1. Exemplar Module Analysis
Grade 9 – Module 1
Grade 11 – Module 1
Grade 12 – Module 1
This second portion of our session this morning will cover the first module of the algebra 1, algebra 2, and precalculus courses. We’ll spend a bit more time in Algebra 1 than the other grades only because I know that is where the emphasis is right now. Algebra 11 and Pre Calculus will likely not be rolled out this year, BUT it is REALLY important to see the coherence, so it is very worthwhile to look at those modules as well.
(1 min.)

2. AGENDA
G9-M1
Topic Exploration – a Sampling of Exercises and Key Concepts
Exercise: Assessment & Scoring Rubric
Overview of Other G9 Modules
Summary of Key Shifts of Instruction
G11-M1
G12-M1
For grade 9 module 1 we will explore the content of each topic through a sampling of exercises and a discussion of key concept and definitions.
We’ll answer an assessment question and grade it using the rubric. Then we’ll speak briefly about the rest of the grade 9 year and then summarize the key shift of instruction in grade 9.
Then we’ll take a similar examination of G11-M1 and G12-M1, witnessing the coherence in the curriculum along the way.
Be aware that it is not possible to do even one entire exercise per topic within such a short time span, so some exercises we will do together, and some we will just look at or discuss and you can do them later today or when you go home.
There is so much in these modules, so much that I won’t even be able to touch on today, we are looking forward to sharing the full lessons with you at the next NTI.
Let’s get started then with Grade 9 module 1
(1 min / 2 min total time passed)

3. What’s in G9-M1?
Topic A: Explore the main functions of the year (linear, exponential and quadratic) through graphing stories (making graphs of situations)
Topic B: Study the structure of expressions, define what it means for expressions to be equivalent
Topic C: Precisely explain each step in the process of solving an equation
Topic D: The modeling cycle – solving problems using equations and inequalities in one variable, systems of equations in two variables
First let’s take a quick peak at the main ideas in each topic.
Find the table of contents and read the name and lessons in topic A. (pause for 30 sec.) So what is topic A about?
(Allow for 1 audience member to say and then click to advance first bullet.)
In topic A students explore all of the functions they will be studying throughout the year. As we shall experience in a moment, their exploration is conducted through graphing stories where student make graphs to describe a situation they witness in a video. Their experiences serve to foreshadow the work of the year ahead.
What is topic B about? (pause for 30 sec, allow for 1 audience member to say and then click to advance 2nd bullet.)
In Topic B, students study deeply the structure of expressions and arrive at a definition of what it means for expressions to be equivalent.
What is topic C about? (pause for 30 sec, allow for 1 audience member to say and then click to advance 2nd bullet.)
In Topic C, they will develop the capacity to articulate with precision the validity of each step in solving an equation.
What is topic D about? (pause for 30 sec, allow for 1 audience member to say and then click to advance 2nd bullet.)
In this topic students will engage in the modeling cycle: What do I mean when I say ‘the modeling cycle?
(ask a participant to contribute, display the modeling cycle flow chart on the document camera)
Modeling is one of the conceptual categories of the CCSS for high school math.
So, in topic d, student will apply the modeling cycle as they work through application problems using equations and inequalities in one variable, systems of equation in two variables, and graphing. They have had experience with each of these areas in grade 8, and in this module, really consolidated and formalized their understanding while expanding their perspective beyond basic linear functions.
(5 min / 7 min total time passed)

4. Topic A Introduction to Functions Studied this Year: Graphing Stories
Sample Exercise:
Describe the motion of the man in the video
Now let’s dive in and experience first hand what students will do in their first week of this Algebra I course. We are going to watch a video of a man and when we’re done, we want to describe his motion.
(1 min (including time to get the video running) / 7 min total time past.)

5. The Video: http://mrmeyer.com/graphingstories1/graphingstories2.mov
(watch only the first 1:08 minutes)
Display ONLY the first 1 min and 8 sec of the video.
(1 min / 9 min total time passed.)

