ENGLISH CURRICULUM GUIDE - Breaking News ... · Web viewThis curriculum guide serves both Mentor Teachers and Teacher-Interns providing context, sequencing and concrete lesson plan

Curriculum Guide

for

Teacher-Interns and Mentor Teachers

Content:

Mathematics

Algebra for Rising 8th GradersAnd

Geometry for Rising 9th Graders

Mathematics Curriculum GuideAlgebra and Geometry

Table of Contents

Introduction 3

Texts For Your Class 3

Reading and Taking Notes in Mathematics 4

Department Meetings 5

Lesson Plans 5

o Attributes of Strong Lesson Planso Bloom’s Taxonomyo Procedureso Assessmentso Homeworko Marzano’s Nine Essential Instructional Strategies

Lesson Observations 9

o Classroom Management Rubrico Instruction Rubric

Curricular Objectives

o Long Term Skill Objectives 11

o Medium Term Skill Objectives (By Course, Unit, and Level)

Algebra 12

Geometry 19

Deadlines 25

Online Resources for Teacher-Interns 26

Contact Information 26

Appendix A: LESS Problem Solving Method 27

Appendix B: Sample Lesson Plans 29

Appendix C: Sample Activities 32

Appendix D: Cornell Notes 36

Appendix E: Student Success Plan 45

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Introduction

Purpose

This curriculum guide serves both Mentor Teachers and Teacher-Interns providing context, sequencing and concrete lesson plan components with a sample of lessons. This material is based on the work of seven Teacher-Interns conducted during Breakthrough New Haven’s six-week Summer Program in 2008.

Source Texts

The Number Devil, Hans Magnus Enzensberger (ISBN-13: 978-0805062991, Holt Paperbacks)

Algebra – Beginner Level

Glencoe Pre-Algebra (2010 edition) (ISBN 9780078885150, Glencoe-McGraw Hill)

Algebra – Intermediate and Advanced

Glencoe Algebra 1 (2010 edition) (ISBN 9780078884801, Glencoe-McGraw Hill)

Geometry

McDougall Littell Geometry (2007 edition) (ISBN 978-0618595402, Houghton Mifflin Harcourt)

Purpose of the Courses

These courses are designed for academically motivated seventh and eighth grade students who plan to attend rigorous college preparatory high schools. The majority of the seventh graders who completed these courses during the Breakthrough New Haven Summer Program will enroll in parochial and independent day and boarding schools.

These courses should be academically rigorous in order to prepare students to thrive for the more challenging academic institutions to which they aspire and instill intellectual confidence that comes with academic success.

The additional exposure to the language of mathematics should also increase students’ social and intellectual confidence as they acquire an increased ability to express their analytical thoughts.

Long-Term Goals for Our Students

Students will be able to: 1. Thrive academically in a mathematics class in a rigorous, college preparatory high school.2. Take reliable notes that will help them to prepare for assessments3. Learn the importance of the process of solving a problem not just supplying the answer

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Context

Breakthrough New Haven Students have all demonstrated the desire and capacity to do well academically. However, typically the student’s schools are both under-resourced with respect to materials and academic support for struggling students. The latter issue often results in teachers’ devoting a significant amount of time addressing behavioral and remediation issues that detract from the academic rigor of their classrooms. Finally, due to resource constraints, and, in some cases, institutional academic expectations and standards of success, the strongest students are not always challenged to the degree necessary to prepare them for rigorous high schools. For many of our students, Breakthrough represents their sole opportunity to be consistently challenged academically and achieve their full scholastic potential in an environment of similarly capable students.

Reading and Taking Notes in Mathematics Efficient reading and note-taking is an integral part of student success at any level. Breakthrough New Haven is committed to providing students with a toolbox with which to do these activities. To this end, Breakthrough Teacher-Interns should use the Cornell (STAR) note-taking system. At Breakthrough note-taking is not optional and should be treated as a serious part of the requirements for good scholarship. Breakthrough teachers should monitor that students are taking notes when appropriate and should give the students periodical notebook checks to assure that they are regularly rereading, revising, and rewriting their notes.

Cornell Notes

Breakthrough New Haven has adapted Cornell note-taking as the strategy we teach students to take notes in their classes. Since the strategy was used during the school year classes, students should already be very familiar with it. The strategy includes four steps (See Appendix D):

Set Up Paper- Before beginning to take notes, students set up their paper. They put an appropriate heading (including their name, the class’s name, the date they are taking the notes, and a title for their notes) in the upper right corner, draw a vertical line about one third of the way from the left edge of the paper, and draw a horizontal line about two inches from the bottom of the paper.

Take Notes- Students take most of their notes in the right hand column of the paper. At the very minimum, students should make sure to write down whatever the lecturer writes on the board. They should also be sure to write down any particular points that the lecturer might indicate are important and to delineate the important points from the lesser details by underlining, highlighting, or another method. If the student has any questions about the material or if any questions are answered in class, the student should write those questions in the left hand column of their notes, preferably right next to the portion of their notes that answers the question. Students should try to ask themselves higher level questions about the material rather than questions which focus solely on recall.

After Class- As soon as possible after class, students should summarize their notes for the day at the bottom of the page. They should also reread them looking for places that require additions of information or that are redundant. Students should also reread their notes with an eye to understanding. If there are any parts of the concepts or information that the student does not understand, the student should note those in order to ask the lecturer about them.

Review- Students should review their notes at least once a week regularly. Students can review their notes by folding their note pages on the long vertical line and asking themselves the questions in the left column.

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Student Success PlansBreakthrough Collaborative provides each of its affiliate sites with the template for Student Success Plans (SSPs). SSPs allow Breakthrough to categorically track the components of a student’s academic competency in a given subject. In the course of conducting the lessons during the Summer Program, Teacher-Interns and Mentor Teachers should be sure to include assessments of the skills addressed in the SSP.

