Supporting Math - Fresno Unified School District · Supporting Math in the AVID Elective 1 Author Acknowledgements If I have seen further it is only by standing on the shoulders of

I M P L E M E N T A T I O N

SupportingMath

in theAVID Elective

Teacher Guide

Copyright ©2011

AVID Center • San Diego, CA

All rights reserved.

Dedicated in Loving Memory to my Father, Walter Bugno

You were the first to instill in me the

importance of education, that it is not

laws, but education that truly frees

humanity. You taught me to seek the

best in myself, and to see the best

intentions in others. Your lessons still

instruct my actions, and your example

guides my thinking.

1Supporting Math in the AVID Elective

Author AcknowledgementsIf I have seen further

it is only by standing on

the shoulders of giants.

-Sir Isaac Newton

I would like to begin by thanking those who have created before me for their brilliant thinking—the talent of our AVID Math, ELL, History, Tutorology, Focused Note-Taking, and original Algebra Tutorial writers is woven through every page. A special thanks to Jim Donohue, who has been a tremendous influence as a mentor, colleague, and friend. I don’t believe that I have ever had a conversation with Jim that did not make me a better educator and thinker. His intelligence, creativity, and kindness are unparalleled, and his sense of doing the right thing for students challenged my thinking to new levels when I might have been content with lesser paths.

To the contributors to The Write Path I: Mathematics and The Write Path II: Mathematics books (Jim Donohue, Tim Gill, Amy Armstrong, Amy Kosler, Anna Lotti, Erich Gott, Kym Butler, Maggie Aleman, Ricardo Gomez, and Sia Lux): you were the first and most valuable resources in the construction of this book. So much of your work will aid AVID Elective students and teachers in their pursuit of excellence in math.

To the teachers and students of Bear Creek High School: you have helped shape me into the educator that I am today. (Sorry, but someone has to take the blame.) You will always hold a special place in my heart. Both Bear Creek High School and Farb Middle School have been crucial piloting grounds for this work. Your input and feedback have greatly aided me through this process.

To the editor, Robyn Samuels, thank you for your diligent efforts in correcting my many mistakes; to Chris Stell, thank you for your amazing graphic work. This book would not have been of the same caliber without you. A special thanks to Roz Hafner and Sara Casey for your amazing eye for detail.

Heartfelt thanks to Mark Wolfe, AVID's National Director of Curriculum, and Rob Gira, AVID's Executive Vice President National Programs who provided me the opportunity to undertake this project. You are true leaders and visionaries. And, to all of those at AVID Center, you make it a joy to come to work every day.

To Christina and my wonderful family, thank you for your love and guidance over the years.

Finally, a special thank you to the airline that gave me the four-hour flight delay. Without your help, I do not know when I would have found the time to finish this book.

Tim Bugno

2 Supporting Math in the AVID Elective

STUDENT HANDOUTS:

Chapter 1: Incorporating Math . . . . . . . . . . . . . . . . . . . . . 7

Math Timeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14

Bookmark Template . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18

Bookmark Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19

Volume Sample Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23

Inverse Proportions Sample Response . . . . . . . . . . . . . . . . . . . . . . .24

Chapter 2: Math Cornell Notes . . . . . . . . . . . . . . . . . . . .27

Cornell Note Graph Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31

Cornell Note Dot Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33

Cornell Note Lined Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35

Cornell Note Blank Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37

Power Notes Blank Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39

Power Notes Graph Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40

Power Notes Dot Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41

Power Notes Lined Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42

Common Math Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43

Generating Essential Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46

Cornell Note-Taking Revision List . . . . . . . . . . . . . . . . . . . . . . . . . . . .50

Vocabulary: Costa's Levels of Thinking and Questioning . . . . . .51

Costa's Levels of Thinking and Questioning: Math . . . . . . . . . . . .52

Application: Solving Equations Cornell Notes . . . . . . . . . . . . . . . .56

Application: Solving Equations Cornell Notes with Question . .57

Application: Parallel & Perpendicular Lines Cornell Notes . . . . .58

Application: Parallel & Perpendicular Lines Cornell Notes with Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59

Compare/Contrast: Solving Systems of Inequalities Cornell Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62

Compare/Contrast: Solving Systems of Inequalities Cornell Notes with Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63

Prediction: Solving Equations Cornell Notes . . . . . . . . . . . . . . . . . .66

Prediction: Solving Equations Cornell Notes with Questions . .67

Chapter 1: Incorporating Math . . . . . . . . . . . . . . . . . .7

Chapter 2: Math Cornell Notes . . . . . . . . . . . . . . . . . .27

Chapter 3: Running Math Tutorials . . . . . . . . . . . . .83

Chapter 4: Math Strategies for the AVID Elective . . . . . . . . . . . . . . . 121

Chapter 5: Higher-Level Math Reflections and Summaries . . . . . . . . . . . . . . . . . . . . 151

Chapter 6: Fun with Mathmetics . . . . . . . . . . . . . . 169

Table of Contents


Prediction: Parallel & Perpendicular Lines Cornell Notes . . . . . .68

Prediction: Parallel & Perpendicular Lines Cornell Notes with Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69

Sample Textbook Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72

Sample Cornell Notes with Gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74

Chapter 3: Running Math Tutorials . . . . . . . . . . . . . . .83

Raising the Level of Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .86

Application of a Mathematical Process . . . . . . . . . . . . . . . . . . . . . . .89

Application of a Process Flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . .92

Compare and Contrast T-Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .96

Compare/Contrast Graphic Organizer . . . . . . . . . . . . . . . . . . . . . . . .98

Venn Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Compare and Contrast Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Index Scramble Cards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Chapter 4: Math Strategies for the AVID Elective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Evaluating the Error: Integers/Combining Like Terms . . . . . . . 128

Evaluating the Error: Factoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Evaluating the Method: Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Evaluating the Method: Quadratic Functions and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

The Golden Mistake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

A Picture Is Worth a Thousand Word Problems! Gazelle . . . . . 142

A Picture Is Worth a Thousand Word Problems! Trains . . . . . . . 143

A Picture Is Worth a Thousand Word Problems! Trees . . . . . . . 144

Sample: A Picture Is Worth a Thousand Word Problems! Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Create Your Own Math Dictionary . . . . . . . . . . . . . . . . . . . . . . . . . 148

Chapter 5: Higher-Level Math Reflections and Summaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

GIST Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Higher-Level Essential Question Template for Math Teachers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Chapter 6: Fun with Mathematics . . . . . . . . . . . . . . 169

AVID Academic Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Charting My GPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Inquiry Cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

What's My Limit? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Post-Socratic Seminar Writing Assignment . . . . . . . . . . . . . . . . . 190

Academic Language Scripts for Socratic Seminar . . . . . . . . . . . 192

The Curve of Forgetting – Socratic Seminar . . . . . . . . . . . . . . . . 195

Four Color Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

Four Color Percent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Four Color Decimal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

Four Color Graphic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Four Color Activity Answer Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . 202


The AVID College Readiness System

I. IntroductionThe AVID College Readiness System is an elementary through postsecondary system that brings together educators, students, and families around a common goal of AVID’s Mission: to close the achievement gap by preparing all students for college readiness and success in a global society. The system is represented by the figure below and shows how AVID’s mission is the foundation of the system.

II. Strategic Use of WICR in the AVID Elective ClassroomIn AVID, we strategically embed WICR (writing, inquiry, collaboration, and critical reading) into our courses to engage students in active learning and critical thinking.

Each strategy in this book incorporates WICR to move students to higher levels of thinking. Whether using these strategies or ones of your own making, it is imperative to consider WICR as their foundation.

W: Writing is a cornerstone in the AVID Elective classroom and is infused in most of the reading strategies throughout this book. AVID focuses on “writing to learn” activities—those activities that allow students the opportunity to make sense of information, to use writing to figure things out, and that jumpstart the process in which students communicate their engaged thinking.

I: Inquiry is the foundation upon which all progress is borne. Scientifically, it begins with a hypothesis, but it is “the question” that moves the learner to action, whether that question is an explicit question or a set of implicit questions that drive the process of working through ideas to a solution. Questioning the text and questioning what is seen, heard or discussed is at the heart of the AVID Elective classroom as is the learner questioning his/her own thinking or learning, making the implicit questions more “visible” in the process. Inquiry is inherent in the act of creating a visual, a written piece, formulating an oral, physical, or musical response. The key is for teachers to establish an environment where it is safe for students to engage in authentic inquiry, where wondering, question-ing, and hypothesizing are fostered and students recognize how to push each other’s thinking to higher levels.

To understand what it means to move to higher levels of cognition, AVID uses Arthur Costa’s Levels of Thinking. Benjamin Bloom’s Taxonomy is also a point of reference and can be used just as well, but students seem to find Costa’s hierarchy easier to remember (three tiers vs. Bloom’s six tiers). Costa’s levels can be described as:

Level 1: Input: This is the level at which we find, gather, identify, and recall information; it requires us to think literally.

Level 2: Processing: This is the level at which we make sense of information, using what we know from our sources to make connections and create relationships; it requires us to think analytically and inferentially.

Level 3: Output: This is the level at which we apply information and try it out in new situations; it requires us to think creatively, evaluatively, and hypothetically.

C: Collaboration in AVID is about working with others toward a common goal or goals, and about tapping into that mammalian side of the brain discussed above to increase motivation and attention to rigor. The strategies in this book demonstrate how to use collaboration to help students learn math content. For collaboration to be truly effective, teachers have to structure such activities to maximize engagement and accountability.

R: Reading is a key primary component of WICR, and the goal is to help students read for meaning vs. reading for identification. To develop the necessary college readiness skills, students have to practice close and critical reading, and teachers have to model and teach the skills using the critical reading process.


ACRS

Elementary Secondary

AVID’s MissionAVID’s mission is to close the achievement gap by preparing all students

for college readiness and success in a global society.

LEADERSHIp SySTEMS INSTRUCTION CULTURE/EqUITy

postsecondary

III. 21st Century Skills and College Readiness In an age where “21st century skills” has become synonymous with “survival skills” for students entering college or a career track, it is important to reference them here. Twenty-first century skills generally refer to a set of interdisciplinary skills that have been identified as important for students to have if they are going to be successful in life and careers in the coming decades. These skills include the ability to:

• use technology to gather, decipher, select and evaluate information in digital, scientific or verbal formats and use the information ethically;

• communicate clearly and to design and share information in diverse environments for a variety of purposes and in multiple formats;

• work effectively in diverse groups, compromising and sharing responsibility;• think critically and solve problems by analyzing and reasoning, asking questions, and

making sound judgments;• think and work creatively, developing innovative and original ideas and using failure as a

stepping stone to success.

(For more information about 21st century skills, see www.p21.org.)

It is important to realize that technology is not just a vehicle for implementing strategies; it actually provides a different way of “seeing” and making sense of the world. Our high-tech students enter our classrooms with a whole new literacy that we want to engage. Students accustomed to Googling a topic for instant research, to texting as a way of creating a shared dialogue, to creating multi-media images as a means of self-expression, have developed very complex literacy skills, some of which teachers might not fully understand. It is incumbent upon us to engage students by bridging their high-tech literacy skills to some of the more traditional literacy skills found in the classroom.Students are poised to engage in rich and complex inter-textual study if teachers are willing to seize the opportunity. For this to be possible, teachers need to be willing to use new technology themselves to know where the technology and text intersections fit.


How to Use This Book

Supporting Math in the AVID Elective is designed as a tool to help the AVID Elective teacher incorporate math strategies into the class. The very word “math” often evokes strong images and feelings long after our own schooling is behind us. Occasionally those feelings are positive, but more likely they reflect a history of confusion, trepidation, and in some cases, fear. This places many of our Elective teachers in a difficult position—they are not entirely comfortable with mathematics, yet find they must offer guidance in the subject because it is one of the most common areas in which AVID students struggle.

The content in this book focuses on fundamental strategies that teachers, tutors, and students can utilize to increase math performance. This book was written to move past pedantic rhetoric that complicates mathematics and focus instead on practical language that, hopefully, clearly outlines what strategies to incorporate and how to implement them.

It is not within the scope of this book to teach the reader mathematics. And it is not necessary for the reader to even know much math. In fact, this book has very little direct instruction on mathematics at all; instead, it uses math as a tool to help students connect ideas. Ideally, the educators who use this book will be able to look past the x and y to see how various problems fit together, how both tutors and students can lead students to deeper levels of thinking, and how to use strategies over time.

In many cases, the book uses a particular math sample to demonstrate a strategy. However, it is important to note that these samples are given only to expose you to the strategy; they are not intended to be single-use lessons. As you progress through this book, you will repeatedly see activities such as Lesson 2.4 “Generating Application of a Process Questions for Cornell Notes,” which has student samples/activities that demonstrate how your students can incorporate process questioning into their notes. A specific content sample tied to algebra is used—for Lesson 2.4 it is solving multiple-step equations—but the strategy is one that can work with virtually any math concept or math course. If you are struggling with how the strategy can be adapted, it is advisable to work with a math teacher on your campus.

The chapters of this book cover some of the main difficulties we face in the AVID Elective around math: Cornell notes, tutorials, summaries, math strategies, and some of the more fun and motivating activities. The content of these chapters is not necessarily sequential in nature, so teachers should carefully select activities to address the specific needs of their students. With the myriad requirements for the AVID Elective, it is unreasonable to expect to use these activities daily or even weekly. However, helping your students generate their Cornell note questions and work through the tutorial process early in their education will help students create successful habits that will pay off in the long run.

The purpose of this book is to enable the reader to do two key things: experiment and adapt. Try the activities out during your AVID Elective class and change the pieces that do not work for your situation. There are suggested timelines regarding when certain activities could/should be done, but they are only suggestions to help tie the Elective to what is happening in the students’ math courses. The most powerful work you can do with your students is to use authentic student samples and discuss how to incorporate the strategies on an ongoing basis. Finally, this book is designed for you to work in conjunction with math teacher(s) on your campus. There are some suggested “Conversations with the Math Teacher” that can help you open these discussions with the math faculty at your school. The only real way to increase student achievement is to work in collaboration with your math teachers to bridge the support structures.

Log on to MyAVID resources frequently, as new items and supplemental materials are available and updated throughout the academic year.

www.avid.org

7Chapter 1: Incorporating Math

1Chapter“A journey of a thousand miles begins with a single step.”

-Confucius

Given the choice between walking a thousand miles and

having to incorporate math into their teaching, some

teachers might ask, “How long do I have to complete the

thousand mile walk?” But there is hope! As the quotation

above points out, it is about moving toward your goal

rather than arriving at a destination.

This chapter focuses on some fundamental math

strategies to incorporate into your AVID Elective class

with ease. It examines some of the dos and don’ts of

mathematics in the AVID Elective for both the teacher

and tutors. As their AVID Elective teacher, you play a very

influential role in your students’ perceptions, which can

reinforce or alter their current beliefs about math. This

chapter also provides a method for quickly assessing

your students’ experiences with math, which can aid in

your ability to support their future success. The chapter

concludes with a method for examining the structure of

multiple choice math problems.

Your students will thank you for taking your first step

along the mathematical journey.

Incorporating Math


1.1 Math Etiquette for the Elective Teacher

The 11 Commandments of Supporting Math

1. Eliminate the phrase, “I’m just not a math person.”

Why do we never hear, “I’m not a reading person” or “I’m just not a writing person,” but for some reason we excuse an ignorance of math? When you are “not a math person,” it allows your students not to be math people.

2. Be positive about math and your approach to the subject.

Not everyone is an artist, but we can appreciate art. We cannot all be Hemingway, but we can appreciate well-written prose. We may not all be professional mathematicians, but we should all see and appreciate the advances in technology and science that stem from the field of mathematics.

3. Challenge yourself!

No one has ever gotten better at running by sitting on the couch. No one has gotten better at cooking by eating at McDonald’s. And no one has gotten better at math by avoidance. Become a math person!

4. If you don’t understand something… ASK! If you are discussing something with a math teacher and you don’t understand, stop and ask for clarification.

When speaking with math teachers on your Site Team, don’t be afraid to stop the conversation and ask them to explain in a different way. Mathematicians often speak “mathese.”

5. It needs to be OK for you and others to make mistakes, especially in mathematics.

In fact, our brains learn much more from correcting mistakes than they do from a right answer.

6. Do not accept a tutorial question as a Level 2 or 3 question just because the student uses a key word.

“Infer how to factor x2 – 4x – 21” is not an inference question, when all the students do is factor like they normally would. As with most things, it is what we do and not what we say!


7. Have you ever said the words, “I wish I were better at math”? If so, do everything you can to ensure that your AVID students don’t have to say those words when they are adults.

Don’t let personal regrets become your students’ regrets. Challenge your students to go as far as they can in math.

8. Be wary of letting your tutors teach (or allow other students to teach) “shortcuts” or “tricks.”

Although tutors are obviously trying to help by demonstrating a “faster” method, these gimmicks can occasionally have unforeseen exceptions that may backfire on the students.

9. Don’t ever FAKE it! (And don’t allow your tutors to fake it either.)

There is a popular mantra regarding tutors, which is, “You don’t have to be a content expert in order to tutor the subject.” This is true…in a way. HOWEVER, you never want a student to leave the tutorial with incorrect information that they think is correct. If you are unsure about a solution, there is nothing wrong with telling the student to check in with his or her math teacher to make sure the answer is correct. If it is, great; if it is incorrect, have them share the solution during the next tutorial session. Celebrate the corrected mistake. Having a student leave with misinformation is MUCH worse than not having done a tutorial at all, because you are further ingraining an incorrect approach to the problem.

10. Make math a priority!

Every teacher believes his or her subject is the most important. However, math becomes a priority in the AVID Elective due to the needs of the students. On any given Tuesday or Thursday, you could walk into any tutorial, at any grade level, anywhere in the country (or world), and a majority of the tutorial sessions would be on mathematics. When a student is failing a subject, it is most often math. If a student fails to complete the college entrance requirements, it is most often in math.

11. Begin the math conversation with your math department.

Unfortunately, this book will not be the solution to all of your problems. At best, it is a collection of tools to help support you in your work. The real solution lies in the collaboration among you, the other elective teachers, and the math teachers. Once you have solved pieces of the puzzle, make sure that you share your discovery with the others.


1.2 Math Etiquette for the AVID Tutor

The 11 Guidelines of Supporting Math

1. Never say, “I’m just not a math person.”

Remember that you are a role model to these students, and they listen closely to what you have to say. When you are “not a math person,” it allows your students not to be math people.

2. Be positive about math and your approach to the subject.

Not everyone is an artist, but we can appreciate art. We cannot all be Hemingway, but we can appreciate well-written prose. We may not all be professional mathematicians, but we should all see and appreciate the advances in technology and science that stem from the field of mathematics.

3. If you think something is wrong... something is probably wrong.

Once you have finished with a question, if you are not certain it is correct, it is likely that a mistake was made. Having a student leave with misinformation is MUCH worse than not having done a tutorial at all, because you are further ingraining an incorrect approach to the problem.

4. Don’t ever FAKE it!

There is a popular mantra regarding tutors, which is, “You don’t have to be a content expert in order to tutor the subject.” This is true…in a way. HOWEVER, you never want a student to leave the tutorial with incorrect information that they think is correct. If you are unsure about a solution, there is nothing wrong with telling the student to check in with his or her math teacher to make sure the answer is correct. If it is, great; if it is incorrect, have them share the solution during the next tutorial session. Celebrate the corrected mistake.

5. If you don’t understand something… ASK! If something comes up that you aren’t sure about, it is ok to say, “I don’t know, but I’ll find out.” Then, follow up with a math teacher.

There is nothing wrong with not knowing everything. It’s pretending to know everything that is so damaging. Students notice a “pretender” very quickly

6. It needs to be OK for you and others to make mistakes, especially in mathematics.

In fact, our brains learn much more from corrected mistakes than they do from a correct answer.


7. Allow mistakes to happen during tutorials.

One of the least helpful things that tutors or fellow students do is to immediately point out that a mistake has occurred. When this happens, you miss out on a golden opportunity for the students to evaluate the validity of their argument. Sometimes you can even let a mistake go until the end before pointing it out. Determining the location of an error is a critical skill we want students to develop.

8. Do not accept a tutorial question as higher-level just because the student uses a key word.

“Infer how to factor x2 – 4x – 21” is not an inference question. As with most things, it is what we do, and not what we say! It is OK if a student begins with a Level 1 question (especially at the lower grades), but it is your job to guide them into deeper thinking.

9. Be wary of teaching (or allowing other students to teach) “shortcuts” or “tricks.”

Although you are trying to help demonstrate a “faster” method, these gimmicks can occasionally have unforeseen exceptions that may backfire on the students (usually in the middle of a test).

10. Have students rank how certain they are that an answer is right.

A great practice is to ask the students, “On a scale of 1–10, how certain are you that your math work is 100% correct?” If you do this after every question, it makes it much easier to say, “Why don’t you follow up with your math teacher and report back to us next week?” Occasionally you can even point out a mistake after they have already gotten to the point where they have ranked their certainty.

11. Make math a priority!

Every teacher believes his or her subject is the most important. However, math becomes a priority in the AVID Elective due to the needs of the students. On any given Tuesday or Thursday, you could walk into any tutorial, at any grade level, anywhere in the country (or world), and a majority of the tutorial sessions would be on mathematics. When a student is failing a subject, it is most often math. If a student fails to complete the college entrance requirements, it is most often in math.


1.3 Math Timeline

Topic• Graphing students’ math experiences

RationaleStudents often have a long history of mathematical experiences by the time they enter your classroom. This activity provides a platform for students to describe their past mathematical successes and struggles, which will allow you to evaluate potential areas of concern. It is an excellent method for assessing a student’s comfort level with mathematics at a glance.

ObjectivesStudents will:

• Depict their mathematical experiences graphically

• Write about areas of problems or successes

Timeline• 10 –15 minutes at the beginning of the year

WICR Strategies• Writing to Learn

• Inquiry

• Collaboration

Materials/preparation• Student Handout 1.3a “Math Timeline”

Instructions• Pass out Student Handout 1.3a “Math Timeline.”

• Have students trace the history of their mathematical experiences, marking their highs and lows. Project on a screen or pass out Teacher Reference 1.3b “Math Timeline Sample” as a sample if needed.

• Instruct students to choose three points on their timeline and write one or two sentences describing why that point is significant.

• Have students share their experiences with a partner.

• Optional: Have students share out (whole group) areas of common struggles.

• Review students’ timelines, noting any common areas of concern for struggling math students.


Conversations with the Math Teacher

• (Bring math timeline of a struggling

student to the math teacher):

We did a math timeline and it seems like

__________ has had some difficulties in

the past. Do you have any suggestions

about how I/We can support him/her?

• Many of my AVID students expressed

having difficulty with ____________

(fill in the blank with common area of

struggle, e.g., fractions, factoring...). Do

you have any suggestions about things we

can do to review this concept?


STUDEnT HAnDOUT 1.3a

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TEACHER REfEREnCE 1.3b

Mat

h Ti

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1

23


1.4 Math Bookmark1

Topic• Developing a study tool

RationaleConstructing a “math bookmark” with reminders about Levels of Questions, a Cornell notes rubric (see Lesson 2.9 “Cornell Notes: Note-Reflecting”), and study suggestions will help students review and personalize study and learning skills. It will also provide a daily reminder of these skills. Vocabulary bookmarks can also be created at the beginning of a new unit or chapter.


