mc-14193-39844713.us-east-1.elb.amazonaws.commc-14193-39844713.us-east-1.elb.amazonaws.com › fil… · Web view2012/09/11 · A Story of Functions- Exemplar Module Analysis Part

A Story of Functions- Exemplar Module Analysis Part 1

Grades 9, 11, and 12-Module 1

Sequence of SessionsOverarching Objectives of this May 2013 Network Team Institute

Participants will be familiar with the structure and components of modules in the A Story of Ratios and A Story of Functions documents giving them an early sense of comfort with how to use the curricular materials.

Participants will understand the focus of module 1 for each grade level 6-12 and the vertical coherence of its related content within the K-12 curriculum, preparing them to describe the coherence of the curriculum to colleagues.

Participants will understand the precise concepts, definitions, and representations for the content in G6-M1 and G7-M1 and be prepared to deliver the content of these modules (or train others to do the same), acknowledging their alignment with the related Progressions document, 6-7 Ratios and Proportional Relationships.

Participants will understand the major shifts in instruction of Geometry content of Grades 8 and 10, preparing them to shift their own instructional strategies and communicate these shifts to their colleagues.

Participants will understand the major shifts in instruction of Algebra and Precalculus content of Grades 9, 11, and 12, preparing them to shift their own instructional strategies and communicate these shifts to their colleagues.

High-Level Purpose of this Session To examine the content of G9-M1 including the key definitions, concepts and representations so that participants can provide instruction

with fidelity to those definitions concepts and representations and train others to do the same. To work an assessment question from G9-M1 and score it using the rubric so that participants are prepared to use the rubrics to grade

assessment questions. To examine the content of G11-M1 including the key definitions, concepts and representations so that participants can provide instruction

with fidelity to those definitions concepts and representations and train others to do the same. To examine the content of G12-M1 including the key definitions, concepts and representations so that participants can provide instruction

with fidelity to those definitions concepts and representations and train others to do the same.

Related Learning Experiences This NTI was preceded by NTI’s in which participants studied the PARCC Model Content Framework for Grades 3-11 This Exemplar Module Analysis was preceded by a similar session on G6-M1, G7-M1, G8-M1, and G10-M1.

Key Points• In Grade 9, students articulate precisely regarding equivalent expressions, justification of a method for solving an equation, and

the graphs of equations in two variables. • In Grade 9 and beyond, graphs of equations are described as pictorial representations of solution sets. • In Grade 9 and beyond, the graph of the function, f, is described as a pictorial representation of the solution set of y = f(x). Students

investigate, ‘How does the graph of the function stay the same or change as we modify the function?’• In Grades 9-12 students experience learning and modeling; concepts are presented beginning with an intuitive notion --> followed by

exploring with examples and an observation of structure --> followed by an exploration of rogue examples and figuring out what to do with them ---> finalized by arriving at a nice definition.

Session Outcomes

What do we want participants to be able to do as a result of this session?

How will we know that they are able to do this?

Participants will explore the content of each topic through a sampling of exercises and a discussion of key concept and definitions.

Participants will complete sample problems and be able to model how to do the problems

Session OverviewSection Time Overview Prepared Resources Facilitator Preparation

Session Introduction 9:30- Review session objectives and agenda

• Session PowerPoint• NYSED Video of Session 2 is strongly

recommended in order to deliver this session • Review session notes, video and PowerPoint

presentation

Grade 9-Module 1 Topic Exploration Overview of other G9

Modules Summary of Key Shifts Assessment & Scoring

Rubric

Explore the content of each topic through a sampling of exercises and a discussion of key concept and definitions.

Complete sample problems from Grade 9


recommended in order to deliver this session • Review session notes, video and PowerPoint

presentation

Grade 11- Module 1 Explore the content of each topic through a sampling of exercises and a discussion of key


Review session notes, video and PowerPoint

Topic Exploration Summary of Key Shifts

of Instruction Assessment & Scoring

Rubric

concept and definitions. Complete sample problems from Grade 11

recommended in order to deliver this session presentation

Grade 12- Module 1 Topic Exploration Supporting Content

from Grades 8-11 Summary of Key Shift

of Instruction Assessment & Scoring A look at the rest of

Grade 12

Explore the content of each topic through a sampling of exercises and a discussion of key concept and definitions.

