EDUC 731 Unit Plan

Rational and Irrational Numbers (9th-Grade Algebra I Unit) Ben Cohen Towson University EDUC 731: Curriculum and Assessment May 20, 2013

Table of Contents Unit Rationale and Objectives............................................................ Section 1

Unit rationale ............................................................................................. Pages 1-2 Objectives ................................................................................................... Pages 3-4

Context for Learning ......................................................................... Section 2 Class profile ............................................................................................... Pages 5-8 Implications for planning ........................................................................ Pages 9-13 Prerequisite content knowledge and skills ................................................... Page 14

Lesson Plans (days 1-3) ..................................................................... Section 3 Day 1 lesson plan .................................................................................... Pages 15-19 Day 2 lesson plan .................................................................................. Pages 20-25 Day 3 lesson plan .................................................................................. Pages 26-29

Unit Outline (days 4-14) .................................................................... Section 4 Introduction to unit ...................................................................................... Page 30 Days 4-14 outline .................................................................................... Pages 31-51

Summative Assessment ..................................................................... Section 5 Summative assessment ..........................................................................Pages 52-53 Answer key and rubric .......................................................................... Pages 54-56 Accommodations for diverse learners ......................................................... Page 57

Section 1 Unit Rationale and Objectives

1 Unit Topic and Rationale This unit will be based on the Common Core topic “The Real Number System” and aims to give students a deeper understanding of the relationship between exponents and radicals, as well as an introduction to irrational numbers and their applications. By the end of the unit, students will understand that it is possible to convert a number expressed as a radical, such as

10 , to exponential form (101/2), and be able to explain how the properties of exponents that make it possible to do this—that is, (101/2)2 must equal 101 = 10 because of rules governing exponent multiplication, so 101/2 is defined as the square root of 10. Students will be able to both rewrite rational expressions in exponential form (e.g.,

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7 = 71/3) and exponential expressions in

rational form (e.g., 281/5 = 5 28 ). Students will also be introduced to irrational numbers, numbers that cannot be expressed as the quotient of two integers. They will be able to classify a given number, as well as the results of addition and multiplication operations on rational and irrational numbers, as rational or irrational, with explanation—for example, they will be able to determine that the product of an irrational number and a nonzero rational number is irrational and explain why this is the case. At the very beginning of the unit, some techniques of elementary set theory, and their applications (such as sample spaces and payoff matrices) will be introduced; these techniques are necessary to lay the groundwork for the study of irrational numbers that follows.

The exponent/radical portion of this unit builds off of knowledge that is introduced during the study of exponents in an elementary-algebra course. The rules of exponent addition and multiplication—(xm)(xn) = xm

+ n

; (xm)n = xmn—are among the most crucial pieces of

knowledge that are taught in elementary-algebra courses, so background knowledge is not a concern for this section of the unit. Knowledge and understanding of this extension of exponent rules to radical expressions is crucial when factoring is introduced in elementary algebra, and, by

2 extension, in higher-level pre-calculus and calculus classes in which proper factoring is vital for success. In fact, in any course in which functions play a central role—including some courses in different disciplines, such as economics and physics—students must understand how to rewrite radical and exponential equations, and much of the material in these courses will presuppose this knowledge. Exponent rules are generally introduced in Algebra I roughly midway through the school year, so radical and exponential expressions that build on these rules follow naturally.

Irrational numbers, though a more advanced topic, should also be covered as part of a standard Algebra I course.

2 and π play key roles in geometry courses during the study of

triangles and circles, respectively; both are also irrational numbers, and knowing this can help put their geometric significance into the proper perspective. Once irrational numbers are defined, it is a logical step to cover operations on them, and this can again reinforce and expand upon students’ knowledge of geometry—for example, given that π represents the ratio of a circle’s circumference to its diameter and is irrational, it must be the case that the circumference of every circle is also irrational. Though seemingly obvious, this is one of the most crucial discoveries in the history of mathematics; the first known references to π date to the 26th century BC, but it was not until the late 1700s that its irrationality was proven. In addition, the coverage of rational and irrational numbers introduces students to the ideas of mathematical closure and group theory, the foundations of abstract algebra. While these topics are too advanced to be covered rigorously as part of a high-school curriculum, a simplified introduction to them helps elementary algebra students gain a deeper understanding of the concepts central to the course, such as why zero added to any number equals that number and why addition and multiplication are associative.

Thus, while certainly advanced, an introduction to irrational numbers helps students understand the “why” of algebra, rather than just the “what,” making it a crucial addition to the curriculum.

3 Unit Goals and Objectives This unit is based on the following Common Core standards: • HSN-VM.C.6: Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

• HSS-CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

• HSN-SN.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.

• HSN-RN.A.2: Rewrite expressions involving radicals and rational exponents using the properties of exponents.

• HSN-RN.B.3: Explain why the sum or product of two rational numbers is irrational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

Lesson 1

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Objectives Given a series of objects, students will form a mathematical set consisting of the objects, or determine that a set cannot be formed, with 100 percent accuracy. (Knowledge) Given an operation on two sets, students will determine the resulting set with 100 percent accuracy (e.g., {1, 3, 5} ∪ {2, 4, 5} = {1, 2, 3, 4, 5}). (Knowledge) Given an experiment with clearly defined outcomes, students will create a sample space and payoff matrix to model the outcomes and events with 100 percent accuracy. (Knowledge) Given a number, students will identify whether it is rational or irrational, with justification, with 100 percent accuracy (e.g., 3/17 is rational because it is the quotient of two integers). (Reasoning) Given two numbers and an operation, students will perform the operation and determine whether the resulting number is rational or irrational, with explanation, with 100 percent accuracy.

(Reasoning) Given a set of numbers and an operation, students will determine whether the set is closed under that operation, with explanation, with 100 percent accuracy. (Reasoning) Given a number expressed in exponential form, students will express it in radical form (and vice versa) with 100 percent accuracy. (Knowledge) Given numbers expressed in exponential and radical form, students will determine whether the two numbers are equivalent with 100 percent accuracy. (Reasoning) Given a number expressed in radical form, students will utilize the

CC Standards HSN-VM.C.6

HSN-VM.C.6 HSS-CP.A.1 HSN-RN.B.3 HSN-RN.B.3

HSN-RN.B.3 HSN-SN.A.1 HSN-RN.A.2

HSN-RN.A.2

4 method of prime factorization to simplify the radical with 100 percent accuracy (e.g., 50 = 10 5 = 5 5 2 = 5 2 ).

(Knowledge)

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Given an encryption algorithm and its applications, students will gain an appreciation for the role of the number-theory topics discussed in this unit in modern-day cryptography. (Dispositional) Given a number expressed in radical form, students will determine, with justification, whether the radical is rational or irrational with 100 percent accuracy. (Reasoning) Given a declarative statement, students will use the reductio ad absurdum method of proof to show that it is true with 100 percent accuracy. (Knowledge) Given examples of the golden ratio and its influences throughout history and in nature, students will gain an appreciation for the significance of the ratio. (Dispositional) Given a compass and straight edge, students will construct a “golden rectangle,” 30-60-90 right triangle, and 45-45-90 right triangle, and identify the steps involved in constructing these figures, with 100 percent accuracy. (Knowledge) Given a set of side lengths, students will determine whether it is possible to construct a 30-60-90 or 45-45-90 right triangle with these side lengths with 100 percent accuracy. (Knowledge) Given a fraction with an irrational denominator, students will rationalize the denominator and express the resulting fraction in simplest form with 100 percent accuracy. (Knowledge) Given an irrational number, students will explain, in writing or verbally, its role in historical mathematics with 90 percent accuracy (e.g., π represents the ratio of a circle’s circumference to its diameter). (Dispositional, Knowledge) Given a mixed set of rational and irrational numbers representing an encrypted phrase, students will use the set as a key to decrypt the phrase with 100 percent accuracy. (Reasoning) Given a substitution cipher, students will use it to encrypt a specific word or phrase with 100 percent accuracy. (Reasoning) Review for summative assessment.

Summative assessment.

HSN-RN.A.2 HSN-RN.B.3

HSN-RN.B.3 HSN-RN.B.3

HSN-RN.A.2 HSN-RN.B.3

HSN-VM.C.6

Not applicable Not applicable

Section 2 Context for Learning

5 Class Profile Abe – Abe is a generally solid student who comes from an upper-class background. He has a vision impairment that requires him to sit in the front of the classroom so he can see the chalkboard properly; he also needs large-print copies of books and handouts. Abe has demonstrated significant talent as a musician, although he has never spoken of any desire to pursue a career in the field.

Anita – Anita is a well-rounded, if undistinguished, student who generally scores in the low-A range in all of her classes. She is the captain of the school soccer team and is generally an active participant in group work in class. However, she cannot be placed in a group with Clarence due to mutual animosity between the two students; if they are paired together, they will verbally harass each other rather than focusing on the assignment.

Anthony – Anthony hails from a single-parent, low-SES household; he lives in a crime-ridden section of the city and is often unsupervised before and after school due to the long hours his mother works. However, he is an extremely motivated student who does not need to be reminded to stay on task in class or do his homework. Anthony is one of 15 students eligible for free and reduced-price meals.

Becca – Becca is a favorite among teachers and students for her witty, dry sense of humor. She does occasionally annoy teachers by cracking jokes in the middle of lectures and sometimes needs to be reminded to stay on task during group work. Becca is also a huge sports fan, and the quality of her homework occasionally dips if one of her favorite teams is playing the night it is assigned. Becca is one of 15 students eligible for free and reduced-price meals.

Blanche – Blanche has never been formally diagnosed with a learning disability but tends to have difficulty paying attention in class. She is often observed doodling in her notebook and passing notes to boys. During group work, she must be placed in all-female groups or she will spend the duration of the exercise talking to the male students in her group instead of doing work.

Despite these limitations, Blanche is not a weak student; she maintains a “B” average in all of her classes. Blanche is one of 15 students eligible for free and reduced-price meals.

Byron – Byron lives with his aunt in a crime-ridden, impoverished section of the city. He was robbed multiple times on the way home from school as a youth but is now unafraid to engage anyone who attempts to cause trouble, including fellow students; he was involved in an afterschool fight with Felix, and the two cannot be placed in a group together due to lingering tensions. Byron is one of 15 students eligible for free and reduced-price meals.

Clarence – Clarence is one of two African-American students in the class. He has been identified as having a learning disability in math, although his written work shows evidence of exceptional talent. Clarence rarely speaks in class and prefers working independently to working in a group.

Due to mutual animosity, he cannot be placed in a group with Anita; this will result in verbal sparring between the two students.

David – David is known throughout the school for his affinity for reading; he was the school’s top reader in a past competition to see which student could read the most books during the school

6 year. He is also somewhat resistant to new technology—he prefers overhead transparencies to PowerPoint presentations, for example—and uses it grudgingly when required to in class. David is one of 15 students eligible for free and reduced-price meals.

Earl – Earl is noticeably talented in language arts (though he has not been identified as gifted and talented); his math skills are on-target for the grade level. Earl sometimes appears uninterested in the material and has been observed sleeping in class multiple times. Although generally popular, Earl was involved in an after-school fight with Felix; as a consequence, the two students cannot be paired together during group work. Earl is one of 15 students eligible for free and reducedprice meals.

Emily – Emily is a hard-working, well-rounded student who is an active participant in the school’s theater program. She has shown exceptional talent when reading and writing short stories, poetry, and plays in English class. Emily is also fairly shy and does not speak often in her classes, although she commands the attention of the entire class when she does due to the profound insights she often expresses.

Esther – Esther is a highly religious Jewish student who has been classified as a gifted and talented learner. She is an extremely hard worker who occasionally pushes herself too hard; past teachers have tried to stress to her that not all of her work needs to be perfect. Although she is very well-liked by her peers, Esther is somewhat shy and reserved and often turns down opportunities to speak in front of the class. Esther is one of 15 students eligible for free and reduced-price meals.

Felix – Felix was born in Germany and emigrated to the U.S. with his family this past summer.

He speaks German at home and is classified as an English language learner in the Speech Emergence Stage. He is very strong in math but is reading below grade level because of the language barrier. Felix is somewhat of a social outcast; he was involved in an after-school fight with Earl and Byron and cannot be placed in a group with them due to lingering tensions. He appears to be most comfortable when working independently.

Hannah – Hannah is another well-rounded student who suffers at times from a lack of selfconfidence. She is often unwilling to speak in class for fear of providing an incorrect answer and being mocked by her peers, although they hold her in high esteem. She does occasionally need to be reminded to stay on task during group work instead of spending the allotted time talking to boys. Hannah is one of 15 students eligible for free and reduced-price meals.

Harry – Harry comes from a low-income, single-parent household. He is a gifted and talented learner in both language arts and mathematics and is a highly motivated student, often speaking to his teachers outside of class about extensions of the material and taking on challenging problems. He is an active participant in all of his classes and is unafraid to share opinions that differ from those of a majority of his classmates. Harry is one of 15 students eligible for free and reduced-price meals.