6. Topic A Introduction to Functions Studied this Year: Graphing Stories
Sample Exercise:
Describe the motion of the man in the video
<< Participants need graph paper.>>
Who will share an ideas on describing the motion of the man.
(Some might speak in terms of speed, or distance he traveled over time, or change of elevation. All approaches are valid. Help participants shape their ideas with precise language.)
Is it possible to describe his motion in terms of elevation?
How high do you think he was at the top of the stairs?
How did you estimate that elevation?
Were there intervals of time when his elevation wasn’t changing? Was he still moving?
Did his elevation ever increase? When?
(Help participants discern statements relevant to the chosen variable of elevation. If students don’t naturally do so, suggest representing this information on a graph.)
Let’s graph in, there is graph paper in the back of your binder under the additional materials tab.
( Ask questions of the sort: )
“How should we label the vertical axis? What unit of measurement should we choose (feet or meter)?”
“How should we label the horizontal axis? What unit of measurement should we choose?”
“Should we measure the man’s elevation to his feet, or to his head?”
“The man starts at the top of the stairs. Where would that be located on our graph?”
“Show me with your hand what the general shape of the graph should look like.”
(Give time for participants to draw the graph of the story. Lead a discussion through the issues of formalizing the diagram: The labels and units of the axes, a title for the graph, the meaning of a point plotted on the graph, a method for finding points to plot on the graph, and so on.)
(12 min at the end of this we have used 21 min)

7. Piecewise Defined Functions (cont.)
A Follow-On Activity:
Here is an elevation vs.. time graph of a person’s motion, can we describe what the person might have been doing?
The next activity might be something like this. We won’t take time this activity in its entirety now, but I encourage you to work on it this evening or when you return home. As the teacher you might ask questions such as:
What is happening in the story when the graph is increasing/decreasing/staying the same?
What does it mean for one part of the graph to be steeper than another? [The person is climbing or descending faster than in the other part.]
Is it reasonable that a person moving up and down a vertical ladder could have produced this elevation versus time graph? [It is not reasonable because the person would be climbing up the later over several minutes. If the same graph had units in seconds then it would be reasonable.]
Is it possible for someone walking on a hill to produce this elevation versus time graph AND return to her starting point at the 10-minute mark? If it is, describe what the hill might look like. [Yes. The hill could have a long path with a gentle slope that would zigzag back up to the top and then a shorter and slightly steeper path back down to the beginning position.]
What was the average rate of ascent of the person’s elevation between time 0 minutes and time 4 minutes? [10/4 ft/min or 2.5 ft/min.]
Such questions help students understand that the graph represents only elevation, not speed nor distance from the starting point. This is an important observation.
(2 min / 23 min total time passed)

8. Topic A Introduction to Functions Studied this Year: Piecewise Defined Functions
Define Real piece-wise defined linear function:
Given a finite number of non-overlapping intervals on the real number line, a real piecewise-linear function is a function from the union of the intervals, to the real number line, such that the function is defined by (possibly different) linear functions on each interval.
Recall yesterday that Scott indicated we would be releasing a definitions document. Since those definitions are still being refined, we haven’t included definitions in this module overview. SO… I want YOU to take a stab at defining what a
real piece-wise defined function is with precision. Share your ideas with a partner at your table if you’d like. I’ll give you 2 minutes.
It’s challenging isn’t it? Let’s reveal what we have for the lessons:
(Click to reveal definition.)
PIECEWISE-LINEAR FUNCTION. Given a finite number of non-overlapping intervals on the real number line, a (real) piecewise-linear function is function* from the union of the intervals to the real number line such that the function is defined by (possibly different) linear functions on each interval.
(3 min / 26 min total time passed)

9. Topic B The Structure of Expressions
Sample Exercises from Lesson 7:
Use this diagram to build an expression using these symbols and operators.
Recall that in Topic B students study deeply the structure of expressions.
Try this exercise from Lesson 7 of this topic. It is printed on a full size sheet of paper and provided as the last page of your slide notes handout.
(Allow 2 min)
How do exercises like this one shape a students understanding of expressions?
(Allow for contributions)
In this way students begin to own expressions and demystify what they are made up of.