In order to ensure consistency of assessment, Breakthrough New Haven’s Student Success Plans are completed by the Mathematics Mentor Teacher and the Director with the input of the Teacher Interns. The assessments of students’ performance are based on the academic demands of the Hopkins School Junior School Mathematics department curriculum.

Refer to Appendix E for a copy of the Mathematics section of the SSP.

Department Meetings

At least once a week, Teacher-Interns in the Mathematics Department will meet as a group with their mentor teacher during one of their preparation periods. Department meetings are not optional and will be scheduled at a regular time and place. Teacher-Interns should use the department meeting to talk about common issues with students or skill development, discuss lesson ideas and possible pitfalls, exchange curriculum materials, and do any necessary department-wide planning.

Department meetings are run by the mentor teacher or the Teacher-Intern that has been designated the “department chair.” Teacher-Interns should email both their department chair and their mentor teacher with any items that they would like included on the meeting’s agenda.

Lesson Plans

Attributes of Strong Lesson Plans

Objectives should be specific, concrete, manageable, and assessable.

1. The objective(s) should clearly define what students will be able to do that they may not have been able to accomplish prior to the class. Avoid vague objectives such as “Students will understand…”, “Students will discuss …”, and “”Students will appreciate …”

2. Collectively, the objectives should be sufficiently comprehensive to ensure that students are not left with knowledge gaps that hinder successful completion of assessments, class work and/or homework.

3. Teacher-Interns should be absolutely sure of what, if any, prior knowledge students need to accomplish the objectives.

4. Objectives should stress the development of critical thinking skills. Please refer to a brief discussion and outline of Bloom’s Taxonomy to creation of lesson objectives below.

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Bloom’s Taxonomy

The following chart organizes cognitive functions into a structure used to assure that classroom instruction supports and develops higher order thinking skills. The taxonomy prevents common pitfalls like teaching only the definition of a concept, instead of how and when a concept is used. Ideally, students participate in activities and discussions distributed over all six levels of the taxonomy.

Bloom’s TaxonomyLevel and Definition Action Verbs for Lesson Objectives

1. KNOWLEDGE Recalling and recognizing factual information

ListStateNameTabulateQuote

MatchExamineDescribeChooseDefine

IdentifyShowLabelcollect

2. COMPREHENSION Translating and interpreting ideas

RepresentExamineIllustrate

ShowSelectExplain

ParaphraseCalculateSummarize

3. APPLICATION Generalizing and applying knowledge to new situations

ModifyPredictSolve

ConstructCombineIntegrate

RearrangeDemonstrateCompute

4. ANALYSIS Break into parts to understand connections

ClassifyCategorize

DistinguishSeparate

CompareContrast

5. SYNTHESIS Combine elements into a new format

DesignDevelopCompose

CreateInventFormulate

HypothesizeInfer

6. EVALUATION Develop judgments and opinions according to criteria

RecommendPrioritizeAssess

AppraiseJustifyJudge

CriticizeDefend

Procedures should be well thought out and reviewed carefully and thoughtfully by the Teacher-Intern, Mentor Teacher and any other appropriately experienced educational professions. A Teacher-Intern should imagine him- or herself as the director of a play in which the actors have significant freedom to improvise and develop their roles.

Procedures should be skills-based, experiential, and designed to appeal to students with multiple learning styles. To that end, Teacher-Interns should be mindful of what students will be doing at any given point of the lesson when devising procedures for their classes. The most engaging classes are those in which the students are actively participating in their own learning at all times. While lecture is sometimes necessary in Mathematics classes, it is much better for lectures to imitate guided conversations wherein students actively ask questions, contribute ideas, and make connections. To assure this kind of participation, Teacher-Interns should try to formulate guiding questions for their lectures and discussions. Furthermore, lesson plans that integrate different numbers of voices (i.e. single voice = teacher only, two voices = teacher and student conversation, and multiple voices = group work) and different stations of learning (i.e. students at desks, students out of their seats, students exploring concepts out on campus) are structurally more likely to be engaging than ones that do not include such differentiation.

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Assessments serve an integral part of any educational program by allowing teachers to easily determine whether or not each student has met lesson or longer-term objectives. Good teachers regularly assess their students’ progress through assignments, homework, projects and tests, and in-class observation. Teacher-Interns should develop their major assessments early on and adjust them as classroom conditions warrant. By planning assessments early, Teacher-Interns give themselves concrete benchmarks by which to gauge their progress toward their medium- and long-term content and skill goals. Each Mathematics class at Breakthrough should have at least two minor assessments (preferably a rubric-based activity) during the course of the summer. A more traditional “post-test” will be scheduled for the last week of classes. (See page 95 of the Breakthrough Collaborative Teacher Guidebook for examples and resources of assessments and rubrics for assessment. The Guidebook may be found at http://teachbreakthroughs.org/teacher-resources/).

While it is important that Teacher-Interns always teach with their assessment in mind, assessments at Breakthrough New Haven do not have to fit the mold of the traditional test/quiz dichotomy. Teacher-Interns are encouraged to develop creative assessments so long as they are objectives-based and those objectives are compatible with the objectives of the Teacher-Intern’s classes. Additionally, Teachers-interns should develop rubrics for their assessments that use clear and concrete guidelines to demonstrate the students’ achievement. If there are any longer-term assignments, Teacher-Interns should provide students with the rubrics that will be used to evaluate the students’ work at the time they assign the assessment.

Homework should be 1) a form of assessment as well as review of the material covered in class or 2) preparatory work for the next class. Satisfaction of the objectives by successful completion of the lesson procedures and thoughtful in-class assessments should allow students to complete the homework independently or in groups, as determined by the Teacher-Intern.

