INTEGRATED COLLABORATIVE MATHEMATICS Charity M

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INTEGRATED COLLABORATIVE MATHEMATICS

Charity M Tanaleon B.A., University of California, Davis, 2002

PROJECT

Submitted in partial satisfaction of the requirements for the degree of

MASTER OF ARTS

EDUCATION (Curriculum and Instruction)

CALIFORNIA STATE UNIVERSITY, SACRAMENTO

SUMMER 2009 C 2009

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INTEGRATED COLLABORATIVE MATHEMATICS

A Project

Charity M Tanaleon

Approved by:

Committee Chair D/ Julta Lambating

au¢t~ .2 6(, 024 q Date 6/

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L I *Student: Charity M. Tanaleon

I certify that this student has met the requirements for format contained in the University format

manual, and that this project is suitable for shelving in the Library and credit is to be awarded for

the Project.

3raduate Coordinator . Julita Lambating Date v

Department of Teacher Education

iv Abstract

INTEGRATED COLLABORATIVE MATHEMATICS

Charity M Tanaleon

Currentmathematical instructionalpractices remain unchanged, allowing educational practicesand curriculum to remain shallow, undemanding and diffuse in content coverage

(NationalResearch Council, 2001). The standardizedmath curriculums are basal programsthat

utilize a spiral approach to instruction, often presenting concepts as isolatedor unrelated skills.

These concepts present themselves too quickly and in a specified order with educatorsfocusing on basic skills due to the amplified stress of high stakes testing and accountability. These curriculums lack relevance to students' lives, inhibitingeffective brainfunctioning and covering more math topics than other countries.

The sources of data used in this project include achievement comparisonsfrom internationalassessments such as the Programfor InternationalAssessment (PISA) and Trends in InternationalMathematics and Science Study (TIMSS). This project grounds itself in constructivist theories of Piaget, Dewey, and Vygotsky, as well as brain research and observational studies examining variationsof problem based learning (PBL) incorporatedinto the classroom.

This project provides gradefive teachers with a supplementary math curriculum entitled,

Integrated CollaborativeMathematics (IC Math). This is a separate curriculumthat incorporateslessons and songs into an individualize unit and is versatile, allowing teachers the

V opportunity to extrapolate individual lessons or songs to supplement the particularmath concept currently taught. IC Math integration of gradefive mathematics and language arts helps students practice these skills while minimizing teacher's time teaching individual standards.

Collaborationengages students toward solving real life scenarios as they engage their prior knowledge, and realize that math is relevant to their lives. IC Math attempts to help students gain flexible knowledge, self-directed learning skills, and higher order thinking skills that are necessary to be successful.

Committee Chair

Datea

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DEDICATION

I dedicate this project, to my family and friends who supported me through this endeavor.

Especially my parents for their assistance on this academic journey; Marie and Grace for being a

sounding board to my ideas; and to Renee, whose love of math, interest in social justice, and

thought provoking ideas helped inspire me to create IC Math.

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h ACKNOWLEDGEMENTS

I appreciate the support and encouragement I received from the faculty at CSU

Sacramento. Especially, Dr. Julita Lambating, for her patience, understanding, and guidance that helped me focus my ideas and complete this project. I would like to acknowledge Dr. Forest

Davis in his assistance toward developing the title, and thank Dr. Frank Lilly for encouraging me to pursue and include my passion for music in this project. Dr. Lilly, your advocacy for

Universal design and enthusiasm to enhance students' ability to think critically and evaluate evidence inspires me to become an agent of change.

I would like to thank eight-year old, gentleman Oman, for continuously cheering me on through this process. Finally, a special thanks to Renee and Sabrina for their willingness to be my other editors. I appreciate your constant encouragement, motivating words, good food, positive energy, and your unwavering belief in me.

viii TABLE OF CONTENTS

Page Dedication...... vii

Acknowledgments...... v...... viii

List of Figures ...... x

Chapter

1. INTRODUCTION...... 1

Overview ...... 1...

Purpose of the Project .... 2...... 2

Statement of the Problem ...... 4

Significance of the Project ...... 5

Limitations...... 6

Definitions of Terms .... 7...... 7

Organization of the Project ...... 9

2. REVIEW OF RELEVANT LITERATURE ...... 10

History of Problem Based Learning ...... 10

Constructivism ...... 11

Brain Research ...... 14

Educational Discrepancy ...... 17

Math Curriculum ...... 23

21st Century: Skills that Matter ...... 24

Summary...... 25

3. METHODOLOGY...... 27

Introduction...... 27

ix Integrated Collaborative Mathematics ...... 27

Implementation...... 31

Summary...... 31

4. SUMMARY, LIMITATIONS, RECOMMENDATIONS ...... 36

Summary...... 36

Limitations...... 37

Recommendations...... 39

Appendix A. Grade five California Subject Standards ...... 41

Appendix B. The Restaurateur Unit ...... 53

Appendix C. Math Songs ...... 111

References ...... 121

x LIST OF FIGURES

Page

1. Figure 1: Math and language arts standards ...... 33

2. Figure 2: The Restaurateurunit overview and integrated standard ...... 34

3. Figure 3: Note/example illustration ...... 35

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Chapter 1

INTRODUCTION

Overview

Current educational practices focus on academic achievement, especially with the goal of all students demonstrating proficiency in math and reading-language arts by 2014 (Yell, 2006).

Bush's 2001 educational legislation, No Child Left Behind (NCLB), set this precedent.

Unfortunately, the emphasis to achieve this goal diminishes time spent on problem-solving, critical thinking and enrichment activities, especially in urban schools where procedural computations and obtaining the correct answer is the sole focus (Sutton & Kruger, 2002).

Teachers are under tremendous pressure to get students performing at proficient levels. Yell

(2006) emphasizes that "100% of a state's students must reach 100% proficiency in both reading- language arts and math by the 2013-2014 academic year" (p. 192).

NCLB transformed standardized tests, designating them as the force behind instructional practices; instead of a guiding tool to supplement curriculum standards. Many states and school districts spend more time focusing on NCLB tested areas - reading and math, as well as testing strategies that may give students an advantage during the test (Abrams & Madaus, 2003). Even though the United States (U.S.) focuses on academic achievement through test score improvement, its students continually perform below average on the Program for International

Student Assessment (PISA). The Pisa is an assessment that examines a student's ability to understand concepts and apply them to various situations within the subject.