Hugo – Hugo comes from an upper-class background and generally appears indifferent about his studies, frequently turning in work late or not at all and grudgingly participating in group work during class. He often attempts to antagonize the teacher as well as those whose views or cultural

7 backgrounds differ from his own. He has been diagnosed as a learning disabled student in both English and math; his frequent lack of effort makes it difficult for teachers to work with him.

Hugo maintains a “C” average and is unpopular among his peers, often being the last student chosen when the class is allowed to form its own groups.

John – John is regarded as one of the school’s strongest and most highly motivated students; however, he was never evaluated for a gifted and talented designation at the wishes of his parents.

Despite this, John’s teachers often attempt to challenge him on assignments and exams with extra-credit questions; more than one teacher has remarked that he is capable of producing college-level work. John is highly regarded among his peers and teachers.

Joyce – Joyce is the daughter of El Salvadorian immigrants and is fluent in both English and Spanish, which she speaks at home. Although not gifted and talented, Joyce has been described by past teachers as a “model student;” she is a well-rounded learner who exhibits strength in reading, writing, and math and holds herself to extremely high standards. Joyce is one of 15 students eligible for free and reduced-price meals.

Lauren – Lauren has been diagnosed with ADHD and is easily distracted during class. However, she is also a gifted and talented student and has displayed particularly strong skills in her English and art classes. She is one of the most popular students in the school and is almost always the first person chosen when students are allowed to select their own class groups; she also has a tendency to sulk if she does not get her way. Lauren is one of 15 students eligible for free and reduced-price meals.

Madeline – Mad comes from an economically disadvantaged background and struggles at times with reading comprehension, partially because of a lack of exposure to reading materials as a child. However, she is an extremely high achieving student in math. Mad is known as a free spirit and helped found a number of grassroots groups at her previous school; many teachers would like to see her take a similar leadership role in class groups. Mad is one of 15 students eligible for free and reduced-price meals.

Pao-Lin – Pao-Lin is originally from Hong Kong but moved to the U.S. at a young age. She was home schooled until high school and is in the Intermediate Language Proficient Stage. She has demonstrated achievement in math on par with gifted and talented students but has not been classified as G/T due to her school’s inability to evaluate English language learners. However, she welcomes opportunities to speak in front of the class and is an active participant during group work.

Ruth – Ruth is the only African-American female in the class. She is an extremely high achieving student, having been identified as gifted and talented in language arts and mathematics.

Ruth is also held in very high regard by her peer students, who particularly admire her leadership qualities; she was handily elected class president at the start of the school year. Ruth is one of 15 students eligible for free and reduced-price meals.

Suzanna – Suzanna is an extremely strong writer who hopes to become a professional journalist.

She is also a solid math student, although not as high an achiever as in English. She is the romantic interest of many boys in the school; for this reason, most teachers prefer to place her in

8 an all-female group to do class work to minimize distractions. Suzanna is one of 15 students eligible for free and reduced-price meals.

Warren – Warren is another student from a low-income background; he is also an exceptionally talented football player and views it as his ticket out of poverty. He maintains a “B” average and is well liked by his peers and teachers, although he can be quick to anger at times. Warren is one of 15 students eligible for free and reduced-price meals.

William – William comes from a low-SES family, having been born and raised in inner-city housing projects and currently residing in an economically disadvantaged area of the city. He is also gifted and talented in math and is a strong achiever in English as well. He carries a 4.0 and is an extremely hard worker with aspirations of receiving a scholarship to an Ivy League university. William is one of 15 students eligible for free and reduced-price meals.

9 Implications for Planning: Gifted and Talented Students •

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Teachers have reported that gifted and talented students can be impatient with the slower pace of other students; this is a particularly acute problem for this class given the number of learning disabled students. There are four gifted and talented students—in addition to Joyce, Pao-Lin, and John, who achieve at G/T level but were never formally identified as such—and three LD students plus Blanche, who exhibits some traits of a learning disabled student. To solve this problem, gifted and talented students can be paired with lower-achieving students, such as those who are learning disabled or are otherwise having difficulty understanding the material. Lauren presents a particular challenge, being both gifted and talented and learning disabled; the best strategy would be to pair her with another G/T student who can help her stay on task.

G/T students are known for asking deep, insightful questions that are sometimes beyond the teacher’s understanding of the content (this has been observed with at least one student in the class, John). These questions can be used to push students to re-examine what they have learned, and also to help me extend the breadth and depth of my knowledge of the topic. For example, I expect to be asked why zero divided by zero cannot be classified as a rational or irrational number, since it appears to be a special case of division by zero, even though any nonzero number divided by zero is undefined. (In fact, the students are partially correct: Zero divided by zero is not undefined, but rather indeterminate.) Although students will not learn the proper techniques to answer this question (limits) until they study differential calculus, I would feel comfortable introducing them in a limited manner to explain to students why zero divided by zero cannot be properly evaluated. In addition, I would use these questions to extend the knowledge of the class as a whole, rather than simply the G/T learners—while they may be the ones asking the questions, they are by no means the only students in the class who can benefit from the answers.

Gifted and talented learners sometimes exhibit a fear of failure and expect perfection from themselves, becoming stressed or anxious when it appears this goal of perfection may not be met. Two of the G/T students in the class have previously exhibited this tendency. Instructors can help out in such cases by encouraging students to set realistic goals and recognize their limitations—for example, in this unit, I would encourage Esther and Joyce to acknowledge that they are dealing with a realm of mathematics that has confounded researchers for over 2000 years, and so there is nothing wrong with not getting a perfect score on the unit test or having trouble understanding a lesson.

G/T learners often dislike routines and busy work in class, placing an onus on teachers to ensure the class is not becoming too predictable. One way to do this is through the use of computer-assisted introduction; for example, students will use iPads to create encryption keys during lesson 7. Discussion questions, which are integrated throughout the unit, provide another way to vary the delivery of content from class to class. Technologybased assessments are also a useful strategy when working with gifted and talented learners. Overall, the integration of technology into the lessons should benefit many of the students (although not all of them—David, for example, prefers more traditional methods of instruction; depending on how many other students in the class share this preference, it may be necessary to abandon this plan rather than aiding some students at the expense of a nontrivial group of others).

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A more traditional method for assessing gifted and talented students is to place extra credit questions on tests. In this instance, the summative assessment could include a bonus question asking students about closure of a set under an operation not covered in class, to provide one example.

A 2008 study attributed part of the dearth of minority gifted and talented learners to African-American students’ perceptions of “acting White” and “acting Black” and a desire to avoid the former to maintain acceptance among their peers; some would-be G/T black students intentionally performed poorly in school so as not to be accused of “acting White.” This must be kept in mind when working with Ruth, an African-American gifted and talented student; although her enormous popularity in the school would seem to preclude any difficulties in this regard, it nonetheless cannot be ignored entirely and should be kept in mind if her academic performance begins to slip.

Implications for Planning: Learning Disabled Students

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If the school is so equipped, speech-to-text software could be used to aid these students.

Each learning-disabled student could be given an iPad that would display instructions or information visually to aid these students. In addition, some schools now have the ability to record lessons (with both audio and video components) and archive them for download by students later; this would be extremely helpful for these students but would also require a substantial financial commitment on the part of the school.

Given that this is a math lesson, students who struggle with reading comprehension (such as Mad) will likely not be put at a significant disadvantage. However, there still will be textbook readings (albeit ones very heavy on numbers); if possible, these students should be given an audio version of these readings. It is unclear whether most high-school textbooks have readily available recordings; however, it is very easy to record someone (a teacher, student, or learning specialist) reading a given section of the book and make the recording available to all students online. (Because some of the lower-income students may not have Internet access—or even a computer—at home, time must be allotted during the school day or after school to allow them to listen to the recording.)

For students like Abe who have poor vision, enlarged-print materials can be provided.

The procurement of a large-print version of the textbook is of course dependent on both its availability and the school’s budget; however, if such a version is not readily available, or the school cannot afford it, I could easily use a copy machine to prepare scaled-up versions of textbook pages. For materials produced by the teacher, the font can be easily increased in Microsoft Word on as many copies as necessary. This tactic can be used for both the instructional and assessment materials.

In much the same way gifted and talented students benefit from working with learning disabled students, LD students can benefit from working with G/T learners. Pair or smallgroup work can reinforce the concepts that these students learned in lecture; in addition, students may find it easier to understand the material when it is delivered by one of their peers or may be more receptive to a lesson from a fellow student than from a faculty member. These group sessions provide another opportunity to reinforce the material.

Again depending on the school’s budget situation, it may be possible to pursue a multisensory teaching strategy. Although number theory is a subject that does not lend itself particularly well to tactile activities, it is possible that learning disabled students could

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determine whether given numbers are rational or irrational by putting together puzzles— pieces would fit together to form a set of rational numbers and a set of irrational numbers.

It is unlikely that such puzzles are available commercially, so this would only be feasible at a school whose finances allowed it to custom-order such a puzzle. Students could also use building blocks (which are presumably more readily available) to match up radical and exponential expressions—for example, they could be given a Lego structure representing a radical expression and be required to convert it to exponential notation by building an identical structure with Legos labeled with exponential expressions; the final exponential expression would be the equivalent of the rational one.

Learning-disabled students could be allowed extra time to complete assessments, or even be allowed untimed testing. Ideally, students should complete the assessment in a room free of distractions when they have been granted extra time. This will be most beneficial to students with slow processing times or those who struggle with reading and writing (such as Mad). As the summative assessment for this unit includes a short-essay component, this strategy should be pursued.

LD students could also be allowed to have a scribe during assessments, which would allow them to focus on developing their ideas and expressing their knowledge rather than writing down their thoughts. This strategy could be used in tandem with extended-time or untimed testing to ensure the learning-disabled students are being adequately assessed.

Implications for Planning: English Language Learners

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Culturally responsive teaching can be of great benefit for English language learners. In this lesson, I would attempt to integrate the contributions of global mathematicians throughout history who have dealt with rational and irrational numbers or related topics, particularly those that share a cultural background with the students. For example, in lesson 9, which includes the golden ratio, I would include references to the contributions of German mathematicians Heinrich Agrippa and Adolf Zeising, both of whom researched the golden ratio and its applications in nature, which I feel will help Felix, my German student, relate to the material.

It is also essential to ensure ELL students are at ease in class to prevent their emotions from interfering with their ability to comprehend the material. This is again most acute for Felix; not only would it be advisable to avoid placing him in groups with Byron and Earl, but I would attempt to ensure the physical distance between these students is maximized at all times to avoid aggravating them and causing Felix’s emotions to cloud his comprehension of the lesson content.

The more background information I have regarding my ELL students’ prior education, the better I will be able to meet their needs. Since Felix has not even been in the U.S. for a year and Pao-Lin was home schooled until high school, this information may be difficult to find. It is possible the school district has information regarding the respective curricula they studied prior to enrolling; if not, I would ask my department chair to give these students a diagnostic test at the start of the school year. I would ask the department chair to do this rather than doing it myself because I fear they might feel exceptional pressure to succeed and subsequently panic if I administered the assessment, and also because the information gleaned from this assessment would be of use to all of the math

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department faculty, and the department chair is in the best position to maintain and disseminate this information.

If possible, I would explore the feasibility of delivering assessments to Felix and Pao-Lin in their native languages (German and Chinese, respectively). I would attempt to use foreign-language faculty at the school to translate both directions and, whenever possible, problems into these students’ native languages. (There are some times where this will not be possible—for example, one question on the unit test asks students to encrypt a given sentence in English using a Caesar cipher; the nature of such a problem makes it impossible to translate it into a different language. However, this problem is also based not on comprehension of the sentence, but on recognition of the characters without regard to their meaning; as such, I do not believe it would pose substantial difficulties for these students.) The students would be allowed to answer the short-answer questions in their native languages as well, and I would have the foreign-language faculty translate their responses into English for me.

If direct translation is not feasible, students could be assessed through non-traditional means that minimize or eliminate the language barrier. For example, lesson 1 asks students to translate set operations into plain English, such as “ 5 ∈A ” into “5 is an element of A.” The ELL students could instead be allowed to represent this visually, such as by drawing a large “A” and drawing a smaller “5” inside it. Similarly, they could express set unions, intersections, and differences through the use of Venn diagrams.

When asked to explain the role of an irrational number in the development of modern geometry, ELL students could be allowed to draw a picture and verbally explain it, such as drawing a picture of a circle and explaining that π represents “this” (run finger around circumference of circle) “over this” (run finger across diameter).

Implications for Planning: Students in High-Poverty Environments

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Students of a low socioeconomic status may lack support at home (this is explicit in the case of Anthony and may also be an issue for Byron and Harry, neither of whom lives in a two-parent household). As their teacher, it is critical that I show my low-SES students that I believe in them, and that I do not believe their impoverished backgrounds in any way compromise their ability to achieve at a high level. Along these lines, it is key to set high expectations for these students—I do not want to send a decidedly mixed message by telling these students I believe in them and subsequently setting the bar lower for them than for the other students.

