# How to Introduce Complex Numbers N.CN.A High School:  Algebra 2

Complex Numbers

This learning progression will be taught to a class that consists of sophomores and juniors in high school who are currently taking Algebra 2.  The Common Core State Standards that will be addressed come from two different domains.  The first domain is from High School:  Algebra-Arithmetic with Polynomials and Rational Expressions.  The CCSS Math cluster that will be addressed is “Understand the relationship between zeros and factors of polynomials.”  The second domain is High School: Number and Quantity-The Complex Number System.  The CCSS Math clusters that will be addressed are “Perform arithmetic operations with complex numbers,” “Represent complex numbers and their operations on the complex plane,” and “Use complex numbers in polynomial identities and equations.”  The third domain is from High School: Number and Quantity-The Real Number System.  The cluster that will be addressed is “Extend the properties of exponents to rational exponents.”  Students will also meet Mathematical Practices 1, 2, 3, 7, and 8.

The textbook that I will use as a resource is Glencoe’s Algebra 2:  Integration, Applications, and Connections.  The lesson will be taught from sections eight through ten of chapter five of this book.  These sections transition students from simplifying expressions including radicals and rational exponents to simplifying expressions containing numbers that are a part of the complex plane.

The central focus of this learning progression is an introduction to complex numbers and the complex plane.    The progression begins with the strategies that are used in simplifying expressions involving radicals.  These strategies will help student s understand how to use complex numbers and how to simplify expressions that contain complex numbers.   Students will be first introduced to what a complex number is and will then learn how to graph them in the complex plane.  The purpose of this learning progression is for students to gain a better conceptual understanding of the complex plane and will lead into solving quadratic equations that do not have real solutions.  This progression is set up so that the entry tasks from each section review a concept or ask students to think critically about a problem that will help them understand the new information that will be taught during the lesson.  How students do on this introductory information will influence where each lesson begins.  This will then influence how far we get in the planned lesson and so the next day’s lesson will also be affected.  Each lesson has been set up to be flexible and to run off of the previous lesson.  Beginning the class with questions that lead students to recall information that they have previously learned and to explore a new way of thinking will help students be more successful during the remainder of the class period and will help students become more interested in what they are learning.  Kubiszyn and Borich state in the book Educational Testing and Measurement that by imbedding a formative assessment into each lesson, “well-constructed performance test can serve as a reaching activity as well as an assessment.  This type of assessment provides immediate feedback on how learners are performing, reinforces hands-on teaching and learning…it moves the instruction toward higher order behavior.”

Common Core State Standards

Extend the properties of exponents to rational exponents.

CCSS.MATH.CONTENT.HSN.RN.A.1
Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.

CCSS.MATH.CONTENT.HSN.RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Perform arithmetic operations with complex numbers.

CCSS.MATH.CONTENT.HSN.CN.A.1
Know there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real.

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.

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

Mathematical Practices

CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.

CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively.

CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.

CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.

CCSS.MATH.PRACTICE.MP8  Look for and express regularity in repeated reasoning.

Learning Progression

Learning Progression-Complex Numbers-164vuer

# Weighing in on linear correlations A.REI.A

When students first learn about how to graph a linear function they are often confused about what the correlation is between the independent and dependent variables. To some students a linear function just seems like a magic trick about how to obtain an x-value when given a y-value. This activity will be able to solidify the concepts of a linear function because all of the pennies will have the same weight. If the students double the amount of pennies then the weight should also double. This can be easily done using the Dual-Range Force Sensor produced by Vernier Software & Programming. This device is compatible with computers, Chromebook, mobile devices, and several additional platforms.

In this activity, students will be working in groups of 3-4 and will be given a Dual-Range Force Sensor and 48 pennies per group. The students will start by using 8 pennies and measuring the weight, then 16 pennies, then 24 pennies, and so on until they weigh all 48 pennies while recording the weight for every 8 pennies. The students will then be plotting the values on the coordinate plane, and by using the slope formula to determine the weight of each penny, and find the function that represents their data. This will allow for students to understand the correlation of x and y-values in a linear function.

Common Core State Standards:

CCSS.Math.A-REI.10- Understand that the graph of an equation in two variables is the set of
all its solutions plotted in the coordinate plane, often forming a curve
(which could be a line).

CCSS.Math.N-Q.1- Use units as a way to understand problems and to guide the solution
of multi-step problems; choose and interpret units consistently in
formulas; choose and interpret the scale and the origin in graphs and
data displays.

# HS Genetics Lesson using Hardy Weinberg Equation N.Q.A This learning progression was used in a 10th grade biology classroom. The students are completing a unit on genetics are learning how to calculate allele frequencies in a population. This unit will focus on Common Core State Standards (CCSS) and Next Generation Science Standards (NGSS).

Standards:

CCSS.MATH.MP5: Model with mathematics

CCSS.MATH.CONTENT.HSN.Q.A.2: Define appropriate quantities for the purpose of descriptive modeling.

NGSS HS-LS3-3. Apply concepts of statistics and probability to explain the variation and distribution of expressed traits in a population.