10. Topic B The Structure of Expressions
DEFINE VARIABLE: A variable symbol is a symbol that is a placeholder for a number from a specified set of numbers. The set of numbers is called the domain of variability. If a variable symbol represents a number from a set of numbers with more than one element, it is typically just called a variable. If the domain of variability is exactly one element, then the variable symbol is called a constant.
I want to take the opportunity now do define variable. I feel like there is one description of variable that is fairly common, but that does not serve students well.
Does anyone want to take a stab at guessing what that is?
(something that varies) It may be true that I can use x as 5 in one problem and 2 in another problem. But within a given problem, the variable doesn’t vary at all. It is simply a placeholder for a number. Sometimes we specify what set of numbers this one number will come from. We call that the domain of variability.
(click to advance 1st bullet).
Typical domains of variability are the whole numbers, integers, rational numbers, real numbers, or (in 12th grade) complex numbers. The domain is usually specified when describing the symbol, for example, “Let x be an integer such that…”
When the domain of variability is not specified, it is common practice to assume that the domain is the set of real numbers.
When would we call the variable a constant? When its domain of variability has only one element in it.

11. Topic B The Structure of Expressions
DEFINE ALGEBRAIC EXPRESSION: An algebraic expression is either a numerical symbol or a variable symbol, or the result of placing previously generated algebraic expressions into the two blanks of one of the four operators (__+__, __-__, __×__, __÷__) or into the base blank of an exponentiation with an exponent that is a rational number. A numerical expression is a algebraic expression that does not have any variables in it and that evaluates to a real number value.
At the end of lesson 7, students will develop a definition for algebraic expression.
How about if you try to define it first
(Allow 1-2 minutes)
Share your definition with a partner at your table.
(Allow 1 minute)
Now let’s reveal what the lessons in this module provide.
(Click to advance through the next 2 bullets, reading as you go.)
We can also define a numerical expression as…. (reveal last bullet and read).

12. Topic C Solving Equations and Inequalities
ALGEBRAIC EQUATION: An algebraic equation is a statement of equality between two algebraic expressions. NUMBER SENTENCE. A number sentence is a statement of equality between two numerical expressions. TRUTH VALUES OF A NUMBER SENTENCE. A number sentence is said to be true if both numerical expressions evaluate to the same number; it is said to be false otherwise.
That is the thrust of topic B.
In Topic C, gets very precise about solving equations and inequalities.
So an algebraic equation is simply a statement of equality between two algebraic expressions.
And a number sentence is a statement of equality between two numerical expressions.
In this curriculum, A Story of Units and A Story of Ratios, students are being told, not “write a number sentence”, but “write a true number sentence.” So already they are aware that (click to advance bullet and read…)

13. Topic C Solving Equations and Inequalities
The vocabulary of Solution Sets:
An equation with variables is often viewed as a question asking for which values of the variables is the equation true.
The equation serves as a filter that sifts through all the numbers in the domain of the variability and sorts those into two disjoint sets: The Solution Set and the set of values for which the equation is false.
In 6-8 solve an equation was find the number for that placeholder variable that would make a true number sentence.
Now, in grade 9, students expand their understanding of what is meant by “solve an equation”
(Click to advance 1st bullet.) That expansion begins by adding to their equation lexicon the term, “Solution set”.
(Click to advance 2nd bullet)
An equation with variables is often viewed as a question asking for which values of the variables is the equation true…
(Click to advance 3rd bullet). the equation serves as a filter that sifts through all the numbers in the domain of the variables and sort those into two disjoint sets: The Solution Set and the set of values for which the equation is false.
Students are taking equations back down to numbers.

14. Topic C Solving Equations and Inequalities
Key concepts and definitions:
“Solve”– identifying all the values of the solution set for a given system of one or more equations.
Solving an equation:
… starts with the assumption that the original equation has a solution, example: x + 5 = x - 3
… and then strategically uses the associative, commutative and distributive properties and “If-then” moves that apply the properties of equalities (or inequalities).
So there are 2 key shifts in their understanding of solving equations.
The first… is that, to ‘Solve’ then, is to identify all the values of the solution set for a given system of one or more equations.
And second, the standards and progressions call for students to recognize that solving an equation….(Click to advance)
starts with an assumption that the original equation has a solution. If I start with the equation x + 5 = x – 3. And I say the value on the left equals the value on the right and therefore if I add two to each side, they will still be equal… the validity of that step depended on there being an x value such that the equation was a true statement. It was IF the two sides are equal… THEN they’ll still be equal when I’ve subtracted 5 from each side. So, we are called to ask that students acknowledge their first assumption, that they be aware of and accountable for the assumptions they make. So we ask students: before they take any steps that depend on their assumption… to say “If there is a solution to this equation, then it must be true that…
It’s a subtlety, but it is time that students be accountable. How does that feel to you?
Here is an example of why I think those subtleties are important. If I ask my student, what is the Pythagorean theorem say….
So, next we (click to advance the next bullet) ask students to deliberately and strategically use …