• Personalize a math bookmark

• Add key vocabulary and formulas to their bookmark

• Use the bookmark to set up and utilize Cornell notes

• Develop the math bookmark as a study aid

Timeline• One 50-minute class period for students to create and begin entries on a personal math bookmark


• Organization

• Collaboration

Materials/preparation• Student Handout 1.4b: “Bookmark Sample”

• Colored markers and pens

• Before class, duplicate Student Handout 1.4a: “Bookmark Template” on card stock; then cut-to-size and three-hole punch the cards following the handout sample.

• Math textbook

1Donohue, J., Gill, T. (2009). 1.2 Math Bookmark. The Write Path II: Mathematics (pp. 35–38). San Diego, Ca: AVID Press


Instructions• Review the importance and value of setting up Cornell notes with a two and one-half inch (6.5 cm) left

margin for questions.

• Discuss the advantages of having a ready reference for important vocabulary, formulas, question starters, and a rubric for self-assessment of Cornell notes.

• Brainstorm as a class how Cornell notes should be graded for the class. Let the students determine the importance of headings, neatness (organization), questions, summary, etc.

• Generate a class rubric, with the number of points for each category.

• Divide students into small groups.

• Distribute/display and review Student Handout 1.4b: “Bookmark Sample.”

• Distribute the precut and hole-punched Student Handout 1.4a: “Bookmark Template.”

• Provide students with time to design and personalize their bookmarks. Make sure that each student writes down the class’s rubric, Levels of Questions, and STAR.

• Have students review the formulas in their book and add any information or formulas that they believe they will need.

• Recognize exemplary bookmarks with a “Gallery Tour” or other sharing activity.

• Remind students to use their bookmarks as an aid in setting up their Cornell notes, when they are writing questions, or when they are doing a self-assessment of their notes. Encourage them to add to their bookmarks when they learn a new vocabulary word or formula.

• Allow the use of the bookmarks from time to time during formative assessments.



Bookmark Template


STUDEnT HAnDOUT 1.4b (1 of 2)

Formulas

Square: A = s2

P = 4s

Rectangle: A = LW

P = 2L + 2W

Triangle: A = 1/2 bh

Circle: C = 2πr

C = πd

A = πr2

π ≈ 3.14

Volume for 3D prisms and cylinders:

Volume = Area of the base x height

(V = Bh)

Volume for 3D pyramids and cones:

Volume = 1/3 Bh

Surface Area:

Rectangular Prism

SA = 2(LW) + 2(WH) + 2(LH)

Cylinder

SA = 2πr2 + 2πrh

Additional Information:

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

Bookmark Sample


STUDEnT HAnDOUT 1.4b (2 of 2) Do your notes have these characteristics?

1. Consistent Cornell physical format, notes dated & titled, readable. . . . . . . . . . . . . . . . . . . . .3 pts.

2. Use of abbreviations, key words/phrases, underlining, starring . . . . . . . . . . . . . . . . . . . . . . .1 pt.

3. Main ideas are easily seen; correct sequence of information. . . . . . . . . . . . . . . . . . . .1 pt.

4. Questions are completed on left-hand side: Level 2 and 3 questions. . . . . . .1 pt.

5. An accurate, complete reflection follows the notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 pts.

CHARACTERISTICS pTS.

1. Consistent Cornell format

2. Use of abbreviations, key words, etc.

3. Main ideas are easily seen4. Questions are completed on left-hand side: Level 2 & 35. An accurate, complete summary

TOTAL pOINTS

RUBRICConsistent Cornell format3 pts Vertical line 2.5” (6.5 cm) from the left-hand

margin hand margin heading is complete with name, date, subject. The notes are titled.

Notes are adequate length.

2 pts Minor problems with format

1 pt No date or no title

0 pts Failed to use Cornell note-taking format or date and title are missing or notes are inadequate in length

Suggestions for terms to use and avoid.Level 1 questioning (needs to be avoided): defining, identifying, naming, label, listing, observing, reciting, solving

Level 2 questioning: analyzing, comparing, contrasting, grouping, inferring, sequencing, synthesizing.

Level 3 questioning: applying a principle, hypothesizing, imagining, judging, predicting, speculating

Set up notes – Name, Date, Warm-up, Terms, and Homework AssignmentTaking NotesAfter Class – Review your daily notes, fill in gaps

from peers or textbook, and generate questions.

Reflect – Underline key learnings and write a summary about what was learned for the day.

Important formulasArea (rectangle) = L • WP = 2L + 2WArea (circle) = πr2

General Equation of a line: y = mx + b

Bookmark Sample


1.5 Test Preparation: Why, Why, Why, Why?2

Topic

• Inquiry-based activity for multiple-choice standardized test preparation

Rationale

In today’s high-stakes testing, standards-based education environment, a significant amount of instructional time is spent on preparing students for state and national tests. “The Four Whys” is an activity that maximizes the impact of the time spent on test prep by asking students to think deeply about the choices given for a multiple-choice question. It is very different from giving students worksheets of practice problems and can be an excellent alternative during dedicated test preparation time.


• Determine why the correct response to a multiple-choice math question is correct and why the distractors are incorrect

• Answer a set of five multiple-choice questions similar to the original question

• Write their own multiple-choice item similar to the original questions

Timeline

• 15–30 minutes to find the correct response to a mathematical problem and understand how the incorrect answers function as distractors


• Inquiry

• Collaboration

• Reading

Materials/preparation

• Cornell note paper

• One sample multiple-choice state test question and five similar questions

• Student Handout 1.5a: “Test Preparation: Why, Why, Why, Why – Volume?”

• Student Handout 1.5b: “Test Preparation: Why, Why, Why, Why – SAT?”

2Donohue, J., Gill, T. (2009). 2.3: Test Preparation: Why, Why, Why, Why?. The Write Path II: Mathematics (pp. 78 – 81). San Diego, Ca: AVID Press


Instructions• Arrange the class into collaborative learning groups of two to four students.

• Distribute or have students create their own Cornell note paper.

• Distribute or project the sample multiple-choice test item.

• Distribute Student Handout 1.5a: “Test Preparation: Why, Why, Why, Why – Volume?” for a sixth grade class or Student Handout 1.5b: “Test Preparation: Why, Why, Why, Why – SAT?” for high school students and discuss expectations for the activity.

• Ask your students to write the problem down in their Cornell notes and to work in their groups to find the correct response.

• When each group has an answer, elicit responses from all of the groups. When consensus is reached on the correct response, ask each student to write a justification of why the response is correct in their Cornell notes.

• Use “Think, Pair, Share” or another Active Learning Methodology to share out a few of the student descriptions. Tip: Keep it moving here; we still have a lot to do with this question!

• In their groups, ask the students to determine the common mistake a student would make to arrive at each of the incorrect choices (distractors). Students should record their thinking in their Cornell notes.

• Distribute or project the five similar practice questions. When everyone has had a chance to write the questions and answers in their Cornell notes, quickly review the correct answer for each question.

• Ask students to create their own version of the multiple-choice test question, complete with correct response and common mistake distractors.

• Have students trade their Cornell notes with a partner and answer each other’s question.

• Have students write a thoughtful reflection at the end of the activity in their Cornell notes.



Test Preparation: Why, Why, Why, Why – Volume?

Volume Sample Response

Sample 6th Grade State Assessment Item

Which answers are incorrect?

What is the correct choice?

Why is the correct choice correct?

What mistake would be made to arrive at the incorrect choices?

A cube-shaped packing box is 30 inches deep. What is the volume in cubic inches of the box?A 90B 900C 27,000D 360The correct choice is C because a cubic box would require use to multiply 30 x 30x 30Choice A. 30 inches x 3 sides (length, width, and height) = 90Choice B. 30 x 30 = 900 (depth times side)Choice D. 30 x 12 (each side of the cube) = 360

Practice Problems (Students would copy and answer four more questions similar to the original question here.)What is the volume of a 5-foot-long, 3-foot-wide, and 2-foot-deep rectangular toy chest?A 10 cubic feetB 15 cubic feetC 30 cubic feetD 60 cubic feet

Write a problem just like the ones above.

(Students create their own problem.)A rectangular prism has a 3-centimeter by 3-centimeter square base and a height of 4.5 centimeters. What is the volume in cubic centimeters of the prism?A 13.5B 40.5C 91D 200

Summary In order to find the volume of a box, you must multiply the base area times the height. Most often they will try to trick you with multiplying by the number of dimensions or cubing the number (since it is a cubic shape). The most important thing is to realize what would be a reasonable answer for the volume of the shape.


STUDEnT HAnDOUT 1.5b

Test Preparation: Why, Why, Why, Why – SAT?

Inverse Proportions Sample Response

Sample SAT Item

Which answers are incorrect?

What is the correct choice?

Why is the correct choice correct?

What mistake would be made to arrive at the incorrect choices?

The variables x and y are inversely proportional. If x = 20 when y = 5, what is the value of x when y = 1?A 100B 99C 20D 16E 4The correct choice is A, because inversely proportional would mean that as x increases y would decrease, meaning x • y.Choice B. Results from multiplying 20 • 5 = x +1Choice C. Possibly a misunderstanding of inversely proportional,

so they may think x remains constant.Choice D. Results from solving the proportions 20:5 = Mx :1Choice E. Results from solving the proportions 20:5 = x:1

Practice Problems (Students would copy and answer four more questions similar to the original question here.)If x and y are inversely proportional, and x = 10 and y = 5, what is y when x is 60?A 1/30B 5/6C 10/6D 8

Write a problem just like the ones above.

(Students create their own problem.)If x and y are inversely proportional, and x = 5 and y = 6, what is y when x = 15?A 2B 18C 1/2D 1/15

Summary Connections, Summary, Reflection, Analysis When dealing with inverse proportions, it is important to understand that it is a number times a number. For example, x times y. They will most often try to trick you by creating answers that use x times 1/y (or the opposite).



27Chapter 2: Math Cornell notes

2ChapterMath Cornell notes“If I were again beginning my studies, I would follow the advice of Plato and start with mathematics.”

-Galileo Galilei

One of the fundamental AVID strategies for supporting college readiness is Cornell note-taking. Developed in the 1950s by Walter Pauk, this form of note-taking is based on research done in the areas of memory and learning theory. It is not simply a format for recording notes; it is a focused method for processing the information. If you are not familiar with Cornell note-taking, please examine 10 Steps of the Cornell Way on the following page, (For a more in-depth look at Cornell note-taking the AVID way, we recommend viewing the “Focused Note-Taking” CD.)

Learning the art of taking notes takes time and requires careful instruction and purposeful feedback to support student success. Cornell note-taking in math can prove especially challenging to both teachers and students, both in recording information and in generating higher-level questions for the left side of the page. This chapter examines the four overarching stages of the Cornell Note Process: Note-Taking (strategies 2.1-2.2), Note-Making (2.3-2.7), Note-Interacting (2.8), and Note-Reflecting (2.9) and provides strategies to make Cornell Notes meaningful from a mathematical point of view.

It is important to note that at AVID we have added an extra component to the Cornell note-taking page: the Essential Question. This is a teacher-generated question, appropriate to a particular lesson, which the students copy at the top of their Cornell notes before beginning a new math chapter and use to guide their thinking during math lectures and presentations for that chapter. You will find in lesson 2.2 not only a strategy for getting students accustomed to utilizing Essential Questions as they take notes, but also some ideas for ways you can get help generating those questions.


10 Steps of the CORnELL WAY

I. NOTE-TAKING:Reading or hearing information for the first time while jotting down and organizing key points to be used later as a learning tool.

C Create Format Step 1: CREATE Cornell notes format and complete heading

O Organize Notes Step 2: ORGANIzE notes on right side

II. NOTE-MAKING:Within 24 hours of having taken the notes, revise these notes, generate questions, and use collaboration to create meaning.

R Review and Revise Step 3: REVIEW AND REVISE notes

N Note Key Ideas Step 4: NOTE key ideas to create question

E Exchange Ideas Step 5: ExCHANGE ideas by collaborating

III. NOTE-INTERACTING:Interact with notes taken by creating a synthesized summary. Use Cornell notes as a learning tool to increase content class achievement.

L Link Learning Step 6: LINK learning to create a synthesized summary

L Learning Tool Step 7: Use completed Cornell notes as a LEARNING TOOL

IV. NOTE-REFLECTING:Use written feedback to address areas of challenge by setting focus goals to improve future notes. The Cornell Note Reflective Log Handout provides the opportunity to reflect on the notes and the learning.

W Written Feedback Step 8: provide WRITTEN feedback

A Address Step 9: ADDRESS written feedback

Y your Reflection Step 10: Reflect on yOUR learning


2.1 Cornell notes: note-Taking

Topic

• Introduction to the Cornell note format

Rationale

Cornell note-taking is one of the cornerstone AVID strategies. Teaching Cornell notes takes time and effort, but will be of tremendous value as the students enter college.


• Become familiar with the Cornell note-taking system

• Utilize various formats to support note-taking efforts

Timeline

• 40 minutes


• Inquiry


• Student Handout 2.1a through 2.1h “Cornell Note Paper”

• Student Handout 2.1i “Common Math Abbreviations”

Instructions

• Brainstorm with students why taking notes might be a good skill to learn.

• Explain to students that there are several skills needed to become an effective note-taker, for example:

Know what to write down.

Be able to listen to what the teacher is saying, look at the board, and write it down at the same time.

Learn how to effectively abbreviate.

Use symbols and/or indentations on the note page to organize notes while writing.

3Donohue, J., Gill, T. (2009). Common Math Abbreviations. The Write Path I: Mathematics (p. 21). San Diego, Ca: AVID Press

(continued on next page)


• Distribute and review samples of Student Handouts 2.1a through 2.1h “Cornell Note Paper” to introduce the structure of the teacher creating Essential Questions and students taking notes on the right, writing questions on the left, and generating a summary at the bottom. Discuss in small groups or as a whole class how the different papers might be useful in different situations.

• Have students take the handouts with them and choose one to use in their next math class. Tell them to write the algebraic information for solving a problem in their math class, and the written-out process that describes (in words) how they solve the problem.

• Have students bring in their Cornell notes and discuss aspects of the notes’ organization and incorporation of abbreviations.

• Distribute Student Handout 2.1i “Common Math Abbreviations” and discuss how abbreviations can be used during the note-taking process.

power Math Notes Instructions• Explain to your students that Power Notes (Student Handouts 2.1e through 2.1h) are a variation of Cornell

notes. Power Notes pull out key pieces of information so that they are easier to find.

• Instruct students to use the Tool Box to record: the objective, formulas, vocabulary, the plan for how the work will be done, and the homework for the night.

• Have students fill out the summary at the top of the page, but instruct them that the summary should still be written after the class is completed (once they are at home).


• Would you be able to incorporate

the Cornell note-taking process into

your math classes?

Provide electronic copies of Student

Handouts 2.1a through 2.1i to the

math teacher. These can be found

on the CD that was included in this

book, or on AVID File Share at

www.avid.org.

• Could you (if you are not already

doing so) write out the process for

solving math problems to the right of

the algebraic steps?


STUDEnT HAnDOUT 2.1a (1 of 2)CORnELL nOTE GRAPH PAPER

CORNELL NOTES TOPIC/OBJECTIVE:

ESSENTIAL QUESTION:

QUESTIONS:

SUMMARY:

NAME:

CLASS/PERIOD:

DATE:


STUDEnT HAnDOUT 2.1a (2 of 2) CORnELL nOTE GRAPH PAPER

QUESTIONS:

SUMMARY:


STUDEnT HAnDOUT 2.1b (1 of 2)CORnELL nOTE DOT PAPER


ESSENTIAL QUESTION:

QUESTIONS:

SUMMARY:

NAME:

CLASS/PERIOD:

DATE:


STUDEnT HAnDOUT 2.1b (2 of 2) CORnELL nOTE DOT PAPER

QUESTIONS:

SUMMARY:


STUDEnT HAnDOUT 2.1c (1 of 2)CORnELL nOTE LInED PAPER


ESSENTIAL QUESTION:

QUESTIONS: NOTES:

SUMMARY:

NAME:

CLASS/PERIOD:

DATE:


STUDEnT HAnDOUT 2.1c (2 of 2) CORnELL nOTE LInED PAPER

QUESTIONS: NOTES:

SUMMARY:


STUDEnT HAnDOUT 2.1d (1 of 2)CORnELL nOTE BLAnk PAPER


ESSENTIAL QUESTION:

QUESTIONS: NOTES:

SUMMARY:

NAME:

CLASS/PERIOD:

DATE:


STUDEnT HAnDOUT 2.1d (2 of 2) CORnELL nOTE BLAnk PAPER

QUESTIONS: NOTES:

SUMMARY:


POWER MATH NOTES TITLE:

ESSENTIAL QUESTION:

TOOL BOX:

NAME:

CLASS/PERIOD:

DATE:

SUMMARY:

Adapted from the Cornell notes system by: James O. Donohue (2003)

STUDEnT HAnDOUT 2.1ePOWER nOTES BLAnk PAPER



ESSENTIAL QUESTION:

TOOL BOX:

NAME:

CLASS/PERIOD:

DATE:

SUMMARY:


STUDEnT HAnDOUT 2.1f POWER nOTES GRAPH PAPER



ESSENTIAL QUESTION:

TOOL BOX:

NAME:

CLASS/PERIOD:

DATE:

SUMMARY:


STUDEnT HAnDOUT 2.1gPOWER nOTES DOT PAPER



ESSENTIAL QUESTION:

TOOL BOX:

NAME:

CLASS/PERIOD:

DATE:

SUMMARY:


STUDEnT HAnDOUT 2.1h POWER nOTES LInED PAPER


STUDEnT HAnDOUT 2.1i

Name _____________________________

Date ______________________________

Period ____________________________

Common Math AbbreviationsCommon Shortcuts for Note-taking—Abbreviations/Acronyms

For 4 Factorial !To 2 Difference/change ΔWith W Therefore ∴Without w/o Congruent ≅Within w/i Mean µAnd & or + Pi ΠMinus - Theta – used for angles ΘEqual/same = Sigma – standard deviation σNot equal ≠ Infinity ∞School Sch Union ⋃Part prt Intersection ⋂Point pt Then – implies →Be B Empty set ⊘Between b/w Sum/summation ∑Reference ref Similar ~Symbols ≥ ≤ > < Approximately equal ≈If and only if IFF, ↔ Perpendicular ⊥

Parallel ||

Additional Suggestions

• Make names and titles into acronyms after writing them the first time.

• Write first few syllables of long words and complete the word when reviewing notes. (coll = collect; comm = communicate)

• Write words without vowels until notes are reviewed. (spk = speak; commnct = communicate; commnty = community)

Think of some of your own shortcuts.

1. _________________________________________ 6. _____________________________________________

2. _________________________________________ 7. _____________________________________________

3. _________________________________________ 8. _____________________________________________

4. _________________________________________ 9. _____________________________________________

5. _________________________________________ 10. _____________________________________________


2.2 Utilizing Essential Questions

Topic• Providing students Essential Questions to help guide thinking

RationaleStudents often have difficulties generating questions for the left side of their Cornell notes and writing meaningful summaries of their lessons. Getting in the habit of using an Essential Question to guide them will help students get to a deeper level of mathematical thinking.

ObjectivesStudents will:• Incorporate Essential Questions to help guide thinking and write note summaries

Timeline• 15 minutes with a math teacher (before they teach a new chapter)• 3–5 minutes (in AVID Elective class)• 3–5 minutes (at home)

WICR Strategies• Writing to Learn• Inquiry

Materials/preparation• One page of Cornell notes • A timer/clock

Materials needed if math department teachers do not all utilize Essential questions (Please note: the teacher-generated Essential Questions may need to be requested several weeks in advance.)• Math textbook• Student Handout 2.2a: “Generating Essential Questions”• Teacher Reference 2.2b: “Generating Essential Questions Sample”

Instructions(If math teachers are not creating Essential questions in class)• Using Student Handout 2.2b: “Generating Essential Questions,” fill in the chapter number and topics for each

lesson using the math textbook. You are simply filling in the Chapter.Lesson and the title (most math textbooks will use

a heading of 2.4 to represent Chapter 2, the 4th lesson).


• Ask a math teacher on your Site Team, or one with whom you have established a relationship, if they can complete Student Handout 2.2a by creating some Essential Questions for each chapter. Use Teacher Reference 2.2b: “Generating Essential Questions Sample” to help guide the math teacher’s thinking about what he/she needs to accomplish.

Ask the math teacher to turn the lesson’s objective into a question a student could write a response to for a few minutes.

The math teacher should focus on creating overarching questions that capture the main point of the lesson.

It is critical to give this to the math teacher several weeks before the students begin the work on the chapter.

For Students in your AVID Class• Copy and distribute the filled-in Student Handout 2.2a: “Generating Essential Questions,” which the math

teacher has created before they begin the chapter. • Instruct students to use the generated Essential Questions for their Cornell notes during the math lecture/

presentation. Students should copy the appropriate Essential Question on the top of their Cornell notes

before the math teacher presents the information.• Students should then take their Cornell notes from the lecture, project, or activity.

After Class• Students should review and revise their notes at home that night for 5 minutes.• Have students then set a timer for 3 minutes.• Ask students to treat the Essential Question as a quickwrite prompt, and have them use the information in their

notes to answer the teacher-generated Essential Question.• The next day, have those finished summaries serve as “Entrance Tickets” to the classroom (collected upon

entry into your AVID class) rather than the more popular “Exit Tickets.”• File Student Handout 2.2a: “Generating Essential Questions” for use with future AVID Elective students.


• Would you be willing to adopt the Essential Questions to

start you lecture, as a method to help guide the students

thinking and ability to summarize?

• In situations where the math department does not largely

adopt the Essential Question, ask a math teacher (with

whom you have established a relationship) to help you

create an overarching “Essential Question” for each

section of the book.

You can then turn these questions into a handout

for all of your AVID students to utilize as quickwrite

prompts throughout the chapter of study.



Generating Essential Questions

Chapter ________

____ .1 ___________________________________________

Essential Question ____________________________________________________________________________

_____________________________________________________________________________________________

____ .2 ___________________________________________


_____________________________________________________________________________________________

____ .3 ___________________________________________


_____________________________________________________________________________________________

____ .4 ___________________________________________


_____________________________________________________________________________________________

____ .5 ___________________________________________


_____________________________________________________________________________________________

____ .6 ___________________________________________


_____________________________________________________________________________________________

____ .7 ___________________________________________


_____________________________________________________________________________________________



Generating Essential Questions Sample

Chapter 2 properties of Real Numbers

2.1 Using Integers and Rational Numbers

Essential Question ______________________________________________________________________________

_____________________________________________________________________________________________

2.2 Adding Real Numbers


_____________________________________________________________________________________________

2.3 Subtracting Real Numbers


_____________________________________________________________________________________________

2.4 Multiplying Real Numbers


_____________________________________________________________________________________________

2.5 Apply the Distributive property


_____________________________________________________________________________________________

2.6 Divide Real Numbers


_____________________________________________________________________________________________

What is the difference between finding the opposite of a number

and the absolute value of a number?