The learning progression and activity is attached below:

edtpa learning progression

Genetics Lab pg 1

# Graphing Linear Equations A.REI

This learning progression is for a high school algebra class. In this unit, students will learn important concepts about graphing linear equations. In the first lesson,  students will learn about the different properties of a graph. For the second lesson, it will be broken down to two days. Students will check whether the set of ordered pairs are  solutions to both the equation and the graph. The next day,  students will be introduced to writing the equation as a function form. For the third lesson, students will  learn how to find the x-intercept and y-intercept  both algebraically and graphically. The following Common Core State Standards will be satisfied in this unit:

The following Mathematical Practice will be satisfied:

Learning Progression is attached:

learning progression for edtpa

# A Beginning Look at Calculus IF.B4

This learning progression is for a high school calculus class. This is the first unit of Calculus, where students will be introduced to the two goals of calculus. The learning progression will start first by students developing concepts of slope and slope functions. Students will show how particular functions change by examining finite differences.

The learning progression aligns the following Common Core State Standards:

MATH.CONTENT.HSF.IF.A.1

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

MATH.CONTENT.HSF.IF.A.2

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

MATH.CONTENT.HSF.IF.B.4

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

The learning progression aligns with the following Mathematical Practices:

CCSS.MATH.PRACTICE.MP1

Make sense of problems and persevere in solving them.

CCSS.MATH.PRACTICE.MP2

Reason abstractly and quantitatively.

CCSS.MATH.PRACTICE.MP8

Look for and express regularity in repeated reasoning.

The learning progression can viewed at the link below

# HSN.CN.A.1: Perform Arithmetic Operations with Complex Numbers CCSS.MATH.CONTENT.HSN.CN.A.1
Know there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real.

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.

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

#### CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.

Learning Progression Overview:

The lesson will open up with a brief recap to refresh the students on the prerequisite knowledge that is required. After this, the instructor is to present the concept of an imaginary number. The presentation follows the same format as described in the textbook, “Algebra 2”. To put imaginary numbers into context, the instructor will provide a quadratic equation, such that an imaginary number is produced, for the students to solve. Upon deriving the complex number, the instructor will introduce the properties of the imaginary number, i and complex numbers. The students will then transition into their first entry task of performing operations in complex numbers, beginning with addition and subtraction. Succeeding this, the students will learn proceed to their second entry task of multiplying complex numbers, employing their knowledge of the distributive property.  The third entry task will center on the utilization of the conjugate to simplify instances of a number being divided by a complex number. Throughout, the instructor is to prompt the class with short “checks” in the form of verbal questions to assess the progress of the class. Finally, the students will be administered an assessment to gauge their overall mastery of the material. The learning progression can viewed at the link below.

Complex Numbers Learning Progression

# Cliff Diving HSN-VM.A.3 The picture above, would be use to teach different an application of quadratic functions as well as using vectors and velocity. We can find the height of the diver when given the time. We can also solve the maximum velocity a diver can reach. In addition we can find how much time in seconds it takes for a diver to reach the water. All these problems, require a vector diagram which can be interpreted using trigonometry.

# Putt Putt Golf and Vectors HSN.VM.A.2 It is often difficult to engage students when it comes to harder concepts in Mathematics. In particular, mathematics involving vectors is difficult to grasp and therefore students are less likely to engage during instruction. To help alleviate the tendency to disengage, teachers can use an image to pull the students into the lesson. Video games tend to be an engaging and relevant aspect to students’ lives.

For this activity we will use the picture above of a golfing game. We can see that the golfer is looking from his ball position (initial point) to the cup (terminal point). We can assign these two points on the Cartesian plane. Using those two points as reference, we can break down the distance from the ball to the cup in to their i and j components on the x and y axis. Similarly, other video games or sports that can be represented in two dimensions can be used to create the engagement with students.

This problem is aligned to :

CCSS.MATH.CONTENT.HSN.VM.A.2

(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

# HSN.Q.A – You said how much!?! The construction of building Science 2 at Central Washington University completed September 2016. As I walked past the building the other day, I thought, how much did it cost in materials for this side of the building? There are many factors to consider when attempting to solve that problem by looking at this picture but for materials there are: brick and mortar, windows, doors and concrete. A possible task of this picture could be for students to determine the cost of the materials in the picture by HSN.Q.A.1 – Reason quantitatively and use units to solve problems. Given how much money per brick, how much money was spent on just bricks? Or it can be solved the other way by giving students the total cost of bricks then find the price per brick. In order to solve those problems, students will have to either count the bricks or do sections of areas. Even cost of labor could be incorporated into the equation for cost of the buildings side.
A teacher can take a picture of just about any construction of say a landscape or a parking lot and relate the problem of cost to unit conversions.

# The Gizmos of Tomorrow -Algebra1.7.A Gizmos is a interactive math tool for both teachers and students. It provides Common Core State Standard aligned activities, lesson plans, and simulations. Gizmos gives visual representations of concepts that aid students in analyzing data and it provides activity sheets that expands the students’ critical thinking.

the-gizmos-of-tomorrow