Students should be given no more than thirty minutes of homework for each class, each day that the class meets. Teacher-Interns may not double the amount of homework they assign because it is to be completed over a weekend! It is important for Teacher-Interns to realize that the summer program is the first time that many of our students will be trying to balance a difficult workload. Teacher-Interns should take the time to complete each homework assignment that they intend to assign well before assigning it to students. Doing this can serve as a good way to anticipate some of the issues that students might have and to gauge how much time it will take the students to complete the assignment. Teacher-Interns should assume that it will take students four to five times longer to complete any assignment than it takes the Teacher-Intern to complete it. In the case of homework, Teacher-Interns should expect to be able to complete their assignments between seven and eight minutes.

Marzano’s Nine Essential Instructional Strategies

Researchers identified nine instructional strategies that are most likely to improve student achievement across all content areas and across all grade levels. These strategies are explained in the book Classroom Instruction That Works by Robert Marzano, Debra Pickering, and Jane Pollock.

1. Identifying similarities and differences2. Summarizing and note taking3. Reinforcing effort and providing recognition4. Homework and practice5. Nonlinguistic representations6. Cooperative learning7. Setting objectives and providing feedback8. Generating and testing hypotheses9. Cues, questions, and advance organizers

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http://teachbreakthroughs.org/teacher-resources/


1. Identifying Similarities and Differences: helps students understand more complex problems by analyzing them in a simpler way

a) Use Venn diagrams or charts to compare and classify items.b) Engage students in comparing, classifying, and creating metaphors and analogies.

2. Summarizing and Note-taking: promotes comprehension because students have to analyze what is important and what is not important and put it in their own words

a) Provide a set of rules for asking students to summarize a literary selection, a movie clip, a section of a textbook, etc.

b) Provide a basic outline for note-taking, having students fill in pertinent information

3. Reinforcing Effort and Providing Recognition: showing the connection between effort and achievement helps students helps them see the importance of effort and allows them to change their beliefs to emphasize it more. Note that recognition is more effective if it is contingent on achieving some specified standard.

a) Share stories about people who succeeded by not giving up.b) Find ways to personalize recognition. Give awards for individual accomplishments.c) "Pause, Prompt, Praise." If a student is struggling, pause to discuss the problem then prompt with

specific suggestions to help her improve. If the student's performance improves as a result, offer praise.

4. Homework and Practice: provides opportunities to extend learning outside the classroom, but should be assigned based on relevant grade level. All homework should have a purpose and that purpose should be readily evident to the students. Additionally, feedback should be given for all homework assignments.

a) Establish a homework policy with a specific schedule and time parameters.b) Vary feedback methods to maximize its effectiveness.c) Focus practice and homework on difficult concepts.

5. Nonlinguistic Representations: has recently been proven to stimulate and increase brain activity.a) Incorporate words and images using symbols to represent relationships.b) Use physical models and physical movement to represent information.

6. Cooperative Learning: has been proven to have a positive impact on overall learning. Note: groups should be small enough to be effective and the strategy should be used in a systematic and consistent manner.

a) Group students according to factors such as common interests or experiences.b) Vary group sizes and mixes.c) Focus on positive interdependence, social skills, face-to-face interaction, and individual and group

accountability.

7. Setting Objectives and Providing Feedback: provide students with a direction. Objectives should not be too specific and should be adaptable to students’ individual objectives. There is no such thing as too much positive feedback; however, the method in which you give that feedback should be varied.

a) Set a core goal for a unit, and then encourage students to personalize that goal by identifying areas of interest to them. Questions like "I want to know" and "I want to know more about . . ." get students thinking about their interests and actively involved in the goal-setting process.

b) Use contracts to outline the specific goals that students must attain and the grade they will receive if they meet those goals.

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c) Make sure feedback is corrective in nature; tell students how they did in relation to specific levels of knowledge. Rubrics are a great way to do this.

8. Generating and Testing Hypotheses: it’s not just for science class! Research shows that a deductive approach works best, but both inductive and deductive reasoning can help students understand and relate to the material.

a) Ask students to predict what would happen if an aspect of a familiar system, such as the government or transportation, were changed.

b) Ask students to build something using limited resources. This task generates questions and hypotheses about what may or may not work.

9. Cues, Questions, and Advanced Organizers: helps students use what they already know to enhance what they are about to learn. These are usually most effective when used before a specific lesson.

a) Pause briefly after asking a question to give students time to answer with more depth.b) Vary the style of advance organizer used: Tell a story, skim a text, or create a graphic image. There are

many ways to expose students to information before they "learn" it.

Lesson Observations

In addition to submitting lesson plans to their Mentor Teacher each week, Teacher-Interns will be observed by their mentor teachers at least twice a week. During the classroom observations, Mentor-Teachers will evaluate the classroom management and instructional quality of the class according to the standardized rubric provided by Breakthrough New Haven.

After each observation, the Mentor Teacher and the Teacher-Intern will have a sit-down meeting to talk about the events of the class. During this meeting, Mentor Teachers and Teacher-Interns should discuss those things that went well in the classroom, those things that might be improved, any particular student-teacher issues, and creative ideas for the future.

The classroom observation rubric below was designed by the Roosevelt Institution at Yale and edited by Master Teacher Martha Combs. They are not designed to pick up on all of the finely grained intricacies of a Teacher-Intern’s teaching. Instead they are designed to measure whether Teacher-Interns are using techniques in the classroom which, when done properly, will result in a learning environment conducive to learning by learners with differing needs.

Classroom Management: Does the teacher “own” the classroom?