Describe the process for adding two fractions together.

What do you notice about subtracting two negative numbers and

subtracting one positive and one negative number?

How is multiplying one negative number related to multiplying

two negative numbers; and three negative numbers?

How can distributing a constant be connected to combining like terms?

How is dividing by fractions related to multiplying by fractions?


2.3 Cornell notes: note-MakingTopic

• Clarifying notes and generating questions

Rationale

Note-making is a critical step, where the students review their notes for points of confusion and generate questions that will support their future studies. During this time, the students are looking for areas of their notes where they might need additional information, or where more study is needed. The students then generate questions, which will help them review, recite, and recall the information that is on the left. Finally, the students need to add information to their notes from their teacher, peers, textbook, or future tutorials. This consolidation of information will aid students in their attempts to review and reflect upon the information.

Objectives

Students will:

• Review and revise their notes

• Generate questions to aid in their review and recall of information

• Use their Cornell notes as a living document, where they continually add to and clarify the recorded information after asking questions of the teacher, studying from the book, or in AVID tutorials

Timeline

• 40 – 50 minutes (broken over multiple weeks)


• Inquiry

• Collaboration

• Reading


• Student Handout 2.3a “Cornell Note-Taking Revision List”

• Student Handout 2.3b “Vocabulary: Costa’s Levels of Thinking and Questioning”

• Student Handout 2.3c “Costa’s Levels of Thinking and Questioning: Math”

• Cornell notes

• Highlighter

• Pencil

• Red pen


Instructions

• Inform students that the revision and questioning steps should all occur after the lecture, activity, or project is finished, and note-taking is completed.

Revision and Revising notes (Step 3)

• Have students take out a page of notes from their math class.

• Distribute Student Handout 2.3a “Cornell Note-Taking Revision List.”

• Give students 3 to 5 minutes to review their notes, using the Cornell Note-Taking Revision List to organize, prioritize, clarify, and delete information.

It is helpful to vary the use of color to note the changes. A supply of colored pens or highlighters will be required here.

noting key Ideas (Step 4)

• Distribute Student Handout 2.3b “Vocabulary: Costa’s Levels of Thinking and Questioning” and Student Handout 2.3c “Costa’s Levels of Thinking and Questioning: Math.”

• Have students generate one to three questions in the left column, which will help them recall the information on the right side.

• Use Student Handout 2.3b “Vocabulary: Costa’s Levels of Thinking and Questioning” to help students create higher-level questions.

Encourage students to create questions around potential test items or areas of confusion.

If you find that students continue to struggle creating Level 2 and 3 questions you can use Lesson 2.4 “Generating Application of a Process Questions for Cornell Notes,” Lesson 2.5 “Generating Comparing and Contrasting Questions for Cornell Notes,” and Lesson 2.6 “Generating Prediction Questions for Cornell Notes” to help students formulate higher-level questions.

• Remind students to generate their question after the lecture is finished. They can work on creating questions during the last few minutes of class or within 24 hours.

Exchange (Step 5)

• Encourage students to add information to their notes as they collaborate with others. Let students know that they should add information to their page of notes when they ask their math teacher a clarifying question, discuss points of confusion with peers, read their textbook, or garner understanding during tutorials.

At a later point, use Lesson 2.7 “Using the Textbook to Fill the Gaps in Cornell Notes” as a tool to support your students’ ability to fill in missing information.


• Would you be able to encourage all of your students to

review and revise their notes, and generate questions after

you finish your daily lecture or activity?

There is generally sometime between the end of the

lecture/activity and when the students begin their

in-class work. This is an ideal time review their notes.

• If an AVID student asks a question, can you encourage

them to add this new information into their notes?



Name _____________________________

Date ______________________________

Period ____________________________

Cornell note-Taking Revision ChecklistDirections: Review and revise notes taken in the right column. Use the symbols below to revise your notes.

COMpLETED SyMBOL REVISION

1, 2, 3...

A, B, C...1. Number the notes for each new concept or main

idea.

Key Word 2. Circle vocabulary/key terms in pencil.

Main Idea 3. Highlight or underline main ideas in pencil.

4. Fill in gaps of missing information and/or reword/rephrase in red.

Unimportant 5. Delete/cross out unimportant information by drawing a line through it with a red pen.

6. Identify points of confusion to clarify by asking a partner or teacher.

7. Identify information to be used on a test, essay, for tutorial, etc.

Visual/symbol 8. Create a visual/symbol to represent important information to be remembered.



Vocabulary: Costa’s Levels ofThinking and Questioning

LEVEL 1Remember Define List Recall Match Repeat State Memorize Identify Name Describe Label Record

Show Give examples Rewrite Review TellUnderstanding Restate Reorganize Locate Extend Discuss Explain Find Summarize Express Report Paraphrase Generalize

LEVEL 2Use Dramatize Use Translate InterpretUnderstanding Practice Compute Change Repair Operate Schedule Pretend Demonstrate Imply Relate Discover Infer Apply Illustrate Solve Examine Diagram Question Analyze Criticize Distinguish Inventory Differentiate Experiment Compare Categorize Select Break down Contrast Outline Separate Discriminate Divide Debate Point out

Create Compose Draw Plan Modify Design Arrange Compile Assemble Propose Suppose Revise Prepare Combine Formulate Write Generate Construct Organize Devise

LEVEL 3Decide Judge Rate Choose Conclude Value Justify Assess Summarize Predict Decide Select Evaluate Measure Estimate

Supportive Prove your Give reasons Explain your Why do you feel Evidence answer for your answer that way? Support your answer Why or why not? answer


STUDEnT HAnDOUT 2.3c

Costa’s Levels of Thinking and Questioning: MathLEVEL 1

• What information is given?

• What are you being asked

to find?

• What formula would you use in

this problem?

• What does ________mean?

• What is the formula for…?

• List the…

• Name the…

• Where did…?

• What is…?

• When did…?

• Explain the concept of…

• Give me an example of…

• Describe in your own words

what _______means.

• What mathematical concepts

does this problem connect to?

• Draw a diagram of…

• Illustrate how ______works.

LEVEL 2

• What additional information is

needed to solve this problem?

• Can you see other relationships

that will help you find this

information?

• How can you put your data in

graphic form?

• What occurs when…?

• Does it make sense to…?

• Compare and contrast ________

to ________.

• What was important about…?

• What prior research/formulas

support your conclusions?

• How else could you account

for…?

• Explain how you calculate…

• formula to the subject you’re

learning?

• What equation can you write to

solve the word problem?

LEVEL 3

• Predict what will happen to

____ as ____ is changed.

• Using a math principle, how

can we find…?

• Describe the events that might

occur if…

• Design a scenario for…

• Pretend you are…

• What would the world be

like if…?

• How can you tell if your answer

is reasonable?

• What would happen to______

if ______(variable) were

increased/decreased?

• How would repeated trials affect

your data?

• What significance is this formula

to the subject you’re learning?

• What type of evidence is most

compelling to you?


2.4 Generating Application of a Process Questions for Cornell notes

Topic

• Applying the process from one example to another

Rationale

The creation of authentic higher-level questions that provide an opportunity for students to examine the relationship between solving one problem and another is often a difficult concept to teach. The strength of using such questions is that they allow the students to utilize the process for solving a previous problem to help guide their thinking as they work out a subsequent problem.

Objectives

Students will:

• Generalize the process for solving mathematical problems

• Create connections between disparate problems

Timeline

• 5 – 10 minutes

WICR Strategies

• Writing to Learn

• Inquiry


• Student Handout 2.4a “Application: Solving Equations Cornell Notes”

• Student Handout 2.4b “Application: Solving Equations Cornell Notes with Question”

• Two to three pages of students’ personal math notes

• Color other than the ink/pencil used to take the notes


Instructions

• Pass out the Student Handout 2.4a “Application: Solving Equations Cornell Notes” and Student Handout 2.4b “Application: Solving Equations Cornell Notes with Question.”

• Have the students examine the application question.

Students should pay special attention to the fact that the Sample 1 process can be applied to Sample 2.

• Divide the class into groups of three to four students.

• Have each student take out several of their own math Cornell notes.

Note: It is critical that in your students’ math classes (or during your tutorials), the processes for solving equations are written out.

• Have each student generate a question modeled after Student Handout 2.4b “Application: Solving Equations Cornell Notes with Question” in their notes.

You may want to have the students begin by rotating their Cornell notes, so that they are generating the question for someone else’s notes.

• Have groups discuss what question they generated and what information they had to add to the right side of the notes.

• Inform the students that for the next three weeks you would like one page of their math notes to contain this question frame and that it will be a part of their Binder Check grade.

Extension:

• If students need additional support with questioning skills, you can use Student Handout 2.4c “Application: Parallel & Perpendicular Lines Cornell Notes” and Student Handout 2.4d “Application: Parallel & Perpendicular Lines Cornell Notes with Questions” during second semester using the same process.



• When will you be covering solving multiple step

equations, e.g., 4(2x –1) = 15x + 10?

This will be when you want to cover Student

Handout 2.4b “Application: Solving Equations

Cornell Notes with Question.”

This is most likely in the second or third month

of school.

• When will you be covering finding perpendicular

and parallel lines passing through a point?

This will be when you want to cover Student

Handout 2.4d “Application: Parallel &

Perpendicular Lines Cornell Notes with

Questions.”

This will most likely be at the end of the first

semester or beginning of the second semester.


STUDEnT HAnDOUT 2.4a APPLICATIOn: SOLVInG EQUATIOnS CORnELL nOTES


APPLICATIOn: SOLVInG EQUATIOnS CORnELL nOTES WITH QUESTIOn STUDEnT HAnDOUT 2.4b


STUDEnT HAnDOUT 2.4c APPLICATIOn: PARALLEL AnD PERPEnDICULAR LInES CORnELL nOTES


STUDEnT HAnDOUT 2.4dAPPLICATIOn: PARALLEL AnD PERPEnDICULAR LInES CORnELL nOTESWITH QUESTIOn


2.5 Generating Comparing and Contrasting Questions for Cornell notes

Topic

• Finding similarities and differences between problems within one lesson

Rationale

Mathematics is a discipline of connections. There is a scaffolding of skills and a logical progression of thinking that connects previous concepts to current ones. These minuscule changes and variations in problems often trip up students on their tests. However, the ability to identify such variations while still seeing the consistency among problems lies at the heart of students’ ability to solve problems effectively.

Objectives

Students will:

• Gain a deeper understanding of how to use the comparative questioning to determine a problem’s common characteristics with another and the critical differences.

• Pull information from a page of notes to create focused higher-level study questions

Timeline


WICR Strategies


• Inquiry

• Collaboration


• Student Handout 2.5a “Compare/Contrast: Solving Systems of Inequalities Cornell Notes”

• Student Handout 2.5b “Compare/Contrast: Solving Systems of Inequalities Cornell Notes with Questions”




Instructions

• Pass out the Student Handout 2.5a “Compare/Contrast: Solving Systems of Inequalities Cornell Notes.”

• Have students underline key concepts.

• Have the students generate a Level 2 or 3 question using the initial notes.

• Pass out Student Handout 2.5b “Compare/Contrast: Solving Systems of Inequalities Cornell Notes with Questions.”

• Have the students compare the question they generated with Student Handout 2.5b “Compare/Contrast: Solving Systems of Inequalities Cornell Notes with Questions.”



• Have each student generate a question modeled after Student Handout 2.5b “Compare/Contrast: Solving Systems of Inequalities Cornell Notes with Questions” in their notes.

It is important to note that students should look to examine the similarities and differences between problems through the use of compare/contrast graphic organizers found in Lesson 3.5 “Critically Comparing Concepts”

• Inform the students that for the next three weeks you would like one page of their math notes to contain the question frame, “Compare how solving ____________ is similar/different from solving __________.” Tell them that it will be a part of their Binder Check grade.

Students should focus on differences between the samples of a day’s notes, but should also compare problems from one day to the next.


• When will you be teaching graphing inequalities, e.g., y ≥ – 4x + 7?

This will be when you want to cover Student Handout 2.5b “Compare/Contrast: Solving Systems of Inequalities Cornell Notes with Questions.”

This is most likely taught near the end of the school year.


STUDEnT HAnDOUT 2.5a COMPARE/COnTRAST: SOLVInG SYSTEMS Of InEQUALITIES CORnELL nOTES


COMPARE/COnTRAST: SOLVInG SYSTEMS Of InEQUALITIES CORnELL nOTES WITH QUESTIOnS



2.6 Generating Prediction Questions for Cornell notes

Topic• Formulating questions focusing on predicting change within one page of notes

RationaleA skill that will greatly help students think more deeply about mathematical concepts is to be able to predict what effect the changing of numbers, variables, or tasks will have on the resulting solution.


• Pull information from a page of notes to create focused higher-level study questions

• Gain a deeper understanding of how change affects the solution

Timeline• 20 – 25 minutes


• Inquiry

• Collaboration

Materials/preparation• Student Handout 2.6a “Prediction: Solving Equations Cornell Notes”

• Student Handout 2.6b “Prediction: Solving Equations Cornell Notes with Questions”



Instructions• Pass out the Student Handout 2.6a “Prediction: Solving Equations Cornell Notes.”

• Have students underline key concepts.

• Have the students generate a Level 2 or 3 question using the initial notes.


• Pass out Student Handout 2.6b “Prediction: Solving Equations Cornell Notes with Questions.”

• Have the students compare the question they generated with Student Handout 2.6b “Prediction: Solving Equations Cornell Notes with Questions.”



• Have each student generate a question modeled after Student Handout 2.6b “Prediction: Solving Equations Cornell Notes with Questions” in their notes.

You may want to have the students begin by rotating their Cornell notes, so that they are generating the question for someone else’s notes.

• Have groups discuss what question they generated and what information they had to add to the right side of the notes.

• Inform the students that for the next three weeks you would like one page of their math notes to contain the question frame “Predict what effect changing ____________ into ___________ would have on our solution.” Tell them that it will be a part of their Binder Check grade.

Extension:

• If students need additional support with these questions, or if you were unable to work with them on questioning when they covered solving multi-step equations, you can use Student Handout 2.6c “Prediction: Parallel & Perpendicular Lines Cornell Notes” and Student Handout 2.6d “Prediction: Parallel & Perpendicular Lines Cornell Notes with Questions” during the second semester.


• When will you be covering solving multi-step equations,

e.g., 4(2x – 1) = 15x + 10?

This will be when you want to cover Student Handout 2.6b

“Prediction: Solving Equations Cornell Notes with Questions.”

This is most likely in the second or third month of school.

• When will you be covering finding perpendicular and parallel

lines passing through a point?

This will be when you want to cover Student Handout 2.6d

“Prediction: Parallel & Perpendicular Lines Cornell Notes with

Questions.”

This will most likely be at the end of the first semester or

beginning of the second semester.


STUDEnT HAnDOUT 2.6a PREDICTIOn: SOLVInG EQUATIOnS CORnELL nOTES


PREDICTIOn: SOLVInG EQUATIOnS CORnELL nOTES WITH QUESTIOnS STUDEnT HAnDOUT 2.6b


STUDEnT HAnDOUT 2.6c PREDICTIOn: PARALLEL AnD PERPEnDICULAR LInES CORnELL nOTES


STUDEnT HAnDOUT 2.6dPREDICTIOn: PARALLEL AnD PERPEnDICULAR LInES CORnELL nOTES WITH QUESTIOnS


2.7 Using the Textbook to fill the Gaps in Cornell notes

Topic

• Using a math textbook to fill gaps in Cornell notes

Rationale

In the AVID Elective, we often discuss the importance of “filling in the gaps” within students’ notes, but what exactly does a “gap” look like, where do they get that information, and where do they write the information? This activity provides several situations that allow students to use a fabricated math textbook to fill in the gaps on a page of pre-written notes. However, the extension of this activity allows the opportunity to cement the students’ understanding by using the strategy with their own textbook to fill in the gaps of their daily notes.


• Learn to identify holes in their notes

• Fill in information from a textbook to complete their notes

Timeline

• 10 – 15 minutes (best used when the Algebra I class is covering Proportions & Cross multiplication)


• Inquiry

• Collaboration

• Reading


• Student Handout 2.7a “Sample Textbook Page”

• Student Handout 2.7b “Sample Cornell Notes with Gaps”

• Transparency or Visual of Teacher Reference 2.7c “Cornell Notes with Filled in Gaps”

Material for Extension of Activity• Students’ Cornell notes and math textbook


Instructions

• Pass out Student Handout 2.7a “Sample Textbook Page” and Student Handout 2.7b “Sample Cornell Notes.”

• Allow students the opportunity to read and analyze the math textbook.

• Have students add any missing information to the sample student notes.

• Have students discuss with an elbow partner or collaborative group about what they added and why they added it.

• Project Teacher Reference 2.7c “Cornell Notes with Filled in Gaps.”

• Ask student volunteers to talk about what they didn’t add. Ask, “Why might that missing information be important?”

Extension• Collect a variety of student notes from their math classes that were all taken from the same lesson. (For example,

collect the notes from the day that every math teacher taught Section 3.2 “Solving One-Step Equations.”) Most likely, the math teachers will teach identical topics several days apart because of pacing.

• Photocopy two or three students’ notes to pass out to the class. Have students fill in the gaps using their math textbooks.

• Discuss where gaps were filled from the text.


• When will you be covering cross multiplication/solving proportions? Could you let me know a week or so in advance?

Generally, proportions are covered near the end of the first quarter or the beginning of the second quarter of an Algebra I elective.

• What topic/pages will you cover this Tuesday?

This will allow you time to collect and copy student sample of the notes.

• Could you look at these notes and these pages in the math book? What information is missing from this student’s notes?

If you are unsure about what information is missing from the student’s notes, have a photocopy of their notes and the textbook. Ask the math teacher to create your “answer key” for what the notes should look like.


3.4 Solving Ratios and proportions

Sample Textbook PageSTUDEnT HAnDOUT 2.7a

Key Concept

If a = c then ad = bc Example: 2 = 8 2 • 20 = 5 • 8 b d 5 20

Sample 1: Use the Cross product property

Solve the proportion 6 = 9 x 3

6 = 9 x 3 Cross multiply proportions CHECK

6 • 3 = 9 • x Multiply constants 6 = 9 2 3

18 = 9x Divide by 9 3 = 3

2 = x Check answer

Sample 2: Cross Multiply and Check Solution

Solve the proportion 5 = 15 x x + 6

5 = 15 x x + 6 Cross multiply proportions 5 = 15

x x + 6

15 • x = 5 (x + 6) Distribute 5 = 15 3 3 + 6

15x = 5x + 30 Subtract 5x 5 = 15 3 9

10x = 30 Divide by 10 5 = 5 3 3

x = 3 Check answer 15 = 15

Sample 3: Write and Solve a proportion

Speed Racer: Devon paid $24 for the 8 gallons of gas she used in a race. How much would Daisy pay for using 5 gallons of gas?

Cost for Devon = Cost for Daisy Gallons of Gas for Devon Gallons of Gas for Daisy

$24 = x 8 5 Write the proportions 24 = 15

8 5

8 • x = 24 • 5 Cross multiply proportions 3 = 3

8x = 120 Divide by 8

x = $15 Check answer

Answer: Daisy would pay $15 for the gas.

Student HelpRemember to check your solution with the original problem. The two ratios should be equal.



STUDEnT HAnDOUT 2.7b SAMPLE CORnELL nOTES WITH GAPS


TEACHER REfEREnCE 2.7cSAMPLE CORnELL nOTES WITH fILLED In GAPS


2.8 Cornell notes: note-Interacting4

Topic

• Summarizing and studying Cornell notes

Rationale

One of the most critical aspects of creating Cornell notes is the summarization of key ideas, and the ongoing studying of that information. It is during the interaction phase that students shift a day’s lesson from their short-term memory to their long-term memory. This greatly increases their likelihood of effective information recall.


• Summarize key concepts and ideas

• Review and recall important information

Timeline



• Inquiry

• Reading


• Cornell notes

• Teacher Reference 2.8a “The Curve of Forgetting”

4Ebbinghaus, H (1885). Memory: A Contribution to Experimental Psychology. Berlin


Instructions

Link Learning – Summarization (Step 6)• Chapter 5 provides a variety of strategies to incorporate as a means to support your students’ ability to

summarize information.

• Instruct students to use the Essential Question as a quickwrite prompt at home that night, to highlight key points of the lesson.

• Students can then use these summaries to reflect on their learning during future periods of study.

Learning Tool (Step 7)• Display Teacher Reference 2.8a “The Curve of Forgetting.”

• Have students discuss how this information connects to their study habits for 3 to 5 minutes. Then, have students share out their realizations.

Optional: Use the Socratic Seminar presented in Lesson 6.7 “The Curve of Forgetting” to open a discussion about memory loss.

• Have students determine a study schedule and understand the importance of reviewing their notes.

• Provide students 5 minutes of class time to take out a page of Cornell notes, cover up the right side, and actively rework the problems on the left side.

This in-class time is critical to model to students how they should study at home.

Remind students that every day, they should review their notes from that day and the previous week.


• Would you be willing to provide a few minutes

during class for students to take out their notes

from the previous day and use them to study?

Let the math teachers know that they don’t

necessarily have to do this every day, but doing

so will help foster the habits of mind that will

support positive study habits.


The Curve of forgetting

Rate of forgetting with Study/Repetition

TEACHER REfEREnCE 2.8a

100%

90%

80%

70%

60%

50%

40%

30%

20%

10%

0

100%

90%

80%

70%

60%

50%

40%

30%

20%

10%

0

1st Repetitionwithin

18 minutes

Percentof Information

Retained


Retained

2nd Repetitionwithin1 day

3rd Repetitionwithin7 days

4th Repetitionwithin

31 days


3 months

2 days = 27.8% 6 days = 25.4%

31 days = 21.1%

1 day = 33.7%

1 hour = 44.2%

9 hours = 35.8%

20 minutes = 58.2%

Time

Time


2.9 Cornell notes: note-Reflecting

Topic

• Providing purposeful feedback for student improvement

Rationale

In order for students to improve their note-taking skills, they must receive clear feedback about what to improve and address that feedback in a meaningful way. It is critical to focus on manageable areas for improvement and set clear student expectations. Too often, we attempt to improve all aspects of note-taking at once, rather than focusing on improving one aspect at a time.

Objectives

Students will:

• Collaborate to design a Cornell note rubric

• Address feedback and adjust note-taking style

• Reflect on their note-taking ability

Timeline

• 20 – 30 minutes for rubric design


• Inquiry

• Collaboration


• Teacher Reference 2.9a “Cornell Note-Taking Checklist” 5

• (Optional): Student Handout 1.4a “Bookmark Template”

5Donohue, J., Gill, T. (2009). Cornell Note-taking Checklist. The Write Path I: Mathematics (p. 23). San Diego, Ca: AVID Press


Instructions

• Determine how many points you are going to designate for your Cornell note grade.