The 2006 PISA results reveal that U.S. students performed below the 498 Organization for Economic Cooperation and Development (OECD) average. These students performed behind

31 European and Asian countries in math literacy. Even isolating U.S.'s high achieving students 2 still left the U.S. trailing behind 29 of the 51 participating countries. No measurable changes occurred from the prior 2003 PISA results (Baldi, Jin, Skemer, Green, & Herget, 2007).

U.S. students may be capable of solving straightforward computational procedures; however, they lack math literacy. Students are unable to apply math skills to basic problems due to a limited understanding of math concepts (National Research Council, 2001). The U.S. education system does a poor job mathematically preparing students in comparison to its global counterparts. Schoenfeld's (2002) analysis of the Third International Math and Science Study

(TIMSS) found countries like Germany and Japan continuously outperforming U.S. math students. Numerous academic standards and current curriculums contribute to the U.S's

mathematical inadequacy.

California presents an example of a math curriculum that perpetuates the cycle of learned helplessness. The current math curriculum creates passive learners. Students already have a

minimal understanding of math concepts and rely on the teacher to provide information, which

reinforces the belief a student needs external help (Parmar & Cawley, 1991); instead, of

collaborating to solve the problem. Another explanation for poor math literacy is due to the gaps

and poor sequencing of lessons within the current curriculum (Witzel & Riccomini, 2007). Poor

curriculum developments with the plethora of math topics force teachers to gloss over topics.

There simply is not enough time to cover the material adequately.

Purpose of the Project

The purpose of developing the Integrated Collaborative Mathematics (IC Math)

supplementary curriculum is to help grade five teachers in California decrease the amount of time

spent teaching individual standards. IC Math accomplishes this by integrating math and language

arts standards into its lessons with songs that fortify math concepts and vocabulary. This

supplementary curriculum also demonstrates the symbiotic relationship between academics and -

the reality of life which current curriculum rarely presents. Providing examples of how skills

transcend out of the classroom allow students to see the relevance of school and learning. Zull

(2002) suggests that helping people learn requires that they must first be able to see how it

matters in their life. This is what IC Math tries to accomplish in its use of plausible scenarios

occurring in real life, infused with math that builds upon prior knowledge, ultimately, assisting in

the building of a strong connection to new information (Peele & Foster, 2001; Smith & Geller,

2004).

This curriculum endorses itself as an agent of change; increasing learning efficiency

through subject integration and music reinforcement; promoting diversity appreciation, and

enhancing student understanding. As discussed earlier, students have more time practicing math

and language arts skills through curriculum standards integration. Providing students

opportunities to discuss ideas and perspectives promotes an appreciation of diversity and

collaboration. Knowlton (2003) suggests that collaboration also enhances student understanding

because students are able to interact with the material. Justification of their methodology and

solutions help to promote equity within the classroom because students are able to effectively

explain and assist peers who have difficulty understanding the material (Boaler, 2006a; Boaler,

2004). IC Math helps achieve a more equitable classroom through its ability to have students

practice skills necessary in a workforce through frequent collaboration. This is especially

significant for students in urban schools because they rarely receive time to enhance their

collaborative communication skills or critical thinking due to the emphasis on correcting math

calculations (Sutton & Kruger, 2002). Computation busy work disables these students from

developing higher-order thinking skills, placing them at a disadvantage in a competitive

workforce that reveres analytical skills.

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Statement of the Problem

Math instructional practices remain stagnant (Mulcahy, 2006), maintaining a consistent

shallow and undemanding curriculum diffuse in content coverage (National Research Council,

2001) which continue to be taught in school. In general, schooling demands basic levels of

convergent thinking, where learners recall information and focus on content acquisition through

rote rehearsal instead of thinking for analysis and synthesis (Sousa, 2006). Teaching dedicated to

memorization, emphasizing storage and recall of unconnected isolated facts, does not facilitate

the transfer of learning and is an unproductive use of the brain (Caine & Caine, 1994). According

to the National Research Council (2001), students have limited understanding of basic

mathematical concepts and have difficulty applying math skills to solve simple problems. This

occurrence is more prevalent in urban schools due to the fixed sequence of math tasks associated

with the current curriculum (McKinney, Chappell, Berry, & Hickman, 2009).

Standardized basal curriculums, like McGraw Hill, cover material as isolated or unrelated

skills, often presenting math concepts quickly in a prescribed order that focuses on basic skills

practice (McKinney, Chappell, Berry, & Hickman, 2009). Sutton and Krueger (2002) imply that

students, particularly in urban schools, "spend much of their time on basic computational skills

rather than engaging in mathematically rich problem solving experiences" (p. 26). Too often,

students view math as an obstacle instead of a tool (Mulcahy, 2006) that working to enhance

reasoning, critical thinking and problem solving skills (McKinney, Chappell, Berry, & Hickman,

2009). This current math curriculum lacks relevance to student's lives, inhibiting effective brain

functioning (Caine & Caine, 1994), and attempts to cover more math topics than what is typical

in other countries. The United States (U.S) expects teachers to cover 30 math topics meticulously

each year; whereas, Germany incorporates 20 topics and Japan introduces 10 math topics yearly

(Schmidt, McKnight, & Raizen, 1996).

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Modifications of current methods are necessary for increasing mathematical success, especially in urban schools. Walker and Chappell (1997), recommend that math teachers working in urban schools develop or utilize teaching strategies that will enhance their students' learning. Teachers should consider lessons that integrate the external environment to that of a classroom since the products of the mind derive from interactions the brain has with the body and physical environment (Zull, 2002)

Significance of the Project

Sizer (1992) speculates, that a good place to begin thinking about curriculum is "how to attract and hold a student's attention and how to instill in them a commitment to think hard" (p.

68). This project is significant because it begins to examine Sizer's comment. This project's curriculum, IC Math, may play a significant role for teachers and the students they educate in grade five. IC Math presents real-life problems that incorporate math concepts; highlighting the importance of acquiring mathematical proficiency and understanding. The curriculum's combination of melodious songs and lessons that integrate math and language arts standards while using a balanced instructional approach makes IC Math unique. Zull (2002) suggests, a balanced approach increases understanding at higher levels than just using one instructional approach of either teacher driven or reform based student centered.