It is also important to establish a positive relationship with these students early, focusing on their strengths rather than their weaknesses. Assuming these students do not want to spend the rest of their lives living in poverty, it is essential that they know that I can be a resource for them and will always support them and help them live a better life, whether it’s by helping Warren get a football scholarship or working independently with William as he pursues his goal of an Ivy League education. Again, the overarching goal is to ensure I do not treat these students any differently, or hold them to a different set of standards, than the other students in class.

Given that I spent 13 years at a wealthy private school, there are many modern conveniences, such as laptops and iPods, that I have tended to take for granted. However, I must be cognizant of the fact that many of my low-SES students have not had the

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fortune to experience such luxuries, and I must refrain from casually referencing them in the context of a lesson. For example, if I were to mention in class that the design of the iPod was based on the golden ratio, this would exclude those students who have never owned an iPod, which may be true of many (if not all) of the students living in poverty.

Instead, it would be wiser to talk about how the design of the Parthenon integrated golden rectangles; all students should be familiar with the Parthenon from their history classes, so such a reference would not be inappropriate for a group of students.

Students living in poverty typically speak casually, but test directions are often given more formally, placing these students at a disadvantage. One goal is to help these students learn to speak in what is known as “formal register”—such as “This assessment will contain thirty items” rather than “This test has 30 questions”—which will improve their performance on such assessments. In addition, much like English language learners, low-SES students can benefit from alternative assessments. Because children living in poverty often do not have access to an abundance of reading materials, their verbal skills tend to be much more highly developed than their written ones. Such students may benefit from having the option of oral assessments, both on the unit test and as part of a lesson objective (such as translating an expression from rational to exponential form verbally). While the goal is to bring these students’ literacy skills up to the same level as those of their peers, their performance in this class should be based on their knowledge and understanding of mathematics, not written expression.

Low-SES students can also benefit from verbal feedback on assignments, given their stronger verbal communication skills. While I would still mark these students’ assessments like those submitted by the rest of the class, I would make time before or after school to meet with these students to discuss their performance and give them feedback. Again, I do not want their ability to achieve in the class to be determined by their proficiency in written expression, and eliminating a barrier to fully understanding the feedback they receive is a necessary component of this.

14 Prerequisite Content Knowledge, Skills, and Understandings Despite the advanced subject matter, this unit is designed to require minimal background knowledge. However, students will still need mastery of some elementary concepts and will also need to recall some concepts that were introduced during earlier units in this class: • • •

• •

Students will need to know the rules of exponent addition and multiplication: (xm)(xn) = xm + n and (xm)n = xmn. Students should be familiar with these rules from a prior unit in the course.

Students will need to know the definition of an integer (a whole number). This is fairly elementary content knowledge to which they have had substantial previous exposure.

Students will need to be familiar with the diameter and circumference of a circle; they will also need to be able to identify a triangle by sight. This is elementary content knowledge, and it can be reasonably assumed that students will be familiar with it. (More advanced topics, such as the irrationality of π and what it represents, will be introduced as the unit progresses.)

Students will need to understand radical notation (that is, x ) and that it represents taking the root of a number. They should be familiar with this from a previous unit in the course; the equivalence of radical and exponential notation is a topic in this unit.

Students will need to be able to perform elementary addition, subtraction, multiplication, and division on whole numbers.

Section 3 Lesson Plans

15

Instructional Lesson Plan: Lesson 1 I. Purpose of the Lesson – What will the students learn? Why is this learning meaningful, important and appropriate? What will the students say or do that will serve as evidence of learning?

Subject Mathematics Date

Unit Rational and Irrational Numbers Grade Class Size 9 24 School

Mentor Standard(s): ____MSC

____CLG

_x__ CCSS

Topic Set Notation and Operations Time 45 minutes Intern

____Other: ___________

HSN-VM.C.6: Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

Big Idea or Essential Question Up to this point, students have only worked with individual numbers; how is a group of multiple numbers expressed mathematically? The answer is by forming a set, one of the most important objects in mathematics. This lesson will teach students how to form a mathematical set and how to perform basic operations on a set. This lesson will prepare students for future units on functions, in which there exist multiple solutions to a problem that must be expressed in set notation.

Alignment with Summative Assessment

Lesson Objectives Given a series of objects, students will form a mathematical set consisting of the objects, or determine that a set cannot be formed, with 100 percent accuracy.

Given an operation on two sets, students will determine the resulting set with 100 percent accuracy (e.g., {1, 3,5} ∪ {2, 4,5} = {1,2, 3, 4,5} ).

Formative Assessment For homework, students will complete a 15-question problem set on WebAssign; six questions will ask students if it is possible to form a set from the given objects, and the remaining nine will ask students to perform basic set operations (three questions for each of the three operations will be provided).

II. Context for Learning – What factors will influence my instructional decisions? How will my instruction remove barriers to learning and/or build on students’ strengths?

Knowledge of Learners (Data-based information – pretest and/or formative assessment, anticipated misconceptions or areas of confusion, student interests/motivation)

►►►►►

Instructional Decisions based on this Knowledge

16 Data-based information (required)

Response to data (required)

Specific Individual or Small Group Needs ►►►►► Differentiated Practices for this Lesson (Ex. IEP/504 accommodations, ESOL, social concerns, multicultural/equity measures, etc.)

• Learning disabled students may have difficulty working with the mathematical (i.e., number-based) definition of a set.

• The instructor will use the items in his lunch to demonstrate how to form a set, as well as to explain that sets can be made from items other than numbers.

• Non-mathematical objects used to form sets should take into account the high proportion of lowSES students in the class and should not include luxuries they may not be fortunate enough to possess.

• The instructor will refrain from using items that may be insensitive to these students, such as his clothing or apps on his iPhone, or what he ate for breakfast or dinner (for these students, their free/reduced-price lunch at school may be the only meal they eat in a typical day).

• Gifted and talented students may quickly finish the set-operation worksheet and become bored.

• Three extra, challenging problems will be included on the handouts to target these learners.

• Care must be taken to include a multicultural component in the lesson.

• This lesson includes a description of the German mathematicians Georg Cantor and Richard Dedekind, who began the study of set theory, as well as Ernst Zermelo (Germany) and Abraham Fraenkel (Israel), in whose honor a branch of set theory is named.

III. Instructional Procedures – What instructional strategies will I use to ensure that every child is a successful learner?

Instructional Materials/Resources A handout will be used in class that contains problems for the students to work out, working individually; this includes three extra problems to challenge gifted and talented students. In addition, the instructor’s lunch will be used as an instructional material to help accommodate learning disabled students.

At the start of the school year, students were given slips reading “TRUE” and “FALSE” to use to respond to the instructor’s questions, rather than simply raising their hands, which could lead to response bias.

These slips will be used during this lesson.

Technology Integration The SMART Board with document camera will be used to project the worksheet onto the board as students are answering questions from it. A PowerPoint presentation will also be used to run through examples of items from which sets can and cannot be made.

Cross-curricular Connections The Prisoner’s Dilemma is a central problem in game theory, a branch of economics. Although ninthgrade students are not eligible to take economics, they may encounter this problem later in their high school careers should they elect to take an economics class.

Management Considerations (Procedures, Transitions, Materials) and Student Engagement (required) • Gifted and talented students may finish the worksheet well before the rest of the class, and may in turn

17 become bored—and subsequently disruptive—while waiting for the rest of the class to finish. To alleviate this problem, three additional, more challenging questions were placed on the worksheet to specifically target gifted and talented learners, although they can also be used to challenge any student who finishes early.

• This lesson is more lecture-heavy than the other lessons in the unit. To ensure that students do not grow weary and tune out the instructor or disrupt the class, students will be asked to come to the front and demonstrate non-mathematical examples of set operations (using the example of the instructor’s lunches). In addition to keeping these students involved, it is expected that this will help ensure other students are actively engaged; they will presumably be more receptive to a message from one of their peers than from the instructor in a lecture-based class.

Instructional Sequence Planned Beginning • • •

Approximate Time

6 minutes

Warm-up Motivation Bridge

Procedure

• Begin with a famous problem in economics known as the “Prisoner’s Dilemma”: Imagine you are a member of an organized crime ring, and you and your partner have both been arrested.

Both of you are in solitary confinement with no way of contacting each other. The police officer makes you the following offer: “We’re going to question you and your partner. If neither of you confesses, you both get one year since we only have enough evidence to convict you on a smaller charge. If you confess and your partner doesn’t, we’ll let you go free and give your partner three years, but if your partner confesses and you don’t, you get three years and your partner walks. And if you both confess, you each get two years.”

• Ask the students to determine what they will do: confess or remain silent. Give them 1-2 minutes to think about their choice, and then have students hold up their response slips to reveal their choice (“TRUE” for remain silent and “FALSE” for confess).

• After reviewing the class’s decisions, inform students that the best possible outcome is that both prisoners agree to talk. How do we know this? The answer lies in mathematical objects known as sets, which are going to be the subject of this day’s lesson.

Development of the New Learning (Procedure will vary with the instructional model used. Ex. 5E lesson plan, scientific inquiry, teacher/student modeling, cooperative learning, discovery, etc.

Focus on active student engagement.)

14 minutes

• Students are familiar with the notation of ordered pairs, such as (0, 1), from graphing functions. However, an ordered pair is considered two distinct numbers—an x-coordinate and a ycoordinate. What if they have a problem that requires expressing more than one number in the solution?

2

• Example: x = 4. Either x = 2 or x = -2, but how can this be expressed? The answer is through a mathematical set: x = {2, -2}.

• Give students the definition of a set: A collection of well defined objects. Sets are most frequently used to denote numbers, but can be used to denote anything. Provide a non-mathematical example to accommodate learning disabled students: Today for lunch, Mr. Cohen brought a turkey sandwich, a bag of Oreos, and an apple. We can express the contents of Mr. Cohen’s lunch as a set: {turkey sandwich, Oreos, apple}.

• What does the “well defined” in the definition mean? Ask each student to individually come up with a set of the five best foods;

18 take 1-2 minutes and then ask a few students to share. Point out any differences in the sets to the class and note that these differences exist because different students have different ideas as to what the five best foods are. Thus, it is impossible to make a set of the “five best foods” since it is not well defined—one person’s five best foods are different from another person’s five best foods. “Well defined” means that there is a concrete definition for it— “students in this class,” “United States presidents,” and “countries of Europe” are all well defined objects, and so sets can be made from them, but “beautiful cities in Canada” and “crazy world leaders” are not well defined, so sets cannot be made from them.

• As a multicultural component, inform students that the study of modern set theory was initiated by two German mathematicians, Georg Cantor and Richard Dedekind. Two other mathematicians, German Ernst Zermelo and Israeli Abraham Fraenkel, laid the foundations for research in branch of mathematics now known as Zermelo-Fraenkel set theory.

• Introduce students to two sets that play crucial roles in mathematics:  (the set of all whole numbers) and  (the set of all natural numbers; that is, nonnegative whole numbers).

• Throughout this portion of the lesson, the instructor will be monitoring students to check for any confused looks or other signs of difficulty understanding the material.

Enrichment or Remediation (As appropriate to lesson)

18 minutes

• Ask students if sets can be made from the following objects, displaying them in a PowerPoint presentation (ask them to respond using their TRUE/FALSE slips): current members of the United Nations (yes), Vietnam War veterans (yes), fun attractions at the Maryland State Fair (no), good things about thunderstorms (no), and dogs currently in the classroom (yes). The last example is intended to show students that it is possible to form a set consisting of no objects, known as the empty set or null set and denoted {}.

• Introduce students to the three most common set operations: union, intersection, and difference. Begin by defining the union of two sets: the set of elements that are in set A or set B. For example, the union of A = {1, 2, 3} and B = {2, 4, 6} (denoted A ∪ B ) is {1, 2, 3, 4, 6}. Provide a second non-mathematical example to accommodate learning disabled students: Mr. Cohen brought a turkey sandwich, bag of chips, and a banana for lunch yesterday, so the union of his two lunches is {turkey sandwich, chips, Oreos, apple, banana}.

• Now define the intersection of two sets: the set of elements that are in set A and set B (denoted A ∩ B ). For example, the intersection of {1, 2, 3} and {2, 4, 6} is {2}. For a non-mathematical example, the intersection of Mr. Cohen’s two lunches is {turkey sandwich}.

• Finally, define the difference of two sets: the set of elements in set B that are not in set A (denoted A \ B ). The difference of {1, 2, 3} and {2, 4, 6} is {4, 6}. In a non-mathematical example, the difference of Mr. Cohen’s lunches is {chips, banana}.

19 • Distribute a worksheet with nine set operations on it (three of each). In addition, three extra problems (one of each) will be placed on the back to provide an extra challenge for students who finish quickly. Ask students to work individually to find the result of each operation, and circulate as students are working to answer and questions and informally assess students’ comprehension.