15. Topic D Creating Equations to Solve Problems
Sample exercise:
Create a piecewise defined function to represent the total effective Federal Income tax due for a married couple filing jointly.
Schedule Y-1 — Married filing Jointly or Qualifying Widow(er)
If taxable income is over--
But not over--
The tax is:
of the amount over-- $0
$17,000
10%
$69,000
$1, %
$139,350
$9, %
$212,300
$27, %
$379,150
$47, %
$102, %
Topic D if you recall was on creating equations to solve problems. Here is an example of one of the modeling problems in Topic D. We will not be doing this problem today, but you are welcome to try it on your own. Where does this task fall in the spectrum of well defined modeling tasks vs. ill defined modeling tasks? (Allow for sharing)

16. Sample from the Assessments:
1. Jacob lives on a street that runs east and west. The grocery store is to the east and the post office is to the west of his house. Both are on the same street as his house. Answer the questions below about the following story: At 1:00 p.m., Jacob hops in his car and drives at a constant speed of 25 mph for 6 minutes to the post office. After 10 minutes at the post office, he realizes he is late, and drives at a constant 30 mph to the grocery store, arriving at 1:28 p.m. He then spends 20 minutes buying groceries. a. Draw a graph that shows the distance Jacob’s car is from his house with respect to time. Remember to label your axes with the units you chose and any important points (home, post office, grocery store).
Turn to the Grade 9 Module 1 tab in your binder, find the page number in the table of contents for the Mid-Module Assessment. What is the page number? Great, go to that page now and work question 1a.
Page 24 has the student work exemplar
Page 19 has the grading rubric

17. Other Grade 9 Modules: G9-M2: Descriptive Statistics
Distributions and their shapes
Measures of center and spread
Modeling relationships of numerical data on two variables
Module 2 of this year covers descriptive statistics; students learn about distributions and their shapes, measures of center and spread and then they study the process of modeling relationships of numerical data on two variables which fits nicely with what the rest of this year is all about – getting familiar with other forms of functions – not just linear, but quadratic and exponential, so again, this module takes a next step in anticipation of the next 3 modules.

18. Other Grade 9 Modules: G9-M3: Linear and Exponential Relationships
Formal function notation – a study of arithmetic and geometric sequences
Rates of change – contrasting linear and exponential
Interpreting graphs of functions – domain, range, maxima, minima
Relating equation notation to function notation
Absolute value function – studying transformations – how graphs change when equations change
Applying functions to real world contexts – systems of equations
A focus of module 3 is the comparing and contrasting of linear and exponential relationships. Students will get comfortable with formal function notation and the ideas of domain and range. They also begin a study of rate of change which will stay with them and serve them all the way into Calculus. They make close ties between functions and their graphs, learning what happens to the absolute value function when we replace x with x-2 for example. Of course, we keep close ties to applications and the modeling cycle throughout.

19. Other Grade 9 Modules: G9-M4: Expressions and Equations
Explaining properties of quantities represented by an expression based on contextual situation
Identify ways to rewrite quadratics and the usefulness of each
Operations with polynomials
Symmetry in quadratic graphs
Solving quadratic equations, deriving the quadratic formula
POLYNOMIAL.  A polynomial is any algebraic expression generated in the following way:  (1) declare all variable symbols and all numerical expressions to be polynomials.  (2)  any algebraic expression created by substituting two polynomials into the blanks of an addition operator or multiplication operator is also a polynomial.  
Module 4 students continue their study of expressions and equations specifically, polynomial expressions, and in particular quadratic expressions. In Module 1, students graphed elevation versus time of a guy who dives into 12 inches of water from a height of 36 feet.  It turns out that the quadratic function that models this fall, s(t)=36-16t^2, is very accurate.  So toward the end of module 3 students are poised to revisit this task with a new goal:  derive the quadratic function of the elevation of the man and verify the accuracy of the function (i.e., compute the time he is in the air, etc.). 
Students will arrive at a precise definition for a polynomial. (click to advance and allow participants time to read).
POLYNOMIAL.  A polynomial is any algebraic expression generated in the following way:  (1) declare all variable symbols and all numerical expressions to be polynomials.  (2)  any algebraic expression created by substituting two polynomials into the blanks of an addition operator or multiplication operator is also a polynomial.  