Implications of evolving educational practices may occur due to the possibility that this project reestablishes the idea that learning can be fun. Infusing fun back into learning enhances the learning method, and makes recalling the information easier (Dweck, 1989; Wilson & Horch,

2002). IC Math will engage students with its catch songs that infiltrate neural connections, leading to a quicker recall (Wallace, 1994; Wilson & Horch, 2002) of math procedures and concepts. 6

This curriculum may also influence educational practices for students in urban areas. The

mathematical focus placed on these students is to use correct computational procedures (Sutton &

Kruger, 2002). Learning math should not confine students to rote memorization and conventional

teacher driven lectures. IC Math attempts to go beyond the confines of traditional techniques,

opting to develop student's problem solving skills with PBL infused lessons. The concept of

making learning fun, along with brain research, educational theories, and practices make this

curriculum practical and relevant. IC Math also endorses students to engage with each other,

providing communication skill practice while developing an appreciation for their peers and the

diversity of their perspectives. Students collaborating with the material forge connections with

the content, increasing neural activity, which leads to a stabilization or strengthening of synaptic

connections (Hinton, Miyamoto, & Della-Chiesa, 2008) and deeper understanding (Dominowski,

1998).

Limitations

IC Math is designed as a supplementary curriculum that benefits all students;

unfortunately, students comfortable with a traditional approach to learning mathematics may be

resistant to change. Students with learning disabilities (LD) may also have a difficult time

engaging fully with this project's curriculum.

Implementation of this project may also be limited due to the time constraints afforded in

a school day. Even though IC Math attempts to decrease the amount of time spent teaching

individual standards, it will take time to establish a positive and cooperative learning

environment. Teachers must spend time modeling appropriate behaviors needed for group work,

like active listening, working out a compromise and coming to a consensus. Time will be spent

acclimating students to cooperative group tasks and obtaining appropriate collaborative skills.

Establishing this environment conducive to problem based learning (PBL) may be challenging if

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teachers are unfamiliar with this approach or have not received training on utilizing cooperative

groups in the classroom.

Definitions of Terms

Adaptive reasoning: Capacity of thought reflection, explanation, and justification.

Balanced instructionalapproach: An approach to teaching that utilizes an equal

combination of direct instruction and student-centered exploration.

Basalprogram: A set of textbooks with accompanying workbooks for students.

Chunking: Dividing large amounts of information and teaching them in smaller portions.

Constructivism: A description of theories that suggest learning based on making

meaningful connections with the world and interacting with other people.

Convergent thinking: A learner's ability to recall previously learned information and

apply it to a question.

Complexity: Thought process that the brain uses to interact with information.

Conceptual understanding:Comprehension of mathematical concepts, operations, and

relations.

Culturally relevant: Recognizing, understanding, and applying practices that is sensitive

to and appropriate for people with diverse cultural socioeconomic educational backgrounds and

abilities.

Difficulty: The amount of effort a leaner expends to accomplish a learning objective.

Divergent thinking: A learner's ability to process information; to formulate new ideas and

perspectives.

Effort: Devoting effective academic learning time to a task.

Equity: A chance to learn free from bias based on race, gender, socioeconomic status and

language.

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Integrated CollaborativeMathematics (IC Math): A problem based curriculum that

includes math songs and reality driven lessons that integrate mathematics with language arts

standards.

Math literacy: An individual's ability to identify, understand, and apply math to make judgments and manipulate it in ways that meet the needs of that individual's life

Mnemonic device: A verbal learning and memory aid to help facilitate a greater recall of

information.

Organizationfor Economic Cooperationand Development (OECD): An

intergovernmental organization of 30 industrialized nations (OECD, 2007).

Programfor InternationalStudent Assessment (PISA): A system of international

assessments that measures 15-year-olds' capabilities in reading literacy, mathematics literacy, and

science literacy every three years (OECD, 2007).

Problem based learning (PBL): A learning approach incorporating a student's current

ability level, appropriate teacher facilitation in order to explore possible solutions to real life

problems.

Proceduralfluency: Skills in carrying out procedures, flexibly, accurately, efficiently,

and appropriately.

Productive disposition: Having a habitual inclination to see mathematics as useful and

worthwhile.

Rote memorization: A learning strategy that places emphasis on the recall of information

without developing understanding of the material.

Strategic competency: The ability to formulate, represent and solve novel mathematical

problems.

Synapse: Point of connection between two nerve cells. -

Organization of the Project

This project consists of four chapters. Chapter 1 provides an overview of the project,

introducing current practices prompting the design of IC Math. Chapter 2 contains the literature

review that examines educational theories, as well as, discusses research and literature that

provide the foundation of IC Math. Chapter 3 briefly restates the problem before explaining the

components of IC Math. The standards and lessons alignment are included within this chapter.

Chapter 4 contains the synopsis of the project, discusses plausible limitations of the curricula and

providing recommendations to further mathematical research. The final section consists of the

Appendix, which contains the lessons and math songs.

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Chapter 2

REVIEW OF RELEVANT LITERATURE

This project creates the Integrated Collaborative Mathematics (IC Math), a

supplementary curriculum comprising of two components: lessons and math songs. This

curriculum includes problem based learning (PBL) lessons with songs infused with math

terminology. This curriculum integrates (I) math and language arts standards allowing students

the opportunity to collaborate (C) on PBL scenarios. IC Math aligns itself with the standards

established in the Mathematics Framework (California Department of Education, 2000) and

Reading/languagearts Frameworkfor Californiapublic schools (California Department of

Education, 1999). This literature review looks at the historical success of PBL and examines

constructivist theories, brain research, and observational studies that provide the foundation for

IC Math.

History of Problem Based Learning

In 1969 at Canada's McMaster University Medical School, project based learning was born

(MacKinnon, 1999). Goodnough (2005) credits Barrows as the founder of PBL. Barrows'

established PBL in order to have medical students acquire, and apply content knowledge in

clinical settings, forcing them to utilize problem-solving abilities. Since then, the field of medical

education uses PBL (McPhee, 2002), and is a successful way of preparing prospective doctors for

situations they will face in their careers. However, PBL no longer confines itself to medical

education; other areas of study including pharmacy, law, business, and education have adapted

this method (MacKinnon, 1999). This project attempts to design a curriculum allowing students

to gain academic knowledge and integrate it into daily life in the hope of replicating the success

of the previously mentioned preparatory programs. This literature review looks at cognitive

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theorists; the relationship of brain research in current educational practices; current math trends

toward enhancing mathematical equity, and the need for critical thinking in today's workforce.