When all or a clear majority of students have finished the front of the worksheet, ask for a volunteer to answer each problem. As an informal assessment strategy, note how many students volunteer for each problem, as well as the accuracy of their answers. If a student answers incorrectly, ask if another student can correct the error.

Assessment/ Evaluation

Ongoing (4 minutes)

• Students will be informally assessed throughout the lesson.

During the lecture component, the instructor will look around the room to see if any students are visibly having difficulty understanding. As students are working individually on their worksheets, the instructor will circulate to answer questions and ensure all students understand and are remaining on topic. The instructor will also note how many students volunteer for each problem—including the three extra questions—and the accuracy of the students’ answers. If multiple incorrect answers are given, the instructor will note whether students are making the same mistakes in multiple problems.

• Set aside time to ask if any students have questions that have not yet been answered.

• Formative assessment: For homework, students will complete a 15-question problem set on WebAssign. The first six questions will ask if a set can be made from the given objects; the remaining nine will ask students to perform a set operation, with three devoted to each operation.

Planned Ending (Closure) • •

Summary Homework

3 minutes

• Briefly summarize and restate the information that was covered during the lesson.

• Assign homework (15-question problem set on WebAssign).

20

Instructional Lesson Plan: Lesson 2 I. Purpose of the Lesson – What will the students learn? Why is this learning meaningful, important and appropriate? What will the students say or do that will serve as evidence of learning?

Subject Mathematics Date

Unit Rational and Irrational Numbers Grade Class Size 9 24 School

Mentor Standard(s): ____MSC

____CLG

_x__ CCSS

Topic Sample Spaces and Matrices Time 45 minutes Intern

____Other: ___________

HSM-VM.C.6: Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

HSS-CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

Big Idea or Essential Question It was mentioned during the prior lesson that the best possible outcome for both individuals in the “Prisoner’s Dilemma” is that both confess, even though they will both serve longer sentences as a result.

How is this possible? The answer lies in special sets known as sample spaces, as well as payoff matrices, which build on the elementary set theory covered in the last lesson. This lesson will show how it is possible to use sets to model outcomes of any action, as well as determining what the best outcome is for everyone involved.

Alignment with Summative Assessment

Lesson Objective Given an experiment with clearly defined outcomes, students will create a sample space and payoff matrix to model the outcomes and events with 100 percent accuracy.

Formative Assessment For homework, students will work on an expansion of a game theory problem originally covered in class.

While the problems covered in class have only two actions, their homework problem will have four. There will also be an extra-credit component (showing the existence or lack of a Nash equilibrium) to accommodate gifted and talented students; to accommodate ELL and low-SES students, all students will be given the option of recording a video describing their answers rather than writing them out, using equipment available in the school library.

II. Context for Learning – What factors will influence my instructional decisions? How will my instruction remove barriers to learning and/or build on students’ strengths?

Knowledge of Learners (Data-based information – pretest and/or formative assessment, anticipated misconceptions or areas of

►►►►►

Instructional Decisions based on this Knowledge

21 confusion, student interests/motivation) Data-based information (required)

Response to data (required)

Specific Individual or Small Group Needs ►►►►► Differentiated Practices for this Lesson (Ex. IEP/504 accommodations, ESOL, social concerns, multicultural/equity measures, etc.)

• Learning disabled students may have difficulty working with the mathematical (i.e., number-based) version of a sample set.

• The instructor will use the items in his lunch to demonstrate a sample space, as well as to define an event.

• The high percentage of low-SES students must be taken into account when providing examples of sample spaces, events, and payoff matrices.

• The instructor will use basic playing cards, a simple item which virtually all children can be reasonably expected to have had access to. In addition, when demonstrating that the empty set can be an event, the instructor will use only items available in every classroom in the school.

• Care must be taken to include a multicultural component in the lesson.

• This lesson includes a description of the contributions of Hungarian mathematician John von Neumann, considered one of the greatest mathematicians of the 20th century, German economist Oskar Morgenstern, who published the first book on the subject of this lesson, and British biologist John Maynard Smith, who won the Crafoord Prize, an annual science prize presented by the King of Sweden, for his work in applying these techniques to biological evolution.

III. Instructional Procedures – What instructional strategies will I use to ensure that every child is a successful learner?

Instructional Materials/Resources As was done in the previous class, the instructor’s lunch will be used as an instructional material to help accommodate learning disabled students. In addition, playing cards and the game “Go Fish,” with which all students can be expected to be familiar, will be used as another accommodation, as will a standard six-sided die.

Technology Integration The SMART Board with document camera will be used to project both players’ hands during “Go Fish” as well as the results of rolls of the six-sided die so that all students can see.

Cross-curricular Connections The techniques discussed in this lesson play a central role in game theory, a branch of economics in which the Prisoner’s Dilemma is based. Although ninth-grade students are not eligible to take economics, should they elect to take an economics class in a later year, they will observe heavy use of these strategies during the game theory unit(s).

Management Considerations (Procedures, Transitions, Materials) and Student Engagement (required) • As with the prior day, this lesson is more lecture-heavy than much of the rest of the unit. To ensure that students do not grow weary and tune out the instructor or disrupt the class, students will be asked to

22 come to the front for multiple demonstrations, including rolling the six-sided die and playing “Go Fish.”

There are also multiple opportunities provided for students to voice their conjectures about possible payoffs and strategies in a theoretical game.

• Rather than wasting students’ time sorting cards at the start of the “Go Fish” demonstration, which would presumably lead to a loss of attention and, potentially, disruptive behavior as they wait, both hands as well as the fishing pool will be prepared prior to the start of class so they are ready for immediate student involvement.

Instructional Sequence Planned Beginning • • •

Approximate Time

5 minutes

Warm-up Motivation Bridge

Procedure

• Begin by revisiting the “Prisoner’s Dilemma” problem introduced in the previous class. Recall that students were informed that the best possible outcome is that both prisoners confess—but how do we know this? The answer is in the set of outcomes known as a sample space. By examining the sample space, we find that neither prisoner can improve his/her situation by changing his/her action—that is, if both prisoners make the decision to confess, either prisoner would be made worse off (i.e., receive a longer sentence) by switching decisions and instead choosing to remain silent.

• The sample space for this problem consists of a series of ordered pairs listing the sentences for each prisoner depending on his/her action: {(1, 1), (0, 3), (3, 0), (2, 2)}. But wait! Wouldn’t it be better for both prisoners to remain silent, since in that case they would both only serve one year rather than two? How can we tell that this is not the best possible outcome for both prisoners? The thought process underlying this finding is the subject of today’s lesson.

Development of the New Learning (Procedure will vary with the instructional model used. Ex. 5E lesson plan, scientific inquiry, teacher/student modeling, cooperative learning, discovery, etc.

Focus on active student engagement.)

16 minutes

• Begin by using a straightforward example of a sample space: rolling a six-sided die. The sample space for this event is {1, 2, 3, 4, 5, 6}, where each number in the sample space (set) corresponds to a number that can result from rolling the die, and each number has an equal chance of occurring.

• Now, invite a student up to the front to roll the die. The result of the die roll, known as the event, can be expressed as {n}, where n is the number rolled (1-6). The event is what is called a subset of the sample space—a set consisting of one or more elements of the original set.

• To accommodate learning disabled students, use the instructor’s lunch once again to model the set. Today, Mr. Cohen brought a peanut butter and jelly sandwich, a package of chocolate chip cookies, and a rotten banana for lunch, so his lunch can be modeled with the set {PB & J sandwich, cookies, banana}.

However, Mr. Cohen doesn’t want to eat the banana since it’s rotten, so what he eats can be modeled with the set {PB & J sandwich, cookies}. This is a subset of Mr. Cohen’s lunch, since it consists of two items that were in the original set. Similarly, if Mr. Cohen randomly chooses one item from his lunch, this will form a subset of the sample space (the original set, his lunch).

• Provide students with another example, this time using a sample

23 space from the card game “Go Fish.” Ask for two student volunteers to come to the front of the classroom and deal each a pre-prepared hand of seven cards; the two hands combined will consist of all twos, threes, and fours in the deck, as well as both jokers. Reveal each student’s hand to the class, and then have one student ask the other for a card (s)he does not have. Now, explain to students that the sample space is the entire deck of cards, excluding the 14 cards that are in each player’s hand—so in this case, it would be the five through ace of all four suits. Ask the student to draw a card at random and then reveal it to the class.

• The card the student has drawn becomes the event—the subset of the sample space of all available cards. In addition, the sample space has decreased by one element, the card that was drawn.

For example, if the seven of clubs was drawn, that card becomes the event and is no longer part of the sample space. Now, ask the second student to ask the first for a card (s)he does not have and repeat the process of drawing a card at random. Again, show students how this card joins the event set and is removed from the sample space. If, for example, the two cards drawn are the seven of clubs and the queen of hearts, the event set is {seven of clubs, queen of hearts}.

• As a multicultural component, briefly describe the contributions of three diverse individuals to the study of sample spaces.

Hungarian mathematician John von Neumann, considered one of the greatest mathematicians of the 20th century, was the first to study this field in depth. German economist Oskar Morgenstern published the first book on the field, and British biologist John Maynard Smith won the Crafoord Prize, an annual science prize presented by the King of Sweden, for his work in applying these techniques to biological evolution.

• Throughout this portion of the lesson, the instructor will be monitoring students to check for any confused looks or other signs of difficulty understanding the material.

Enrichment or Remediation (As appropriate to lesson)

17 minutes

• When there are two individuals involved in an event, there is a special way to demonstrate all of the events formed from a sample space, known as a payoff matrix. This shows all of the possible outcomes depending on which action each person takes.

• For example, take a soccer player attempting a penalty kick. For simplicity, assume his only choices are to shoot to the left or the right, and the goalkeeper’s choices are to dive to the left or the right (use the perspective of the player taking the kick for simplicity). In this case, the payoff for the player is 1 if (s)he scores and 0 otherwise; the payoffs are reversed for the goalkeeper. These payoffs can be expressed using the following payoff matrix (the player’s payoff is listed first):

24 • Now, let’s use a payoff matrix to model the Prisoner’s Dilemma.

In this case, the values listed are the years each prisoner will have to serve based on each person’s action, and not the payoffs:

• Return to the question posed at the beginning of class: Why is the best outcome that both prisoners confess, rather than both remain silent, when if they both remain silent they will each serve one fewer year? We can examine the payoff matrix to understand this. Initially, if both prisoners are silent, both will serve one year.

However, prisoner 1 knows that if his/her partner stays silent, (s)he can end up serving no time by changing his/her strategy to confessing. Prisoner 2 knows this as well, so in fact both will end up confessing and thus serving two years. So why is this the best outcome? Once both prisoners have decided to confess, neither has an incentive to switch his/her strategy to silence since this would result in a longer sentence. This is what is known as a Nash equilibrium, after the economist John Nash, and is a special type of subset of a sample space.

• Now pose a question to the class as a whole: Can the empty set be an event? Put another way, can the empty set be a subset of a sample set? Give students a few moments to think about it, and then ask for a few students to give their opinions and justifications.

After a few students have volunteered, inform them that yes, the empty set can be an event—it is possible that nothing will happen.

For example, if I pick up the phone in the classroom to call the school office and the line is dead (i.e., nothing happens), that would be an event that can be described by the empty set. By extension, then, the empty set is a subset of any set.

• As before, the instructor will be monitoring students throughout this part of the lesson to check for any confused looks or other signs of difficulty understanding the material. The instructor will also note how many students volunteer to guess whether the empty set is an event, and the accuracy of their answers, as an informal assessment.

Assessment/ Evaluation

Ongoing (4 minutes)

• Students will be informally assessed throughout the lesson.

During the lecture components, the instructor will look around the room to see if any students are visibly having difficulty understanding. The instructor will also note how many students volunteer to answer the question about the empty set being an event, as well as the accuracy of their answers. If multiple incorrect answers are given, the instructor will note whether the answers are incorrect for the same reason or if different students have different inaccurate rationales.

• Set aside time to ask if any students have questions that have not yet been answered.

• Formative assessment: For homework, students will determine the events and payoff matrix for an event with two actors and four

25 choices. The problem is an extension of the soccer penalty kick question mentioned in class; this time, the player taking the kick also has the option of aiming high or low in addition to left or right, and the goalkeeper can either jump (i.e., guard high) or dive (i.e., guard low) in addition to choosing a side. The payoffs (0 and 1) are the same as those in class. To provide a special challenge for gifted and talented students, students will receive extra credit if they can determine whether or not a Nash equilibrium exists in this problem and justify their answer.

Planned Ending (Closure) • •

Summary Homework

3 minutes

• Briefly summarize and restate the information that was covered during the lesson.

• Assign homework (extension of soccer game theory problem covered in class). To accommodate low-SES and ELL students, all students will be given the option of recording a video describing their solution to the problem rather than writing it out, using equipment available in the school library.