20. Other Grade 9 Modules: G9-M5: Quadratic Functions
Key features of quadratic and non-quadratic graphs
Exponential vs.. quadratic growth
Modeling with quadratic functions
So in Module 1 students got a taste of 3 types of functions: linear which they had experience with, quadratic and exponential.
In Module 2 they saw the application of these types of functions at it relates to statistical analysis of data on two quantitative variables.
In Module 3 students focused on comparing and contrasting linear and exponential functions as they gained comfort with function notation.
In Module 4 they spend a good deal of time with quadratic functions,
Now, In module 5 they will compare and contrast exponential vs. quadratic growth and spent a lot of time modeling with non-linear functions.

21. Key Shifts of G9 Curriculum:
A focus on the solution set.
How does the solution set stay the same or change as we modify the equation.
Graphs of equations are pictorial representations of solution sets.
The graph of the function, f, is a pictorial representation of the solution set of y = f(x).
How does the graph of the function stay the same or change as we modify the function.
Students experience learning and modeling:  Start with an intuitive notion --> play with examples and look for structure --> find rogue examples and figure out what to do with them ---> arrive at a nice definition.
So, we’ve seen a taste of what G9-M1 is all about. Let’s point out some of the key shifts between this implementation of the standards and what is a typical Algebra 1 course.
Would anyone like to share their observations before we go through the list that I prepared?
<< Allow for participants to contribute. >>
Thank you all for sharing. Here are the ones I made note of.
(Click to advance first bullet.)
First, it is standard for curricular materials to present a definition and then practice using the definition. In A Story of Functions, students use a set of experiences to contemplate a definition before arriving (through scaffolded instruction) at a definition with grade-level appropriate precision.
(Click to advance 2nd bullet.)
Next, our lessons are designed so that students truly experience learning and modeling. We want to make sure students DON'T feel like they are passive observers of a curriculum that is being dictated to them. Even though everything is kind of scripted, and everyone knows this, we still want to give students the sense of this being a story that they OWN. That there is flexibility to explore ideas, ask questions, go on tangents.

22. AGENDA
G9-M1
G11-M1
Topic Exploration – a Sampling of Exercises and Key Concepts
Summary of Key Shifts of Instruction
G12-M1
Ready for Grade 11 – Algebra II

23. What’s in G11-M1?
Topic A: POLYNOMIALS: extending from, and analogous to, base 10 arithmetic; division with polynomials
Topic B: FACTORING: Its use and its obstacles, leading to modeling with polynomials
Topic C: Solving and applying polynomial and rational equations
Topic D: A surprise from geometry: complex numbers overcome all obstacles; radical equations
First let’s take a quick peak at the main ideas in each topic.
In topic A students pick up on their fundamental definition of a polynomial as an expression formed from numerical symbols and a single variable symbol, under the operations of addition, subtraction and multiplication. They develop their experience base with polynomials as they continue to explore them as forming a system analogous to base 10 arithmetic, with similar ‘standard form’ and operations.
In Topic B students move from dividing polynomials and recognizing factored form of a polynomial to actually finding a factored form without knowing one of the factors first. They see the use of or the beauty of what factored form does for you in your capacity to draw a graph of the points of the function.
In Topic C students solve and work with applications of solving polynomial and rational expressions,
What is a rational expression …. How is it analogous to the integers? Notice that just as the integers are closed under add, sub and mult, but not division because ½ produces something that’s not an integer, similarly …
Lastly, in Topic D students discover a geometric understanding of complex numbers relating back to their understanding of rotations in Geometry, and discover that complex numbers resolves their last obstacle to factoring… what to do when there is no solution in the reals.
<<add other obstacles>>