• Ask students, “What are the important aspects of Cornell notes that we need to have in place?”

• Allow students to generate the items on their own (e.g., heading, questions, summary).

• Use Teacher Reference 2.9a “Cornell Note-Taking Checklist” as a reference for a potential rubric design.

• Have students pick three to five items from their brainstorm that can be incorporated into the note grade.

• Tell students that your Cornell notes will be worth a certain number of points. Then have students assign point values to each of the three to five items.

Important: As the teacher choose one area of focus where students have struggled where you want to see an improvement (e.g., revising notes, summaries, evidence of study). Assign this as the last category.

• Use Lesson 1.4 “Math Bookmark,” which students can use as a reference of the rubric.

Written Feedback (Step 8)• Use the student-generated Cornell note rubric for a 10 to 20 week period.

• Collect one to three random pages of notes each week and grade them using the rubric.

• At the end of the 10 to 20 week period, change the area of focus to a new problem area.

Address Feedback (Step 9)• After you hand back the graded notes, have students write a focused goal that addresses what adjustments

they will make in the coming week.

your Reflection (Step 10)• After students have finished a chapter or unit of study, have them collect all of their notes from those weeks.

• Instruct students to reread all of their summaries and highlight key ideas.

This is also the perfect time for students to compare their notes with problems that were missed on any quizzes or tests.

• Have students reflect on any areas where they believe there might be lingering confusion.


• This quarter/semester, our AVID

class is focusing on improving

________________________.

Would you be able to reinforce

this skill in your math class?

(focus area of improvement)


Name _____________________________ Cornell note-Taking Checklist Period ____________________________

Do your notes have the following characteristics? 1. Consistent Cornell physical format, notes dated and titled, readable . . . . . . . . . . 2 points 2. Use of abbreviations, key words/phrases, underlining, starring . . . . . . . . . . . . . . . . 1 point 3. Main ideas are easily seen; correct sequencing of information . . . . . . . . . . . . . . . . . 1 point 4. Questions are completed on left hand side; Level 2 and 3 questions . . . . . . . . . . . 2 points 5. An accurate, complete summary follows the notes . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 points 6. Focus of the quarter _______________________________ . . . . . . . . . . . . . . . . . . . 2 points

CHARACTERISTICS

DATE

SCORE1. Consistent Cornell physical format, notes dated and titled, readable2. Use of abbreviations, key words/phrases, underlining, starring3. Main ideas are easily seen; correct sequencing of information4. Questions are completed on left hand side; Level 2 and 3 questions5. An accurate, complete summary follows the notes6. Focus of the quarter _______________________________TOTAL POInTS

RubricConsistent Cornell physical format, notes dated and titled, readable 2. Vertical line drawn 2.5 inches from the left margin. Heading is complete with name, date, subject.

The notes are titled.

Notes are adequate in length. 1. Minor problem with format. 0. Fails to use Cornell note-taking or date and title or notes are inadequate in length.

Use of abbreviations, key words/phrases, underlining, starring 1. Techniques used throughout. 0. Too much verbiage.

questions are completed on left hand side; Level 2 and 3 questions 2. A substantive number of higher-level questions are written in the left margin and are answered in the

notes to the right. 1. Level 1 questions only. 0. No questions in the left hand margin.

An accurate, complete summary follows the notes 2. Detailed summary covers the main topics of the notes. 1. Summary is generic or incomplete. 0. Summary missing.

Focus of the quarter __________________________________________________ 2. ______________________________________________________________________________________ 1. ______________________________________________________________________________________ 0. ______________________________________________________________________________________



83Chapter 3: Running Math Tutorials

3ChapterRunning Math TutorialsIt’s not that I’m so smart, it’s just that I stay with problems longer.

-Albert Einstein

This chapter is intended for any AVID Elective teacher who has looked across a sea of whiteboards on Tutorial Day, seen the myriad x’s, y’s, and z’s and thought, “They must be working hard!” Coaching effective tutorials and supporting your students to be able to formulate good questions are among the most important (and difficult) aspects of the AVID Elective. Presenting how questions should look is one thing, but actually getting meaningful higher-level questioning and thinking is another.

In the math tutorial, we must move beyond a focus on the incorporation of a Costa’s word in front of a mediocre question and create opportunities to think more deeply about mathematical problems. Overall, we examine three main thinking styles based on Costa’s Levels: Comparing/Contrasting, Applying/Generalizing a Process, and Predicting Change of a Problem. These three stems will help your students move past focusing on a single math question into the realm of generating connections between problems.

The single most important thing to take away from this chapter is not to let your students off the hook simply because they finish a single problem. Math is about connections between problems. Math is about relationships of ideas. Just as there is a common thread that ties ancient Greek theater to Shakespeare and modern television dramas, so are there common properties to numerical, algebraic, and trigonometric fractions. It is our job to help paint the pictures that will help students discover and understand how mathematical processes are tied together.


3.1 Upping the Level of QuestionTopic

• Combining questions and information to generate higher-level questions

Rationale

Often when teachers instruct students to bring in a higher-level question, the students simply cram a higher level “word” in front of a lower-level question. The combination of multiple lower-level questions can result in students delving into deeper levels of comprehension as they begin examining commonalities, differences, reasonability, and generalization of processes.


• Discover the commonalities among seemingly disparate concepts

• Practice formulating higher-level questions

Timeline

• One 50-minute tutorial period


• Inquiry

• Collaboration


• Student Handout 3.1a “Raising the Level of Question”

• Lesson 3.2 “Application of a Process” and Lesson 3.3 “Critically Comparing Concepts”

Instructions

• Print out Student Handout 3.1a “Raising the Level of Question” and discuss with the Algebra I (Pre-Algebra) teacher when they will cover solving multiple-step equations.

Have the math teacher cross out any problems they will not teach during the beginning of the year.

• Once students have covered all aspects of solving multiple-step equations covered in Student Handout 3.1a “Raising the Level of Question,” cut up the questions cards.

• Pass out one card to each student.

• Have each student determine which of Costa’s Levels his or her question addresses


• Then, have students write a two- to three-sentence description of what would happen during tutorial in order to answer their question.

• As a whole-class discussion, have students share the levels of their questions and descriptions of what would be done during the tutorial process.

One of the most important things that students should garner from this discussion is that something additional should be happening in order for the question to be truly higher-level.

• Ask the students, “Have you ever studied for a math test and thought you understood what to do, but then when you took the test you were asked a question that you did not recognize?” (Pause) “One reason this happens is that when we go through tutorial, we often ask the question ‘How do I solve for x in the equation 6x – 18 = 12?’, but on the test they don’t ask that question. They ask some variation like ‘How do you solve 3(2x – 6) = -24?’ So today, we are going to practice creating connections between various mathematical concepts.”

• Place students randomly into groups of seven.

• Have students present the questions from their cards and work through them.

You can use Lesson 3.2 “Application of a Process.”

• Ensure that all students are taking Cornell notes on the material.

• Allow enough time for multiple students to have the opportunity to go to the board. Stop the tutorial process with about 20 minutes remaining in the class.

• Have students take two of the questions from the tutorial and use Lesson 3.3 “Critically Comparing Concepts,” collaboratively filling in one of the graphic organizers.

Some of the interesting questions that students can compare:

Solving 4(5x – 10) = 5(2 + 3x) – 15 vs. 3(2x – 5) + 4(6 – 5x) = 23

Solving – 4p – 8 = 12 vs. 5t – 8 = 7

3 Solving 2x − 9 = 3x + 8 vs. -3(2x – 5) = 7(2x + 5)

-3 5


• When will you be working on solving multiple-

step equations in class? (Student Handout 3.1a

“Increasing a Question’s Level of Thinking”)

• Are there any of the problems from Student

Handout 3.1a “Increasing a Question’s Level of

Thinking” that are not part of your unit of study?


Raising the Level of Question


Infer how to solve for

5t – 8 = 7

Analyze the solution of

4(5x – 10) = 5(2 + 3x) +15

Solve for f:

-5 = 9 – 7f

Predict the solution to

-51 = -9 + 7w

Hypothesize a possibleanswer for

2x − 9 = -3x + 8 -3 5

Solve and justify the steps:

3(2x – 5) + 4(6 – 5x) = 23

Solve and justify the steps

p – 5 = -3 1 12 4 2

Solve for s:

9 = 8s – 16 -1 5s – 10

What is x?

3(2x – 8) + 7x = 5x – 8(2x – 3)

Hypothesize a solution for

-200 – y = 1000

Speculate the solution for

3(2x +5) = 45

How do you solve

36 = 2x 54 6

Apply a process to solve

4m – 8 = 6m + 6

Compare

8 – (4c – 12) = 0and

10(3c + 1) = 70

– 4 p – 8 = 123

5s – 11 = 2s + 10 -3(2x – 5) = 7(2x + 5) t – 4 = 8 4


3.2 Application of a Process

Topic

• Breaking out a process from one question and applying it to others during an AVID tutorial

Rationale

In mathematics, it is critical that students understand how the approach to one type of problem directly interrelates with other types of problems. Too often, students focus on how to solve a single type of problem and do not see how various pieces from a chapter (or unit) can fit into a larger process for approaching problems.

Objectives

Students will:

• Utilize graphic organizers to solve and delineate the process

• Apply the thinking from one problem onto others

• Continually refine their (collaborative) mathematical process

Timeline

• 20 – 35 minutes during a tutorial

WICR Strategies


• Inquiry

• Collaboration


• Student Handout 3.2a “Application of a Mathematical Process”

• Student Handout 3.2d “Application of a Process Flowchart”

• White board

• Various colored dry erase pens


Instructions

• During a tutorial, have the first student come to the board and write his or her question using the format in Student Handout 3.2a “Application of a Mathematical Process”

Use the sentence frame: Apply the process for solving (write first student’s question here) .

• Have the student show the work for solving the problem on the left and write out the steps to the process on the right.

Have students leave space between steps on the right side for later.

• Have the other students record the solution and process in their Cornell notes.

The tutor should be recording the work for the student at the board.

• Once all work has been recorded, have the student erase the solution (on the left hand side) but keep the sentence frame at the top and the process on the right hand side.

• Have a second student (or the first can stay at the board) present a similar type of problem, but with some variation (such as parenthesis in one equation but not the other, different method needed to solve, etc.).

The student should write the second question into the sentence frame:

“Apply the process for solving (first student’s question) to the question (write second student question here).”

• Have the student use the process to help guide working through the new problem on the left.

• If a new step is needed in order to solve the problem, have the student add it to the right hand side (in the appropriate place) with a different colored dry erase or chalk.

• Once the second student has finished, erase only the second student’s question. Have a third, fourth, fifth, etc. student write their problems (covering similar topics) and use the refined process to guide their work.

• Fill in steps with the different colored pen when needed.

• Once all questions on the group of topics has been exhausted, have the students write the process into the Student Handout 3.2d “Application of a Process Flowchart.” Write one step in the form of a question into each box.


• What types of problems are you covering during

your classes?

• Are there any key differences between these types

of problems that are likely to show up on a test?

For example: Solving one-variable equations

when there is distribution, combining terms,

fractions…



Application of a Mathematical Process

Math question:

Apply the process for (solving/factoring/simplifying)

_____________________________________to the question _____________________________.

Solution (Show Work) Mathematical Steps (Written Out)



Application of a Mathematical Process (Question 1)

Math question:


_____________________________________to the question _____________________________.


23 = 5m – 3m + 1

23 = 5m – 3m + 1

23 = 2m + 1 -1 -1

22 = 2m2 2

11 = m

Check Solution

23 = 5m – 3m + 1

23 = 5(11) – 3(11) + 1

23 = 55 – 33 + 1

23 = 22 + 1

23 = 23

Combine like terms.

Subtract to eliminate the constant from both sides of the equation.

Divide both sides by the number in front of the variable.

Substitute what the variable equals into the original equation.

Multiply the terms.

Add/subtract from left to right.

Make sure both sides are equal. (If not, a mathematical error was made.)


TEACHER REfEREnCE 3.2c

Application of a Mathematical Process (Question 2)

Math question:


_____________________________________to the question _____________________________.


4 + 4(p - 5) = 32

4 + 4p - 20 = 32

4p – 16 = 32

+16 +16

4p = 48 4 4

p = 12

Check Solution

4 + 4(p - 5) = 32

4 + 4(12 - 5) = 32

4 + 4(7) = 32

4 + 28 = 32

32 = 32

Distribute number outside the parenthesis.

Combine like terms.

Add/subtract to eliminate the constant from both sides of the equation.

Divide both sides by the number in front of the variable.

Substitute what the variable equals into the original equation.

Add/subtract numbers in the parenthesis.

Multiply the terms.

Add/subtract from left to right.

Make sure both sides are equal. (If not, a mathematical error was made.)

23 = 5m – 3m + 1 4 + 4(p + 5) = 32 .


STUDEnT HAnDOUT 3.2d

App

licat

ion

of a

Pro

cess

flo

wch

art


TEACHER REfEREnCE 3.2e

Wha

t doe

s it

mea

n if

the

num

bers

ar

en’t

equa

l?

App

licat

ion

of a

Pro

cess

flo

wch

art S

ampl

e

Are

ther

e pa

rent

hese

s th

at w

e w

ould

ne

ed to

use

di

strib

utio

n to

el

imin

ate?

Can

we

com

bine

lik

e te

rms?

A

re th

ere

varia

bles

/co

nsta

nts

on

the

sam

e si

de

of e

qual

sig

n?

Do

we

need

to

add/

subt

ract

to

elim

inat

e th

e co

nsta

nt fr

om

both

sid

es o

f th

e eq

uatio

n?

Is th

ere

a nu

mbe

r in

fron

t of t

he

varia

ble?

Can

we

subs

titut

e ou

r ans

wer

to

che

ck o

ur

solu

tion?

Can

we

sim

plif

y ou

r che

ck?


3.3 Critically Comparing Concepts

Topic

• Utilizing comparison in tutorials and notes

Rationale

In mathematics, students are often very concerned with “how to solve” an equation or problem. However, just finding out that “x = 3” is not enough to help the student in the long run because on a quiz or test, the answer will not always be “x = 3.” What the student really needs to discover is how the approaches to one type of problem are similar to another, and how they are different. This will help the students create schemata for approaching a broader body of mathematical problems.

Objectives

Students will:

• Compare the similarities and differences in solving two problems

• Culminate their understanding using a graphic organizer

Timeline


WICR Strategies


• Inquiry

• Collaboration


• Cornell notes (from same or consecutive days)

• Math book

• Tutorial Request Form

Instructions

• Have three to four students go through the first portion of the tutorial, asking semi-related questions (possibly using the Application of a Process work). Generally, closely related topics are covered within four to five days of each other.


• After several students have asked their questions, choose one of the compare/contrast graphic organizers and have the students choose two of the problems they have worked out that seem like they would be interesting/important to compare.

Hint 1: If you are not sure what type of questions to compare/contrast, open the book to the chapter the students are learning. Often, each section of the chapter (e.g., 4.2 Solving Multiple-Step One-Variable Equations) opens with a series of sample problems worked out (e.g., Sample 1: Solving an equation using the Distributive Property; Sample 2: Solving an equation by combining like terms, etc.). These sample problems are often the critical components that students need to compare.

Hint 2: Many books have an “Extra Practice” page at the end of the book that has every chapter, with bold headings that break up the chapter into its important sections. For example the bold headings might read, “Solve the inequality and graph,” “Solve the absolute value inequality and graph,” etc. These sections of the chapter are often closely related and are valuable for helping the student indentify the similar components.

• Have students collaboratively fill in a comparison graphic organizer. It is often valuable to focus on how the processes for solving the problems are similar, and how they are different.

• At the conclusion of the tutorial, have the students use the comparison organizer to write a higher-level reflection. (See 3.3a through 3.3h.) These graphic organizers can help students determine characteristics that are similar and different between problems.


• In your current unit of study (or chapter), what are the most

important pieces of information?

• Are there any problems in this chapter that are especially tricky

and will appear on your quizzes and tests? (For example: systems

of equations with no solution, solving one variable equations

with fractions in them, or anything that students often struggle

with solving)

It is especially important that students know how solving

these “trick” problems is different from solving other similar

problems.

• Could you write down an example of one of these “trick”

problems, with the solution?


Compare and Contrast T-Chart


Level 2 Question: ______________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

SIMILARITIES DIFFERENCES


Compare and Contrast T-Chart Sampley = 1 x −4 and y > -3 −6 2


Level 2 Question: ______________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

SIMILARITIES DIFFERENCES

• The first equation is only a line,

and the second is a line that

would require shading.

• When graphed, the first equation

will have a solid line, and the

second will have a dotted line

(because the > sign means that

the points on the line are not

solutions).

• The first equation has a positive

slope and the second has a

negative slope.

• Both equations can be graphed

by starting at the y-intercept and

using slope to graph.

• Both of these equations have two

variables.

• Both of these equations have

multiple solutions.

• Both of these equations are

linear.

• Both of these equations have

negative y-intercepts.

Compare how the graphs for:are similar and different in their look, function, and purpose.


Compare/Contrast Graphic Organizer

How Alike?

How Different?with regard to


1. ____________________________________________________________________________________

____________________________________________________________________________________

2. ____________________________________________________________________________________

____________________________________________________________________________________

3. ____________________________________________________________________________________

____________________________________________________________________________________

4. ____________________________________________________________________________________

____________________________________________________________________________________

1. ___________________________________

___________________________________

___________________________________

2. ___________________________________

___________________________________

___________________________________

3. ___________________________________

___________________________________

___________________________________

4. ___________________________________

___________________________________

___________________________________

1. ___________________________________

___________________________________

___________________________________

2. ___________________________________

___________________________________

___________________________________

3. ___________________________________

___________________________________

___________________________________

4. ___________________________________

___________________________________

___________________________________


Compare/Contrast Graphic Organizer Sample

How Alike?

How Different?with regard to

TEACHER REfEREnCE 3.3d

1. ____________________________________________________________________________________

____________________________________________________________________________________

2. ____________________________________________________________________________________

____________________________________________________________________________________

3. ____________________________________________________________________________________

____________________________________________________________________________________

4. ____________________________________________________________________________________

____________________________________________________________________________________

1. ___________________________________

___________________________________

___________________________________

2. ___________________________________

___________________________________

___________________________________

3. ___________________________________

___________________________________

___________________________________

4. ___________________________________

___________________________________

___________________________________

1. ___________________________________

___________________________________

___________________________________

2. ___________________________________

___________________________________

___________________________________

3. ___________________________________

___________________________________

___________________________________

4. ___________________________________

___________________________________

___________________________________

The area above the line is shaded

showing that the region above the line

represents solutions to the equation.

The line has a negative slope.

The y value would need to be larger

than the right side of the equation.

The line is dotted, showing that the

points on the line are not solutions.

Both of these equations are linear.

Both equations have an infinite number of solutions.

Both lines have a negative y-intercept.

We can use the y-intercept and the slope to sketch a graph for

both equations.

y = 1 x – 4 2 y › –3x – 6

There is no shading on the graph

showing that the solutions are only

on the line.

The line has a positive slope.

Solutions are found by substituting an

x and determining the y.

The line is solid, demonstrating the

solutions are on the line.


Com

pare

/Con

tras

t _

____

____

____

____

____

___

to

_

____

____

____

____

____

___

Venn

Dia

gram

STUDEnT HAnDOUT 3.3e


Com

pare

/Con

tras

t _

____

____

____

____

____

___

to

_

____

____

____

____

____

___

Venn

Dia

gram

Sam

ple

TEACHER REfEREnCE 3.3f

y =

1 x

– 4

2

y › –

3x –

6

Both

equ

atio

ns

are

linea

r.

Both

hav

e an

infin

ite

num

ber o

f sol

utio

ns.

Both

line

s ha

ve a

nega

tive

y-in

terc

ept.

Ther

e is

no

shad

ing

on th

e

grap

h, s

how

ing

that

the

solu

tions

are

only

on

the

line.

The

line

has

a po

sitiv

e sl

ope.

Solu

tions

are

foun

d by

sub

stitu

ting

an x

and

det

erm

inin

g th

e y.

The

line

is s

olid

, sho

win

g th

at th

e

solu

tions

are

on

the

line.

The

area

abo

ve th

e lin

e is

sh

aded

, sho

win

g th

at th

e

regi

on a

bove

the

line

re

pres

ent s

olut

ions

to

the

equa

tion.

The

line

has

a ne

gativ

e sl

ope.

The

y va

lue

wou

ld n

eed

to b

e

larg

er th

an th

e rig

ht s

ide

of

the

equa

tion

to b

e a

solu

tion.

The

line

is d

otte

d to

sho

w th

at th

e po

ints

on

the

line

are

no

t sol

utio

ns.


Com

pare

and

Con

tras

t Bub

bles

Com

pare

/Con

tras

t _

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

__

to

___

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

_

STUDEnT HAnDOUT 3.3g

103

Com

pare

and

Con

tras

t Bub

bles

Sam

ple

Com

pare

/Con

tras

t _

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

__

to

___

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

_

TEACHER REfEREnCE 3.3h

the

char

acte

ristic

s of

the

grap

h y

= 1

x –

4

2

the

char

acte

ristic

s of

y ›

–3x

– 6

.

y =

1 x

– 4

2

y › –

3x –

6

No

shad

ing

on g

raph

, so

solu

tions

are

on

ly o

n th

e lin

e.

Both

equ

atio

ns

are

linea

r.

The

area

ab

ove

the

line

is s

hade

d,

show

ing

the

solu

tions

.

Subs

titut

ing

an x

val

ue w

ill

dete

rmin

e th

e

y va

lue.

Both

equ

atio

ns

have

an

infin

ite

num

ber o

f so

lutio

ns.

The

y va

lue

on

the

left

nee

ds

to b

e la

rger

than

th

e va

lue

on

the

right

.

Gra

ph o

f G

raph

of

The

poin

ts o

n th

e lin

e ar

e so

lutio

ns, s

o th

e lin

e is

sol

id.

You

can

use

y-in

terc

ept a

nd

slop

e to

ske

tch

each

equ

atio

n.

The

poin

ts o

n th

e lin

e ar

e no

t a

solu

tion,

so

the

line

is d

otte

d.


3.4 What Happens When...?Tutorial follow-up QuestionsTopic

• Follow-up questions after a tutorial

Rationale

Often during tutorial, we let the students leave the board after a tutorial question without any follow-up investigation about understanding. Occasionally, we ask the student at the board to explain what he or she did to solve the question. While this is a great method to examine if the student understands the problem, it is still a re-examination of the same problem. This activity examines a simple method to have students make predictions and assumptions about connected problems, thus forcing the students to delve deeper and utilize predictive and critical thinking skills.


• Answer questions connecting understanding to other concepts

• Examine similarities among concepts

Timeline

• 1 – 3 minutes at the conclusion of a question

WICR Strategies• Inquiry

• Collaboration


• Student question

• Math textbook


Instructions

• Before the tutorial period begins, discuss with the tutors the idea of asking follow-up questions with student presenters.

• Have a student presenter come to the board, present a question, and answer it using the standard inquiry format.

• Following the question, ask the student to predict how to solve a modified question.