26

Instructional Lesson Plan: Lesson 3 I. Purpose of the Lesson – What will the students learn? Why is this learning meaningful, important and appropriate? What will the students say or do that will serve as evidence of learning?

Subject Mathematics Date

Unit Rational and Irrational Numbers Grade Class Size 9 24 School

Mentor Standard(s): ____MSC

____CLG

_x__ CCSS

Topic Intro to Irrational Numbers Time 45 minutes Intern

____Other: ___________

HSN-RN.B.3: Explain why the sum or product of two rational numbers is irrational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

Note: This standard is not perfectly aligned with the lesson, since operations on irrational numbers are not introduced until a later lesson. However, while irrational numbers are not introduced in the Common Core curriculum until high-school algebra, there is no specific standard dealing with their recognition and definition. In the instructor’s judgment, it is key to devote a full lesson to teaching students the definition and recognition of irrational numbers, since they have had no previous exposure to this topic, rather than introducing both irrational numbers and their operations in a single lesson.

Big Idea or Essential Question Our understanding so far has been that any decimal can be converted into a fraction. However, this is not a universal property—what about numbers like π that do not repeat? Much of our previous studies have focused on rational numbers, numbers that can be expressed as the ratio of integers. Now we will begin studying irrational numbers, which cannot be expressed as the ratio of integers.

Alignment with Summative Assessment

Lesson Objective Given a number, students will identify whether it is rational or irrational, with justification, with 100 percent accuracy (e.g., 3/17 is rational because it is the quotient of two integers).

Formative Assessment Students will complete a 10-question problem set on WebAssign, determining whether a given number is rational or irrational, for homework.

II. Context for Learning – What factors will influence my instructional decisions? How will my instruction remove barriers to learning and/or build on students’ strengths?

Knowledge of Learners (Data-based information – pretest and/or formative assessment, anticipated misconceptions or areas of

►►►►►

Instructional Decisions based on this Knowledge

27 confusion, student interests/motivation) Data-based information (required)

Response to data (required)

Specific Individual or Small Group Needs ►►►►► Differentiated Practices for this Lesson (Ex. IEP/504 accommodations, ESOL, social concerns, multicultural/equity measures, etc.)

• Learning disabled students may have trouble grasping the difference between rational and irrational numbers, but gifted and talented students will get bored quickly if too much time is spent on review.

• Students will be divided into heterogeneous groups and will be assigned a set of numbers to decide whether they are rational or irrational, with explanation—G/T students will be able to help guide their LD peers through this portion of the lesson. In addition, complex numbers (which were introduced in an earlier unit) will be included among these numbers as an accommodation for gifted students.

• Care must be taken to include a multicultural component in the lesson.

• The lesson will include brief descriptions of historical mathematicians whose work focused on irrational numbers, including Maryland native Benjamin Banneker, the son of a former slave.

III. Instructional Procedures – What instructional strategies will I use to ensure that every child is a successful learner?

Instructional Materials/Resources Links containing more information about the irrational number research mentioned in class will be posted to the course website after class. Students will also utilize a handout during group work in class to determine whether a given number is rational or irrational; this handout will be posted on the course website after class.

Technology Integration Students will be given the opportunity to use the SMART Board to demonstrate that a given number is rational or irrational.

Cross-curricular Connections It is likely that the students either have covered or will soon cover some of the contributions of the mathematicians mentioned in this lesson during their world history class, particularly Archimedes and Kepler. Obviously, the history classes will focus more on the historical significance of these mathematicians, but this lesson is also designed to help students further grasp this historical significance and how the mathematicians’ discoveries continue to shape contemporary developments in mathematics.

Management Considerations (Procedures, Transitions, Materials) and Student Engagement (required) • It is expected that students will talk when they are getting into their problem-solving groups regardless of the teacher’s instructions to the contrary. Rather than attempt in vain to have them remain silent, students will be asked to discuss in their groups who will be the “spokesperson” for the group; this person will be in charge of sharing the group’s answers to the questions posed.

• A number of students must be placed in special groups: Anita and Clarence must be kept separate, Felix cannot be placed in a group with Byron or Earl, and Blanche and Suzanna must be placed in allfemale groups. This must be taken into account when assigning problem-solving groups.

28

Instructional Sequence Planned Beginning • • •

Approximate Time

6 minutes

Warm-up Motivation Bridge

Procedure

• Begin by showing students the instructor’s t-shirt that includes the first 5,000 digits of π . Remind students of one key quality of π : It does not repeat. Mention to them that the current world record for memorized digits of π is over 67,000!

• Building off of this, ask the students a question: Is it possible for a baseball player to have an average of π ? Give students a few moments to think about, and then ask for a student to provide his/her thoughts. Ask the class if anyone agrees/disagrees with this student’s rationale, and if so, why. (For example, if the first person says that it is possible, since π is a decimal and all batting averages are decimals, ask if any students would like to add to this, or perhaps propose a different theory.)

• Answer: It is not possible to have a π average, since a baseball player’s average is equal to hits divided by at-bats, both of which must be integers. π , on the other hand, cannot be expressed in this manner. π is what is known as an irrational number—one that cannot be expressed as a fraction.

• This classification of rational and irrational numbers will be the focus of today’s lesson.

Development of the New Learning

10 minutes

(Procedure will vary with the instructional model used. Ex. 5E lesson plan, scientific inquiry, teacher/student modeling, cooperative learning, discovery, etc.

Focus on active student engagement.)

• Begin by asking for a student volunteer to come to the SMART Board and show if a given number is rational. The first number is 0.5; the expectation is the student will identify this number as rational since it can also be expressed as 1/2, the ratio of two integers. Ask if the class agrees or if anyone has a different answer, or if anyone has any questions.

• Ask for another volunteer to come to the SMART Board. This time, the number is 17. Once again, this number is rational (it can be expressed as 17/1). Again, ask if the class agrees or if anyone has either a different answer or a question.

• Ask for a third volunteer. Now, the number is e, Euler’s number (students are familiar with this number from a previous unit). In this case, the number is irrational since, like π , it does not repeat.

Follow the same procedures as before for surveying the class.

• Ask for a fourth and final volunteer. For this question, the number is 0. This is a rational number since it can be expressed as the ratio of 0 and any integer; it is also one likely to trick students since they often do not think of 0 as an integer.

• During this portion of the lesson, the instructor will note how many students are volunteering to come to the board as an informal assessment tool. The instructor will also monitor the class to see if any students seem to be having trouble with the material.

Enrichment or Remediation

22 minutes

• Break students into six groups of four, arranged heterogeneously and taking care to keep separate students who cannot be placed together (as described previously).

29 (As appropriate to lesson)

• Pass out sheets with 12 rational and irrational numbers on them to each group. Groups will be responsible for determining whether each number is rational or irrational and explaining it. While the first few numbers will be straightforward (e.g., 0.2), others will be more difficult, such as .406 or 2.5/7. The goal is for students to recognize that a decimal can be expressed as a fraction—and is thus rational—if the decimal terminates ( π and e, for example, go on forever), and also that a fraction that includes only one integer is not necessarily irrational—in the example above, 2.5 can be expressed as 5/2, yielding the rational fraction 5/14. The last two will be radicals of non-perfect squares to present a special challenge for gifted students; radicals will be covered in greater detail later in the unit.

• Ask a spokesperson from each group to give an answer to each question with a brief explanation, and ask if the rest of the class is satisfied or has questions or a different answer. As an informal assessment technique, the instructor will note how many students volunteer for each problem, the accuracy of their answers, and whether multiple students are making the same mistake. The instructor will also circulate as students are working in groups.

• Why are irrational numbers important? The work of many mathematicians throughout history either focused on or integrated irrational numbers, such as Pythagoras (30-60-90 and 45-45-90 right triangles), Archimedes ( π ), and other less well known mathematicians such as Johannes Kepler (who improved previously proposed ideas of a heliocentric solar system) and Benjamin Banneker, the son of a former slave who was born in what is now Baltimore County and used irrational numbers to publish an almanac predicting solar eclipses and tides.

Assessment/ Evaluation

Ongoing (4 minutes)

• Students will be informally assessed throughout the lesson. The instructor will note how many students are volunteering for each problem, as well as the accuracy of these answers. Any patterns in incorrect answers provided by students will be noted as well.

The instructor will circulate during group work to ensure all students are actively engaged, are remaining on topic, and are not having any difficulties understanding the material.

• Set aside time to ask if any students have questions that have not yet been answered.

• Formative assessment: For homework, students will complete a 10-question problem set on WebAssign that will ask them to determine if a series of numbers are rational or irrational.

Planned Ending (Closure) • •

Summary Homework

3 minutes

• Briefly summarize and restate the information that was covered during the lesson.

• Assign homework (10-question problem set on WebAssign).

Section 4 Unit Outline

30 Unit Outline The following is an outline of lessons 4-14 in a 15-day lesson. (Full lesson plans are available for days 1-3; day 15 is the summative assessment.) Included are the objectives for each lesson; materials and resources that will be used during the lesson, as well as after it to enhance students’ understanding; an outline of instruction and activities for the day; and that day’s informal and formative assessments. All solutions posted on the course website show all work so students can see how the answer is derived instead of just the answer.

Many homework assignments make use of WebAssign, a website that allows for online submission of homework. The number of submissions allowed for a problem can be adjusted— students will generally be allowed two submissions for problems involving calculations and one for all other problems—and the instructor can easily view data on each problem, including what percentage of students got it right, what the most common incorrect answers were, and, in the case of calculation problems, what percentage of students needed both submissions to correctly answer the problem. Students who do not have Internet access at home will be provided time before and/or after school to use the school’s computers to complete the assignments. Oral assessments will also be available in lieu of written problem sets (both those completed through WebAssign and those distributed in class) to accommodate students who are not comfortable with written assignments.

Accommodations for individual lessons are included in each lesson outline. However, there are also some accommodations that are used throughout the unit. All lessons will be recorded (with audio and video components) and posted to the course website after class to accommodate learning-disabled students; in addition, to accommodate students with reading-comprehension difficulties, MP3 files of all sections in the textbook that correspond to this unit have been placed on the course website. As with WebAssign, students may take time before and/or after school to use the school’s computers to access these items. In addition, one student in the class is visually impaired; enlarged copies of all handouts distributed in class will be prepared for him.

This unit is based on the following Common Core standards: • • • • •

HSN-VM.C.6: Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

HSS-CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

HSN-SN.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.

HSN-RN.A.2: Rewrite expressions involving radicals and rational exponents using the properties of exponents.

HSN-RN.B.3: Explain why the sum or product of two rational numbers is irrational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

31 Day 4 Lesson Objective Given two numbers and an operation, students will perform the operation and determine whether the resulting number is rational or irrational, with explanation, with 100 percent accuracy.

Materials/Resources: The instructor will set up a practice problem set on WebAssign involving operations on rational and irrational numbers. Since this is for practice only, students will be allowed unlimited submissions, and it will not be graded. However, the instructor will still have access to the same data set as with graded assignments.

Lesson Outline • Begin by reviewing the definition of irrational numbers (those that cannot be expressed in the form p/q, where p and q are integers), with examples.

• Introduce operations (addition and multiplication) on irrational numbers. Students will need to classify the results of four different operations as rational or irrational: o Addition of a rational number and an irrational number o Addition of two irrational numbers o Multiplication of a rational number and an irrational number o Multiplication of two irrational numbers • Begin by asking students to consider the case of multiplication: Is the product of two irrational numbers rational or irrational? (Guide students through the thought process.) o Ask students for their thoughts, with justification—for example, “Rational for the same reason that the product of two negative numbers is positive.” o First, take the case of 2 ∗ 2 . Since the product of these numbers is 2, which is rational by definition, this seems to suggest that the product of two irrational numbers is indeed rational. o Now, examine the case of 2 ∗ 3 . The product of these numbers is 6 , which is irrational. This contradicts the earlier assumption that the product of irrational numbers is rational. Thus, there is no blanket rule for the classification of the product of irrational numbers!

• Use a similar series of problems when guiding students through the case of a rational number multiplied by an irrational one.

• Break students into their problem-solving groups and ask them to consider the two addition cases. Instruct them to consider several examples of their choosing and then determine what the outcome is based on these examples. As an accommodation for gifted and talented students, suggest that groups can try going from specific examples to the general case, using variables instead of actual numbers, and see if they can prove their conjecture regarding the rule. Circulate while students are working in groups to monitor student progress, answer any questions, and help guide students who are stuck (e.g., help them work through an example).

• Bring the class back together and ask a few students to share what their groups found, with justification. Ask students from other groups to comment on these findings before explaining to the class the addition rules.

• Take time at the end of class to see if students have any questions.

32

Informal Assessment The teacher will circulate to monitor students during group work and ensure students’ understandings. During lecture portions of the lesson, the teacher will check to see if any students look lost or appear to be having difficulty comprehending the material. Time will be allotted at the end of class for students to ask any additional questions they may have; the instructor will make note of the content of these questions, since it is likely the material with which students are struggling the most, as well as whether questions are being asked by many students or just a few.