24. Topic A Polynomials from Base Ten to Base X
Exercise:
Write the number 8943 in base 20
Let’s be as general as possible – not identify which base we are in, just call it x.
1 x x x x x x x 1
We typically write numbers in base 10. Why do we humans have this predilection for base 10? What do I mean when I say we write numbers in base 10?
(If many participant’s already know what base 10 means, and can already write numbers in another base, you can skip this portion below.)
(Use the document camera to demonstrate)
What is the ones digit in relation to 10? (If participant’s struggle to answer this, move on to the next question and then come back to it.)
What is the tens digit? - How many 10s.
What is the hundreds digit in relation to 10? How many 10 squared’s.
What is the thousands digit in relation to 10? How many 10 cube’s
(Now come back to the ones digit question… it’s how many 10^0th power’s.)
(4 min for above here)
(Click to advance 1st bullet.)
Could we write the number 8943 in base 20?
(Allow participants 2 min to work on it and compare with their neighbor then present the solution using the document camera or allow someone else to present their solution.)
(4 min for above, then give solution 3 min)
(Click to advance 2nd bullet.)
Let’s be as general as possible here, and just say we are in base x. Recall that x is a placeholder, waiting for us to assign it a number. Once we assign it a number, then we know what base we are in and what number we are representing in that base.
So for example, what number does this polynomial represent if we let x be 10? (Pause briefly and allow participants to state aloud.)
What number does this polynomial represent if we choose x to be 5? (Allow participants a minute to compute.)
(3 min)
So this concept of the analogy between base 10 arithmetic and polynomials will shape students thinking not just about polynomials, but later about rational expressions (a polynomial divided by a polynomial) and will prepare students to meet the (+) standard A-APR.7, understanding the rationals as being a closed system analogous to the rational numbers.
Recall we worked with polynomials in grade 9 as well.. How is the grade 9 treatment different? In grade 9 it was a natural extension of our work in forming various algebraic expressions by combining numerical and variable expressions with operators.
(15 min total for this slide)

25. Topic A Polynomials from Base Ten to Base X
Two models for division with polynomials:
Long division algorithm that is analogous to the division algorithm in base 10 arithmetic.
Reverse Galley method that is analogous to the area model
Extensive exploration with division serves as a precursor to factoring:
What happens when we divide x2 – a2 by x – a?
If I think of polynomials as an analogous to base 10 arithmetic, just picking a generic base of x. Then division of polynomials can occur with an algorithm that is analogous to the long division algorithm of integers.
So we present two models for division with polynomials, both of which show coherence with the math of PK-5:
1st, the long division algorithm and 2nd the area model.
Extensive exploration with division that serves as a precursor to factoring.

26. Topic B Factoring: Its Use and Its Obstacles
Exercise: Write 501 as a product of prime numbers.
1st Obstacle: What if a factor is not given to us first… how can we achieve factored form when we don’t know what to divide by?
Motivated by graphing (A-APR.3):
Can we write an equation whose graph would look like this?
So factoring is an extension of division with a problem, how do I divide if I don’t know what to divide by. We face the same problem when we are asked to find the factors of 501 right? How do I know what to divide by first? It’s not that easy… we know a couple of tricks, if it’s an even number, then divide by 2, if the sum of the digits is divisible by 3, then it’s divisible by 3, know your multiplication tables for other primes like 5, 7, 11…. But if I get to primes beyond 11, it’s gets tricky, yes?
So this topic is called Factoring: Its use and its obstacles. (Click to advance animation.)
The first obstacle is knowing what to start dividing by…. Factoring by grouping, recognizing structure that was seen in our experience with dividing by (x-a)
The remainder theorem and testing things out through division.
The motivation is that knowing the factors of a polynomial helps us create a quick sketch of the graph of the polynomial equation. This is connected to the zero factor property and the remainder theorem.