Important! It is critical to change the question so that it relates to another type of problem that they are currently working on in class.

Some aspects to consider changing are positives to negatives, adding parentheses, adding a variable, or changing inequalities to equalities.

• Have the student verbalize a response to this question, but do not have the student write down a solution.

• It is critical that this is a small modification, one that simply has the student think on a deeper level about the problem.


• (If tutors are really having difficulty

adapting questions)

Would you be able to work with my

AVID tutor(s) about modifying a

math tutorial question?

Have the tutors bring in sample

questions and Teacher Reference

3.4a “Predicting Change, Follow-

up Questions.”



Predicting Change, follow-up Questions

Initial Tutorial question:Find the mean, median, and mode for the data points: 84, 92, 88, 90, 67, and 89.

predictive Follow-up question: What would happen to the mean, median, and mode if you were to get 100 as your next point?

Initial Tutorial question:Explain how to find x in the equation 3(2x – 4) = 24.

predictive Follow-up question: What would we have to do differently if the equation were 3(2x – 4) + 2x = 24?

Initial Tutorial question: Explain how to solve and graph the inequality 3x + 5 > 5x – 15.

predictive Follow-up question: What would happen to our graph if we changed the 3x into a -3x?

Initial Tutorial question:Explain the process for factoring x2 + 5x – 45

predictive Follow-up question: What would be different about the process for factoring x2 + 5x – 45 = 0?

Initial Tutorial question:Tell whether the point (3, -1) is a solution for x + 8y ≥ 10.

predictive Follow-up question: (After the student finds out it is not a solution): What would be a solution for x + 8y ≥ 10?

Initial Tutorial question:What is the cube root of 27?∛27

predictive Follow-up question: What is different about finding 27

1? What is different about finding 27-1

? 3 3

Initial Tutorial question:Evaluate the expression – y – (-x) when x = 1.5 and y = -4.

predictive Follow-up question: What would change about this question if the x = -1.5?



Questioning Prompts for the AVID Tutor/Students

At the start of the tutorial session• Who has the most pressing question?

• Does anyone have a test or quiz coming up?

At the beginning of a student’s question• What do we know already about_________________________?

• Do we have any notes on this topic?

• Do we have our books/binders out?

• Could we make a prediction about a reasonable answer?

Remember that a prediction is based on some type of reasoning, but does not have to be exactly correct. It is merely a starting place for the students to look at the larger picture.

• What strategies have you tried so far?

• Where are you getting stuck?

• What would be our first/next step?

• Is there a similar problem we could reference from our notes/book?

If students get stuck• Let’s go back to our notes/book; is there anything that can help us?

• Remind us, what are we trying to find?

• What strategies/principles can we apply to this step that might move us closer to a solution?

• Can you think of a simpler problem that we could work through that might give us insight into this question?

Very important: This is not to disregard the current question, but if students can work through a problem that they know how to do, it often allows them to reconnect with the more difficult problem. Once you have solved the simpler problem, apply the same strategy to the original problem.

At the conclusion of the problem• What were the steps we used to get us to this point?

• Does this answer seem reasonable? (Ask the students to tell you why it is or is not reasonable.)

• Can we substitute our answer in to see if it is correct?

• How can we check to see if this answer is correct? (This goes beyond substituting x back into the equation. Encourage students to think outside the box about how they can check solutions.)


3.5 The Power of Prediction

Topic

• Utilizing logical reasoning to estimate and justify a reasonable solution

Rationale

Students often treat a mathematical problem like an isolated event that eventually leads to a mystical “answer.” However, students have enough conceptual knowledge to make reasonable approximations and, with use, their reasoning skills will undoubtedly improve. A critical distinction is that the students should be making predictions, not guesses! A guess is an opinion based on little or no evidence, whereas a prediction is a foretelling made on the basis of observation, experience, or scientific reasoning. The same reasoning skills that enable a student to make predictions can also be used to reflect on a solution to see if the answer makes sense.


• Use logic to estimate a reasonable solution to a mathematical problem

• Refine their method of estimation to more closely reflect the answer

Timeline

• 1 – 2 minutes as the students begin their tutorial question

• 1 – 3 minutes refining the prediction


• Collaboration


• Student question


Instructions

• At the beginning of a new tutorial question, have the students make a prediction about their answer.

This could be incredibly tough, especially initially. However, it is important that students express what they think about the answer…

• Will it be a big number or a small number?

• Is it positive or negative?

• Is it a whole number, a decimal, a fraction?

• Will it be a number, a graph, an expression?

• Will it have units: if so, are they feet, ounces, m2?

It is very likely that they will be “wrong,” and it is important that that be OK! This is about thinking through the process of what they are going to be doing and what might be a reasonable answer.

• Have the students write their prediction in the top right corner of the board.

• Once they have arrived at their answer, have them revisit their prediction.

Were they close?

How could they have refined their prediction?

• As a wrap-up to a student’s question, change some aspect of the question and have the students explain what would change about the way they would solve the problem.

For example: If the student’s question was originally 3x – 4 > 2x + 6, once the students have worked through this problem, don’t let the student sit right away. Instead ask, “What would change about our solution if the question was -3(x – 4) > 2x + 6?”


• Are any topics coming up

for which estimation will

be valuable?

• Are you going to be teaching

word problems in the near

future?


EVALUATION as it is most often used is NOT a Level 3 question!!

Students will often try to get away with bringing in questions like…

Evaluate: 3 + 5x + 10 – 2x when x = 6.

THIS IS A LEVEL 1 QUESTIOn, nOT A LEVEL 3!

Students should never bring in a math question that begins with

“evaluate” and only requires you to substitute in a value for a variable. In

such cases, the word “evaluate” just means “find the solution.”

The only true case of evaluation is if a student brings in a quiz or test

on which they made a mistake and the group collaboratively works to

evaluate the work, identify the mistake, and clarify the misconception.

For an in-class activ ity on evaluation, see

Lesson 4.4 “The Golden Mistake” or

4.2 “Evaluating the Error.”

Warning!!!Content May Contain Low-Level Questions


3.6 four Corner Comparisons

Topic

• Comparing important distinctions within a unit of study

Rationale

One of the most frustrating aspects of mathematics for students stems from when they believe they understand a concept for a math test only to have something that seems very new show up on the test. The reality is that when an instructor teaches any new concept (for example, section 5.3), there are generally three to five sub-concepts that students must understand before they can demonstrate mastery on a quiz or test. Fortunately, authors write in a way that promotes a progression of difficulty, which allows for easy comparison when you know where to look.


• Identify variations in similar problems

• Predict questions that will appear on upcoming quizzes or tests

• Describe the similarities and differences among problems

Timeline

• 10 minutes during a tutorial


• Inquiry

• Collaboration

• Reading


• Math textbook (specifically the “Extra Practice” pages in the back of the book)


Instructions

• Have students work through their tutorial using the normal process (possibly utilizing the procedures from Lesson 3.2 “Application of a Process” or 3.3 “Critically Comparing Concepts”).

• Once students have a base level of understanding around a concept, have the students/tutor turn to the “Extra Practice” pages in the very back of the book. These pages are generally located around pages 700 or 800.

• Turn to the appropriate chapter and find the section that you have worked on during tutorial.

• Have students locate the four corners of that section (for this sample we will use Chapter 4, Section 4). See Teacher Reference 3.6a “Extra Practice Text Sample.”

• These four corners often demonstrate the key distinctions that a student will need to comprehend in order to demonstrate mastery on test or quizzes.

• Have students use one of the graphic organizers from the previously mentioned strategy, Lesson 3.3 “Critically Comparing Concepts,” and collaboratively choose two corners to examine similarities and differences.


• What is the hardest type of problem you

will ask on your next chapter test?

Some examples you can give are

systems of equations with no

solution, solving an equation where

you need to distribute a fraction, or

any other particularly tricky concept

for students to master.


Solve the equation, and then check your solution

4.1 1. x + 7 = 14 2. 12 = t – 11 3. p + 7 = -5 4. q – 5 = -9

5. 6g = 42 6. -5y = -55 7. f = 7 3

8. r = 4 -9

4.2 9. 5w + 8 = 33 10. 7s – 8 = 13 11. 31 = 4k – 5

12. v – 2 = 5 3

13. d + 2 = -3 5 14. 5y – 2y = -21

4.3 15. 7x + x + 7 = 47 16. 11t – 2 – 4t = 40 17. 3r – 4 – r = -20

18. 5y + 4(y + 3) = -15 19. 8x – 4(x – 2) = 24 20. 34(2k – 12) = 9

4.4 21. 6d – 3 = 2d + 5 22. 15 – 3x = 2x – 15 23. 7 – 7x = 27 – 2x

24. 5(3h – 2) = 3h – 34 25. 6 + 3s = 3(5 – s) 26. 8x – 4 = 4 (3x + 9) 5

27. 10b + 4 = 2(5b + 4) 28. 4(6p – 2) = 24p – 8 29. 4 (15f – 33) = 8f – 8 5

Solve the proportion, and then check your solution

4.5 30. 5 = x

2 8 31. t = 18

3 27 32. s = 45

5 50 33. -9 = c

4 8

34. 2g = 12

7 14 35. 15r = 45

5 3 36. 4 + x = 48

6 24 37. 12 = a – 2

4 3

Solve the equation for y

4.6 38. 5x + y = 13 39. 6x – 2y = 24 40. 14 = 2x + 7y 41. 8x + 4y = 12 + 16x

42. 11 – 5x = 8 + 3y 43. 6x + 5y = 31x – 25 44. 8x – 13 = 6y – 3 45. 12 – 24 = 6y – 12

TEACHER REfEREnCE 3.6aExTRA PRACTICE TExT SAMPLE

Chapter 4


3.7 Tag Team Tutorials

Topic

• Collaboratively creating questions

Rationale

During the initial stages of teaching a student to generate questions that lead to deeper levels of thinking, it can be helpful to have them work with a partner to find similar areas of confusion.


• Work collaboratively with a partner to generate higher-level questions

• Identify common areas of confusion

• Use graphic organizers to determine critical similarities and differences

Timeline

• 10 – 15 minutes the day before a tutorial

• 20 – 25 minutes the day of the tutorial


• Inquiry

• Collaboration


• Math textbook

• Teacher Reference 3.6a (as a reference if needed)

• Tutorial Request Form


Instructions

The Monday or Wednesday before tutorial day

• Have students bring in their math textbooks and turn to the “Extra Practice” section in the book for the chapter they are studying.

This will generally look like Teacher Reference 3.6a.

• Have each student select two problems that are giving them difficulty and record them into their notes.

• Move students into different corners of the room based on the math they are studying. (Pre-Algebra, Algebra, Geometry...)

• Instruct students to share the problem they selected as their trouble areas.

• Once all students have read their questions, have students partner up with students that have selected a similar type of problem.

• Give students 5 minutes to generate a comparison question that resembles the questions in Lesson 3.5 and have them record their collaborative questions on individual Tutorial Request Forms.

• Collect all of the Tutorial Request Forms.

The following Tuesday or Thursday tutorial

• Break them into small groups (doing your best to keep tag teams together).

• Have the partners present their problem to the board one at a time.

• After both have presented their problems, have the group use one of the graphic organizers from Lesson 3.3 to distinguish between similarities and differences.


• Are there any connected math concepts

that students should focus on?

For example, will students need to: solve

systems of equations with solutions and

without solutions, factor quadratics

with an x2 coefficient and without, or

find a perpendicular and a parallel line

for an equation?


3.8 Index Card Scramble

Topic

• Sequencing the process to solve mathematical problems

Rationale

There are two primary reasons that having students re-sequence their work is a valuable use of their time. The first is that the students do not get nearly enough practice revisiting their problems and thinking about them in a different way. This strategy forces the student to revisit every question, and quickly re-sequence the steps. The second is an answer to the students’ age-old question, “We’re done with tutorial, what now?”

Objectives

Students will:

• Sequence the process for solving their question

• Work collaboratively with a partner for reviewing the work for the tutorial

Timeline

• 5 – 10 minutes at the end of a tutorial

WICR Strategies


• Inquiry

• Collaboration


• Four to ten index cards (or post-it notes) per student

(Optional – Student Handout 3.8a “Index Scramble Cards”)

• Tutorial sheet (with notes on the process)


Instructions

• After students have exhausted all of their questions (especially after using the application of a process system Student Handout 3.2d “Application of a Process Flowchart”), give each student a small stack of index cards.

• Have students write each individual step to the solution on one colored index card/post-it/paper. On a different colored index card/post-it/paper, write the process of what was done.

(Optional) Distribute Student Handout 3.8a “Index Scramble Cards” and have students cut out individual cards.

• Have the students mix up their stack of cards, pass the set counter-clockwise, and have the next person order the algebraic solution and written-out process as quickly as possible.

• Once all students in the tutorial have completed their sets, rotate each set counter-clockwise again.

• Continue rotating the sets of cards until the sets come back to their initial position or it is time to move on to reflections.

Note: When a group does this exercise, it is advisable to collect (and rubber band) each set of questions. This will give you the opportunity to take them out as a Friday activity to review for future quizzes or tests.


• Could you look through these sets

of questions and tell me which will

be most valuable for students to know

in the future?

This will help you determine which sets

will be the most helpful to review before

a future test.


Index Scramble CardsFront Back



Index Scramble Cards SampleFront Back

3(x – 4) + 18 = 2(3x + 3)

3x – 12 + 18 = 6x + 6

3x – 6 = 6x + 6

-6 = 3x + 6

-12 = 3x

-4 = x

Given

Distribute

Combine Terms

Subtract x from both sides of the equation

Subtract constants from both sides

Divide by coefficient



121Chapter 4: Math Strategies for the AVID Elective

4ChapterMath Strategiesfor the AVID ElectiveThere has been a shift in math instruction away from an emphasis on computation and drill, and more an emphasis on problem solving and critical thinking... The changes to our math curriculum force students to do more analysis.

-Stephen Foster

The incorporation of mathematical strategies into the AVID Elective

can greatly support your students’ academic success and ability

to approach mathematical problems effectively. Two of the most

critical skills that a successful math student needs to possess are

the ability to analyze multiple methods for finding a solution (and

choosing the most efficient one) and the ability to analyze their

own work for mathematical errors. However, these skills are not

developed intuitively; they need to be taught explicitly over time.


4.1 Sentence frames6

Topic• Acquisition and use of academic language

Rationale

Fluency in the academic language of mathematics is required for accessing rigorous text and assessment items. Students rarely have oral practice with new concepts and vocabulary. “Sentence Frames” can be adapted for any content and are utilized for the following:

• Acquisition of the academic language of mathematics

• Oral language practice in context

• Development of logical connectors

• Integration of concepts

• Aural practice

Objectives

Students will:

• Create and practice using academic language orally

• Acquire academic language through aural practice

Timeline

• 5 – 15 minutes to complete sentence frames of increasing complexity and to check for correctness

WICR Strategies

• Collaboration

• Reading to Learn

6Donohue, J., Gill, T. (2009). 4.7 Sentence Frames. The Write Path I: Mathematics (pp. 181-183). San Diego, Ca: AVID Press



• Teacher Reference 4.1a “Sentence Frames”

• Prepare sentence frames electronically or as a visual artifact in advance so that students see only one sentence at a time. Sentence frames should be developed as follows:

Each frame starts easy and increases in complexity with each iteration.

Include two or more sentences.

Fill in the blanks with numbers to work on new vocabulary.

Fill in the blanks with vocabulary to review concepts.

Focus on the academic language of mathematics. (See Teacher Reference 4.1a “Sentence Frames.”)

Instructions

• Show students the first sentence frame and give them time to fill in the blanks individually.

• In pairs or larger table groups, ask students to read their sentence frames to check for correctness and fluency.

• Ask each student to read his or her sentence frame. Students may repeat responses if they do not have a unique response. The goal is to have each student say the sentence once and to hear the sentence 25 or more times.

• Repeat the process with each sentence frame in the series.

• Keep it moving, but be prepared to discuss any sentences that are mathematically incorrect or have problems with academic language fluency.


• When will you be covering: combining like terms,

square roots, or slopes of lines?

• Could you look at www.mathsentenceframes.wikispaces.com

and let me know which topics you will be covering?



Sentence frames

Sample 1

1. “ _____________________ and ___________________ _____________ are like terms.”

2. “ _____________________ and __________________ _____________ are like terms.

However, _____________ and __________________ are not like terms.”

3. “ _____________________ plus __________________ equals ________ .” (Like Terms)

4. “ _____________________ plus __________________ equals ________ .” (Unike Terms)

Sample 2

Please complete this sentence frame with a linear equation of the form y = mx + b. Sample: y = 2x + 5

1. “The slope of the linear equation (y = mx + b) is .”

Please complete this sentence frame with a linear equation of the form ax+ by = c. Sample: 2x + y = 5

2. “The slope of the linear equation ( ax + by = c) is .”

Higher-Level questions

Why is the second sentence frame more difficult than the first?

What would happen in the second sentence frame if a = 0? If b = 0?


4.2 Evaluating the Error7

Topic

• Determining and correcting mathematical errors

Rationale

Proofing a mathematical argument is a critical skill that students need to master over time. You can often hear students lamenting in the hallways after a test was handed back, “Man, I can’t believe I made that mistake.” When a student is studying, doing the work, and taking good notes, it is often their propensity to make minor mistakes, rather than a gap in their math knowledge, that leads to low test scores.

Objectives

Students will:

• Proof mathematical arguments

• Determine points of error

Timeline


WICR Strategies


• Inquiry

• Collaboration


• Student Handout 4.2a “Evaluating the Error: Integers”

• Teacher Reference 4.2b “Evaluating the Error: Integers”

• Student Handout 4.2c “Evaluating the Error: Factoring”

• Teacher Reference 4.2d “Evaluating the Error: Factoring”

• Extension “The Golden Mistake” Activity

7Ching, K., Gill, T. (2003). Algebra Tutorial: A Guide for AVID Coordinators and Tutors (pp.80-81). San Diego, CA: AVID Press


Instructions

• Pass out Student Handout 4.2a “Evaluating the Error: Integers” (during first quarter).

• Pass out Student Handout 4.2c “Evaluating the Error: Factoring” (during third quarter).

• Have students individually identify the error from the mathematical argument.

• Have students work with an elbow partner to describe the error on the right column.

• Have students share out their responses as a whole class.

• Use Teacher Reference 4.2b “Evaluating the Error: Integers Sample” and Teacher Reference 4.2d “Evaluating the Error: Factoring Sample” to assist in correcting responses.


• When will you be studying:

Combining like terms? (Generally, first quarter)

Factoring quadratics? (Generally, third quarter)




Evaluating the Error:Integers/Combining Like Terms

Find the mistakes students made (if any) in simplifying the following algebraic expressions.

a) 3x2 + 4x – 5x2 + 2x – 6 ____________________________________________________

7x2 – 3x2 – 6 ____________________________________________________

4x2 – 6 ____________________________________________________

____________________________________________________

b) 5a2b + 4ab – 3ab + 6a2b2 – 2a2b ____________________________________________________

(5a2b + 6a2b 6a2b – 2a2b) + (4ab – 3ab) ____________________________________________________

9a2b + ab ____________________________________________________

____________________________________________________

c) -2(3x + 1) – 4(-2x – 5) ____________________________________________________

-6x – 2 + 8x + 20 ____________________________________________________

(-6x + 8x) + (-2 + 20) ____________________________________________________

2x + 18 ____________________________________________________

____________________________________________________

d) 5x – 3y + 6z – 8x + 2y – 6z ____________________________________________________

(5x – 8x) + (3y + 2y) + (6z – 6z) ____________________________________________________

-3x + 5y ____________________________________________________

____________________________________________________

e) -3r3s + 9r3s – 5rs2 + 7rs2 ____________________________________________________

(-3r3s + 9r3s) – (5rs2 + 7rs2) ____________________________________________________

-6r3s – 12rs2 ____________________________________________________

____________________________________________________



Evaluating the Error:Integers/Combining Like Terms Sample

Find the mistakes students made (if any) in simplifying the following algebraic expressions.

a) 3x2 + 4x – 5x2 + 2x – 6 ____________________________________________________

7x2 – 3x2 – 6 ____________________________________________________

4x2 – 6 ____________________________________________________

____________________________________________________

b) 5a2b + 4ab – 3ab + 6a2b2 – 2a2b ____________________________________________________

(5a2b + 6a2b 6a2b – 2a2b) + (4ab – 3ab) ____________________________________________________

9a2b + ab ____________________________________________________

____________________________________________________

c) -2(3x + 1) – 4(-2x – 5) ____________________________________________________

-6x – 2 + 8x + 20 ____________________________________________________

(-6x + 8x) + (-2 + 20) ____________________________________________________

2x + 18 ____________________________________________________

____________________________________________________

d) 5x – 3y + 6z – 8x + 2y – 6z ____________________________________________________

(5x – 8x) + (3y + 2y) + (6z – 6z) ____________________________________________________

-3x + 5y ____________________________________________________

____________________________________________________

e) -3r3s + 9r3s – 5rs2 + 7rs2 ____________________________________________________

(-3r3s + 9r3s) – (5rs2 + 7rs2) ____________________________________________________

-6r3s – 12rs2 ____________________________________________________

____________________________________________________

This student did not recognize that the x2 and x terms cannot be combined. The simplified form of this expression is -2x2 + 6x – 6

This student combined the 6a2b2 term with the two a2b terms. The simplified form of this expression is 3a2b + 6a2b2 + ab.

This expression has been simplified correctly. Notice that the student used the distributive property before combining like terms.

This student grouped the like terms correctly but changed the -3y to 3y. The simplified form of this expression is -3x – y.

This student incorrectly left the negative sign in front of the parenthesis for the rs2 terms. The negative sign needs to stay with the 5rs2 term. The simplified form of this expression is 6r3s + 2rs2.



Evaluating the Error:factoring

The factoring problems below are done incorrectly. Factor them correctly (if factoring is possible), and then explain what the students may have been thinking when they made the mistakes they did.

a) x2 – 5x – 6 ________________________________________________________________________

(x +2) (x – 3) ________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

b) x2 – 6x + 8 ________________________________________________________________________

(x + 2) (x + 4) ________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

c) x2 + 9 ________________________________________________________________________

(x + 3) (x + 3) ________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

d) x2 – 10 ________________________________________________________________________

(x – 5) (x + 5) ________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

e) 6x2 + 7x – 3 ________________________________________________________________________

(x + 9) (x – 2) ________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

f ) 5x2 + 8x+12 ________________________________________________________________________

(5x + 2) (x + 6) ________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________



Evaluating the Error:factoring Sample

The factoring problems below are done incorrectly. Factor them correctly (if factoring is possible), and then explain what the students may have been thinking when they made the mistakes they did.

a) x2 – 5x – 6 ________________________________________________________________________

(x +2) (x – 3) ________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

b) x2 – 6x + 8 ________________________________________________________________________

(x + 2) (x + 4) ________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

c) x2 + 9 ________________________________________________________________________

(x + 3) (x + 3) ________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

d) x2 – 10 ________________________________________________________________________

(x – 5) (x + 5) ________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

e) 6x2 + 7x – 3 ________________________________________________________________________

(x + 9) (x – 2) ________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

f ) 5x2 + 8x+12 ________________________________________________________________________

(5x + 2) (x + 6) ________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

x2 – 5x – 6(x + 1)(x – 6)

x2 – 6x + 8 (x – 2)(x – 4)

x2 + 9Not factorable

x2-10Not factorable over rational numbers, although (x – M10)(x + M10) is used in higher math.