Constructivism

"Education proceeds by participation of the individual," (Dewey, 1897, p.3). Other

cognitive learning theorists support Dewey's statement. Piaget is one of the more prominent

psychologists. His theories are the foundation of educational practices which current brain

research validates.

Piaget's cognitive theory considers individual development from birth to adulthood, the

stages of maturation, as well as the types of reasoning and thinking abilities demonstrated by an

individual. Piaget was also one of the first theorists to conclude that learning is an active process

involving the interaction one has with their environment. Environmental interactions are not the

sole contributor toward increasing internal brain structures; maturation and social interactions are

also factors.

An individual's experience can fit into one of two categories - concrete or abstract. At

each stage of development, concrete explorations allow individuals to build upon their

background knowledge. Children as young as one year old, operating in the sensorimotor stage,

interact with their environment using objects to solidify their knowledge base. Teachers utilize

manipulatives to help their seven to fourteen year old students, functioning in the concrete

operational stage, enhance their logical reasoning. As students mature into adulthood; older than

fourteen years, they move into the formal operational stages where they are able to reason and

think more abstractly (Hassard, 2007).

During a student's elementary career, peer interactions help students gain a better

understanding of an academic concept (Boaler, 2002; Boaler, 2004; Boaler, 2006b; Jamar & Pitts,

2005) as well as navigating within a school's surroundings. Frequent interactions with the

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environment and peers, along with a developing maturity level enhances neural structures

(Hassard, 2007).

Piaget also established a learning theory regarding the concept of assimilation,

accomodation and equilibrium. According to Piaget (1971), "intellectual adaptation is a process

of achieving a state of balance between [the] assimilation of an experience into deductive

structures and the accommodation of those structures to the data of experience" (p. 14). Students

that engage with their environment go through this process. Barman's (1990), An expanded view

of the learning cycle, expands on Piaget's theory of assimilation and accommodation while

including his own parallel teaching theory of exploration. Like Piaget, Barman establishes the

importance of learning as an active process - exploration with manipulatives and objects allows

students to achieve disequilibrium and begin to question what, why, or how an event occurs.

Disequilibrium forces a student to invent a rationale for a discrepancy while engaging socially

with peers and teachers until they obtain understanding. The student may now move toward

discovery, applying their new schema and knowledge into other ideas, thereby extending their

knowledge to other applications (Hassard, 2007).

Accommodating discrepant events and interacting with peers leads to another cognitive

theorist - Vygotsky and his zone of proximal development theory (ZPD). Morris (1998) quotes

Vygotsky, suggesting that ZPD is,

The distance between the actual level of development as determined by

independent problem solving [without guided instruction] and the level of

potential development [,] as determined by problem solving under adult guidance

or in collaboration with more capable peers. (¶ 4)

Interacting with more capable peers allow students the ability to amplify their prior

knowledge and create connections between new information with their familiar background

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knowledge (Geller-Ketterlin, Chard, & Fien, 2008; Schwartz & Bransford, 1998; Zull, 2002).

This idea is illustrated when students begin to internalize word meanings when they discuss their

findings using mathematical terminology. Steele (1999) employs Vygotsky's theory, explaining

that students learn new math vocabulary via reflecting upon the word, visualizing the definitions

and, most importantly verbalize their thoughts with peers. Through discussions, children begin

developing their ability reason. Social interaction is an important aspect of ZPD; the National

Council of Teachers of Mathematics (NCTM) (1989) affirms that,

Communication plays an important role in helping children construct links

between their informal, intuitive notions and the abstract language and

symbolism of mathematics; it also plays a key role in helping children make

important connections among physical, pictorial, graphic, symbolic, verbal, and

mental representations of mathematical ideas. (p. 26)

Some educators utilize a combination of these two theorists in their classroom - allowing

students the opportunity to explore new concepts using manipulatives as concrete models;

collaborating with peers; or utilizing scaffolding activities with the teacher or capable peers

guiding the way. These theories, and the practices associated with them enable students to

progress according to their developmental level while in a safe environment. A nurturing

environment allows learning to take place, and is what Dewey describes as the fundamental

principle of school (Dewey, 1897).

Other theories absorb Dewey's concept of a nurturing environment, like Wiener's

attribution theory. When students feel safe, they are more readily able to learn (Hinton &

Miyamoto, 2008; Mulcahy, 2006), especially when they feel good about themselves. Factors

influencing student motivation are: ability, complexity, effort and luck with the underlying

principle, "a person's own perceptions or attributions for success or failure determine the amount

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of effort the person will expend on that activity (Attribution Theory and Motivation p. 2). A

student's current ability level, with appropriate teacher facilitation and having students explore

possible solutions is the basis of PBL. Personal exploration and problem solving have students

believing that their own behavior and effort promotes success or failure rather than external

circumstances, which Lefcourt(1976) defines as an internal locus of control.

Many educational practices derive from Piaget, Vygotsky, and Dewey's theoretical

conjectures; however, the evolution of brain research provides emperical evidence that support

these men's theories.

Brain Research

The brain is a major organ that assists in the accruement of knowledge - converging al

experiences into neurological distributions of chemicals through synapses and developing new

dendrite connections (Sousa, 2006; Zull, 2002). The brain is capable of perceiving and

generating patterns through this neuronal network (Caine & Caine, 1994). Neural stimulation

comes from an individual's interaction with the external environment. This implies that learning

is physical - deriving from a sequence of experiences, reflections, abstractions and active testing

(Zambo & Zambo, 2008; Zull, 2002). Since learning is physical, one's emotional state, as well

as, the school's and classroom's climate influence one's ability to learn.

Sousa (2006) implies that under certain conditions, emotions can enhance memory

through the release of hormones to various brain regions that help strengthen memory; hence, the

importance of developing a nurturing environment. Learning is more effective when educators

minimize stress and fear at school; thus, creaing a postive learning environment (Hinton,

Miyamoto, & Della-Chiesa, 2008; Zull, 2002) which students find motivating. PBL is a tool that

may assist toward the development of this positive climate because students are able to

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collaborate and support each other's learning, especailly when faced with challenging math

problems.