Practice (Yes) (No)Often Sometimes Rarely Never

Teacher checks in on all students individually (e.g., calls on them randomly, looks over their shoulders when doing individual work)Teacher moves around the room during a lessonTeacher affirms students’ contributions with positive feedback, especially when students ask questionsTeacher addresses inappropriate behavior with little or no disruption of instructionWhen doing so…

Teacher rejects behavior, but not studentTeacher explains why behavior is wrong

Behavioral rules/expectations are posted on the wall

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Teacher effects quick transitions between lessons/activities

Teacher knows students by name

Instruction: Does the teacher engage all students and communicate clearly?

Practice (Yes) (No)Often Sometimes Rarely Never

Lesson plan objective(s) are posted and visibleTeacher reviews lesson objectivesTeacher ensures all students are actively involved in their learning by:Instruction clearly supports and substantiates posted objectivesTeacher differentiates instruction to accommodate students’ learning stylesTeacher engages students in the lesson by using a variety of teaching methodsIncluding… Group/pair work

Role playingIndividual work/reflectionWhole-class discussionDirect instruction/lectureDebate

Teacher encourages higher levels of levels of thinkingIncluding… Makes students summarize information (i.e. “who?”

“what?” and “when?”)Asks questions that promote critical thinking (i.e. “why?” and “how?”)Asks students to compare/contrast ideas

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Curricular Objectives

Long Term Skill Objectives

By the end of the summer, students will be able to: 1. Develop reliable notes, using Cornell Notes, which will help them to prepare for assessments;2. Effectively communicate in the language of mathematics;3. Effectively use a variety of problem solving strategies;4. Explain their thought process in working through a problem (i.e. meta-cognition);5. Accurately and thoroughly review their work for errors and correct those errors; and6. Translate “real world” situations and verbal expressions into algebraic sentences.

Medium Term Skill Objectives (By Course, Unit and Level)

The Unit objectives herein for Beginner, Intermediate and Advanced Level students are based on each student’s performance on the placement test. Teacher-Interns will be provided with copies of their students placement tests with the Mentor Teachers comments and corrections included.

All students should master the objectives listed in the “Beginner Level” section of a unit. Intermediate and advance level students should have already mastered some of the “Beginner Level” skills. As a result, those concepts should be embedded in the lessons, problems & assignments provided to Intermediate and Advanced students.

The Levels vary in content and pace. For example, Beginners studying the distributive property may be asked to simplify the expression “3(x - 1)”. Simplifying this expression requires the most basic understanding of the distributive property and multiplication of integers. In contrast, Advanced students who are studying the distributive property may be asked to simplify the expression “x – ½ (3x – 0.5)”. Simplifying this expression requires understanding of the distributive property with respect to a negative distributor, subtraction of unlike and mixed fractions, multiplication of fractions, conversion of fractions to decimals or vice versa, and combining like terms. In the case of the Advanced Students, it is assumed that they have a greater degree of retained prior knowledge and are capable of significantly complex, multi-step problems, regardless of the concept covered in class.

Teachers should strive to provide examples and or problems that require students to apply the mathematical concepts to “real world” situations and applications. For example, the basic contractual terms of a cell phone contract can be described in a linear equation: Total Charges over the Life of the Contract = Monthly Fee * Number of Months + Basic Charge for Usage.

Lastly, it is critical to stress the importance of students’ using mathematical vocabulary. Naming something requires a context, and middle school students (like most human beings) find context appealing. Furthermore, during the course of their academic careers, students will need to interpret a host of mathematical terms as they make their way through their high school and college curriculums.

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Algebra - Unit 1: Introduction – Basics of Algebra

Beginner Level be able to add and subtract integers be able to use the proper order of operations (PEMDAS) to simplify expressions that contain integers use the distributive property with natural numbers use the distributive property with integers be able to solve one-step equations using the properties of equality be able to solve one-step inequalities using the properties of inequalities be able to simplify expressions by combining like terms use the distributive property over algebraic expressions

Intermediate Level add and Subtract Integers use the proper order of operations (PEMDAS) to simplify expressions that contain Integers Use the Distributive Property with Natural numbers Use the Distributive Property with Integers Be able to solve one-step equations using the properties of equality Be able to solve one-step inequalities using the properties of inequalities Be able to simplify expressions by combining like terms Use the Distributive Property over Algebraic Expressions

Advanced Level: Be able to use the proper order of operations (PEMDAS) to simplify expressions that contain Integers Use the Distributive Property with Integers Be able to solve one-step inequalities using the properties of inequalities Use the Distributive Property over Algebraic Expressions

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Algebra - Unit 2 – Fractions, Decimals and Percents

Beginner Level: Be able to Add and Subtract fractions with like denominators Be able to Add and Subtract Fractions and mixed numbers with like denominators Be able to Add and Subtract fractions with unlike denominators Be able to Add and Subtract Fractions and mixed numbers with unlike denominators Be able to Multiple and Divide Fractions and mixed numbers Be able to Add and subtract decimals Be able to Multiply and Divide decimals Convert a decimal to a fraction Convert a fraction to a decimal Convert a decimal to a percentage Convert a fraction to a percentage Define a proportion Determine when proportions can be used to solve a problem Set up and solve proportions

Intermediate Level: Add and Subtract Fractions and mixed numbers with like denominators Add and Subtract fractions with unlike denominators Add and Subtract Fractions and mixed numbers with unlike denominators Multiple and Divide Fractions and mixed numbers Multiply and Divide decimals Convert a decimal to a fraction Convert a fraction to a decimal Convert a decimal to a percentage Convert a fraction to a percentage Define a proportion Determine when proportions can be used to solve a problem Set up and solve proportions

Advanced Level: Add and Subtract Fractions and mixed numbers with unlike denominators Multiple and Divide Fractions and mixed numbers Convert a decimal to a fraction Convert a fraction to a percentage Define a proportion Determine when proportions can be used to solve a problem Set up and solve proportions