6x2 + 7x – 3 6x2 + 9x – 2x – 33x(2x + 3) – 1 (2x + 3)(3x – 1)(2x + 3)

5x2 + 8x + 12 Not factorable

The students broke up the 6 into factors correctly, but they added the factors 2 and 3 to get 5, when they should have subtracted the factors 1 and 6 to get 5.

The student broke up the 8 and added the factors 2 and 4 to get 6, but they ignored the negative in front of the 6.

The student attempted to use difference of two squares, but there is an addition sign, not a subtraction sign. It is therefore not factorable.

The student recognized something that resembles difference of two squares but, not being able to find the square root of 10, simply divided 10 in half.

The student knew to multiply the 6 and 3 to get 18, and then to break it up to get that the factors 2 and 9 subtract to 7. But then they reverted back to the short method and simply put the factors 2 and 9 in the final answer.

The student ignored the 5 in front of the x2, factored the 12, and then used the factors 2 and 6 to add to 8. They then noticed the 5, and stuck it in.


4.3 Evaluating the Method

Topic

• Determining the best mathematical method

Rationale

Determining which mathematical method to utilize is a critical skill that students need to master over time. There are many occasions when identical solutions are found using different methods, but one method is easier to utilize than the other. This activity shows students how to evaluate the structure of various algebra problems and determine the best way to proceed.

Objectives

Students will:

• Proof mathematical arguments

• Determine the best method for finding a solution

Timeline


WICR Strategies


• Inquiry

• Collaboration


• Student Handout 4.3a “Evaluating the Method: Systems of Linear Equations”

• Teacher Reference 4.3b “Evaluating the Method: Systems of Linear Equations Sample”

• Student Handout 4.3c “Evaluating the Method: Quadratic Functions and Equations”

• Teacher Reference 4.3d “Evaluating the Method: Quadratic Functions and Equations Sample”

8Ching, K., Gill, T. (2003). Algebra Tutorial: A Guide for AVID Coordinators and Tutors (pp. 68-69). San Diego, Ca: AVID Press


Instructions

• Pass out Student Handout 4.3a “Evaluating the Method: Systems of Linear Equations.”

• Have students individually identify the best method for solving the problem.

• Have students work with an elbow partner to justify why that is the best method in the right column.

• Have students share out their responses as a whole class.

• Use Teacher Reference 4.3b “Evaluating the Method: Systems of Linear Equations Sample” to help determine the reasoning for choosing that method.

• Use the same instructions as those for using Student Handout 4.3a “Evaluating the Method: Systems of Linear Equations” and Teacher Reference 4.3b “Evaluating the Method: Systems of Linear Equations Sample.”


• When will you be covering:

Combining like terms?

(Generally, first quarter)

Factoring quadratics?

(Generally, third quarter)



Evaluating the Method:Systems of Linear EquationsDetermine which method (graphing, substitution, or elimination) would be best suited to determine the point of intersection for the following:

a) y = 5x – 4 y = 3x + 2 Best Methods: ________________________________________________________________________ Why: ________________________________________________________________________ ________________________________________________________________________ b) 3x – 4y = 7 5x + 4y = 9 Best Methods: ________________________________________________________________________ Why: ________________________________________________________________________ ________________________________________________________________________ c) y = 3x – 1 2x – 3y = -18 Best Methods: ________________________________________________________________________ Why: ________________________________________________________________________ ________________________________________________________________________ d) y = -2x + 1 y = 3x – 8 Best Methods: ________________________________________________________________________ Why: ________________________________________________________________________ ________________________________________________________________________ e) y = 5x – 3 y + 2x = 10 Best Methods: ________________________________________________________________________ Why: ________________________________________________________________________ ________________________________________________________________________ f ) 2x + 3y = 6 5x – 4y = 4 Best Methods: ________________________________________________________________________ Why: ________________________________________________________________________ ________________________________________________________________________


Graphing or substitution The equations are set up in a y = mx + b format, which makes the two equations ideal for either graphing the two lines or doing a direct substitution.

EliminationThe addition of the first and second equations will result in the elimination of the y variable.

SubstitutionThe y variable has already been isolated, which allows for an easy substitution into the second equation.

SubstitutionBoth equations are set up into a y = mx +b, but the point of intersection is a decimal, so graphing is inadvisable.

SubstitutionThe first equation has the y isolated making substitution easy.

EliminationThese equations will require manipulation, but the elimination method would work best, since isolating an x or y would be difficult.


Evaluating the Method:Systems of Linear Equations SampleDetermine which method (graphing, substitution, or elimination) would be best suited to determine the point of intersection for the following:

a) y = 5x – 4 y = 3x + 2 Best Methods: ________________________________________________________________________ Why: ________________________________________________________________________ ________________________________________________________________________ b) 3x – 4y = 7 5x + 4y = 9 Best Methods: ________________________________________________________________________ Why: ________________________________________________________________________ ________________________________________________________________________ c) y = 3x – 1 2x – 3y = -18 Best Methods: ________________________________________________________________________ Why: ________________________________________________________________________ ________________________________________________________________________ d) y = -2x + 1 y = 3x – 8 Best Methods: ________________________________________________________________________ Why: ________________________________________________________________________ ________________________________________________________________________ e) y = 5x – 3 y + 2x = 10 Best Methods: ________________________________________________________________________ Why: ________________________________________________________________________ ________________________________________________________________________ f ) 2x + 3y = 6 5x – 4y = 4 Best Methods: ________________________________________________________________________ Why: ________________________________________________________________________ ________________________________________________________________________



Evaluating the Method:Quadratic functions and EquationsDetermine which method (factoring, completing the square, quadratic formula, or graphing) would be best suited to factor the following.

a) x2 – 3x + 2 = 0 All Methods: ________________________________________________________________________ Best Methods: ________________________________________________________________________ Why: ________________________________________________________________________ ________________________________________________________________________

b) x2 – 25 = 0 All Methods: ________________________________________________________________________ Best Methods: ________________________________________________________________________ Why: ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________

c) x2 – 30 = 0 All Methods: ________________________________________________________________________ Best Methods: ________________________________________________________________________ Why: ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________

d) 2x2 + 3x + 12 = 0 All Methods: ________________________________________________________________________ Best Methods: ________________________________________________________________________ Why: ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________

e) x2 – 4x + 19 = 0 All Methods: ________________________________________________________________________ Best Methods: ________________________________________________________________________ Why: ________________________________________________________________________ ________________________________________________________________________

f ) 2x2 + 3x + 1 = 0 All Methods: ________________________________________________________________________ Best Methods: ________________________________________________________________________ Why: ________________________________________________________________________ ________________________________________________________________________



Evaluating the Method:Quadratic functions and Equations SampleDetermine which method (factoring, completing the square, quadratic formula, or graphing) would be best suited to factor the following.

a) x2 – 3x + 2 = 0 All Methods: ________________________________________________________________________ Best Methods: ________________________________________________________________________ Why: ________________________________________________________________________ ________________________________________________________________________

b) x2 – 25 = 0 All Methods: ________________________________________________________________________ Best Methods: ________________________________________________________________________ Why: ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________

c) x2 – 30 = 0 All Methods: ________________________________________________________________________ Best Methods: ________________________________________________________________________ Why: ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________

d) 2x2 + 3x + 12 = 0 All Methods: ________________________________________________________________________ Best Methods: ________________________________________________________________________ Why: ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________

e) x2 – 4x + 19 = 0 All Methods: ________________________________________________________________________ Best Methods: ________________________________________________________________________ Why: ________________________________________________________________________ ________________________________________________________________________

f ) 2x2 + 3x + 1 = 0 All Methods: ________________________________________________________________________ Best Methods: ________________________________________________________________________ Why: ________________________________________________________________________ ________________________________________________________________________

Factoring, completing the square, quadratic formula, graphing Factoring This trinomial is easily factorable.

Factoring, undoing, quadratic formula, graphing Factoring or undoing If the student recognizes difference of two squares, the polynomial is easily factorable. Otherwise, it is easy to move the 25 to the other side and take the square root of both sides.

Undoing, quadratic formula, graphing Undoing It is simple to move the 30 to the other side and take the square root of both sides.

Completing the square, quadratic formula, graphing Quadratic formula Since this trinomial is not factorable, and the roots are not real, the only ways to solve it are by completing the square and using the quadratic formula. The quadratic formula is easier in this problem because to complete the square, we would first have to divide the whole equation by 2 which would give us a fraction.

Completing the square, quadratic formula Completing the square Completing the square would be easy to do because of the implied 1 in front of the x2.

Factoring, completing the square, quadratic formula, graphing Factoring or graphing Although the trinomial has a 2 in front of the x2 term, it still is an easy factoring problem. For those who do not like to factor, this would be a good graphing problem.


4.4 The Golden MistakeTopic

• Reviewing missed quiz or test questions

Rationale

Students often receive a test back, and never take the opportunity to review and reflect on the feedback, especially the mistakes! These are golden opportunities for students to identify their errors and learn from them. Additionally, the practice of reflecting on a completed problem will very likely condition the students to reflect on future problems after they have “found the answer.”

Objectives

Students will:

• Reflect on recent test performance

• Identify mathematical errors

• Describe the mistakes that were made

Timeline

• 5 – 30 minutes (depending on the number of problems)

WICR Strategies


• Inquiry

• Collaboration


• A collection of your AVID students’ last quiz or test

• Yellow highlighters

Instructions

• After students have received a quiz or test from their math class, ask students if you may borrow the test for the night.

• Photocopy sample pages where students have made mistakes.

• Cut out the missed problems and student work.

• Paste several problems onto the left hand column of Student Handout 4.4a “The Golden Mistake.”

• Pass the sheet out to your students.

• Have students highlight the mistakes on the student’s work.

• Have students write a description of the errors that were made in the middle column.

• Have students correctly work out the problems on the far right column.


• On your last quiz/test, did you

notice any particular questions

many students missed?



Name _________________________

The Golden Mistake Date __________________________

STUDENT WORK(INCLUDING THE ERROR) DESCRIpTION OF MISTAKE CORRECTED WORK


4.5 A Picture Is Wortha Thousand Word Problems

Topic• Creating non-linguistic representations of mathematical situations

Rationale

Ask any student in the United States what they dislike most about math, and near the top of the chart would be word problems. The fear of word problems often comes from the student’s inability to conceptualize the information into a meaningful context. This exercise in drawing pictures of the problems focuses on a fundamental skill for the student to use in contextualizing a problem, a skill that is often overlooked but can actually make word problems … fun. In addition, it gives students another tool in their toolbox as they approach a mathematical problem. This is an ideal sponge activity or end-of-the-week problem. The idea is to encourage wonderful pictures, and although you need the students to record all critical pieces of information, you are focusing on the non-linguistic representation, not the mathematical mechanics.


• When is the next time you are covering word problems?

It is highly advisable to introduce these word problem

pictures in coordination with their algebra class, ideally

a day or two before the same type of word problems will

be covered in math class.

• What type of word problems will you be covering?

Some of the more common types of problems are rate,

mixture, investment, and system of equations (some

commonly used systems of equation problems involve

coins or cost of tickets at a game).

• Could you give me some examples of word problems that

you will be using in class?

It is definitely preferable to use the word problems that

the math teacher provides rather than the samples that

follow. This will allow your AVID students to preview

a problem in your class, and follow up by learning the

algebra in their math class.



• Use reading strategies to determine critical pieces of information

• Use non-linguistic representation to depict a mathematical problem

• Develop a prediction or hypothesis about a possible solution

Timeline• 15 – 20 minutes per word problem


Transferring a written word problem into an appropriate visual representation

Justifying predictions or hypotheses

• Inquiry

Investigation of the connection between linguistic and non-linguistic representation of a mathematical problem

• Reading

Practicing highlighting and text annotation skills

Using signal words to identify text patterns and structure

Materials/preparation• Student Handout 4.5a “A Picture Is Worth a Thousand Word Problems – Gazelle”

• Student Handout 4.5b “A Picture Is Worth a Thousand Word Problems – Trains”

• Student Handout 4.5c “A Picture Is Worth a Thousand Word Problems – Trees”

• Teacher Reference 4.5d “A Picture Is Worth a Thousand Word Problems – Trees”

• Colored pens/pencils

• Highlighters

• Look over “Highlighting and Annotating a Math Text” (Write Path II: Mathematics, pg. 180)

Instructions• Distribute any of the Student Handouts “A Picture is Worth a Thousand Word Problems.” It is a good idea to discuss the best order with the math teacher

• Ask students to read through the word problem and highlight (or underline) critical pieces of information.

• Have students focus on their drawing, rather than the “complicated” math problem. However, tell the students that their drawings will need to accurately depict the situation that the math problem describes and they must contain all of the “critical pieces of information.”

One way to phrase this is, “I should be able to look at your picture and know everything that I need to solve this problem.”

• Have students predict the answer to the problem and justify their answers to a neighbor. It is important here that you do not celebrate someone who successfully uses algebra. This is an exercise of logic rather than following steps to produce an answer.

• Optional: Conclude this activity with a “gallery walk” or “carousel.”


A Picture Is Worth a Thousand Word Problems!

GazelleA gazelle is grazing on the African plains, and suddenly spots an approaching cheetah. The gazelle starts running away at 15 meters per second. The cheetah is 120 meters behind and starts running at a rate of 30 meters per second to catch the gazelle. How long will it take the cheetah to catch the gazelle?

STUDEnT HAnDOUT 4.5a Directions: Read the following word problem and highlight the key pieces of information you would need to solve the problem. Draw as accurate and descriptive a picture as possible and write in all the key informational pieces you would need to solve this problem. Once you are satisfied that you have the perfect picture, use the drawing (and information) to predict an answer.

Your Prediction: ________________

Your Reasoning:



TrainsJill boards a train in Sacramento heading towards San Diego at a rate of 70 miles per hour. At the same time, Catherine boards a train heading (in the opposite direction) towards Seattle, Washington. If the trains are 375 miles apart after three hours, what is the speed of the train headed towards Seattle?

STUDEnT HAnDOUT 4.5bDirections: Read the following word problem and highlight the key pieces of information you would need to solve the problem. Draw as accurate and descriptive a picture as possible and write in all the key informational pieces you would need to solve this problem. Once you are satisfied that you have the perfect picture, use the drawing (and information) to predict an answer.


Your Reasoning:



TreesA five foot tall boy is standing outside on a sunny day, and notices that he is casting a shadow that is 3 feet long. He then notices that a tree is casting a shadow that is 15 feet long. How tall would the tree have to be in order to cast the 15 foot long shadow?

STUDEnT HAnDOUT 4.5c Directions: Read the following word problem and highlight the key pieces of information you would need to solve the problem. Draw as accurate and descriptive a picture as possible and write in all the key informational pieces you would need to solve this problem. Once you are satisfied that you have the perfect picture, use the drawing (and information) to predict an answer.


Your Reasoning:



TreesA five foot tall boy is standing outside on a sunny day, and notices that he is casting a shadow that is 3 feet long. He then notices that a tree is casting a shadow that is 15 feet long. How tall would the tree have to be in order to cast the 15 foot long shadow?

STUDEnT HAnDOUT 4.5dDirections: Read the following word problem and highlight the key pieces of information you would need to solve the problem. Draw as accurate and descriptive a picture as possible and write in all the key informational pieces you would need to solve this problem. Once you are satisfied that you have the perfect picture, use the drawing (and information) to predict an answer.


Your Reasoning:

25 feet

The boy is a little less than twice the size of his shadow. So if we double the length of the trees shadow we would get thirty feet. Since it is a little less than double, I am predicting 25 feet.


4.6 Math Dictionary Topic

• Developing the academic vocabulary of mathematics

Rationale

Clear understanding of the academic language utilized in mathematics is a foundational skill. Providing students with opportunities to master new vocabulary and concepts is critical. The “Math Dictionary” activity provides students with a variety of representations and engages students using a range of different modalities.

Objectives

Students will:

• Increase the familiarity and fluency of their mathematical vocabulary

• Create a personalized mathematics dictionary

Timeline

• 10 – 20 minutes for students to set up their personal math dictionary and make the first entries

WICR Strategies


• Collaboration

• Reading to Learn


• Student Handout 4.6a “Create Your Own Math Dictionary”

• Index cards

• 1.5-inch metal ring

• Hole puncher

• Optional:

Dictionaries

Glossary from text

Spanish/English dictionaries

Word wall

9Donohue, J., Gill, T. (2009). 4.10: Math Dictionary. The Write Path I: Mathematics (p. 194-196). San Diego, Ca: AVID Press


Instructions

• Distribute index cards and metal rings.

• Distribute Student Handout 4.6a “Create Your Own Math Dictionary.”

• Ask students to create a cover card for their dictionary. The card should include their name and class period. Allow students time to creatively decorate their cover cards.

• Ask students to hole-punch their cover card and several blank cards in the top left-hand corner.

• Show students how to attach the hole-punched cards to the metal ring or to a ring in their notebook.

• Provide students with a list of vocabulary words and concepts.

• Encourage students to incorporate humor into illustrations and definitions. Humor, color, and creativity will improve memory.

• Model the creation of a vocabulary card. Use creativity, humor, and color.

• Ask students to work collaboratively or individually to construct their dictionary cards.

• Encourage students to include time in their planning agenda to study and construct cards weekly.

• Ask students to alphabetize their cards for easy access.

• Display exemplar cards on a class word wall.

• Collect cards and grade them once every one to two weeks.


• Would you be able to provide me with

a list of math vocabulary words for the

current chapter you are studying?

When this is not possible, the math

vocabulary words for the chapter are

generally at the beginning of the

chapter.


Create Your Own Math Dictionary


Parallel LinesParallel Lines are two lines in the same plane that do not intersect.

The bike lanes on each side of my street are parallel lines and do not intersect.

A parallel line will have the same slope of the line to which it is parallel.The parallel line can be found by using the general equation y = mx + b, and substituting in the known slope and the x and y values of the point it will pass through.

Cover Card

Lynn Garza

Geometry

Period 4

Front of Card

Back of Card

Parallel Lines



151Chapter 5: Higher-Level Math Reflections and Summaries

5ChapterHigher-Level Math Reflections and Summaries

Our language is the reflection of ourselves. A language is an exact reflection of the character and growth of its speakers.

-Cesar Chavez

Summarizing and reflecting are two very different sides of the same important coin. The act of summarizing focuses on the ability to cull out the most important information from a lesson and develop a coherent blending of that information. The summarization process occurs within the 24 hours following a lecture and ties to the Cornell note-taking process. The ability to gather that information and determine main ideas is a skill requiring both time and guidance.

In contrast, reflecting is the “What?”, “So what?”, and “Now what?” that directly follow a learning activity. It is a metacognitive activity that allows the learner to assess how they learned, what they still need to learn, and how they will learn it. Through this constant reflective process, your students will become better thinkers and problem solvers.


Summary Reflection

What • Condenses and paraphrases main point and key information of lecture, text, video

• Gives the GIST, main ideas presented in notes and questions

• Addresses the Essential Question of the lesson

• Includes important content and lesson-based vocabulary

• Addresses the “What?”, “So what?”, and “Now what?” of the learning

• Relies on purposeful thinking, reasoning, and examining one’s own thoughts/feelings and experiences to process learning.

• Includes important content and lesson-based vocabulary

Where • On Cornell notes • On learning logs

• On Tutorial Request Forms

Why • To highlight the major points from the original text and to process information from the notes

• By synthesizing information from the text and notes, the students internalize the learning and help move it to long-term memory

• To connect learning to prior learning, self, or real world

• To find solutions and draw conclusions, resulting in a better understanding of content/information

• It is not our experiences we learn from, but rather reflecting on the experience

When • Within 24 hours • Immediately following the learning, experience, or activity


5.1 GIST Summary10

Topic

• Creating summaries that capture critical content

Rationale

Often, students can effectively describe what occurred during a class, but are then unable to cull the important concepts into a well-constructed summary. The GIST method helps students determine key concepts, and synthesize the ideas into an effective summary.

Objectives

Students will:

• Identify key concepts

• Synthesize thinking into a summary

Timeline


WICR Strategies


• Inquiry

• Reading


• Cornell notes

• Colored pen or highlighter

• Student Handout 5.1a “GIST Summary”

10Donohue, J., Gill, T. (2009). GIST. The Write Path II: Mathematics (p. 23). San Diego, Ca: AVID Press


Instructions

• Have students take out a page of math Cornell notes.

Copying one student’s notes and using it as a handout would help during the modeling stages of this activity.

• Provide students 5 to 10 minutes to review the page of notes and underline the most important concepts from the lesson.

• Have students focus on aspects the teacher said were important, areas of confusion, and key differences between problems.

• Once all students have completed the underlining, have students discuss with an elbow partner for 3 to 5 minutes what they underlined and why.

• Provide pairs 5 minutes to generate a 20-word summary that integrates the important concepts that they underlined.

• Select students share their GIST summaries with the entire class.

Moving Towards Independent GIST Writing:

• Encourage students to use the same method of 5 to 10 minute review and underlining key ideas, and creating 20-word summaries.

• It is helpful to devote a portion of their weekly binder check grade to the quality of their summarization.


• (If you are using a model page of Cornell

notes) Could you review this page of

Cornell Notes and underline the pieces of

information that you believe to be critical?

Then write a sample 20-word summary

using those concepts?

Read the sample GIST (generated from

the math teacher) following the student

share out.



GIST SummaryInclude the following in a GIST summary:

A. Explain what you are summarizing.

B. Describe the concept you are learning about.

C. Highlight or list five key phrases/words that encompass what the notes are about.

D. Use your five key phrases/words to write three to five complete sentences summarizing your notes.

E. Check your summary to be sure the details support the topic and the concept in your notes.

Topic: _______________________________________________________________________________________

_____________________________________________________________________________________________

Concept: ____________________________________________________________________________________

_____________________________________________________________________________________________

Highlight or list five key phrases/words:

1. ________________________________________________________________________________________

________________________________________________________________________________________

2. ________________________________________________________________________________________

________________________________________________________________________________________

3. ________________________________________________________________________________________

________________________________________________________________________________________

4. ________________________________________________________________________________________

________________________________________________________________________________________

5. ________________________________________________________________________________________

________________________________________________________________________________________

Write three to five sentences using the key phrases/words:


5.2 10-2-2 note-Taking Structure

Topic

• Creating ongoing summaries

Rationale

Having students continually reflect upon previous learning helps them begin to process information in manageable chunks of information. Occasionally, teachers feel the pressure of trying to get as much information to the student as quickly as possible. As a result, lectures can often turn into runaway freight trains that can leave behind some of the students in the room. This method breaks longer periods of lecture into more manageable periods of information delivery and information processing.