Amos cites Jensen's (1998) book, Introducation to Brain-CompatibleLearning:

The human brain is social and that learning is enhanced when both children and

adults are given the chance to learn in an environment free of threat. Learning is

maximized when an exchange of emotions, feelings, sharing, discussion,

brainstorming, and problem solving take place (Amos, 2007, p. 69).

PBL provides educators the ability to foster collaboration and nurture inquisitiveness, like

Amos and Jensen suggest, instead of constant emphasis on multiple-choice exam performance.

In this manner, students can focus on obtaining skills needed to enhance understanding and

correct any prior misconceptions regarding math. The reinforcement of correct content is what

Ylvisaker and Feeney (1998) coin as errorless learning - repetitive use of correct connections.

These researchers found this approach highly effective in assisting brain-damaged children learn.

This approach emphasizes the importance of enhancing and strengthening appropriate correct

neuronal networks because it decreases, and eventually eliminates the synaptic connection

harboring incorrect knowledge (Zull, 2002). Synapses become stronger when it reinforced with

practice.

Cognitive conflict also yields to brain enhancement. Jensen (1998) suggests that "the

single best way to grow a better brain is through challenging problem solving" (p. 35) and

defending one's perspective against questions and conflicting ideas of others (Hiebert, et al.,

1997). Engaging the brain in different tasks, helps alter it. Jensen (2006) discusses various

studies that suggest the plasticity of students' brains. He establishes the argument that the brain is

not a fixed mechanism, but an organ that has the ability to change based on environmental

influences. Jensen refers to Hellemans's, Benge's and Olmstead's research claiming that social

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contact can alter genetic expressions (Jensen, 2006). Therefore, teachers have the capacity to

alter students' physical brains with their classroom structure (Caine & Caine, 1994; Zull, 2002),

curriculum choice, and the presentation of curriculum.

Environmental factors contribute to neural alterations; for example, taking different

classes alters the brain more than taking less challenging classes. In a quantitative dendrites'

analysis, brain researchers, Jacobs, Schall, and Scheibel confirm Piaget's and Vygotsky's

conjectures that the brain is designed to interact with the world and make alterations accordingly

(as cited in Jensen, 2006).

Even though challenging the brain with difficult classes helps alter the brain, a University

of Arizona 1986 study demonstrates that brain alteration does not necessarily facilitate an ability

to transfer learning to novel contexts. Students who participated in this study had at least six

years of science content knowledge, but did poorly to demonstrating their scientific

understanding and reasoning skills outside of the classroom. Perhaps the transfer of knowledge

did not occur because students choose to focus on recalling unconnected facts, which does not aid

in a transfer of learning (Caine & Caine, 1994). This 1986 study implicates the importance of

combining knowledge with hands on experience. "Schools would do well to focus on much

more real-world learning. Field trips, simulations, role plays.. .and away-from-school activities

that use school knowledge and skills make much more sense than a focus on field-independent

classroom learning" (Jensen, 2006, p.21).

Field trips provide the brain with novel experiences. Strother (2007) examines the work

of researchers like Bunzeck and Duzel (2006), whom suggest that when the environment lacks

novelty, the brain turns inward for stimulation, and Maquire, Frith and Morris (1999) suggesting

that experience provides relevance, which in turn creates higher value; therefore, becoming more

meaningful and better incorporated into the neurological pathways. Ultimately, "everyone is

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biologically equipped to learn from experience. Learning primarily means making sense of

experience and all people must learn from experience to survive. People are born as virtual

"learning machines" (Strother, 2007, p. 20).

Even though people are capable and programmed to learn, variations of learning styles

and aptitudes create challenges for educators. Curriculum in the U.S. does not often take into

consideration students who may not conform into middle-class, white America.

Educational Discrepancy

Even though justification and reasoning of one's perspective are intrinsic math practices

that helps promote equity (Martino & Maher, 1999), schools should also have an equitable

distribution of material and human resources. Schools that present students with quality

instruction using intellectually stimulating material, and provide educational experiences that

build on prior knowledge prepare students for critical thought in a democratic society (Lipman,

2004, p. 3). NCTM's (2000) equity principle summarizes the distribution of equity as the ability

for "all students.. .a chance to learn mathematics free from bias based on race, gender,

socioeconomic status, and language (p. 12). Unfortunately, a discrepancy of mathematical

proficiency and performance still exist among minorities and socioeconomic classes within the

U.S., even after the 1954 Supreme Court ruling of Brown vs. Board of Education. African and

Hispanic Americans along with low socioeconomic students continue to perform lower on

college entrance exams and tests of basic skills and problem solving tests than their white and

Asian counterparts (Jamar & Pitts, 2005; Tate, 1997). The National Assessment of Educational

Programs (NAEP) reveals 70% of eighth grade students scored below proficient in mathematics,

further analysis shows that 59% of African American students and 50% of Hispanic students

scored below basic (Geller-Ketterlin, Chard, & Fien, 2008, p. 33). Students living in high socio-

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economic levels have more access to knowledge and perform 60% higher on standardized exams

than the average low socioeconomic student (Lee & Burkam, 2002).

The opportunity for all students to learn mathematics is being heralded as the new civil

right (Moses & Cobb, 2001), especially since students attending urban schools perform

significantly lower than their suburban counterparts. U.S. schools are still as segregated today as

it was back in the 1960s Civil Rights Era with 63% of white students attending schools where the

student population is 90- 100% white (Borman & Maritza-Dowling, 2006; Gamoran & Long,

2006). School characteristics of academic tracking and perceived teacher bias favoring more

attentive middle-class students attributes to the variation of student achievement which Borman

(2006) concluded after analyzing the 1966 Coleman report.

Standardized curriculums also play a role in this mathematical disparity. State

standardized curriculum pay little to no attention to the diverse needs of students such as

students' background experience, knowledge and entry skills within a particular grade level. This

may be due to the lack of diversity in curriculum development groups since most curriculum

developers tend to be white and have middle-class American values. These developers often omit

concerns and struggles of ethnic minorities or, if included, they do not relate mathematics to

minority concerns; instead, curriculum developers prefer to integrate non-controversial problems

and neglect issues of inequality and social justice (Norwood, 2009). Current math curriculums

lack cultural relevancy, making little to no connections regarding situations relevant to a student's

life. A standardized curriculum like the one developed by McGraw Hill focuses on procedural

knowledge - the regurgitation of facts through rote memorization. This practice does not permit

students the opportunity to consider why a math procedure works let alone have the ability to

explain it. Havis and Yawkey (2001) recommend that "important mathematics principles

[should] not be taught for the sake of facts, but for organization, reorganizing and systematizing

ho I 19

mathematical thinking structures" (p. 135). Learning mathematics should not be based on rote

memorization; mathematical proficiency is too immense and complex to simply be taught by a

teacher lecturing in front of the classroom explaining the math process using the "left hand pages

in the textbook while students practice with problems on the right hand side" of textbooks

(Gardner, 2001, p. 75).