27. Topic C Solving and Applying Equations: Polynomial and Rational
Rational Expressions are analogous to rational numbers
Solving rational equations in one variable – IF there is an answer, what must the answer be?
1 𝑥 =0
Solving systems of equations in real-world contexts
Deriving the equation of a parabola
In Topic C, students study rational expressions. What is a rational expression? (a polynomial divided by a polynomial). So just the same way that rational numbers are integers divided by integers, …
(Click to advance next bullet.)
The process of solving a rational equation has the potential to give extraneous solutions. This is an opportunity to drive home the point, that the steps of arithmetic are only valid if there IS a number x for which the solution is true. Who said there was? Students are vary likely to multiply both sides by x and say x = 0. UNLESS they said to themselves… IF this has a solution then it might make since that x times the left side is x times the right side, but wait that would mean that x = 0, and 1 / 0 does not evaluate to a number.
Again, when we solve systems of equations, there is that: IF there is a solution, what must that solution be?
(Click to advance next bullet) That same thoughtful statement of an assumption serves students well as they solve systems of equations, some of which do not have any solutions
(Click to advance bullet)
The last item of this topic has students derive the equation of the parabola and decide (based on their foundation from geometry) whether or not all parabolas are congruent? Similar? Or neither?
Are all parabolas congruent? Through a series of rotations and translations, all parabolas can be brought into a position where the vertex is at the origin, and the vertex the focus is at (0, L) and directrix, (y = -L) for some value L, but
No, for different values of a we get non-congruent parabolas
Through dilation we can show that all parabolas are similar to y = x^2.
DANGER: IS EVERY U-SHAPED CURVE SIMILAR TO y = x^2 No. (Hanging power line example.)

28. Topic D A Surprise from Geometry—Complex Numbers Overcome All Obstacles
Some quadratic equations do not have solutions, why?
Does every cubic have a solution?
A surprising boost from geometry – transformations on the number line:
What is the geometric effect of adding 2 to all the numbers on the number line?
… to multiplying all the numbers by -1?
What transformation would create a 90°rotation?
(Advance bullet 1 and read, allow for participant to answer.)
(Advance bullet 2 and read, allow for participant to answer.)
(Advance bullet 3 and read, then change to the document camera and explore …)

29. Summary of Key Shifts in G11-M1
Polynomials are analogous to the integers (foundational from G9-M4’s A-APR.1)
Rational expressions are analogous to rational numbers (precursor to G12-M3’s A-APR.7)
Graphs of equations are pictorial representations of solution sets.
The graph of the function, f, is a pictorial representation of the solution set of y = f(x).
Students experience learning and modeling:  Start with an intuitive notion --> play with examples and look for structure --> find rogue examples and figure out what to do with them ---> arrive at a nice definition.

30. AGENDA
G9-M1
G11-M1
G12-M1
Topic Exploration – a Sampling of Exercises and Key Concepts
Supporting content from grades 8-11
Summary of Key Shifts of Instruction

31. By the end of G12-M1 students:
Use matrix notation to define and interpret transformations of the coordinate plane

32. How do they get there? Topic A: A Question of Linearity
Investigate the qualities of linear transformations (from reals to the reals)
Topic B: Complex Number Operations as Transformations
Geometric representations of complex numbers and operations thereon (in the complex plane)
Extend linearity to transformations from coordinate plane to coordinate the plane (aka, functions that take an (x, y) pair as an input and produce another (x, y) pair as an output.)
Connect to the geometric effect of dilation and/or rotation
Topic C: The Power of the Right Notation
This slide lists out the process of getting students there, but rather than read it, let’s just do it. Shall we?

33. Topic A A Question of Linearity
Wouldn’t it be lovely if functions were “nice” and just did what we expected them to do? Exercise: Here are some common student mistakes: (𝑎+𝑏) 2 = 𝑎 2 + 𝑏 2 1 𝑎+𝑏 = 1 𝑎 + 1 𝑏 log 2𝑎 =2 log 𝑎 a. Substitute in some values to show these statements are not in general true. b. Are there any values for which these statements, by coincidence, happen to work? Find all such values a and b for which these statements are true.
Wouldn’t it be lovely if functions were “nice” and just did what we expected them to do?
Look at this exercise. Let’s take a few minutes to work them out.
Some participants may notice that the second example has no real number solutions but does have complex number solutions. We’ll stick with real number thinking for now, but the idea of going to complex numbers is a good one… and we shall (soon)!

34. Topic A A Question of Linearity
If only all functions behaved in this way:
𝐿 𝑥+𝑦 =𝐿 𝑥 +𝐿(𝑦)
𝐿 𝑘𝑥 =𝑘 𝐿(𝑥)
Functions that behave in this way are called:
LINEAR TRANFORMATIONS
Are all linear functions also linear transformations?
No, but functions of the form 𝑓 𝑥 =𝑚𝑥 are linear transformations.
Notice that we have been dreaming / wishing that all functions behaved in this pattern.
L(x + y) = L(x) + L(y)
And L(kx) = k L(x) for all real numbers k.
(Click to advance 3rd bullet).
We say these functions classify as linear transformations …is this the same as linear functions?... We shall see.
Let’s take an example of a linear function and test it. Who can supply me with a linear function…<<switch to doc camera and work at least one of the form f(x) = mx + b and one of the form f(x) = mx.
(Return to slide show and click to advance last bullet.)
So we come to find that only functions of the form f(x) = kx satisfy the conditions of a linear transformation.