Objectives

Students will:

• Process information during lectures

Timeline

• Length of lecture

WICR Strategies


• Inquiry

• Collaboration

• Reading


• Cornell notes

• Pen/pencil

• Teacher Reference 5.2a “10-2-2 Note-Taking Structure”


Instructions

• Present information for 10 minutes (lecture/presentations).

• Have students record information in Cornell note format.

• After 10 minutes, have students share notes with an elbow partner for 2 minutes.

Have students share notes, revise information, and add any information that they might have missed.

• Continue the lecture or presentation following the pattern of 10 minutes of lecture, 2 minutes of sharing, 2 minutes of independent summarization.


• Do you think a 10-2-2 lecture

style would help your students

retain information?

• Is it something that you could

utilize in your classroom?


10-2-2 note-Taking Structure

10-2-2 Structure & Rationale

• The structure involves the following:

10 minutes: Presenting information/note-taking

2 minutes: Processing information

2 minutes: Summarizing information

• Allows students the necessary time to process information and concepts presented in whole-group instruction

• The structure allows for:

Greater retention of information

Improvement in the quality of notes, question, and summaries

10 Minutes: Whole-Group Instruction

• The instructor lectures/presents information or gives and audio-visual presentation for 10 minutes while the students take Cornell notes.

• Encourage students to use abbreviations and short-cuts while taking notes.

2 Minutes: partners/Small Groups

• The instructor then pauses for 2 minutes while the students take time to process the information by working collaboratively in partners/small groups to do the following:

Share notes

Revise/refine notes

Fill in gaps in notes

Clarify information/concepts presented

Create questions on the left side

• During this time, students are not allowed to ask the instructor questions; students should rely on the support of peers to assist them in processing the information.

2 Minutes: Independently

• The students then take 2 minutes to individually and silently process the information and create a one-sentence summary to be placed across the page just below the chunk of notes.

• The teacher may choose to have students share out their summaries as a way to check for understanding.

Repeat the process • Repeat the process until all information is presented.

Last 5 Minutes of Class: Whole Group

• Reserve the last 5 minutes of the period for the students to interact with the teacher. Students can ask questions to:

Resolve unanswered questions

Get clarification about information presented

Sort out misconceptions/gaps



5.3 Higher-Level Essential Questions and Summaries

Topic

• Creating higher-level questions to support higher-level summaries

Rationale

While strategies such as exit ticket summaries or 10-2-2 summaries are excellent methods to ensure that summaries are completed and that students have adequate in-class time to summarize, the ultimate goal is to enable students to reflect on the information and do the work independently. The strategy described here can help guide students to a greater sense of self-sufficiency.

Objectives

Students will:

• Reflect on their notes using higher levels of thinking

Timeline

• 3 – 5 minutes (at home)

WICR Strategies


• Inquiry


• One page of Cornell notes

• A timer/clock


Instructions

• Use the instructions for “Essential Questions Guiding the Summarization” to help students perform an at-home quickwrite.

Creating Essential questions

• Fill in the chapter number and topics for each lesson in a math textbook on Student Handout 5.3a: “Higher-Level Esssential Question Template for Math Teachers” (e.g., Chapter 2 Properties of Real Numbers).

• Ask a math teacher with whom you have established a relationship to create some higher-level Essential Questions for each chapter.

Ask the math teacher to turn the lesson’s objective into a question that a student could write a response to for a few minutes.

Have the math teacher create questions that focus on comparing two different concepts within a lesson, how yesterday’s lesson applies to today’s learning, how today’s concept relates to other concepts, or how changing coefficients, signs, or process for solving will affect the solution.

Use Teacher Reference 5.3c “Sample Higher-Level Essential Questions.”

• Copy and distribute these Higher-Level Essential Questions to your AVID students with the instructions that they should use these questions only if their own math teacher is not utilizing the Essential Question in class.

• File these questions for future use.


• Could you help me generate higher-level Essential Questions for each of the (# of sections) in this chapter?

I am hoping to create questions that help students focus on:

Comparing the two most important skills within a section.

Applying the previous lesson to the current lesson (e.g., Lesson 2.3 to 2.4).

Applying a similar example to a more complex example (e.g., in Lesson 4.3, how can we apply the process to solve Sample 1 to Sample 5?).

Finding relationships between concepts.

Synthesizing various concepts into a single process.




Higher-Level Essential Question Template for Math Teachers

Chapter _________

____ .1 ___________________________________________


_____________________________________________________________________________________________

____ .2 ___________________________________________


_____________________________________________________________________________________________

____ .3 ___________________________________________


_____________________________________________________________________________________________

____ .4 ___________________________________________


_____________________________________________________________________________________________

____ .5 ___________________________________________


_____________________________________________________________________________________________

____ .6 ___________________________________________


_____________________________________________________________________________________________

____ .7 ___________________________________________


_____________________________________________________________________________________________



Higher- Level Essential Question Template for Math Teachers

Chapter 2 properties of Real Numbers

2.1 Using Integers and Rational Numbers


_____________________________________________________________________________________________

2.2 Adding Real Numbers


_____________________________________________________________________________________________

2.3 Subtracting Real Numbers


_____________________________________________________________________________________________

2.4 Multiplying Real Numbers


_____________________________________________________________________________________________

2.5 Apply the Distributive property


_____________________________________________________________________________________________

2.6 Divide Real Numbers


_____________________________________________________________________________________________

What is the difference between finding the opposite of a number

and the absolute value of a number?

Generalize the process for adding two fractions together.

What do you notice about subtracting two negative numbers and

subtracting one positive and one negative number?

How is multiplying one negative number related to multiplying

two negative numbers; and three negative numbers?

How can distributing a constant be connected to combining like terms?

How is dividing by fractions related to multiplying by fractions?



Sample Higher-Level Essential Questions

How does finding a parallel line passing through a point contrast with finding a perpendicular line passing

through a point?

(Comparison)

What similarities do you see between adding/subtracting expressions with variables and adding/subtracting

expressions with radicals?

(Synthesis)

What skills for solving a multiple-step equation would you use to solve inequalities?

(Application)

How can you apply the method for solving equalities to help solve absolute value equalities?

(Justify)

How do you know whether to solve a system of equations using substitution or elimination?

(Predict)

What would happen to a line if we changed the slope from a positive coefficient to a negative one?

(Metacognition) (After a review day)

Which concepts am I still struggling with, and what will I do to find support?


5.4 Higher-Level Reflections During Tutorial11

Topic

• Generating higher-level reflections

Rationale

Teachers may often find themselves frustrated at the low level of thinking that occurs following a tutorial session. Students generally focus on a recall of events or lists of steps to solve an equation. This activity helps tutors/teachers generate writing prompts that lead the students to deeper levels of thinking as they reflect upon their learning.

Objectives

Students will:

• Reflect upon their learning and understanding

• Create written reflections of tutorial sessions using sophisticated prompts

Timeline


WICR Strategies


• Inquiry

• Reading


• Tutorial Request Forms (TRF)

• Cornell notes

• Pen/pencil

• Teacher Reference 5.4a “Higher-Level Reflection”

11Donohue, J., Gill, T. (2009). Higher-Level Reflections. The Write Path II: Mathematics (p. 24). San Diego, Ca: AVID Press


Instructions

• Before the beginning of class (or in an after school meeting/tutor training), discuss the need for tutors to generate a higher-level quickwrite prompt at the end of the tutorial.

• Approximately 10 minutes before the end of class, have the tutor generate a higher-level prompt based on the information covered during the tutorial session.

• Ideas that the tutor might consider:

How does the concept of _____________ compare with ______________?

Generate a real world sample that demonstrates the use of ______________.

What distinguishes the way we solve _____________ from solving ___________?

If the concepts of today’s tutorial were on a test, I believe the areas where I would struggle the most are_______.

Generalize the method to solve _________________ so that a younger grade (i.e., 5th grader if the students are 7th grade) would understand.

How does the concept of _____________ relate to _____________, which we covered during our previous tutorial session?


• As you are supporting today’s tutorial, could you identify some of the important concepts covered and create a writing prompt for the students?



Higher-Level Reflection Just as it is important to bring higher-level questions to the tutorial, it is equally important to write a higher-level reflection at the conclusion of the tutorial.

Costa’s Levels of Thinking

Level 1 Level 2 Level 3complete compare evaluatedefine contrast generalizedescribe classify imagineidentify sort judgelist distinguish predictobserve explain (why?) speculaterecite infer if/thenselect analyze hypothesize

forecast

Student Samples

Level 1 Reflection

Today I learned that the perimeter of a polygon is the sum of the lengths of all its sides. Since a rectangle has 4 sides, and the opposite sides of a rectangle have the same length, a rectangle with sides 5 cm and 8 cm would have a perimeter of 26 cm. When I write my answer to a perimeter problem, I need to remember to indicate the specific units I’m using. (Describe)

Level 2 Reflection

The perimeter of a polygon is the sum of the lengths of all its sides while the area of a figure measures the size of the enclosed region of the figure. Area is expressed as square units whereas perimeter is not. For example, the perimeter of a figure would be centimeters while the area would be described as square centimeters. If a polygon has sides that measure 5 cm and 8 cm, the perimeter (5 + 5 + 8 + 8) would be 26 cm while the area of the polygon (5 x 8) would be 40 square cm. (Compare and Contrast)

Level 3 Reflection

The perimeter of a polygon is the sum of the lengths of all its sides while the area of a figure measures the size of the enclosed region of the figure. Area is expressed as square units whereas perimeter is not. For example, the perimeter of a figure would be centimeters while the area would be described as square centimeters. If a polygon has sides that measure 5 cm and 8 cm, the perimeter (5 + 5 +8 + 8) would be 26 cm while the area of the polygon (5 x 8) would be 40 square cm. In my own life, I needed to know the perimeter of my poster paper for my science project when I was making a special border for it. My father asked me to help him calculate the area of our kitchen floor at home when he needed to find out how many tiles to buy. (Evaluate/Generalize)


169Chapter 6: fun with Mathematics

6Chapterfun with Mathematics

Mathematics should be fun.

-Peter J. Hilton

For some people, “doing math for fun” may very

well be the quintessential oxymoron. That could

be because there is a beauty to mathematics that,

sadly, many do not see. Most never think of using

math to paint a picture; few delight in searching

for a “more elegant proof,” or notice the beauty

of mathematics in nature, or its ability to see

strings of disparate events and create efficiency.

It is unclear whether those who have a good time

solving puzzles of logic will develop a love of math

or whether it’s those who love math in the first

place who tend to enjoy such puzzles. What is clear

is that using some of the engaging activities in this

next section will help your students examine and

discuss their reasoning for a solution, and help

them become more effective thinkers. It is a nice

bonus if they have a little fun while they are at it.


6.1 Algebra Aerobics

Topic

• Graphing functions and transformations for linear, quadratic, absolute value, square root, cubic, logarithmic, exponential and rational parent functions

Rationale

“Algebra Aerobics” provides an alternative, kinesthetic assessment and practice for students learning parent functions and their transformations. Visual monitoring will provide clear evidence of students’ mastery of graphs of functions.

Objectives

Students will:

• Learn parent functions and their transformations through a hands-on activity

Timeline


WICR Strategies

• Inquiry

• Collaboration


• (Optional) PowerPoint provided by AVID Center

• A list of several parent equations and the equations of the transformations

• Teacher Reference 6.1a “Algebra Aerobics”

12Donohue, J., Gill, T. (2009). Algebra Aerobics. The Write Path I: Mathematics (pp. 143-146). San Diego, Ca: AVID Press


Instructions

• When you sense a lull in attention or need to energize your students, have every student in the room stand up.

• Write the equations on an overhead/visual presenter and reveal them one at a time.

• Call out equations and have the students map out the equations using their arms like the figures on the Teacher Reference Sheet to demonstrate what the equation will look like.

• A version of the game “Simon Says” can be used for variety.

• Ask two or more student to work together to represent the same function. This will require that they discuss which quadrant they will represent and what the behavior of the function is in that quadrant.


• What types of functions (graphs) are you covering at the moment? (Linear, Parabolic, Absolute Value)

• Could you write down five or six equations you are working on and sketch a quick graph for me?


TEACHER REfEREnCE 6.1a (1 of 2)

Algebra Aerobics

y = x y = x + 1 y = x – 5

y = 2x y = 12 x y = -x


TEACHER REfEREnCE 6.1a (2 of 2)

Algebra Aerobics

y = x2 y = 3x2 y = -x2

y = |x| y = x3 y = ± Mx


6.2 Bagel, Pico, fermi

Topic

• Using logic to determine a number

Rationale

The use of logic is critical in many areas of mathematics. Geometric and algebraic proofs exemplify the need for utilizing information and justifying the next step. Like many of the brain’s abilities, logical reasoning is strengthened through a variety of brain exercises. Bagel, Pico, Fermi is a logic game that forces students to synthesize information to make logical predictions about potential solutions.

Objectives

Students will:

• Combine pieces of data to predict a number

• Justify their prediction

Timeline


WICR Strategies


Write a reflection of current knowledge

Write a prediction about possible solutions

Justify what you know about the mystery number

• Inquiry

Evaluate the validity of potential solutions

Use information to determine logical predictions

Determine potential solutions

• Collaboration

Work with a partner to determine current knowledge

Discuss potential solutions and what solutions can be eliminated



• Chalkboard or overhead

• Post-it note or index card

Instructions

• Pick a “secret” two digit number (the digits can’t repeat and the number can’t start with a zero). Write the number down, but do not tell anyone the number.

• Draw three columns on the board

The first column is labeled “Guess.”

The second column is labeled “Digits.”

The third column is labeled “Places.”

• Ask students to guess a number.

• Write the guess in the first column.

Write down how many of the digits are correct, regardless of whether or not they appear in the same place as in your number.

Write down how many of those correct numbers are in the correct place.

• After three or four guesses, have the student discuss with a partner what information they know, and using that information come to a conclusion about a logical prediction for the number.

• After the number is guessed, have students write a few sentences reflecting upon if and how they could have guessed the solution earlier.

Sample: (If the number is 38)

Trial Number Guessed Digits Place1st Guess 49 0 0

Trial Number Guessed Digits Place1st Guess 49 0 02nd Guess 13 1 0

Trial Number Guessed Digits Place1st Guess 49 0 02nd Guess 13 1 03rd Guess 37 1 1


**Optional piece if you want to have your AVID students write a proof.

What I know Why I know it

The number does not have a 4 or a 9 in it. When I guessed 49 I had 0 numbers and 0 places

The number is either _1 (something one) or 3_.

When I guessed 13 I had one digit right but it was in the

wrong place. So either the 1 is in the ones place or the 3 is

in the tens place.

The number has to be 3_. When we guessed 37 we had one number that was right

and it was in the right place. The number could not be _7

because we had one number correct in the last guess.


These questions are designed for if you have geometry

students and would like to help support the concept of

geometric proof. You do not need to use the proofing

questions in order for the game to be valuable.

• (If students are enrolled in a geometry course.) How

do you want your students to show proofs? (Some use

T-Charts other balloon proofs.)

• Could you help me convert a mock game into a proof?


6.3 Calculating Your GPA

Topic

• Tracking students’ GPAs over time

Rationale

Having students record and monitor their progress over the course of multiple weeks is extremely valuable because it:

• Gives students the opportunity to monitor their own progress, celebrate their successes, and alert them if their grades begin to fall.

• Encourages students to set goals for themselves and discuss these goals with their parents or guardians.

• Allows you to monitor each student individually and look for trends or sudden changes in a student’s success.

• Alerts you to potential difficulties and provides a door to discuss what might be causing sudden dips in performance.

• Generates an informative item for you to insert in Student Portfolios and your school’s certification binder.

Objectives

Students will:

• Monitor their GPAs over the course of a quarter

• Generate, track, and evaluate data points over a three-month period

Timeline

• 10 – 15 minutes in class to learn, then 10 – 15 minutes at home weekly or bi-weekly for three months

WICR Strategies


• Inquiry


• Student Handout 6.3a “AVID Academic Report”

• Student Handout 6.3b “Charting My GPA”

• Teacher Reference 6.3c “Charting My GPA Sample”

Instructions

• Have students collect their grades in each of their academic classes once a week (or every other week).

• Have students record their progress on Student Handout 6.3b “Charting My GPA.”


• (Only if this activity doesn’t make sense to you) Could you help me work through the math of students finding averages and graphing data?


STUDEnT HAnDOUT 6.3a (1 of 2)

TOTAL POINTS ___________ NAME: ___________________________________

Week of: _________________________________ AVID Academic Report

period Subject Goal Grade

Current Grade

Teacher Signature Comments

0

1

2

3

4

5

6

GOAL GPA: ____________ ACTUAL GPA: ____________ pOINTS: _______

ACTION pLAN:For each class, assess what you need to do this week by reflecting on what transpired last week. Set a goal that you can accomplish.

Period 0 Plan/Goal:

Period 1 Plan/Goal:


Period 2 Plan/Goal:

Period 3 Plan/Goal:

Period 4 Plan/Goal:

Period 5 Plan/Goal:

Period 6 Plan/Goal:

Student Signature: ____________________________________________

Parent/Guardian Signature: _____________________________________ pOINTS: _______



Inst

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4.5

4.0

3.5

3.0

2.5

DAT

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GRADE pOINT AVERAGEN

ame _

____

____

____

____

____

____

____

____

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____

____

____

____

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5.0

4.5

4.0

3.5

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2.5

DAT

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GRADE pOINT AVERAGEN

ame _

____

____

____

____

____

____

____

____

_

Qua

rter

___

____

____

____

____

____

____

____

_

9/17

9/2

10

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10/8

10

/15

10/2

2 10

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11/5

11

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11/1

9 11

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Char

ting

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Sam

ple


6.4 Inquiry Cube

Topic

• Inquiry-based activity involving the slopes of perpendicular lines

Rationale

The AVID math classroom should provide a variety of opportunities for students to experience the Inquiry method of teaching and learning. The “Inquiry Cube” is a fun, collaborative way to engage students in the Inquiry method while formulating theories and drawing connections between information.

Objectives

Students will:

• Ask each other questions to determine what should be on the blank side of the “Inquiry Cube”

• Practice inquiry as a study method

• Access prior knowledge of slope and perpendicular lines to solve the puzzle

Timeline

• One 30 – 40-minute class period to find the missing side of the “Inquiry Cube”

WICR Strategies

• Inquiry

• Collaboration


• Student Handout 6.4a “Inquiry Cube”

• Cornell note paper

• Prepare the cubes before class or have students cut out the net and tape the cubes together.

• Prepare some guiding questions to ask groups that are not making progress on the problem. Sample: “What form of a line might help you to recognize certain characteristics of the line?”


Instructions

• Cut out Student Handout 6.4a “Inquiry Cube” and tape together.

• Arrange the class into collaborative groups of four students.

• Tape the side with Francene to the desk.

• Distribute Cornell note paper or ask students to create their own.

• Each group should record its ideas, questions, and thoughts that arise while completing the activity.

• Assign someone in the group to record (on Cornell note paper) the questions, ideas, and thoughts that came up as they worked on the activity. This will help them in summarizing their thinking at the end of the activity.

• Explain that the task is to find what should be on the missing side of the “Inquiry Cube.” The group needs to identify the color, top right number, bottom left number, and the name on the taped side.

• Circulate as students are working to answer clarifying questions or to ask guiding questions.

• If students fail to make initial progress, you can prompt them with a question such as, “Do you notice anything about the opposite faces of the cube?”

• When students have a solution, ask them to write a reflection in their Cornell notes. Once students have summarized their findings in words, you may want to help them make the connection of how they naturally used inquiry to complete this task.

Inquiry Cube Answer

The color of the bottom side is blue because the opposite sides of the cube are identical colors.

The name on the bottom will be female, because the opposite sides have a girl’s name and a boy’s name. Since Frank is on the opposite side, you know the name is female.

The top right number represents the total number of letters in the person’s name. You know the number is 8 because the top right number begins with 3 (Rob), and progresses sequentially (4, 5, 6, 7) so the next side would require 8 letters in the name.

The bottom left number represents the number of letters that are shared by the names on both sides of the cube. You know that the first four letters are FRAN____.

Putting this all together you know that you are looking for a female name that is 8 letters long, and the first four letters are FRAN.

This leads to the conclusion of Francene. (Spelling is not critical, as long as it is eight letters long.)


Inquiry Cube


8

Francene

4

4

Alma

2

7

Roberta

3 3

Rob

3

6

Alfred

2

5

Frank

4


6.5 Math-Go-Round

Topic

• Reviewing for an upcoming math test

Rationale

This is an interactive twist on an AVID staple…the carousel walk. When there is a need to review for a big test or quiz that is coming up in their math class, this is a wonderful method to get the students to practice in a non-threatening environment. Each group of students is encouraged to do only a single step, which minimizes the risk of being intimidated about not knowing how to solve the entire problem.

Objectives

Students will:

• Work collaboratively to solve problems as a review of key learnings they will need for an upcoming quiz or test

Timeline


WICR Strategies


• Inquiry

• Collaboration


• Poster paper (or whiteboards on multiple walls) with review math problems from their math teacher

• Different color pens (one different color pen for each group)

Instructions

• Write a single problem on each piece of poster paper. Label each poster, starting at A and moving to Z. Make sure that there are at least three more problems than there are groups (e.g., if there are eight groups, have 11 to 12 problems written out).

• Break students into random groups three to four students.

• Give each student a different color pen.


• Assign each group to a poster.

It is advisable to skip letters as you are assigning. For example, placing groups at A, B, D, E, F, H, J.

• Give each group 1 to 3 minutes to do only a single step.

• After it appears most students are finished, have every group rotate to the next letter.

It is OK if not every group has completely finished.

• Tell the students to correct any mistakes made by the previous group with their (different color) pen. If the last step was correct, have them write the next step.

• Keep the groups rotating around until all problems have been solved.

Tell students that if their problem is already finished, they should circle or draw arrows to illustrate what has been done in each step.

• Once all posters are complete, have the students do a gallery walk.

Tell them to pay close attention to whether their group might have made mistakes.

• As a culminating task, have students write individual reflections about what they learned, and mistakes that they will need to avoid on the next test.


When there is an upcoming test

• (With a math book in hand) Could you give me 10 sample

questions that are the sort likely to show up on the upcoming

algebra test?

Ask for a range of problems (easy, middle, and hard) as

well as sample problems from every section that will be

covered in the test.

• What is the most difficult problem that you have on the test?

Could you give me an example of that question?

It is a good idea to star the questions that the teacher

deemed “hard” or “tricky” so that students can identify

them on their gallery walk.


6.6 What’s My Limit?12

Topic

• A Socratic Seminar that examines the concept of limits

Rationale

As students progress in the field of mathematics, they will be introduced to the concept of limits. The idea of limits is a relatively complex one, since it involves the difficult-to-grasp notion of something infinitely approaching a discrete number; however, this activity introduces students to the concept at an introductory level. Although limits are used heavily in Algebra II, Pre-Calculus, and Calculus, students are introduced to the concept in Algebra I when they examine the graph of the radical expression y = 1/x.