Complete sample problems from Grade 12


recommended in order to deliver this session Review session notes, video and PowerPoint presentation

Session Roadmap 9:30-12:00Section: Introduction Time: 9:30-[2 minutes] In this section, you will…

• Review the focus of this session and the objectivesMaterials used include:

• Session PowerPoint

Time Slide #/ Pic of Slide Script/ Activity directions GROUP

30 sec. Slide 1 Welcome and introductionThis 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.

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(1 min.).

Slide 2 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)

Section: Content: Grade 9- Module 1 Time: Minutes] In this section, you will… Materials used include:

Review the content of Grade 9 Module 1 in order to be prepared to train colleagues

Session PowerPoint


30 sec. Slide 4 First let’s take a quick peak at the main ideas in each topic. 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.In Topic B, students study deeply the structure of expressions and arrive at a definition of what it means for expressions to be equivalent.In Topic C, they will develop the capacity to articulate with precision the validity of each step in solving an equation. Lastly, in Topic D 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.

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1 min Slide 4 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.

3 mins. Slide 5 Display ONLY the first 1 min and 8 sec of the video.(1 min / 9 min total time passed.)

Slide 6 << 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)

1 min. Slide 7 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)

1 min. Slide 8 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)

5 min. Slide 9 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 student’s understanding of expressions?(Allow for contributions)In this way students begin to own expressions and demystify what they are made up of.

2 min. Slide 10 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.

Slide 11 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).

Slide 12 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…)

Slide 13 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.

Slide 14 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 …

Slide 15 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)

Slide 16 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 exemplarPage 19 has the grading rubric

Slide 17 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

Slide 18 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

Slide 19 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.

Slide 20 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.

Slide 21 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. Rather than being passive observers…

Section: Grade 11- Module 1 Time: minutes] In this section, you will…

explore the content of each topic through a sampling of exercises and a discussion of key concept and definitions in order to be prepared to implement the module

Materials used include:Session PowerPointVideo


30 secs.

Slide 22 For grade 11 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 speak briefly about the rest of the grade11 year and they summarize the key shift of instruction in grade 11 and conclude by answering one of the

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assessment questions and scoring our work with the rubric.Let’s get started then with Grade 11 module 1Then we’ll review G12-M1, witnessing the coherence in the curriculum along the way.

Slide 23 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>>

1 min. Slide 24 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)

1 min. Slide 25 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.

2 min. Slide 26 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.

Slide 27 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 very 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 parabolasThrough 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.)

Slide 28 (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 …)

Slide 29 Review the key shifts

Section: Grade 12- Module 1 Time: minutes] In this section, you will…

explore the content of each topic through a sampling of exercises and a discussion of key concept and definitions. In order to implement the module

Materials used include:Session PowerPointNYSED video


30 secs.

Slide 30 For grade 12 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 speak briefly about the rest of the grade 12 year and then summarize the key shift of instruction in grade 12 and conclude by answering one of the assessment questions and scoring our work with the rubric.

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1 min. Slide 31 Read the bullet

2 min. Slide 32 This slide lists out the process of getting students there, but rather than read it, let’s just do it. Shall we?

Slide 33 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)!

Slide 34 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.

Slide 35 1. What ‘happens’ in the geometric representation when we add/sub/mult/div with complex numbers (10 min)

Slide 36 Read the slide as you click through the bullets

Slide 37 Read bullet 1So in theory we should be able to rewrite all of our work in terms of points x, y instead of complex

numbers x + iy.

Slide 38 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.

Slide 39 Read the bullets.



Use the following icons in the script to indicate different learning modes.Video Reflect on a prompt Active learning Turn and talk

Turnkey Materials Provided PPT Facilitators Guide Handouts

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