

Tape

XXII. Motivation (5 mins) Begin class by asking students to work through the first problem individually. This is a routine,

easy problem for precalc students, so most should finish it within a minute or two. Go through

the answers as a class. Then, work through the next graph as a class, asking students to supply

answers (What is the domain? What is the range?) as in previous lessons. When it gets to the

point where students are solving for the roots of x2+1, first stop at “Nonreal roots”, to show

them where they would usually stop the problem. Then continue as shown in the notes,

pointing out to the class that now we have a way to describe these nonreal roots more precisely

by using complex numbers.

Take time to make sure that the idea of the domain being “What I plug in to my

function” and the range being “What I get out of my function” is clear. Ask students to

explain how they got the domain they listed, or why a certain point isn’t included in

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their range. Highlight and praise any students who provide explanations that develop

this idea.

Once you do get the roots i and –i, ask the motivating question for today’s lesson: Where are

these roots on the graph? In the one above we saw the roots as x-intercepts…what’s going on

here?

XXIII. Lesson Procedure (47 mins – 10 mins for intro, 37 mins for groupwork) 1. Tell students that this question is your challenge for them for the next two days: finding

a way to link these roots to a graph; to make them visible. Pass out the activity sheet

that Introduce students to project layout.

2. Go over the expectations for this project. All of the supplies will be located at the front

of the room; groups can get what they need once groupwork starts. Explain each of the

roles: The recorder will write down all of the group’s work in a master copy; the

proofreader will check their math for accuracy; the constructor will be in charge of the

creation of the graph (do not say, at this time, that it is a 3D model!); the group

representative will be the one who reports the group’s findings when we have class

discussion, and is also the only person who is allowed to ask the teacher questions (this

is to encourage them to ask questions to each other first, and to work as a team in

deciding that a question should be asked).

o Also explain that, even though the recorder is creating a group copy, each group

member is going to need their own individual copy of the work being done for

the quiz tomorrow

o Ask them to work with their partner from the first day and one other pair.

3. Go over the first section of the worksheet. “Your first task is to work with the equation

we just had, x2+1. Note that I’ve replaced the x with an a, but it’s still the same

equation. In the first part, I want your group to do the calculations yourself for how to

find the roots. Then, fill in the table of values. Note that you’re given values for f(x)

instead of x. Finally, I want you to find a way to graph all the points you get in the table,

and then begin that process.

o Since this part requires more verbal explanation, make sure to check in on the

ELL students and make sure they also understood the directions and clarify if

needed.

4. Allow them to begin. Most of the period will probably require this time for group-work.

5. During group-work:

o Keep reinforcing the norms set by the group roles; if a group isn’t on task, ask

who their group representative is, or what the recorder has written down so far,

or where they are stuck (Groups are often off-task when they feel like they can’t

easily make progress with what’s at hand).

o During the algebra for the table of values, students might need reminding that

they are setting F(a) equal to something instead of a. It is acceptable for

students to use a graphing calculator to get some of the values; the values that

require complex solutions won’t turn up on a calculator, so the main purpose of

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this activity as an application of this arithmetic isn’t harmed by the use of the

graphing calculator.

o The most difficult part of this task is coming up with a method to graph these

points all on one graph. Help guide groups toward the idea of combining both

the complex plane (x/imaginary axis) and the x/y plane into a 3D system. If you

hear a group moving in this direction, it might be beneficial to call a “Time-out”

and have that group’s representative share their method, so other groups can

begin the more time-intensive task of actually constructing these graphs

o Students will be creating graphs that have 2 parabolas – the parabola in the real

plane, and one underneath it that extends into the complex plane. The teacher

should monitor how groups plan on using their materials. Constructions that

work better involve using two sheets of card stock with graph paper already on

them (having axes drawn and labeled before constructing will also be easier),

and one can cut a half-slit into each sheet of cardstock so they meet each other

at 90 degree angles and form an “X”. Again, if a group comes up with a

particularly useful method for physical construction, share it with the class

o Algebraically, students may also have trouble wading through the quadratic

formula, or realizing that they can use the formula in the third problem. If a

student is only having trouble with the algebra, have them check their work

with another student who is computationally strong. Point out, when solving for

H(x) = -2, that they can turn this into a problem they know (solving for a zero) by

moving the 2 over.