Objectives

Students will:

• Examine data and defend a position using logical arguments

• Explore the mathematical notion of limits

• Develop higher-level questions in an effort to clarify their thinking and begin the Socratic Seminar

• Collaborate in an effort to clarify thinking and question others’ understanding

• Critically read and evaluate the validity of an argument

Timeline

• One class period

WICR Strategies


• Inquiry

• Collaboration

• Reading


• Student questions

• Overhead/document camera or LCD projection of “What’s My Limit?” PowerPoint

12Donohue, J., Gill, T. (2009). Common Math Abbreviations. The Write Path I: Mathematics (pp. 101-102). San Diego, Ca: AVID Press


Instructions

• Begin by projecting the “What’s My Limit?” PowerPoint, which contains a graphic representation of the world records for 200-meter Freestyle Swimming.

• Have the students do a quickwrite on the question, “How long can swimmers continue breaking the 200-meter world record? Is there a speed at which it is not humanly possible to swim faster?”

• Have students move into a Socratic Seminar circle, and pass out Student Handout 6.6a “What’s My Limit?”

• As students read the text, have them underline key pieces of information that they can use during the discussion and generate at least one Level 2 or Level 3 question.

• Begin the Socratic Seminar using one of the student questions or the question “How long can swimmers continue breaking the 200-meter world record?”

• Do not forget to include a debrief question at the conclusion of the Socratic Seminar.

post-Socratic Seminar Writing Assignment• Have students complete Student Handout 6.6b “Post-Socratic Seminar Writing Assignment” in the form of an

email predicting the world record in 20 years.


• (For algebra teachers) When will you be covering

non-linear graphs (y = x 3, y = 1/x, y = x 4, etc.)?

Specifically the graph of y = 1/x?

Most likely this will only be one day, but if possible,

it would be great to use this Socratic Seminar a few

days before they discuss non-linear graphs.



What’s My Limit?

So what is a limit? For most of us, limits seem to have no “real world” application. Often, when we think of limits, we conjure up images of those annoying 65 mile per hour speed limit signs, which don’t really stop us from driving faster.

Let us examine the swimming world records for the 200-meter freestyle, which demonstrate this idea of limits. Otto Scheff set the earliest record in 1908, when he swam the 200-meter pool in 2 minutes and 31.6 seconds. Over the next century, this record was repeatedly broken, sometimes taking nine years and at other times repeatedly within the same year. During the 2008 Beijing Summer Olympic games, Michael Phelps shocked the world when he once again broke the world record for the 200-meter freestyle swim, finishing in 1:42.96. However, this record was broken yet again when Paul Biederman of Germany swam the 200 meters in 1:42.00 a year later. The following graph shows the 200-meter world record over the past 100 years.

As you examine the graph, ask yourself the question, “Can human beings continue breaking the 200-meter world record forever, or is there a speed at which it is not humanly possible to swim faster?”

200 Meter Freestyle Swimming World Record

TIME (in minutes)

2.5

2.0

1.5 | | | | | | | | | 1/0/1900 9/8/1913 5/18/1927 1/24/1941 10/3/1954 6/11/1968 2/18/1982 10/28/1995 7/6/2009

DATE


Post-Socratic Seminar Writing AssignmentTo begin our exploration of this mathematical phenomenon, imagine that you are standing up directly facing a wall. Now, walk to exactly halfway between you and that wall. For example, if you are initially 20 feet away, you would move to only 10 feet away. Once in your new spot, walk halfway to the wall again. Then, split your distance again. And again, etc. Write an email (at least one page) to a friend explaining the answers to the questions: Is there a speed at which it is not humanly possible to swim faster? Using the data table, is it possible to predict the world record in 20 years? Use the distance to the wall example above to help make the concept of limits clear to your friend. Include at least one graph in your email.

DATE OF NEW WORLD RECORDTIME

(mins:secs.millisecs)

DATE OF NEW WORLD RECORDTIME

(mins:secs.millisecs)

November 11, 1908 2:31.6 August 12, 1967 1:55.7

September 9, 1910 2:30.0 August 30, 1968 1:54.8

March 28, 1911 2:25.4 August 30, 1968 1:54.3

November 24, 1916 2:21.6 September 4, 1971 1:54.2

April 10, 1920 2:19.8 September 10, 1971 1:53.5

May 26, 1922 2:15.6 August 29, 1972 1:52.78

December 9, 1925 2:15.2 August 23, 1974 1:51.66

April 5, 1927 2:08.0 June 18, 1975 1:51.41

April 12, 1935 2:07.2 June 19, 1975 1:50.89

February 12, 1944 2:06.2 August 21, 1975 1:50.32

September 20, 1946 2:05.4 April 7, 1979 1:49.83

March 31, 1950 2:04.6 April 11, 1980 1:49.16

February 27, 1954 2:03.9 July 19, 1982 1:48.93

March 4, 1955 2:03.4 June 21, 1983 1:48.28

May 5, 1958 2:03.2 August 22, 1983 1:47.87

August 22, 1958 2:03.0 June 8, 1984 1:47.55

January 16, 1959 2:02.2 July 29, 1984 1:47.44

July 26, 1959 2:01.5 September 19, 1988 1:47.25

June 24, 1961 2:01.2 August 15, 1989 1:46.69

August 6, 1961 2:01.1 August 24, 1999 1:46.34

August 20, 1961 2:00.4 August 25, 1999 1:46.00

April 21, 1963 2:00.3 May 14, 2000 1:45.69

June 27, 1963 1:58.8 May 15, 2000 1:45.51

August 17, 1963 1:58.5 September 17, 2000 1:45.35

August 24, 1963 1:58.4 March 27, 2001 1:44.69

May 24, 1964 1:58.2 July 25, 2001 1:44.06

August 1, 1964 1:57.6 March 27, 2007 1:43.86

July 29, 1966 1:57.2 August 12, 2008 1:42.96

August 19, 1966 1:56.2 July 28, 2009 1:42.00

July 29, 1967 1:56.0




for Teachers’ Use to Understand Limits

When examining the questions:

Is there a speed at which it is not humanly possible to swim faster? Using the data table, is it possible

to predict the world record in 20 years? Strangely, the answer is “yes” to both.

First, let us examine the idea that there is a finite swimming speed that humans cannot physically exceed. Again, we start with a question, “Do you think that anyone will ever be able to swim 200 meters in one second?” To put that in perspective, the question is asking if a human being can swim a little over two football fields in a second. Clearly the answer is no! Regardless of how strong, a human being would need some distinct amount of time in order to swim 200 meters. As a result, we can be relatively certain a time barrier exists that is not humanly possible to exceed, which leads us to our first real image of a limit. This barrier might be 10 seconds or a minute, but we know that it exists. A human (under normal conditions without steroids or jetpacks) cannot possibly break this world record barrier. For the sake of our continued argument, let us say that the limit for swimming 200 meters is 90 seconds.

Currently the 200-meter freestyle world record is 102.2 seconds. If we could use a crystal ball to map out the future, we would most likely see a continued trend where the amount of time shaved off the record decreases and the number of years it takes for the record to be broken increases. For example, perhaps the current record will not be broken for another five years, and only one second will come off. Then 10 years will pass for another second to come off. Then 15 years might pass before 60 milliseconds come off. Finally, we will only be breaking the record by microseconds, then nanoseconds, then picoseconds, and it might take 100 years to break a previous record. As the world record approaches this human limitation (90 seconds for this sample), the decrease in the world record will become smaller and will take much longer for the new record to be set. This approach to our 90-second limit will go on forever (or in mathematical terms, it will approach infinity).

Clearly, a true “mathematical” limit is not some sign on the side of a road that a motorist can decide to break with total anonymity. Rather, a limit is a barrier than can be approached forever, but will never be reached.


STUDEnT HAnDOUT 6.6d (1 of 2)

Academic Language Scripts for Socratic Seminar

Clarifying• Could you repeat that?• Could you give us an example of that?• I have a question about that: …?• Could you please explain what _________________ means?• Would you mind repeating that?• I’m not sure I understood that. Could you please give us another example?• Would you mind going over the instructions for us again?• So, do you mean . . . ?• What did you mean when you said …?• Are you sure that …?• I think what _______________ is trying to say is….• Let me see if I understand you. Do you mean _________________ or _________________?

probing for Higher-Level Thinking• What examples do you have of . . . ?• Where in the text can we find…?• I understand . . ., but I wonder about. . . .• How does this idea connect to . . .?• If _________________ is true, then . . .?• What would happen if _________________?• Do you agree or disagree with his/her statement? Why?• What is another way to look at it?• How are _________________ and _________________ similar?• Why is _________________ important?

Building on What Others Say• I agree with what _________________ said because . . . .• You bring up an interesting point, and I also think . . . .• That’s an interesting idea. I wonder . . .? I think. . . . Do you think . . .?• I thought about that also, and I’m wondering why . . .?• I hadn’t thought of that before. You make me wonder if . . . ? Do you think . . .?• _________________ said that. . . . I agree and also think. . . . • Based on the ideas from _________________ , _________________ , and _________________, it seems like

we all think that….”


Expressing an Opinion• I think/believe/predict/imagine that . . . . What do you think?• In my opinion . . . .• It seems to me that . . . .• Not everyone will agree with me, but . . . .

Interrupting• Excuse me, but . . . . (I don’t understand.)• Sorry for interrupting, but . . . . (I missed what you said.)• May I interrupt for a moment?• May I add something here?

Disagreeing• I don’t really agree with you because . . . .• I see it another way. I think . . . .• My idea is slightly different from yours. I believe that . . . . I think that . . . . • My interpretation is different from yours…

Inviting Others into the Dialogue• Does anyone agree/disagree?• What gaps do you see in my reasoning?• What different conclusions do you have?• _________________ (name), what do you think?• I wonder what _________________ thinks?• Who has another idea/question/interpretation?• _________________ (name), what did you understand about what _________________ said?• We haven’t heard from many people in the group. Could someone new offer an idea or question?

Offering a Suggestion/Redirecting the Seminar• Maybe you/we could . . . .• Here’s something we/you might try: . . . .• What if we . . . ?• We seem to be having a debate instead of a dialogue, can we . . . .• Who has another perspective to offer that will help us re-focus the conversation?• Let’s look at page __________ and see what we think about. . . .

STUDEnT HAnDOUT 6.6d (2 of 2)


6.7 The Curve of forgetting

Topic• Examining the concept of forgetting

RationaleHaving your students revisit and review their notes can be an exhausting task. Even though we know that it is a critical exercise to support information recall, students do not always see the vital nature of this study time. This Socratic Seminar uses the graphical data recorded by Hermann Ebbinghaus regarding the nature of forgetting, and lets students analyze how often and for how long they should review vital information.


• Discuss the nature of forgetting

• Interpret graphical data

• Apply learning to their own study habits



• Collaboration

• Reading

Materials/preparation• Student Handout 6.7a “The Curve of Forgetting – Socratic Seminar”

• Pencil

Instructions• Pass out Student Handout 6.7a “The Curve of Forgetting – Socratic Seminar.”

• Have students read through Student Handout 6.7a (1 of 2) “The Curve of Forgetting – Socratic Seminar,” underlining key information or claims made by the author.

You can just pass out Student Handout 6.7a (2 of 2) “The Curve of Forgetting – Socratic Seminar” and have students interpret the graph.

• Have students formulate higher-level questions based on the graph and text.

• If students do not generate good introductory Socratic Seminar questions you can use:

Do you believe that the author is right about his claims regarding forgetting?

How does this information apply to our lives (study habits)?


• What do these graphs mean?

Only if you are having

difficulty interpreting Student

Handout 6.7a (2 of 2)

“The Curve of Forgetting –

Socratic Seminar”



The Curve of forgetting – Socratic Seminar

Why Do I Have to Take Notes? The Brain-Note Connection It is interesting that so many people lament having a poor memory, when the reality is that our brains are designed to perform one simple task... to forget stuff! The brain is a huge deleting machine. Need proof? Just ask yourself, “What did I have for dinner last night?” How about two nights ago?... A week ago?... A month ago?... A year ago? Most likely you could get the first one or two, possibly you can even recall last week. But over the course of time, your brain dumps any information that it deems unnecessary in order to store the new incoming information. It might be helpful to think of your short-term memory as a huge tank that holds on to a ton of information bits to keep us from minor mistakes, like eating chicken for dinner night after night because we do not remember we ate it last night. It holds on to those pieces of information for very short periods of time (generally about 24 hours), and then it will release those brain cells so that they can store new information.

In contrast, our long-term memory stores only that information that it believes to be highly valuable, like how to tie your shoes, a special birthday dinner you had with your family or friends, and how to do simple arithmetic. So how do we remember information? Primarily by revisiting the information often enough that we move it from the very unreliable short-term memory into the much more reliable long-term memory. Think about it – at some point your brain remembered every meal you’ve eaten, every outfit you’ve ever worn, the name of everyone you’ve ever met, and even how you solve systems of linear equations. But over the course of time, sometimes a very short time, our brain hits the delete button.

So where do Cornell notes fit into this? The reality is that without a system for revisiting information, our brains are incredibly efficient at removing extraneous information from our brain. In 1885, Hermann Ebbinghaus did extensive research around the idea of forgetting. In his book, Memory: A Contribution to Experimental Psychology, he mapped out the rate at which the average human forgets information over time.

This information might make a teacher incredibly sad. It is the realization that after a brilliant discourse on graphing inequalities, your students will only retain about a quarter of the lecture for the test.

But there is hope! Ebbinghaus goes on to examine how frequently a student would need to revisit information in order to regain near-perfect recall. His research evinces the brain’s ability to retain information when we revisit the information during key times.

The essence of this research demonstrates the need for students to revisit information in order to kick the information out of the short-term memory and store it in the much more reliable long-term memory. This is the power of Cornell note-taking – it primarily focuses on how we process the information rather than simply on recording the information.



The Curve of forgetting

Rate of forgetting with Study/Repetition

100%

90%

80%

70%

60%

50%

40%

30%

20%

10%

0

100%

90%

80%

70%

60%

50%

40%

30%

20%

10%

0

1st Repetitionwithin

18 minutes


Retained


Retained

2nd Repetitionwithin1 day

3rd Repetitionwithin7 days


31 days


3 months

2 days = 27.8% 6 days = 25.4%

31 days = 21.1%

1 day = 33.7%

1 hour = 44.2%

9 hours = 35.8%

20 minutes = 58.2%

Time

Time


6.8 four Color fractionsTopic• Reviewing percent-fraction-decimal conversion

RationaleOften, AVID Elective teachers seek out motivational and collaborative activities that help students build community but retain the educational backbone. This is a great activity to encourage students to work together while reviewing a mathematical area of need...fractions. Although this particular activity is designed for approximately a 6th grade level, you can adapt this activity for statistics (mean, median, and mode) for 8th grade, or functions (data, equation, and description) for 9th graders.

ObjectivesStudents will:• Work collaboratively to problem-solve• Use critical thinking skills to determine solutions• Evaluate work


WICR Strategies• Inquiry• Collaboration• Reading

Materials/preparation• Colored paper• Envelopes• Student handouts

Instructions• Print out Student Handout 6.8a “Four Color Fraction,” Student Handout 6.8b “Four Color Percent,” Student

Handout 6.8c “Four Color Decimal,” and Student Handout 6.8d “Four Color Graphic.” Make sure that there is one of each color for a group of four to six people.• Print out Student Handout 6.8e “Four Color Activity Answer Sheet” for each student.• Cut out the colored paper into cards.• Place a set of cards F 1-10, P 1-10, D 1-10, and T 1-10 in an envelope.• Divide students into groups of four to six.• Give each student a copy of Student Handout 6.8e “Four Color Activity Answer Sheet.”• Tell students they need to group the fractions, decimal, percentage, and representation.• Have students record the equivalent fractions, decimal, percentage, and representation on Student Handout

6.8e “Four Color Activity Answer Sheet.”• Check student answers against Teacher Reference 6.8f “Four Color Fraction Answer Sheet.”



four Color fractionF1

1 2

F2

3 5

F3

1 3

F4

2 9

F5

10 25

F6

5 2

F7

5 11

F8

22 88

F9

3 27

F10

16 20



four Color PercentP1

40%

p2

80%

P3

22.2%

p4

45.45%

P5

60%

p6

50%

P7

25%

P8

133 3 %

P9

250%

P10

11.1%



four Color DecimalD1

.45

D2

2.5

D3

.6

D4

.22

D5

.5

D6

.33

D7

.4

D8

.1

D9

.25

D10

.8


four Color GraphicG1

G2

G3

G4

G5

G6

G7

G8

G9

G10

STUDEnT HAnDOUT 6.8d


four Color Activity Answer Sheet

Fraction percent Decimal Graphic

F1

F2

F3

F4

F5

F6

F7

F8

F9

F10

STUDEnT HAnDOUT 6.8e


four Color Activity Answer Sheet

Fraction percent Decimal Graphic

F1 P6 D5 G7

F2 P5 D3 G6

F3 P8 D6 G9

F4 P3 D4 G10

F5 P1 D7 G4

F6 P9 D2 G5

F7 P4 D1 G1

F8 P7 D9 G8

F9 P10 D8 G2

F10 P2 D10 G3

TEACHER REfEREnCE 6.8f


6.9 Estimation Station

Topic

• Predicting various quantities

Rationale

One critical skill that students need is the ability to apply reasonable estimations to a problem. In the math world we call this number sense, and it is underdeveloped in many students.

Objectives

Students will:

• Predict a reasonable estimate

• Justify their reasoning

Timeline


WICR Strategies

• Inquiry

• Collaboration


• Mason jars, water bottles, or containers

• Small candies, Goldfish snacks, or beans

Instructions

Method 1 – predicting quantity• Fill a container with small candies (such as Skittles). Read the back of the package to determine (approximately)

how many candies are in the bag.

• Have students enter their predictions at the beginning of class.

• Have students discuss the method they used to make their prediction.

• Reveal the “winner” who predicted the number closest to the actual number.


Method 2 – predicting Length• Fill a container with pretzel sticks, sour straws, Red Vines, or Pixie Sticks.

• Ask students to approximate the total length of the pretzels, red vines, pixie sticks, etc., if they were laid out end to end.

• Have students discuss the method they used to predict the total length.

Make sure that you tell the students what units of length they should use to make the guess (i.e., inches or centimeters).

• Reveal the “winner” who predicted the length closest to the actual number.

How the teacher determines the correct total length

• Measure the length of one item (in inches or centimeters).

• Count the total numbers of items in the jar.

• Multiply the length of one item times the total number of items in the jar.

Method 3 – predicting Weight• Fill a container with Goldfish crackers, hard candies, or other snacks.

• Have students predict the total weight of the snacks

Make sure that you tell the students what units they should use to guess (i.e., grams, ounces, pounds, etc.).

• Have students discuss the method they used to predict the total weight.

• Reveal the “winner” who predicted the weight closest to the actual.

How the teacher determines the correct total weight

• On the back of the box, there will be the weight of the contents.

• Use a small kitchen scale; generally, you can borrow one from a chemistry/science teacher.

Extension

Use a calculator to determine the average (mean) of all of the students’ guesses. Generally, the average will be much more accurate than any individual guess. There are multiple resources online that will help you with these calculations such as: http://www.easycalculation.com/statistics/standard-deviation.php

Have the students compare the average of their data to their individual data points.


6.10 What’s in the Bag?

Topic

• Exploration of one-variable equations

Rational

One aspect of algebra that many students struggle with is the concept of solving for x in an equation. Initially this concept can be confusing for students regarding what we subtract from which side, and even the overall purpose of isolating x. This activity walks students through an inquiry process that has them begin to explore some of the fundamentals of one-variable equations. The final questions and difficulty level will lead students past trying to plug in numbers, and to generalizing the approach for solving these questions.

Objectives

Students will:

• Predict reasonable solutions to a problem

• Collaboratively develop a solution

Timeline

• One 50-minute class period

WICR Strategies


• Inquiry

• Collaboration

Materials

• Five to ten brown paper lunch bags

• Multiple “counters” that are identical (nuts, suckers, cookies, beans, cupcakes, pens, etc.)


Instructions

• Start by placing the same number of “counters” on two different tables.

The example demonstrated below has 14 cupcakes on each table.

• Place a different number of empty lunch bags on the two tables.

The example demonstrated below places three empty bags on the first table and two empty bags on the second.

• Put the same number of “counters” into each bag (e.g., if you put two cupcakes into the first bag you must put two cupcakes into every bag).

The example places four cupcakes into every bag.

• As you start the class, the students should only see the following (with four cupcakes inside every bag).


• Have the students group into a circle and work together to determine the number of cupcakes in each bag.

Have students discuss how they figured out their answer?

How do they know if they are right?

Once the students have determined that there have to be four cupcakes in each bag, prompt the discussion to continue using the following problem (and have them discuss the questions that follow):

“Now imagine 23 bags (with seven items in every bag) and 37 loose on one table, and 30 bags (seven in each bag) and two loose on the other.”

• If you were to take away a bag from each table, would we still have an equal number of cupcakes on each desk? What is your reasoning?

• I f you were to take away two individual cupcakes from both desks, would we still have an equal number of cupcakes on each desk? What is your reasoning?

• If you were to take away a bag from only one table, would we still have an equal number of cupcakes on each desk? What is your reasoning?

• If you were add three bags to each table, would we still have an equal number of cupcakes on each desk? How do you know?

• What would be a general principle that we could say about adding and subtracting bags, if we wanted to keep the total number of cupcakes equal?

After they have determined a principle (something like: “We can add/remove bags from one desk as long as we add/remove the same number of bags to the other side.”) Walk them through the following example.

• Can we apply our principle to make this problem easier?


Mathematical principle

• If we have three bags (with an unknown number of cupcakes in them) and two loose cupcakes on one desk, we could write that as:

3x + 2

• If we have two bags (with an unknown number of cupcakes in them) and six loose cupcakes on the second desk, we would write the equation as:

2x + 6

• And since there are equal numbers on both sides, we can write this as:

3x + 2 = 2x + 6


• When will you be covering solving for a

one-variable equation?

You will want to do this activity about two to

three weeks before they cover the algebraic

method for solving one-variable equations.

Generally, students learn this at the

beginning of the year.


ReferencesBibliography

Brookhart, S. (2010). How to Assess Higher-Order Thinking Skills. Alexandria: ASCD.

Ching, K., & Gill, T. (2003). Algebra Tutorial: A Guide for AVID Coordinators and Tutors. San Diego: AVID Press.

Donohue, J., & Gill, T. (2009). The Write Path II: Mathematics. San Diego, Ca.: AVID Press.

Ebbinghaus, H. (1885). Memory: A Contribution to Experimental Psychology. Berlin.

Larson, R., Boswell, L., Kanold, T., & Stiff, L. (2008). Math: Algebra 1. Evanston: McDougal Littel.

Pauk, W., & Owens, R. J. (2008). How to Study in College. Boston: Houghton Mifflin Company.

Wormeli, R. (2005). Summarization in Any Subject. Alexandria: ASCD.


Notes


Notes

Documents

Supporting Math - Fresno Unified School District · Supporting Math in the AVID Elective 1 Author Acknowledgements If I have seen further it is only by standing on the shoulders of