Unfortunately, the most prominent circulation of knowledge in the U.S. utilizes direct

instruction and practice problems within basal programs. District, county and state achievement

tests measure the progress of each student and their mathematical understanding. These basal

programs utilize a spiral approach, quickly introducing skills in a particular grade level with

constant repetition of the same material in future grades. This spiral approach to instruction is

supposed to add depth to math topics while students progress in grade levels. However, the

reality is a superficial coverage of math concepts and skills making mathematical mastery

difficult. Teachers introduce new skills too quickly in an attempt to cover all the topics provided

in the textbook (Miller & Mercer, 1997). Curriculum in the U.S. is "a mile wide, [and] an inch

deep" (Schmidt, Houang, & Cogan, 2002, p. 3), lacking a clear focus. An introduction of five

math concepts begins a student's first year of math education. Each year afterwards, the

difficulty level of the initial five concepts increases while teachers present more math topics

constantly, regardless of whether a student has a solid foundation of whole number operations,

meaning, or unit measurement (Schmidt, Houang, & Cogan, 2002). Sadly, there are not enough

practice opportunities to solidify new math skills and recognize how it correlates to daily life.

Basal programs also have detrimental effects on students with learning disabilities. In several

case studies, young adults with learning disabilities tend to drop out of school because they feel

that "academic efforts would be anxiety provoking and humiliating" (Lichtenstein, 1993, p. 345).

Research implies that most high school math teachers emphasize textbook problem solutions

L 20

(Cawley & Miller, 1989), rather than novel or real life problems that brain research finds

beneficial.

A better approach to mathematical instruction would use graduated instructional

sequences, moving from the concrete use of manipulatives to representational pictorials and

finally into abstract problems (Geller-Ketterlin, Chard, & Fien, 2008). The ideal curriculum

would introduce three math topics that third graders would master before adding more topics.

This approach provides students a foundation of whole number meaning, whole number

operations, and measurement units, like Japan and Germany (Schmidt, Houang, & Cogan, 2002).

Other educational researchers have other recommendations such as developing an

interdisciplinary unit (Pinzker, 2001); or an intentional curriculum that is content driven,

developmentally appropriate, culturally diverse, considers English language learners while

developing their social and regulatory skills (Klein & Knitzer, 2007). Along with acquiring

necessary math skills, NCTM recommends that students in grades three through five have math

literacy - understanding mathematical relationships to other academic areas and have

computational fluency to be able to see the broad spectrum of applied mathematics and

communicate mathematical thinking using academic language (NCTM, 2000).

NCTM (1991) encourages teachers to focus on new innovative techniques and strategies

that may improve the quality of math instruction and move beyond the 'empty vessel' and

'banking' pedagogy (NCTM, 1991). A strong instructional program needs to support

mathematics learning among all students. Various ideas for a better approach include

interdisciplinary units and intentional curriculums, which strays from the current educational

practice of direct instruction. The goals of the aforementioned practices hope to increase student

engagement, mathematical equity, and develop mathematical proficiency by providing students

the opportunity to learn more about their world and its relationship to mathematics (Pinzker,

I -

2001). Making connections to students' lives enhances a student's background knowledge (Peele

& Foster, 2001). Hirsch (2002) states, "when children share a common base of knowledge,

classroom instruction can be far more effective" (p. 6). Creating base knowledge and aligning it

with daily life creates interest and motivation for students. This enhances their prior knowledge

and leads them toward developing further in math. Students are more likely to be stimulated

toward learning complex mathematics, like trigonometry or calculus later on in life.

Alleviating the inequity

Educators must be aware of the social, economic, and political contexts of schooling that

may hinder or facilitate math learning for underrepresented students (Snider-Allexsaht, 2001) and

address the racism and classism that currently exists in our country. Utilizing a PBL approach

may help minimize the social construction of current racism and classism beliefs. This particular

approach also considers the diversity of a learner's experience, skills, and prior knowledge

making math more relevant and providing a way toward developing equity in classrooms.

Educational researchers find that students in disadvantaged groups, especially minority and low

socioeconomic students who participate in experimental math approaches, such as PBL benefit

academically. These students show higher levels of achievement and challenged themselves by

enrolling in advanced math courses at higher rates than students who enrolled in traditional

mathematics (Horn, 2006). Boykin (1994) suggests, African American students work well in

collaborative groups and do better in mathematics when allowed to work collaboratively.

Stanford researchers found a highly collaborative math department using a reform-

oriented curriculum successful in educating their diverse urban student populations. These

educators adapted PBL to make meaningful to their students. These students learned more,

enjoyed math more, and progressed to higher levels of mathematics through their relevant math

L 22

experience at their high school (Boaler, 2004; Boaler, 2006b). These results stem from the four-

year study undertaken by these researchers.

The study followed 700 students in three different California urban high schools through

their mathematical experiences during their four years attending school. Two of the high schools

utilized traditional textbook math curriculum, placing students in the appropriate traditional math

classes of pre-algebra, algebra, geometry, whichever track the student fell into in accordance with

the traditional math tracking system. The tested high school, pseudo named - Railside High,

utilized the reform-oriented curriculum that used heterogeneous groupings. The student

population breakdown is 38% Latin, 23% African American, 20% white, 16% Asian or Pacific

Islander, with 3% consisting of other ethnic backgrounds. Railside's demographics also include

students with limited financial resources, more English language learners and students with lower

academic achievement upon entering high school than the other two traditional high schools. The

study documented the teaching practices utilized, along with 600 hours of classroom

observations, assessments, and questionnaires given to students during their first three years and

160 individual interviews with students.

The reform-oriented curriculum used at Railside combines problems from the Interactive

Mathematics Program (IMP) and College Preparatory Mathematics (CPM) curriculums that the

collaborative Railside math department redesigns. The math department organizes the curriculum

into unifying themes, incorporating problems the math department felt where, "group worthy" -

problems where solutions required multiple perspectives of different students and math methods.