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Algebra - Unit 3: Multi Step Equations and Word Problems

Beginner Level: Solve two-step equations by working backwards Solve multi-step equations by using order of operations to simplify and then working backwards Turn phrases into algebraic expressions Create and solve equations from a word problem using the LESS Method (Library, Equation, Solve it, Solution in

the form of an English sentence) (See Appendix)

Intermediate Level: Solve two-step equations by working backwards Solve multi-step equations by using order of operations to simplify and then working backwards Turn phrases into algebraic expressions Create and solve equations from a word problem using the LESS Method (Library, Equation, Solve it, Solution in


Advanced Level: Solve two-step equations by working backwards (Review) Solve multi-step equations by using order of operations to simplify and then working backwards Turn phrases into algebraic expressions Create and solve equations from a word problem using the LESS Method (Library, Equation, Solve it, Solution in


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Algebra - Unit 4: Linear Equations

Beginner Level: Plot ordered pairs Graph a Linear Equation from a table of values Define “Slope” Find the slope between two points on a graph Create an equation in Point-Slope Form Graph a Linear Equation by using the slope Define and identify the intercepts of a Linear Equation on a graph Create an equation in Slope-Intercept Form Graph a Linear Equation from Slope-Intercept Form Derive the intercepts of a Linear Equation Graph a Linear Equation by using the intercepts

Intermediate Level: Graph a Linear Equation from a table of values Define “Slope” Find the slope between two points on a graph Create an equation in Point-Slope Form Graph a Linear Equation by using the slope Define and identify the intercepts of a Linear Equation on a graph Create an equation in Slope-Intercept Form Graph a Linear Equation from Slope-Intercept Form Use the relationship between parallel lines (equivalent slopes) to create an equation that is parallel to another Derive the intercepts of a Linear Equation Use General Linear Form Graph a Linear Equation by using the intercepts Graph Linear Inequalities

Advanced Level: Define “Slope” Find the slope between two points on a graph Create an equation in Point-Slope Form Graph a Linear Equation by using the slope Define and identify the intercepts of a Linear Equation on a graph Create an equation in Slope-Intercept Form Graph a Linear Equation from Slope-Intercept Form Use General Linear Form Derive the intercepts of a Linear Equation Graph a Linear Equation by using the intercepts Create an Equation in Intercept form Use the relationship between parallel lines (equivalent slopes) to create an equation that is parallel to another Use the relationship between perpendicular lines (negative reciprocal slopes) to create an equation that is

perpendicular to another Graph Linear Inequalities

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Algebra - Unit 5: Systems of Linear Equations

Beginner Level: Recognize a System of Linear Equations Recognize that the solution to a System of Linear Equations is a coordinate point Solve a system of Linear Equations by Substitution Solve a System of Linear Equations by Linear Combination (Elimination)

Intermediate Level: Recognize a System of Linear Equations Recognize that the solution to a System of Linear Equations is a coordinate point Solve a System of Linear Equations by Graphing Solve a system of Linear Equations by Substitution Solve a System of Linear Equations by Linear Combination (Elimination)

Advanced Level: Recognize a System of Linear Equations Recognize that the solution to a System of Linear Equations is a coordinate point Solve a System of Linear Equations by Graphing Solve a system of Linear Equations by Substitution Solve a System of Linear Equations by Linear Combination (Elimination)

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Algebra - Unit 6: Quadratics

Beginner Level: Recognize a Quadratic Equation Properly use the Quadratic Formula Simplify a Radical into Simplified Radical Form Recognize when not to use the Quadratic Formula Define the Zero Product Property Recognize when to use the Zero Product Property

Intermediate Level: Recognize a Quadratic Equation Properly use the Quadratic Formula Simplify a Radical into Simplified Radical Form Recognize when not to use the Quadratic Formula Define the Zero Product Property Recognize when to use the Zero Product Property Multiply Binomials to create quadratic binomials and trinomials

Advanced Level: Recognize a Quadratic Equation Properly use the Quadratic Formula Simplify a Radical into Simplified Radical Form Recognize when not to use the Quadratic Formula Define the Zero Product Property Recognize when to use the Zero Product Property Multiply Binomials to create quadratic binomials and trinomials Multiply Polynomials to create other polynomials

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Algebra - Unit 7: Additional Topics

Beginner Level: Use Triangle Numbers and the Fibonacci Sequence Use Basic Probability to find the chance that an event will occur Use Combinations to find the chance that an event will occur

Intermediate Level: Use Triangle Numbers and the Fibonacci Sequence Use Basic Probability to find the chance that an event will occur Use Combinations to find the chance that an event will occur

Advanced Level: Use Triangle Numbers and the Fibonacci Sequence Use Combinations to find the chance that an event will occur Simplify expressions that contain exponents by using the properties of Exponents Simplify radical expressions using the properties of Radicals Use a shape to find Probability

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Geometry - Unit 1: Introduction to Geometry

Beginner Level: Communicate using the basic language and symbols (Including):

o Differentiate between lines, line segments and rayso Name angleso Basic ideas of congruencyo Recognize congruent lines, angles and shapes o Recognize parallel lines and transversals

Intermediate Level: Communicate using the basic language and symbols (Including):

o Differentiate between lines, line segments and rayso Name angleso Basic ideas of congruencyo Recognize congruent lines, angles and shapes o Recognize parallel lines, transversals, and perpendicular lines

Use Pascal’s Triangle to find patterns, and permutations Use Triangle Numbers and the Fibonacci Sequence