Formative Assessment For homework, ask students to provide the outline of a proof as to why an irrational number multiplied by a nonzero rational number is always irrational. In addition, they will receive extra credit on the assignment if they can prove the two rules regarding addition of rational and irrational numbers (building on the challenge introduced in class). To accommodate low-SES and ELL students, who may be uncomfortable writing out their proof, all students will be given the option of recording a video in which they explain their proof outline in lieu of producing a written piece; this can be done using equipment available in the school library.

33 Day 5 Lesson Objective Given a set of numbers and an operation, students will determine whether the set is closed under that operation, with explanation, with 100 percent accuracy.

Materials/Resources A bucket and a series of wooden blocks with numbers on them will be used as part of an accommodation for learning-disabled and ELL students. If numbered blocks are available, these will be used; if not, alphabet blocks will be used. If alphabet blocks are used, each will be covered up with a piece of paper with a number written on it.

Lesson Outline • Begin by reviewing students’ findings from the homework (take approximately five minutes). Remind students that the set of all real numbers  includes both rational and irrational numbers (this was covered on day one).

• Because multiplication and addition (and, by extension, subtraction) of two real numbers yields another real number,  is considered closed under these operations. Provide students with the definition of closure: A set is closed under an operation if performing that operation on two elements of the set yields an element of the set.

• As an accommodation for learning-disabled and ELL students, use a visual representation to illustrate this: Take a bucket and place a series of wooden blocks in it, with a number on each block. Shake the bucket back and forth in such a manner that no blocks come out, and explain that the blocks are closed under the operation of shaking the bucket—the result of this operation does not yield an element of a different set. Now, gradually turn the bucket upside down so that one block falls out, and then set it on the table right side up. The set of blocks is not closed under the operation of turning it upside down, since the result yields an element of a different set (the set of blocks not in the bucket).

• Break students into their problem-solving groups and ask them to think about whether the set of all integers  is closed under the following operations: addition, multiplication, division, square root, and trigonometric function (e.g., sin(x)). Write these on the board.

• After time for discussion, ask groups to share their thoughts on whether or not  is closed under these operations. Give other groups the chance to comment as well if their answer differed from the one provided.

• Briefly introduce students to two special cases of closure under operations on  . Zero is known as the additive identity because zero added to any real number is that number.

Similarly, 1 is known as the multiplicative identity because one multiplied by any real number yields that number. The identities are a more restrictive case of closure.

• Take time at the end of class to see if students have any questions.

Informal Assessment The instructor will circulate during the group-work portion of the lesson to check students’ comprehension of these concepts; the instructor will also look around the room during the lecture portion of the lesson to monitor this. In addition, students’ answers for closure of  under the given operations will be used to informally assess students’ understanding—whether students

34 were able to correctly determine closure, and if not, what their justification was and how many other groups (if any) volunteer a different answer.

Formative Assessment For homework, students will be instructed to consider whether  , the set of all rational numbers, is closed under the operations that were discussed in class. As they did in class, students will also be asked to justify their answer for each operation. This will not be done through WebAssign; rather, students will be instructed to write out their answers and turn in the assignment at the start of the next class. To accommodate low-SES and ELL students, who may be uncomfortable writing out their answers, all students will be given the option of recording a video in which they explain their conjecture in lieu of producing a written work; this can be done using equipment available in the school library.

35 Day 6 Lesson Objectives Given a number expressed in exponential form, students will express it in radical form (and vice versa) with 100 percent accuracy.

Given numbers expressed in exponential and radical form, students will determine whether the two numbers are equivalent with 100 percent accuracy.

Materials/Resources A handout with 10 numbers expressed in radical and exponential form will be utilized in class; students will be asked to determine whether the two expressions are equal (there are seven correct expressions and three incorrect ones). This handout, as well as solutions with explanations, will be posted to the course website after class.

Lesson Outline • Begin by writing

2 , the simplest irrational radical, on the board. Students are already 1

•

familiar with this notation. Now write 2 2 on the board—they likely have not seen a radical expressed like this previously.

Explain to them that radical expressions can also be expressed as exponents, such as in the case of 2 . This is because of the properties of exponent multiplication: (21/2)2 must equal 21 = 2, so 21/2 is the square root of 2; recall that ( 2 )2 = 2 by definition. In fact, any radical expressed in exponential notation is a number raised to a fractional power, with the power of the radicand as the numerator and the nth root as the denominator. For the 1

•

•

1

same reason, then, 3 n = n 3 , 4 n = n 4 , and so on.

To accommodate learning disabled and ELL students, use a visual representation with painted puzzle pieces. Initially, two pieces will be together, one with “n” painted on it and one with “ 3 ” on it. Other pieces will be available with various fractions, including 1 . Separate the two pieces and invite one of these students up to the front; ask him/her 3 to find which piece fits with the n piece—this shows the notations are equivalent. If the student gets the correct piece immediately, ask him/her to show the class that the other fractional pieces do not fit.

Now ask students to think about a slightly more challenging case. Ask the class as a 3

•

•

whole how to express n 3 in exponential notation ( n 2 ).

Write three radicals on the board and ask the class as a whole how to convert them to exponential notation, as shown above. Students should not be in groups for this portion of the class; they can work individually. Ask for students to give the answer (they can answer from their seats, but they must explain how they derived the exponent).

Now, ask students to consider the case of going from exponential to radical notation, 2

using the rules previously described. Write n 5 on the board and show the class how to convert this to radical notation ( 5 n 2 ). As with before, invite an LD or ELL student to the front of the class to express this using puzzle pieces as well.

36 • •

Write three exponents on the board and ask the class as a whole how to convert them to radical notation. As before, students should solve the problems and answer individually.

Distribute a handout with 10 numbers expressed in both radical and exponential form— the students are to determine whether the two expressions are equal. The handout will include seven correct expressions and three incorrect ones. In addition, to provide an extra challenge for gifted and talented learners, three problems are included at the end asking students to express a root of a perfect square in simplest form (e.g., since 25 = 5 ,

•

•

1

25 = (5 2 ) 3 = 5 2 3 ). Circulate as students are working to answer any questions.

Go down the list of problems and ask for a student to answer each question (students can remain at their seats for this part of the activity but must justify their answers). Ask if any students were able to complete the final three problems; if so, have them share and explain their answers to these questions as well. If not, ask for a student to come to the board and work through the problem, explaining where (s)he gets stuck, and guide him/her to the correct answer from this point. Also explain this method of expression to the class after demonstrating the three problems.

Take time at the end of the class to see if students have any questions.

3

Informal Assessment The instructor will circulate as students are completing the handouts to see if any students are stuck or are having difficulty understanding. (This will also be checked by glances around the room during the lecture at the start of class.) In addition, the instructor will note how many students volunteer to answer the practice problems in the first part of class as well as the first 10 problems on the worksheet.

Formative Assessment Students will complete a 15-question problem set on WebAssign for homework: five problems requiring conversion from radical to exponential form, five requiring conversion from exponential to radical form, and five asking if two quantities expressed in radical and exponential form are equal.

37 Day 7 Lesson Objectives Given a number expressed in radical form, students will utilize the method of prime factorization to simplify the radical with 100 percent accuracy (e.g., 50 = 10 5 = 5 5 2 = 5 2 ).

Given an encryption algorithm and its applications, students will gain an appreciation for the role of the number-theory topics discussed in this unit in modern-day cryptography.

Materials/Resources During the group work portion of the lesson, students will use iPads to visit the following website and design their own encryption keys: http://people.eku.edu/styere/Encrypt/RSAdemo.html. A link to this page will also be posted on the course website after class.

Lesson Outline • Begin by asking students how they would find the square root of a large number—write 7500 on the board. Impossible to do without a calculator, right?

• Show students that by using the method of prime factorization, they can break down a large number into a more manageable one. Ask a student to volunteer to give a factor of 7500 and begin factoring the number based on their response. o For example, suppose a student gives 500 as the first factor of 7500. Then 7500 becomes 500 15 . 15 breaks down to 5 3 . Ask for a factor of 500— suppose 100 is given. We now have 100 5 5 3 . We know 100 = 10 and 5 5 = 5 by definition, so 7500 can be simplified to 10 ∗ 5 ∗ 3 , or 50 3 .

• Ask a student to provide another nonprime radical; this time, ask for student volunteers to guide the class through the process of factoring it in the manner shown.

• To accommodate learning disabled and ELL students, compare the method of prime factorization to organizing spare change. If I have 689 pennies, I can break that down into larger denominations. 600 pennies form six dollars, 75 more pennies can be exchanged for three quarters, and 10 more for a dime. Now I have six $1 bills, three quarters, one dime, and four pennies. I can simplify this further by exchanging five $1 bills for a $5 bill; this will give me the simplest arrangement of coins and bills.

• Discuss the applications of prime factorization to computer science, specifically cryptography and network security: Data is encrypted through the use of large prime numbers that cannot be factored in a reasonable amount of time (for example, it took computer scientists two years, utilizing a supercomputer cluster containing hundreds of machines, to factor a 232-digit number). The question of whether decrypting this data is as difficult as factoring the key is one of the most significant unsolved problems in computer science (the RSA problem). (Note: Applications to cryptography will be discussed in greater detail in lesson 13.)

• Guide students through the process of creating their own encryption keys, using the website listed above. Each student will subsequently work individually on an iPad to come up with his/her own key; the instructor will circulate to answer questions and ensure students are staying on task.

38 •

Take time at the end of class to see if students have any questions.

Informal Assessment During the first part of class, the instructor will observe how many students volunteer to help factor the second radical, as well as if they run into trouble at any particular spots. The instructor will also look around the room to see if any students appear to be having difficulty comprehending the process. As students are working individually to create encryption keys, the instructor will circulate to ensure all students understand the topic and also to get a feel for the complexity of the keys students are creating, which is a useful barometer of their level of comfort with the topic.

Formative Assessment For homework, students will complete a 15-question WebAssign problem set. Five questions will ask students to factor a radical into simplest form; there will be three more asking students to determine whether a given radical is in simplest form (and if it is not, to rewrite it as such).

Another three will ask them if a given expression is factored properly (e.g., 50 = 5 3 ) and to correct it if it is not (in this case, 50 = 5 2 ). The final four questions will be multiple-choice questions assessing their knowledge of the cryptographic techniques discussed in class.

39 Day 8 Lesson Objectives Given a number expressed in radical form, students will determine, with justification, whether the radical is rational or irrational with 100 percent accuracy.

Given a declarative statement, students will use the reductio ad absurdum method of proof to show that it is true with 100 percent accuracy.

Materials/Resources A link to the following website, which contains an interactive tutorial on the RAA proof techniques introduced in class, will be posted on the course web page after class: http://www.cs.colostate.edu/~cs122/tut_4.php.

Lesson Outline • One of the first examples of an irrational number used in the class was 2 . Ask the class as a whole if they think other roots of non-perfect squares (such as 3 ) are also irrational. Call on a few students to see their answers and reasoning.

• Inform students that every radical not involving a perfect square is irrational (all radicals of perfect squares yield integers and thus are rational by definition). With this established, the question now becomes how to prove it.

• Begin by demonstrating for the class as a whole the proof that 2 is irrational: p o Suppose that 2 is rational. Then it can be expressed as a fraction in simplest q form with p,q ∈ .

•

•

p2 o 2 = 2 ; 2q 2 = p 2 . So p must be even since it is divisible by 2. q o By definition, then, we can express p as 2k 2 , k ∈ . Substitute p = 2k 2 into the original equation.

(2k)2 4k 2 o 2 = 2 = 2 . Rearrange to find q 2 = 2k 2 . So q is divisible by 2 and must also q q be even. p o So p and q are both even. But this is impossible, since was taken to be in q simplest form. This is a contradiction (also written “ ⇒⇐ ”). o Thus it is impossible for 2 to be rational, and so 2 must be irrational.

The proof technique used here is known as “reductio ad absurdum” (RAA) and is one of the most frequently used proof techniques in abstract mathematics. Explain to students that the technique shows that it is impossible for a given statement (e.g., 2 is rational) to be true, and so it must be false. Run through an RAA proof that the world is round (i.e., demonstrate that the world is not flat) to provide additional exposure to the technique.

After asking for questions, divide students into four groups, placing the gifted students together (these are different from the standard problem-solving groups). Assign three

40

• •

•

•

groups the task of creating a proof to show that 3 , 5 , and 6 , respectively, are irrational. As an accommodation, ask the gifted and talented students to work together and attempt to come up with a proof that square roots of all non-perfect squares are irrational.

Circulate as students are working to answer any questions they may have. Guide them if they are having trouble with the proof.

Going in numerical order, ask a representative from each of the first three groups to come to the board and share their proof of the irrationality of their given radical. Once this is complete, ask one of the gifted and talented students to demonstrate their proof for all non-perfect squares. (If they have not completed it, ask a student to illustrate how far the group was able to go, and then guide the student to the completion of the proof.)