The math teachers are responsible for establishing the precedent for students to work as a

collaborative in the math classroom. Teachers emphasize student responsibility - making

students responsible for their own learning and that of their peers.

L 23

During classes, these teachers educate the heterogeneous groups of students on working together, contributing their ideas via questions, and developing conjectures. Block scheduling provide math classes more time to cover material. Longer class periods help teachers foster positive group dynamics - countering differences in social and academic status due to the diverse dialogue of the groups. The results of this approach at Railside transformed students' lives, helping them see mathematics as a part of their future and providing them, the quantitative reasoning capabilities needed to function in an increasingly technological and global economy.

Students appeared to enjoy math more while developing more respect for each other and appreciating diversity. These alterations in perspectives are due to the heterogeneous grouping and collaborative instruction used at Railside (Boaler, 2006b).

Math Curriculum

"Tasks should leave behind important residue" (Brayer, 2007 p. 183). The following studies depict the residue educators leave upon their students. Observing Melissa Jones, a kindergarten teacher for three months allowed researchers to see students mimic the same questioning technique of their teacher. A young girl assisting her peer suggested that her peer begin counting, before asking a leading question of, "what comes next?" Thus, allowing her peer to think and utilize the counting skills she developed to answer the question (Cooke & Buchholz,

2005). During a lesson, Ms. Jones guided her students' interest of the seesaw, making connections to an academic concept. Students were eager to participate in the math lesson, as they continually selected classroom items to put on the balance scale (Cooke & Buchholz, 2005).

Another observational study of a kindergarten class depicts Penny Silver, and her use of daily routines to develop problem solving and communication skills among her students. She would ask leading questions such as: "What do you notice about who is here today? And "How do you know?" before she had them record their thoughts in a journal. Over time, Ms. Silver 24 discovered that this routine enabled her students to transition their mathematical thought process from concrete (drawing people in the classroom and counting them) to more symbolic thought

(using tally marks) account for students in attendance (Brayer, 2007).

The effects of PBL do not confine themselves to a kindergarten classroom. Ahlfeldt,

Mehta, & Sellnow, (2005) examine Hake's (1998) physics study of 6000 students. The study tests 62 introductory physics courses and the effectiveness of PBL's interactive engagement. The study found that interactive engagement allowed students to gain an average of 0.48 with a 0.14 standard deviation above students who were taught in the traditional course; who only gained

0.23 with a standard deviation of 0.04 (Ahlfeldt, Mehta, & Sellnow, 2005 p.6).

215' Century: Skills that matter

Mathematical proficiency is foundational when developing students' ability to reason logically, develop higher order thinking skills, and increase their cognitive capacity (The building blocks of success, 2008). The National Research Council (2001) defines mathematical proficiency as having: conceptual understanding, procedural fluency, strategic competency, adaptive reasoning, and have a productive disposition.

A students' knowledge must exceed that of basic computations. Unfortunately, U.S. students are not learning, nor are they acquiring the math they need (National Mathematics

Advisory Panel, 2008). Employers in high technology, medical, and other fields have a difficult time finding employees with skills necessary to function well on the job and meet expectations

(The building blocks of success, 2008). CEO of Marriott International, Willard Marriott Jr. remarks, "young people need substantial content knowledge and information technology skills; advanced thinking skills; flexibility to adapt to change; and interpersonal skills (Casner-Lotto &

Benner, 2006, p. 11). Students need the ability for complex reasoning and the cognitive capacity to educate themselves. A 2005 survey of the National Association of manufactures found 84% of 25 employers think K-12 schools do not do a good job in preparing students for the workforce.

Another 2005 survey from The American Diploma revealed, 40% of high school graduates feel unprepared for college or the work force (Fletcher, 2007). Students are not the only ones who feel unprepared.

The manufacturing industry is concerned about the preparation of its workers. A survey of four hundred employers across the United States noted that high school and college graduates are entering the workforce lacking proficiency in applied skills, such as critical thinking, self- direction, work ethic, and effective communication (Casner-Lotto & Benner, 2006). PBL can nurture deficiencies in interpersonal skills because it necessitates small group collaboration, emphasizes self-directed learning, and provides the ability present one's findings through oral presentations (Schmidt, Vermeulen, & Van der Molen, 2006).

To be a competitive candidate, workers need math skills to succeed in this evolving global economy. PBL increases a student's ability to acquire more of the necessary skills employers seek. PBL provides math literacy - allowing students to make mathematical connections to other disciplines of study as well as enhance collaborative skills through group discussions of multi-part problems. PBL proved beneficial for international students attending the Budapest Business School. A survey given to the sixty students enrolled in the program found that the students taking PBL courses engaged more with the curriculum and felt they increased their knowledge because they were able to discuss current real-life issues as well as link theoretical knowledge to the problem (Tick, 2007).

Summary

As discussed in this review, current math curriculum utilizes basal programs that continue to bombard students with numerous math topics making skill mastery nearly impossible.

Constant reintroduction of the same skills hinders students' motivation, minimizing interest or 26 willingness to excel in the difficult subject. Students with learning disabilities lack a math curriculum that fits their learning needs.

Global studies discuss coherent math curriculums that introduce minimal math topics in order to achieve greater conceptual understanding before introducing new topics and concepts.

The U.S seem to allow publishers to dictate the number of math topics while the Federal and state government attempt to align academic standards and expectations together, instead of reversing the process. The U.S.'s curriculum does not heed the words of Piaget, Dewey, or Vygotsky regarding the importance of activating prior knowledge; engaging students with the material; or having students interact with the environment. Students should have the opportunity to collaborate and discuss concepts with peers to enhance their neural development. Brain research supports these theorist's conjectures.