Advanced Level: Communicate using the basic language and symbols (Including):

o Differentiate between lines, line segments and rayso Name angleso Basic ideas of congruencyo Recognize congruent lines, angles and shapes o Recognize parallel lines, transversals, and perpendicular lines

Use Pascal’s Triangle to find patterns, and permutations Use Triangle Numbers and the Fibonacci Sequence

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Geometry - Unit 2: Angle Relationships

Beginner Level: Recognize basic types of angles including: Acute, Obtuse, Straight and Right Recognize basic angle relationships and find the values of them using algebra: Adjacent, Vertical,

Complementary and Supplementary Recognize complex angle relationships and find the values of them (based on parallel lines): Alternating Interior,

Alternating Exterior and Corresponding Angle

Intermediate Level: Recognize basic angle relationships and find the values of them using algebra: Adjacent, Vertical,

Complementary and Supplementary Create and solve equations based on complementary and supplementary angles. Recognize complex angle relationships and find the values of them using algebra (based on parallel lines):

Alternating Interior, Alternating Exterior and Corresponding Angle

Advanced Level: Recognize basic angle relationships and find the values of them using algebra: Adjacent, Vertical,

Complementary and Supplementary Create and solve multi-step equations based on complementary and supplementary angles. Recognize complex angle relationships and find the values of them using algebra (based on parallel lines):

Alternating Interior, Alternating Exterior and Corresponding Angle Create and solve equations based on combinations of complex angle relationships and find the values of them

using algebra (based on parallel lines): Alternating Interior, Alternating Exterior and Corresponding Angle

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Geometry - Unit 3: Triangles

Beginner Level: Identify a triangle based on the types of angles that it has (Acute, Obtuse, and Right) Identify a triangle by the number of congruent sides it has (Equilateral, Isosceles, Scalene) Apply the fact that the sum of the angles in a triangle is 180 degrees Create and solve basic multi-step equations based on this concept Show that two triangles are congruent Use the fact that Corresponding Parts of Congruent Triangles are Congruent Use basic paragraph proofs to prove this Explain why Side-Side-Angle and Angle-Angle-Angle do not prove congruency

Intermediate Level: Identify a triangle by the number of congruent sides it has (Equilateral, Isosceles, Scalene) Apply the fact that the sum of the angles in a triangle is 180 degrees Create and solve complex multi-step equations based on this concept Show that two triangles are congruent Use the fact that Corresponding Parts if Congruent Triangles are Congruent Use basic paragraph proofs to prove congruency Explain why Side-Side-Angle and Angle-Angle-Angle do not prove congruency

Advanced Level: Identify a triangle by the number of congruent sides it has (Equilateral, Isosceles, Scalene) Create and solve complex multi-step equations based on this concept Show that two triangles are congruent Use the fact that Corresponding Parts if Congruent Triangles are Congruent (CPCTC) Use basic paragraph proofs to prove congruency Use Coordinate Geometry to prove congruency Explain why Side-Side-Angle and Angle-Angle-Angle do not prove congruency Use the Unit Circle to apply the basics of Trigonometry: Sine, Cosine and Tangent Use SOH-CAH-TOA to find the length of the sides of a triangle Use SOH-CAH-TOA to find the area of a triangle Use the Sum of Angles formula Use the Law of Cosines

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Geometry - Unit 4: Similarity

Beginner Level: Understand what it means for two triangles to be similar Understand what it means for two polygons to be similar Understand how to use similar ratios and proportions

Intermediate Level: Understand what it means for two triangles to be similar Understand what it means for two polygons to be similar Understand how to use similar ratios and proportions

Advanced Level: Understand what it means for two triangles to be similar Understand what it means for two polygons to be similar Understand how to use similar ratios and proportions Use Coordinate Geometry to show this

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Geometry - Unit 5: Quadrilaterals

Beginner Level: Identify different types of quadrilaterals based on the number of pairs of parallel sides (Quadrilateral,

Parallelogram, Trapezoid) Identify different types of parallelograms based on the number of congruent sides or congruent angles

(Parallelogram, Rectangle, Square, Rhombus) Understand that the sum of the angles in a quadrilateral is 360 degrees Create and solve basic multi-step equations based on this concept

Intermediate Level: Identify different types of quadrilaterals based on the number of pairs of parallel sides (Quadrilateral,


(Parallelogram, Rectangle, Square, Rhombus) Understand that the sum of the angles in a quadrilateral is 360 degrees Create and solve basic multi-step equations based on this concept Create and solve basic multi-step inequalities based on this concept Name other regular polygons and the measure of the angles in them

Advanced Level: Identify different types of quadrilaterals based on the number of pairs of parallel sides (Quadrilateral,


(Parallelogram, Rectangle, Square, Rhombus) Understand that the sum of the angles in a quadrilateral is 360 degrees Create and solve basic multi-step equations based on this concept Create and solve basic multi-step inequalities based on this concept Name other regular polygons and the measure of the angles in them

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Geometry - Unit 6: Perimeter, Area, and Volume

Beginner Level: Find the perimeter of a triangle or a quadrilateral given the sides are constant values Find the perimeter of an oddly shaped figure given the sides are constant values Find the perimeter of a triangle or a quadrilateral given the sides are algebraic expressions Find the perimeter of an oddly shaped figure given the sides are algebraic expressions Find the circumference of a circle given the radius or the diameter Create and solve equations given the perimeter and that the sides of a shape are algebraic expressions. Use the formula for the area of a triangle, rectangle, and a circle Use the formula to find the surface area of a rectangular box and a circular right cylinder Create and solve equations based on the above area formulas Use the formula for the volume of a rectangular box and a circular right cylinder