Inform students that while they can determine whether smaller radicals are rational or irrational by sight, they must use the method of prime factorization to make a determination on larger ones. For example, while 729 may appear to be irrational at first glance, 729 = 272, so it is in fact rational.

Take time at the end of class to see if students have any questions.

Informal Assessment During the lecture portion of the class, the instructor will check to see if any students are visibly encountering difficulty with the material being presented; this will also be done as groups are presenting their proofs of the irrationality of non-perfect squares. The instructor will also circulate as students are working in groups to evaluate students’ understanding of the concepts and note if there is any particular step on which multiple groups are getting stuck. The instructor will also informally note the thoroughness and accuracy of each standard group’s proof of the irrationality of its assigned radical, as well as the progress of the gifted and talented group on its universal proof.

Formative Assessment Students will complete a written problem set (not one through WebAssign) containing 10 problems for homework. They will first be asked to determine whether five radicals are rational or irrational using the method of prime factorization; if a radical is irrational, they must show this using the proof techniques outlined in class. (Students will be required to show all work to receive credit to ensure they are using prime factorization and not just their calculators. Two of the five radicals will be irrational.) The other five problems will ask them to construct simple RAA proofs, such as the proof that there is no largest prime number.

41 Day 9 Lesson Objective Given examples of the golden ratio and its influences throughout history and in nature, students will gain an appreciation for the significance of the ratio.

Materials/Resources During the group work portion of the class, students will use iPads to research applications of the golden ratio to science, architecture, and art. In addition, the class begins with a description of how the golden ratio inspired the design of the Parthenon; a link to the following site, which covers this in great detail, will be posted to the course website after class: http://www.goldennumber.net/parthenon-phi-golden-ratio/.

Lesson Outline • Begin by showing the class a picture of the Parthenon. Explain that its distinctive architecture is a product of one of the Greeks’ greatest contributions to mathematics (and fields beyond), the golden ratio.

1+ 5 • Inform students of the value of the golden ratio: ϕ = . Although the first 2 approximation of the value of the golden ratio did not appear until the late 16th century, it has been observed in architectural and other works for over 2400 years. o To accommodate multicultural learners, references should also be made to German mathematicians Heinrich Agrippa and Adolf Zeising, who researched the prevalence of the golden ratio in nature.

• Break students into their problem-solving groups. Assign each group one of the following applications of the golden ratio to research: o Naqsh-e Jahan Square, a temple in Iran o Leonardo da Vinci’s “Vitruvian Man” o Connections between the golden ratio and human genome DNA (proposed by several researchers) o Claude Debussy’s composition Reflets dans l’eau • Groups will be allotted 20 minutes to research how the golden ratio impacted the development and design of their assigned topic, or, in the case of human DNA, how it has been observed. The architecture of Naqsh-e Jahan Square, for example, was based on “golden rectangles,” rectangles whose sides are in the same proportion as the golden ratio.

• Circulate throughout the room as groups are working to answer any questions and ensure students are staying on task.

• After they have concluded their research, groups will take turns giving five-minute presentations to the class on their findings.

• Take time at the end of class to see if students have any questions. If necessary, move one presentation to the beginning of the following class.

Note: There are significant contemporary applications of the golden ratio as well, most of which are based in technology—such as the design of the iPod and some tablet computers. While it can be argued that these may be more applicable and appropriate for the class, this lesson focuses on historical applications both as a means of integrating a cultural component and as an

42 accommodation for low-SES students, who may feel uncomfortable in a class discussion about iPods and other luxuries they may not necessarily own. The formative assessment gives those students who want to focus on applications of the golden ratio to electronic devices a chance to do so without unnecessarily excluding those from lower socioeconomic backgrounds.

Informal Assessment The instructor will circulate as students are working in groups to ensure student understanding and will also note if any particular students are avoiding participating. As groups are presenting, the instructor will note if any students appear to be having difficulty understanding. The instructor will also informally note the thoroughness of each group’s presentation.

Formative Assessment For homework, students will write 1-2 pages describing how they would integrate the golden ratio into a work of their choice. It can be in any field—music, art, technology, architecture, or anything else of their choosing. To accommodate learning disabled and ELL students, all students will have the option of submitting a detailed sketch, or similar work (such as musical staffs), in lieu of a written report. However, the sketch must clearly demonstrate the integration of the golden ratio (such as by labeling side lengths or clearly identifying certain musical notes).

In addition, to accommodate students who may be uncomfortable with the substantial writing component of this assignment, all students will be given the option of recording a video in which they explain their creation in lieu of producing a written piece; this can be done using equipment available in the school library.

43 Day 10 Lesson Objectives Given a compass and straight edge, students will construct a “golden rectangle,” 30-60-90 right triangle, and 45-45-90 right triangle, and identify the steps involved in constructing these figures, with 100 percent accuracy.

Given a set of side lengths, students will determine whether it is possible to construct a 30-60-90 or 45-45-90 right triangle with these side lengths with 100 percent accuracy.

Materials/Resources All students will be given a compass and straight edge to perform the constructions in this lesson, as well as a handout with lengths of triangle sides on it. This handout will be posted to the course web page after class. In addition, the concept of a “golden spiral” (a spiral constructed from a series of golden rectangles) is introduced in class; a link to the following website, which includes more details and examples of golden spirals in nature, will be posted to the course web page: http://www.goldennumber.net/spirals/.

Lesson Outline •

•

• •

•

1+ 5 ). In addition, use 2 this time at the beginning of class to finish any presentations for which there was not time in the previous lesson.

Introduce students to the concept of a “golden rectangle,” a rectangle with a side ratio that is equal to the golden ratio (i.e., l = ϕ and w = 1). A unique property of a golden rectangle is that a square section can be removed to yield another golden rectangle; this process can continue infinitely. Similarly, two golden rectangles can be joined together to yield a third golden rectangle.

Review with students the method of finding a midpoint using a compass and straight edge and ensure all students understand—explain that this plays a central role in the construction of a golden rectangle.

Distribute a compass, straight edge, and blank piece of paper to each student (they will work individually for this portion of the lesson). Guide students through the steps to construct a golden rectangle: o Construct a simple square (using previously learned techniques). o Find the midpoint of one side of the square. o Draw a line from this midpoint to an opposite corner. o Use this line as the radius of a half-circle that gives the height of the rectangle. o Draw in the other sides of the rectangle using the straight edge.

After going through the construction, give students the option of progressing on to the construction of a “golden spiral,” a spiral drawn in a series of golden rectangles, or working through another teacher-guided example. This accommodates both gifted and talented and learning disabled students. For those who would like another example of the construction of a golden rectangle, use as an example the flag of Togo, which was designed as a golden rectangle. Project the flag at the front of the room and run through Begin by reminding students of the value of the golden ratio (

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•

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the construction techniques on the board. (Students who have progressed to the golden spiral will work independently to not distract other students.)

Use the remaining 10-15 minutes to go over triangle constructions. Distribute another sheet of paper with a series of lines on it—one with side lengths x, x, x 2 and one with side lengths x, x 3,2x . These will be used to construct a 45-45-90 and 30-60-90 right triangle, respectively, building on students’ knowledge of applications of irrational numbers.

Explain to students the triangle construction techniques, and that because there is only one way for the sides to fit together, they do not need to know the angle measures (or use a protractor) to construct these special right triangles. Explain also that because of this, it is possible to determine from the given side measures whether a triangle with these sides can be constructed. For this portion of the lesson, students will again work individually with the teacher leading at the board.

Take time at the end of class to see if students have any questions.

Informal Assessment During the lecture and construction portions of the lesson, the instructor will visually monitor the class to see if any students appear to be encountering difficulty. The instructor will also note how many students move on to golden spirals and how many request another run-through on construction of golden rectangles, exclusive of gifted/talented students (all of whom it is expected will move on to golden spirals) as well as those who are learning disabled, who will likely request a second golden rectangle example.

Formative Assessment For homework, students will complete an eight-question problem set on WebAssign. The first three questions will ask students to place the five steps to constructing a “golden rectangle” and each of the two special types of right triangles in their correct order, and the following five will list three side measures and ask if it is possible to construct a right triangle from the given measures. (Three questions will focus on 30-60-90 right triangles, and two will ask about 45-4590 triangles; two of the five will feature impossible side measures.)

45 Day 11 Lesson Objective Given a fraction with an irrational denominator, students will rationalize the denominator and express the resulting fraction in simplest form with 100 percent accuracy.

Materials/Resources This class includes a handout with 10 problems asking students to rationalize the denominator; this worksheet will be put on the course website, along with solutions, after class. In addition, as was done following lesson 4, a practice WebAssign allowing unlimited submissions will be made available to students after the lesson.

Lesson Outline 25 • Write on the board and ask students how to simplify it. In the past, they have learned 3 5 that this equation expressed in simplest form is . Inform students that while this is 3 technically not incorrect, it is preferable to have a rational denominator—that is not the case with this fraction, since 3 is irrational.

• Explain to students that the steps to rationalizing the denominator are the same as when 3 5 3 adding fractions with different denominators. In this case, multiply by to yield , 3 3 which is the “proper” way to express the fraction.

6 2 • Write another fraction with an irrational denominator on the board: . This time, ask 5 for a student volunteer to come to the board and show the class how to rationalize the denominator. If the student gets stuck or encounters difficulty, ask if another student can help him/her.

• Distribute a worksheet with 10 fractions with irrational denominators on it to each student. (Students will work individually on this worksheet, not in their problem-solving groups.) The first four problems will be of the form above; the next four will involve 6 2 fractions that can be simplified beyond rationalizing the denominator, such as , 3 which reduces to 2 6 . To challenge gifted and talented students, the final two problems 3 will include mixed-number addition in the denominator, such as . Circulate while 2+ 2 students are working to monitor progress and answer any questions.

• Ask for students to volunteer to come to the board and demonstrate problems. It is expected that there will be a volunteer for each of the first eight problems; if one of these volunteers runs into trouble, ask for another student volunteer to help him/her. If no students were able to solve the final two problems, ask for a student to come to the board and work through the problem, explaining where (s)he gets stuck. At this point, the

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instructor will guide him/her to the correct answer (it is necessary to multiply by the conjugate to rationalize the denominator).

Take time at the end of class to see if students have any questions.

Informal Assessment The instructor will circulate as students are completing the worksheets to see if any students are stuck or are having difficulty understanding. (This will also be checked by glances around the room during the lecture at the start of class.) In addition, the instructor will note how many students volunteer for the first eight problems on the worksheet, as well as how many students offer to help if a volunteer gets stuck. The instructor will also note which steps tend to give students trouble on these problems.

Formative Assessment Students will complete a 10-question WebAssign problem set for homework. The first five questions will involve fractions with denominators that must be rationalized and cannot be simplified further; the next three ask students to rationalize the denominators and reduce the resulting fraction. The final two problems are the special cases introduced at the end of class that require multiplying by the conjugate to rationalize the denominator.

47 Day 12 Lesson Objective Given an irrational number, students will explain, in writing or verbally, its role in historical mathematics with 90 percent accuracy (e.g., π represents the ratio of a circle’s circumference to its diameter).

Materials/Resources During the group work portion of this lesson, students will use iPads to research the historical role of their assigned irrational number. After the students have turned in their responses during the following class, links containing more information on each of the three numbers will be posted to the course web page.

Lesson Outline • Break students into three heterogeneous groups (unlike in lesson 8, do not place all of the gifted and talented students together). Each group will be assigned one of the following irrational numbers to research: o π (the ratio of a circle’s circumference to its diameter) o e (Euler’s number, used in calculating natural logarithms) o 2 (the length of a diagonal of the unit square; also plays a significant role in right-triangle geometry) • Groups will be allotted 20 minutes to research the historical role and applications of their assigned number, placing it in its proper context—for example, π was extremely controversial at the time it was first posited because it was believed God would not have created an imperfect (i.e., irrational) number. While this independent research project is designed primarily to accommodate gifted and talented students, the entire class can benefit from learning about the historical role of irrational numbers.

• Circulate throughout the room as groups are working to answer any questions and ensure students are staying on task.

• After they have concluded their research, groups will take turns giving 5-7 minute presentations to the class on their findings. Since each group consists of eight students, groups will be asked to choose one person to give the presentation.

Informal Assessment The instructor will circulate to monitor students during group work, answering any questions that may arise and noting if any students are not participating or appear to be having difficulties. The instructor will also note if any groups’ presentations were missing key details and mention these after the presentation has concluded.

Formative Assessment For homework, students will write a one-page response discussing their thoughts on one of the other group’s findings, including the advances in mathematics the chosen number permitted, its social ramifications, and its applications to other fields. Students will be made aware of this assignment at the beginning of class to allow them the opportunity to take notes on other groups’ presentations, if they desire. To accommodate low-SES and ELL students, who may be uncomfortable producing a written response, all students will be given the option of recording a

48 video in which they explain their proof outline in lieu of writing out their thoughts; this can be done using equipment available in the school library. In addition, as is the case with all lessons, students will be given access to a recording of this lesson with both audio and video components if they are not comfortable writing down notes during class.