35. Topic B Complex Number Operations as Transformations
What ‘happens’ in the geometric representation
When we add two complex numbers?
When we multiply a complex number by -1?
When we subtract one complex number from another?
When we multiply two complex numbers?
When we divide a complex number by another?
What ‘happens’ in the geometric representation when we add/sub/mult/div with complex numbers (10 min)

36. Topic B Complex Number Operations as Transformations
Can we come up with a function that takes any point in the complex plane and transforms it to another point in the complex plane based on the rule of the function? YES! We have seen that 𝑇 𝑥+𝑖𝑦 = 𝑥+𝑖𝑦 +(𝑎+𝑏𝑖) translates in the plane. We could write this as 𝑇 𝑧 =𝑧+𝑤. And that 𝑇 𝑥+𝑖𝑦 =(𝑥+𝑖𝑦)(𝑎+𝑏𝑖) rotates and dilates in the plane. We could write this as 𝑇 𝑧 =𝑤𝑧.
Read the slide as you click through the bullets.

37. Topic B Complex Number Operations as Transformations
Can we agree that complex numbers, can be matched with or thought of as 𝑥, 𝑦 points? Consider our function that rotates and dilates about the origin: 𝑇 𝑧 =𝑤𝑧, 𝑜𝑟 𝑇 𝑥+𝑖𝑦 =(𝑥+𝑖𝑦)(𝑎+𝑏𝑖) What are the coordinates of the resulting point in the complex plane in terms of a, b, x and y? ( 𝑎𝑥−𝑏𝑦 , 𝑏𝑥+𝑎𝑦 )
Read bullet 1
So in theory we should be able to rewrite all of our work in terms of points x, y instead of complex numbers x + iy.

38. Topic C The Power of the Right Notation
Doesn’t this look ugly?
𝐿 𝑥, 𝑦 =(𝑎𝑥 −𝑏𝑦, 𝑏𝑥+𝑎𝑦) 
After 70 years of struggle, they came up with this:
𝐿 𝑥 𝑦 = 𝑎 −𝑏 𝑏 𝑎 𝑥 𝑦
We know that: 𝑎 −𝑏 𝑏 𝑎 𝑥 𝑦 has to end up as: (𝑎𝑥−𝑏𝑦, 𝑏𝑥+𝑎𝑦)
Can we decipher what −2 𝑥 𝑦 would end up as?
Now what we did here, does not just apply to the complex plane… every point in the complex plane, could be associated with a point in the x-y plane, and every point in the x-y plane could be associated with a point in the complex plane.
This was a real struggle for mathematicians. In the mid 1800s and all through the early 1900s various situations arose where formulas like the ones above kept appearing, and people were hunting, and struggling, to find a friendlier way to express them. (After all, who wants to be bogged down by difficult notation?)
We don’t want to go through 70 years of struggle here, but will take advantage of what they – eventually – found to be a wonderful way to express these things, even though it may be a little strange to us at first.

39. Supporting Content of Grade 9-11:
Motivates a discovery of complex numbers as students graph polynomials and thus look for their zeros (G11-M1)
Complex numbers are connected to the complex plane via, “If multiplying by -1 has the effect of rotating the number line 180-degrees, what could I multiply by to rotate 90-degrees” (G11-M1)
Provides a study of the trigonometric functions (G11-M2)
Read the bullets.

40. Supporting Content of Grade 9-11:
In depth study of basic rigid motions: rotations, translations, reflections (G10-M1)
In depth study of similarity transformations (dilations) (G10-M2)
Grade 9
Introduces function notation;
Gives focused attention to interpreting the structure of expressions
Read the bullets.

41. Key Shifts of G12-M1: Introduction motivation for use of matrices:
The need for notation of transformations in the plane (their primary mathematical use) – every type of 3 dimensional game uses this type of rotation and dilation matrices.
Introduces the idea of an inverse function.
Read the bullets.