Intermediate Level: Find the perimeter of a triangle or a quadrilateral given the sides are constant values Find the perimeter of an oddly shaped figure given the sides are constant values Find the perimeter of a triangle or a quadrilateral given the sides are algebraic expressions Find the perimeter of an oddly shaped figure given the sides are algebraic expressions Find the circumference of a circle given the radius or the diameter Create and solve equations given the perimeter and that the sides of a shape are algebraic expressions. Use the formula for the area of a triangle, rectangle, and a circle Use the formula to find the surface area of a rectangular box and a circular right cylinder Create and solve equations based on the above area formulas Use the formula for the volume of a rectangular box and a circular right cylinder

Advanced Level: Find the perimeter of a triangle or a quadrilateral given the sides are algebraic expressions Find the perimeter of an oddly shaped figure given the sides are algebraic expressions Find the circumference of a circle given the radius or the diameter Create and solve equations given the perimeter and that the sides of a shape are algebraic expressions. Use the formula for the area of a triangle, rectangle, and a circle Use the formula to find the surface area of a rectangular box and a circular right cylinder Create and solve equations based on the above area formulas Use the formula for the volume of a rectangular box and a circular right cylinder

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Geometry - Unit 7: Review and Additional Topics

Beginner Level: Factor a quadratic trinomial Use the Zero Product Property

Intermediate Level: Factor a quadratic trinomial Use the Zero Product Property Simplify Radicals to Simplified Radical Form

Advanced Level: Factor a quadratic trinomial Use the Zero Product Property Be able use the Completing the Square algorithm to solve an equation. Simplify expressions with Positive, Negative or Zero Exponents Simplify Radicals to Simplified Radical Form Rationalize an expression Use Combinations to find the chance that an event will occur Use a shape to find Probability

Deadlines

Lesson Plans

Lesson plan rough drafts are due every Wednesday at 5:00pm the week before they are to be carried out. Revisions are due by noon on the Saturday before they are to be carried out. Teacher-Interns should turn their lesson plans via email directly to the Mentor Teacher (with a carbon-copy to the Dean of Faculty).

Week 1- Rough Drafts Due 5:00pm, Wednesday, June 25, 2009; Revisions Due Noon, Saturday, June 27, 2009 Week 2- Rough Drafts Due 5:00pm, Wednesday, July 1, 2009; Revisions Due Noon, Saturday, July 4, 2009 Week 3- Rough Drafts Due 5:00pm, Wednesday, July 8, 2009; Revisions Due Noon, Saturday, July 11, 2009 Week 4- Rough Drafts Due 5:00pm, Wednesday, July 15, 2009; Revisions Due Noon, Saturday, July 18, 2009 Week 5- Rough Drafts Due 5:00pm, Wednesday, July 22, 2009; Revisions Due Noon, Saturday, July 25, 2009 Week 6- Rough Drafts Due 5:00pm, Wednesday, July 29, 2009; Revisions Due Noon, Saturday, August 1, 2009

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Online Resources for Teacher-InternsThe following websites might be helpful as you plan your lessons this summer.

Breakthrough Collaborative Teacher Guidebook (http://teachbreakthroughs.org/teacher-resources/) – An introduction to the theories of teaching along with some handy tools to use this summer.

Study Skills

Cornell Notes PDF Generator (http://eleven21.com/notetaker/) – This website generates custom Cornell note pages in .pdf form

Mathematics Resources

Math Warehouse: A Warehouse of interactive Math Lessons with definitions and examples, worksheets, interactive Activities and other Resources (http://www.mathwarehouse.com/) This site has a collection of interactive problems and lessons that could be used for additional review or alternative ways to teach an concept.

Ed Helper Geometry http://www.edhelper.com/geometry.htm This website has a good collection of elementary Geometry worksheets

Ed Helper Geometry http://www.edhelper.com/geometry_highschool.htm This website has a good collection of high school level Geometry worksheets

Ed Helper Middle School Math http://www.edhelper.com/middle_school_math.htm This website has a good collection of elementary Algebra worksheets

Ed Helper Middle School Math http://www.edhelper.com/algebra.htm This website has a good collection of high school level Algebra worksheets

The Math Page: Arithmetic, Algebra, Trigonometry http://www.themathpage.com/index.html An excellent website that contains detailed lessons in Arithmetic, Plane Geometry, Algebra Trig, and Elementary Number Theory

Mathematics for Liberal Arts http://math.fau.edu/richman/mla/contents.htm This site has some good that sparse lessons. Though at the same time it has some very good web applets to show ideas.

Kuto Software http://www.kutasoftware.com/freemain.html This website has a spectacular array of free worksheets for varying levels of Algebra

Art of Problem Solving Forum http://www.artofproblemsolving.com/Forum/ A good source for more Challenging problems.

Math Forum @ Drexel University http://mathforum.org/ An excellent source for additional resources as well as a help forum with problems done in great detail.

For more information about this guide, please contact:

Adam Sperling – Mathematics Mentor Teacher, Breakthrough New Haven, 2009 Summer [email protected]

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mailto:[email protected]

http://mathforum.org/

http://www.artofproblemsolving.com/Forum/

http://www.kutasoftware.com/freemain.html

http://math.fau.edu/richman/mla/contents.htm

http://www.themathpage.com/index.html

http://www.edhelper.com/algebra.htm

http://www.edhelper.com/middle_school_math.htm

http://www.edhelper.com/geometry_highschool.htm

http://www.edhelper.com/geometry.htm

http://www.mathwarehouse.com/

http://eleven21.com/notetaker/

http://teachbreakthroughs.org/teacher-resources/


Michael Van Leesten – Director, Breakthrough New Haven [email protected]

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mailto:[email protected]

Mathematics Curriculum Guide Appendix

The LESS MethodBelow is an example of how to complete a word problem using the LESS Method of solving a word Problem. The acronym LESS stands for the following things:

L – Library – where we define our unknowns

E – Equation – where we take the information from our Library and turn it into an equation that we can solve.