49 Day 13 Lesson Objectives Given a mixed set of rational and irrational numbers representing an encrypted phrase, students will use the set as a key to decrypt the phrase with 100 percent accuracy.

Given a substitution cipher, students will use it to encrypt a specific word or phrase with 100 percent accuracy.

Materials/Resources All students will receive a handout containing a modified Caesar cipher for use during the group work portion of this lesson; this handout will be posted to the course web page after class. In addition, the following link, which includes additional information on Caesar ciphers, as well as techniques to make them more difficult to crack, will be posted to the course web page after class: http://www.cs.trincoll.edu/~crypto/historical/caesar.html.

Lesson Outline • As was covered in lesson 7, prime factorization is widely used in computer science, specifically to encrypt sensitive information. However, this is not the only encryption method in use today. Introduce to students the concept of a substitution cipher, and explore the special case of a Caesar cipher, in which each letter is replaced with another one a fixed number of places down the alphabet. o Provide an example of a word encrypted using a Caesar cipher: Ask a student to provide a sample number; use this number as the shift and encrypt the student’s full name using the cipher.

• To accommodate learning disabled students, provide a non-technical example of a cipher.

The United States Secret Service issues codenames to the president, vice president, and their families—for example, Barack and Michelle Obama are known as “Renegade” and “Renaissance,” respectively. In Secret Service communications, these codenames replace the individuals’ real names. Similarly, in the use of a cipher, a given letter is replaced by one a specified number of spots in the alphabet away from it.

• Ask if students can see any problems with a Caesar cipher (the expected response is its simplicity—students can be guided in this direction if necessary). Once the topic of simplicity is broached, explain how numbers can be used to make the algorithm more complex—the shift can be multiplied by, or added to, a different number to make it more difficult to decrypt the cipher.

• Current ciphers use extremely sophisticated methods, such as ciphers containing hundreds of digits. Since that level of sophistication is impractical in this class, students will work with a modified cipher to solve an encrypted phrase (“Remember, my Eliza, you are a Christian”). For simplicity, students will receive a set of numbers rather than an encrypted phrase. The cipher will use the following rules: o The base number for the shift is 5. o Rational numbers correspond to lowercase letters; irrational numbers correspond to capital ones. o In each case, the shift is multiplied by a number corresponding to the initial letter’s place in the alphabet—so, for example, the “z” in “Eliza” would be

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assigned a value of 26, giving a cipher value of 130 for that letter. In the case of capital letters, the square root will be used ( 3 , 5 , and 18 = 3 3 are all irrational).

Break students into their problem-solving groups for this activity. Distribute a handout containing the set of cipher numbers to each student and project it on the board as well.

Circulate as groups are working to answer questions and assess student progress.

Ask a group to volunteer to share what it worked out for the actual phrase, as well as the cipher. If no groups were able to work out the phrase, ask groups to share the letters they were able to work out; the instructor will guide the class to the solution based on this.

Take time at the end of class to see if students have any questions.

Informal Assessment During the group work portion of the lesson, the instructor will circulate to see if any students are having trouble understanding the material. The instructor will visually monitor this during the lecture portion of the lesson and will also note the problems with Caesar ciphers besides simplicity that the students identify, which can be evidence of either comprehension difficulties or, ideally, in-depth thinking. The instructor will informally note how many groups were able to decrypt the phrase, and if none were able to do so, how many letters each group was able to figure out.

Formative Assessment Students will complete a 10-question problem set on WebAssign for homework. The assignment includes four words and short phrases to decrypt based on a provided set of numbers, four words and short phrases to encrypt based on a provided cipher, and two multiple-choice questions about general cryptographic techniques.

51 Day 14 Note: This lesson will consist of review for the unit test in the following lesson. Because of the nature and format of this lesson, there are no specific accommodations for students; the lesson is appropriate for all learners.

Materials/Resources A Jeopardy! PowerPoint will be used in class as a review game for students; this PowerPoint will include answers and be posted on the course website after class so students can continue to use it to review. In addition, a buzz-in system will be provided for students during the game.

Lesson Outline • Begin by taking approximately five minutes to go over the format of the unit test as well as exam rules (including a prohibition on the use of calculators).

• Break students into their problem-solving groups. The class will play a Jeopardy! game designed in PowerPoint to review material for the unit test; each group will be provided with a pencil and a few blank sheets of paper to use if necessary. The game is designed to take approximately 30 minutes.

• The final 10 minutes of the class will be devoted to answering any additional questions from students regarding the format or material of the exam.

Informal Assessment The game itself serves as an informal assessment, allowing the instructor to monitor what material or types of questions students are either struggling with or shy away from. The instructor will also make note of the questions asked by students following the game—this is likely the material that is giving them trouble. During the game, the instructor will note if any students are not participating in their groups or appear to be having difficulty understanding the material.

Formative Assessment There will be no formative assessment due to the nature of the lesson.

Note: Day 15 is the summative assessment. There is no outline for this day.

Section 5 Summative Assessment

52 Summative Assessment TOTAL: ___/50 PART I. Select the best answer from the choices given. (2 points each) 1. Irrational numbers are (a) Those that can be expressed as the ratio of integers (b) Finite (c) Those that cannot be expressed as the ratio of integers (d) Both (b) and (c) 2. In geometry, π represents (a) The ratio of a circle’s circumference to its radius (b) The ratio of a circle’s radius to its diameter (c) The ratio of a circle’s diameter to its circumference (d) The ratio of a circle’s circumference to its diameter 3. Which of the following is NOT irrational? (a) The sum of a rational number and an irrational number (b) The sum of a rational number and a rational number (c) The product of a rational number and an irrational number (d) None of the above 3

4. Express (a) x

6

(b) x

2

(c) x

17

(d) x

11 15

x

2

5

in exponent notation.

5 15 5

5. Sets can be formed from all of the following EXCEPT: (a) All natural numbers (b) The math teachers at Anytown High School (c) Movies in which Denzel Washington has appeared (d) Good reasons to watch ESPN PART II. Write an “R” next to each rational number and an “I” next to each irrational number. (1 point each) ___ 2

___ 7/19

___ e

___ π

___ 3

___ 4.5

___ 752,000

___ 0

53 PART III. Encrypt the following statement using a Caesar cipher. Be sure to identify your cipher. (4 points) THE QUICK BROWN FOX JUMPED OVER THE LAZY DOG. PART IV. Perform the given set operations. (2 points each) 1. {0,1,2, 3} ∪ {2, 3,5, 7} 2. {2, 3,5,8,11} ∩ {2, 4,6,8,10} 3. {a,b,c,d} \ {a,c,e, g} 4. {Boston, Seattle, Memphis} ∪ {Seattle, Houston, Chicago} 5. {Homer, Marge, Bart, Lisa, Maggie} ∩ {Homer, Odysseus, Menelaus, Agamemnon} PART V. Write out a payoff matrix for the following game. (8 points) You are playing quarterback in the fourth quarter of the Super Bowl, and your team needs a touchdown to win. You have time to run one more play from the one-yard line. You can either throw the ball or try to run it in yourself. The opposing defense will either guard against a pass or guard against the run. Assume your payoffs are 1 if you score a touchdown and 0 if you don’t; the payoffs for the other team are reversed.

PART VI. Answer ONE of the following short-answer questions in 1-2 paragraphs. (10 points) 1. Recall that Ancient Greece was a deeply religious society, and it was universally believed that God would not create something imperfect. Pythagoras, whose theorem on right triangles has been previously covered in this class, was also an influential religious and philosophical leader.

Explain why his renown in these fields presented a dilemma for him upon his discovery of the properties of 30-60-90 and 45-45-90 right triangles.

2. The networking website LinkedIn recently came under fire after it was revealed following a successful hacking attempt that it did not “hash” its users’ passwords—that is, randomly insert additional characters as a security measure. Using your knowledge of the cryptographic techniques we have covered in this unit, explain how hashing can add an extra level of security to encrypted data.

BONUS. Do not attempt the bonus until you have answered all of the other questions on the test. No partial credit will be given for the bonus question. (3 points) Determine whether the set of complex numbers  is closed under multiplication. Justify your answer.

54 Rubric/Answer Key for Summative Assessment Part I (2 points each, no partial credit) 1. (c) Those that cannot be expressed as the ratio of integers 2. (d) The ratio of a circle’s circumference to its diameter 3. (b) The sum of a rational number and a rational number 2

4. (b) x 15 5. (d) Good reasons to watch ESPN Part II (1 point each, no partial credit) R 2

R 7/19

I e

I π

I

R 4.5

R 752,000

R 0

3

Part III (4 points) • 1 point for identifying Caesar cipher • 3 points for correct translation of phrase o Subtract one point if 1-2 letters are shifted incorrectly, two points if 3-5 letters are shifted incorrectly, and three points if six or more letters are shifted incorrectly.

Example: Cipher = 6 Phrase: ZNK WAOIQ HXUCT LUD PASVKJ UBKX ZNK RGFE JUM. Part IV (2 points each) 1. {0, 1, 2, 3, 5 7} • Subtract one point if one element is missing and/or duplicated; give no credit if two or more elements are missing and/or duplicated or if student found the intersection: {2, 3}.

2. {2, 8} • Subtract one point if one element is missing and/or duplicated; give no credit if two or more elements are missing and/or duplicated or if student found union: {2, 3, 4, 5, 6, 8, 10, 11}.

3. {e, g} • Subtract one point if student answered {b, d}; give no credit for other answers.

4. {Boston, Seattle, Memphis, Houston, Chicago} • Follow the same grading conventions as question 1. If the student found the intersection for both question 1 and this question, give full credit for this question for consistent error.

55 5. {Homer} • Follow the same grading conventions as question 2. If the student found the union for both question 2 and this question, give full credit for this question for consistent error.

Part V (8 points)

• • • •

1 point for correctly listing both actors 2 points for correctly listing both choices for both actors 1 point for listing payoffs as 0 and 1 4 points for listing payoffs in proper order (if order is reversed, subtract two points)

Part VI (10 points) Question 1: • 2 points for mentioning side measures of 30-60-90 ( x, x 3,2x ) and 45-45-90 ( x, x, x 2 ) triangles; 1 point for mentioning irrationality of x 2 and x 3 .

• 6 points for mentioning core idea: Irrational numbers, which played a central role in Pythagoras’ mathematical research, were considered imperfect; this put Pythagoras’ mathematical research at odds with his religious and philosophical studies. o Give 4 points if student does not mention that this caused a conflict between Pythagoras’ two research fields. o Give 4 points if student does not explicitly mention significance of irrational numbers in Pythagoras’ research. o Give 3 points if student does not mention that irrational numbers were considered imperfect.

• 1 point for proper spelling, grammar, and punctuation (students are allowed one total “freebie” from these categories combined).

Question 2: • 2 points for acknowledging that hashing makes encrypted data more secure, or that the lack of hashing makes data easier to decrypt.

• 7 points for mentioning core idea: The addition of extra characters essentially scrambles data; there is no way of knowing which characters are part of the actual data and which are a result of hashing, making it more difficult to decrypt the data. o Give 6 points if student mentions “extra letters” rather than “extra characters.” o Give 2 points if student states that this can create a more complicated cipher— note that hashing does not affect the cipher itself.

• 1 point for proper spelling, grammar, and punctuation (students are allowed one total “freebie” from these categories combined).

56 Bonus (3 points, no partial credit) • The set of complex numbers  is closed under multiplication. • The product of any two complex numbers is another complex number: While i2 = -1, there will be another number with i as a coefficient in the product. o Students MUST show this applies to all complex numbers. Do not give credit if they only provide examples with numbers; they must use complex numbers composed of variables only and demonstrate that the product is also complex. It is expected they will use the FOIL method to do this; evaluate other methods on a case-by-case basis. They will receive full credit as long as their chosen method demonstrates the results for all complex numbers.

57 Accommodations for Diverse Learners • • •

•

•

An extra-credit question is included as a means of challenging gifted and talented students in the class.

Learning-disabled students, as well as Lauren (who has been diagnosed with ADHD), will be given the option of untimed testing in a separate room. They will also be allowed to have the assistance of a scribe during the assessment if they desire it.

The services of foreign-language faculty at the school will be enlisted to translate the assessment into German and Chinese to accommodate English language learners Felix and Pao-Lin. These students will also be given the option of responding in their native language; the foreign-language faculty will translate their responses. Note that the phrase to be encrypted in part III (“THE QUICK BROWN FOX JUMPED OVER THE LAZY DOG”) will remain in English since this question focuses on recognition of patterns and not reading comprehension.

All students will be given the option of an oral assessment if they do not feel comfortable taking a written test (this is primarily intended to accommodate low-SES and ELL students). Students will be notified of this option during lesson 14, which takes place the day before the test.

A special enlarged-print copy will be prepared for Abe due to his poor vision, unless he chooses to be assessed orally.