6. Once groups have had a chance to make significant progress or complete the first graph,

come together as a class to discuss the following questions:

o What is the domain and range for this new graph? How does this compare to

the graph when we only looked at a domain of the reals? (Students might say

the domain is all complex numbers, but this isn’t quite true – they just took a

single line from the complex plane. Steer them toward noticing this by asking

them where they plotted the point for, say, 5-7i. For the change in range, they

should notice that now the f(x) or y-value can now take on any real number,

whereas for any parabola in the real plane, we always have y values we can’t

reach!)

o What angle is formed between the two parabolas? (90 degree angle). Where

have we seen that angle before (multiplying by i)

XXIV. Closure (3 mins)

“So before we head out today, I’d like to go around the room and have each group describe what they discovered when trying to create these graphs. You can tell me what your graphs looked like, what was something difficult you ran into when making them, or something new you’ve learned from today’s activity”

XXV. Extension

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While it is unlikely that groups will be able to complete this entire construction activity in the time allotted, if there was a group that was extremely efficient in their constructions, they could begin the “Project Quiz” that was planned for the next day early. Groups could also be asked to look for the zeroes and the “Phantom Graph” associated with other polynomials, such as x4+1.

XXVI. Assessment

As in previous lessons, groupwork allows for teacher observation of student progress. The final models students are constructing also allow the instructor to gauge the depth of understanding of a group. Students will be providing verbal feedback on their understanding through the class discussion. They will also be keeping a record of all their computations on their activity sheets, which will be handed in the following day with the “Project Quiz” for a grade.

XXVII. Standards

Common Core Math Standards

o CCSS.Math.Content.HSN-CN.C.7 Solve quadratic equations with real

coefficients that have complex solutions.

o CCSS.Math.Content.HSN-CN.C.9 Know the Fundamental Theorem of

Algebra; show that it is true for quadratic polynomials.

2. Common Core Practice Standards o CCSS.Math.Practice.MP1 Make sense of problems and persevere in solving

them.

Students are being presented with a nontrivial problem: how to graph a function with complex inputs and real outputs?, and must try several approaches, as well as be willing to work through the arithmetic, in order to accurately portray this concept.

o CCSS.Math.Practice.MP5 Use appropriate tools strategically.

In this lesson, students are told what tools they have at their disposal, but not how to use them. The groups must decide what methods would make sense to model the problem, and how calculators, graph paper, etc could help them progress in the activity.

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Where are the complex roots hiding?

F(x) = x2 - 1

x F(x)

Domain:

Range:

Roots:

F(x) = x2 + 1

x F(x)

Domain:

Range:

Roots:

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Name:

Your challenge: How can we graph complex roots?

Your tools:

- Graph paper - Scissors

- Card stock - Markers

- Tape - Pipe cleaners

- Scissors - Graphing calculators

Your roles:

Recorder: _____________________ Constructor: _______________________

Proofreader: ___________________ Group representative: ________________

Your equations:

1. F(a) = a2 + 1

(i) Solve for the roots of F(a)

(ii) Fill in the following table of values:

a F(a)

5

5

2

2

1

0

0

-3

-3

-8

-8

(iii) How can we graph all the points above?

Useful ideas: the complex plane, the x-y plane

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2. G(a) = (a-2)2 + 9

(i) Solve for the roots of G(a)

(ii) Fill in the following table of values:

a G(a)

13

10

9

10

13

0

0

5

5

2+ i

2 - i

(iii) Graph all the points above

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3. H(a) = x2 + 2x + 3

(i) Solve for the roots of H(a)

(ii) Fill in the following table of values:

a H(a)

6

6

3

3

2

0

0

-2

-2

-1 + 3i

-1 – 3i

(iii) Graph all the points above

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Part III: Assessment

Note: Descriptions of assessment for individual lessons is provided at the end of each day’s plan.

Rationale: For a more formal assessment of all the content goals for these lessons, I decided to supplement the activity planned for the 3rd lesson with a “Project Quiz” that they would complete on the 4th day of finishing up their group project. I created this quiz for several reasons. I built the project specifically so it couldn’t be done without using the arithmetic they have been learning the previous two days. So, it acts as a good summative assessment for many topics. However, instead of just accepting the project itself as the assessment, I wanted a “quiz” that the students complete in order to make the students aware of how they’ve been applying the math, and require them to show me the work that highlights their understanding. Finally, there are certain concepts that are not covered in this project (powers of i and division), and so I provided two choice problems that give the students a chance to show me their skills. I made these choice problems because I want students to have the best chance to show me what they’re learning, and they will feel a bit less pressure on the quiz if they can choose the problem they feel more comfortable with.

While this quiz would be for a grade, almost all of it is simply based on completing the project successfully, so students shouldn’t be penalized too much if they are still learning to put concepts together. For all the 2 pt problems, students receive a point for work/completion, and a point for accuracy. For the 4 pt problems, students would receive 1 pt for completion, 2 pts for work, and 1 pt for accuracy.

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Name:

Project Quiz:

1. (2pts) Circle someplace where you: used i = √(-1)

2. (2 pts) Circle someplace where you: used i2 = -1

3. (4 pts) Box someplace where you: added 2 complex numbers

4. (4 pts) Put a where you: multiplied 2 complex numbers

5. (2 pts) Pick one and simplify: i7 or (-i)3

6. (2 pts) Pick one and divide:

2 + 6i or 3 + 2i i + 4 7 + 4i