S – Solve it!! – in other words here is where we show the work that we have done to solve our unknowns

S – Solution in the form of an English sentence – Because it’s always important to see if the answer makes sense in the context of the problem, we do it here.

Example Problem #1: (Done for the Students)

Together Ms Rowny and Mr. Van Leesten have $99. Ms. Rowny has $3 more than seven times as much money as Mr. Van Leesten. How much money does each have?

L - > m = the amount of money that Mr. Van Leesten has

7m + 3 = the amount of money that Ms. Rowny has

E –> m + (7m+3) = 99

S -> m + (7m+3) = 99

8m + 3 = 99

8m = 96

m = 12

Check:

7m + 3 = 7(12) + 3 = 87

S -> Ms. Rowny has 87 dollars and Mr. Van Leesten has 12 dollars.

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Example Problem #2: (Done with the Students)

Find three consecutive EVEN integers whose sum is 162.

L - > n = the first even integer

n + 2 = the second integer

n + 4 = the third integer

E –> n + (n + 2) + (n + 4) = 162

S -> n + (n + 2) + (n + 4) = 162

3n + 6 = 162

3n = 156

n = 52

n + 2 = (52) + 2 = 54

n + 4 = (52) + 4 = 56

S -> The three integers are 52, 54, and 56.

Example Problem #3: (Done with the Students)

At the end of the first half of a basketball game, Hopkins and Hamden Hall are tied. During the second half Hopkins scored 48 points and Hamden Hall scored twice as many points as they had in the first half. What was the final score if Hopkins won by 2 points?

L –>

E –>.

S –>

S –>Page 29 of 37


Sample Lesson: Function NotationObjectives:

To be able to use Function Notation to find value and solve equations

Warm Up:

Find four solutions to the following equations:

y = 6x – 3 -2x + y = 4

Review:

Definition for DomainThe Set of x values --- The set of InputsRangeThe Set of y values ---The set of Outputs

Discuss Functions as an input / output machine: Plug in the value as the input, out comes the outputFunction Notation:

Take the Equation y = 3x +4

Well it’s the same thing as saying f(x) = 3x + 4What does this mean?

There is a function named f, whose input is x , which is defined as 3x + 4

In plain English, function f says that you take a number, multiply it by 3 and thenadd 4 to it

Example:f(3) = ? – what does this mean? What is the value of the output when the input is 3?

Or what do you get when you plug 3 into function f?

f(3) = 3(3) + 4

= 9 + 4 = 13

so, f(3) = 13 So as a coordinate point we have (3, 13)

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Together:

f(-5) = ?f(-5) = 3(-5) + 4 = -15 + 4 = -11so, f(-5) = -11 Giving us the coordinate point (-5, -11)

Example:

f(k) = 1, find the value of k – what does this mean? When k is the input, the out put is 1.So how do we find the value of k?

f(k) = 3k + 4. We also know that when you plug k into the function the output is 1.

So f(k) = 3k + 4 = 1

Or

3k + 4 = 1. From here we can solve the equation by working backwards.

So we now know that if we plug -1 into the equation we get out 1. Or (-1, 1)

Together:

f(w) = 10, find the value of w

f(w) = 3w + 4.

So f(w) = 3w + 4 = 10

Or

3w + 4 = 10. From here we can solve the equation by working backwards.

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So we now know that if we plug 2 into the equation we get out 10. Or (2, 10)

On their own: As the students are working, walk around the room to help students.

h(x) = -4x + 1

1. Find h(-3) (=13) 2. Find h(3) (=-11) 3. Find h(0) (= 1)

4. h(q) = 9, find the value of q. (q = -2) 5. h(z) = -19, find the value of z. (z = 5)

Review the problems by having the students put their work on the board. When everyone is finished, have each student explain the problems that they have done.

Finish the lesson by taking stock of the students understanding. Using their finger, on a scale of 1 – 5 (one being very confused, five being I could explain this to someone in another class)

Ask generally, anyone under 4, where they are still having problems. And use further examples if necessary.

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Sample Activity 1 – Deriving the Distance Formula1. Determine the length of the line segment , where point A = (0, 0) and point B = (8, 6), by completing the

following steps.

a) Graph

b) Draw a vertical line through point A, and then draw a horizontal line through point B.c) Label the intersection of those lines point C. Determine the coordinates of point C.d) What geometrical figure have you formed?

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e) Determine the length of the base and height of figure ABC.

f) Use the Pythagorean Theorem to determine the length of .

2. Using similar steps to number 1, determine the distance (or length) of the segment between and

.

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Sample Activity 2 – Using the Quadratic FormulaThe Quadratic Formula is used to solve Quadratic Equations. A Quadratic Equation is one that can be written in the form:

, where a, b, and c are each Real Numbers

All Quadratic Equations can be written in this form. To solve them, we can use the Quadratic Formula. The Quadratic Formula is:

When using the Quadratic Formula, it is very important to start with the Quadratic equation in its proper form, as shown above. It is also imperative that you be sure to use the proper signs for a, b, and c.

Example: Solve for x in the equation:

Step 1: Write the quadratic in its proper form.

Step 2: Write down the values for a, b, and c. (This Step should be done every time!)

a = 1

b = -2 (NOTE THE NEGATIVE)

c = -24 (NOTE THE NEGATIVE)

Step 3: Plug the values for a, b, and c into the Quadratic Formula and Solve!

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So x = 6, -4

Practice:

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1. 2.

3. 4.


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ENGLISH CURRICULUM GUIDE - Breaking News ... · Web viewThis curriculum guide serves both Mentor Teachers and Teacher-Interns providing context, sequencing and concrete lesson plan

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