Observational research studies depict positive residual attributes of PBL. Professional educational institutions incorporate PBL to train medical, business, and educational professionals; therefore, elementary classrooms can do the same. Elementary students should see how their academic experience relates and pertains to their daily life; challenging them to think critically and enhance their reasoning skills

There needs to be a math curriculum that takes into consideration all the aforementioned issues. This project is that curriculum. This project dares to make mathematics fun, engaging, and motivating, while building upon students' background knowledge and sharpening their critical thinking skills. This project attempts to unite the foundational theories of cognitive psychologists, the best educational practices derived from brain research, to be inclusive and universally designed- "an approach to creating environments and products that are usable by all people to the greatest extent possible" (Welch, 1995, p. 1). 27

Chapter 3

METHODOLOGY

Introduction

This chapter outlines the purpose, goals, and components that combine to form the

Integrated Collaborative Mathematics (IC Math) supplementary math curriculum. The curricula

in this project align themselves with the California Mathematics and Language Arts Framework

(California Department of Education, 2000; California Department of Education, 1999). A chart

of the standards, which ground IC Math, and overview of its correlating lessons integrate

themselves into this chapter, while each detailed lesson and math song are in the appendices. IC

Math intends to combine school's academic knowledge, especially math, with aspects of daily

life; making connections to students' lives and enhance their background knowledge (Peele &

Foster, 2001) so that students learn to use math to make sense of the stories they encounter in

daily life (Boaler, 2006b).

Integrated Collaborative Mathematics

The attributes of constructivism, brain research, and problem-based learning (PBL), as

discussed in chapter 2, provide the foundation of IC Math. As noted in the prior chapter, a PBL

curriculum stimulates students to become highly motivated, self-directed learners who know how

to integrate information across subjects with existing knowledge, think critically, work

collaboratively (MacKinnon, 1999), demonstrate understanding (Capon & Kuhn, 2004) and are

better prepared to enter the global workforce (Casner-Lotto & Benner, 2006).

The philosophy behind IC Math is based on integrating (I) various subject standards (see

Figure 1) together while having students collaborate (C) on real life scenarios; hence the name,

Integrated Collaborative Mathematics, preferably referred to as IC Math. I

Combining the different subject standards together and integrating it with real life, IC

Math helps students connect new information with their prior knowledge - increasing relevancy

and meaning, generating a greater probability of storing information (Sousa, 2006). IC Math's

inventive way of using a learner's existing neural network changes their brain with increased

synaptic connections (Sousa, 2006; Zull, 2002) and provide opportunities for teachers to include

higher order thinking activities in a high stakes testing environment (Sousa, 2006).

The population

Instead of using "the antithesis of constructivist principles when working with minority

students" (Jamar, 2005, p. 129) and opting for traditional approaches that consist of structured

lectures and teacher direct instruction (McKinney, Chappell, Berry, & Hickman, 2009), IC Math

provides a creative supplementary curriculum that teachers in diverse urban schools can utilize in

their classrooms. The combination of lessons and math songs that encompass IC Math generates

a fun, yet challenging approach to teach math concepts that engage students in critical thinking,

problem solving activities since "many mathematic students spend much of their time on basic

computational skills rather than engaging in mathematically rich problem-solving experiences"

(Sutton & Kruger, 2002, p. 26). While engaging students in real-world problem solving

scenarios, IC Math utilizes 5th grade standards as the guide in developing lessons in The

Restaurateurunit.

California expects grade five teachers to cover a multitude of standards in math, language

arts, science; social studies (see Appendix A). Unfortunately, due to time constraints, these

teachers emphasize math, language arts, and science standards and disregard any of the fine arts

like, dance, theater, visual arts, and music, because they are "confronted with the realities of

preparing students for high stakes testing" (Sousa, 2006, p. 247). The goal and hope is that 5h

grade students perform at proficient or advanced levels on yearly California Standards Tests 29

(CST) in mathematics, language arts, and science. Combining math and language arts like IC

Math does, allows synaptic connections to strengthen due to the increased practice time (Zull,

2002) students have on math and language arts skills as they progress through the 200 standards and 3,093 benchmarks that the Mid-continent Research for Education and Learning (McREL) database tabulated (Marzano, Kendall, & Cicchinelli, 1999, p. 15).

The unit

The IC Math curriculum integrates a handful of mathematics and language arts key standards (see Figure 1) into a supplementary unit entitled, The Restaurateur. The lessons in this unit have students collaborating to find solutions regarding problems they may encounter in life while learning math songs that reinforce mathematical procedures. Figure 2 provides the unit overview and chronological order of lessons, while the list of possible student job roles, and descriptive lessons, for significant assignments are in the appendix.

Each lesson takes a mathematical concept and integrates it into a viable life scenario. For example, the lesson Feeding an army forces students to determine what products to purchase for a dinner party at the grocery store on a limited budget, and Recipes around the U.S. lets students discover regional cuisines and their corresponding recipes. Even though IC Math emphasizes mathematical concepts, language arts components incorporate themselves into the lessons via journal reflections, narrative and expository writing assignments, group discussions, and presentations made to the class. IC Math provides flexibility - allowing lessons to diverge into other academic areas of study outside of math and language arts; for example, the lesson, The

Gardner'sDilemma, can be used to segue into introducing students to basic plant structures necessary for respiration, a science standard, since Garland the gardener works with flowers and plants in this math scenario. 30

The Restaurateurconcludes with a culminating experience - using all the skills, lessons, and concepts covered in the unit. Students will develop a business proposal and present it to the class; drawing parallels to a business plan entrepreneurs develop and present to prospective investors.

The music

The second component of this project includes an array of math mnemonic devices in the form of songs in order to facilitate a quicker recall of math procedures, vocabulary, and definitions. Our auditory system is exquisitely tuned for music (Sacks, 2007), implying that learning is enhanced when set to music (Rainey & Larsen, 2002), especially when the information is associated with a simple symmetrical melody, or a melody of a well-learned song. Hearing a melody repeat across verses enhances the recollection of information (Wallace, 1994) since the brain automatically registers familiar stimuli (Caine & Caine, 1994). Experimental research studies with over 1,000 students suggest mnemonic instruction in the classroom is an effective strategy which students demonstrate better retrieval of information from their memories

(Mastropieri, Sweda, & Scruggs, 2000).

All the songs in IC Math contain verses infused with math explanations while the chorus reiterates the discussed concept. The lyrics integrated with the simplistic melody work into the musical phrasing, rhythmic patterns, and structured rhyme scheme which Wallace (1994) suggests make information more likely to be recalled. Introducing math songs prior to teaching the math concept allows students a preview of the material they will soon encounter.

Teach songs in chunks. Research in learning and cognitive psychology suggests that information is easily remembered when small bits of information build into a larger base of knowledge (Bodie, Powers, & Fitch-Hauser, 2006). This chunking technique makes learning a song and math concept easier, especially since song verses depict a single math explanation. 31

After previewing the song, provide explicit examples of what each verse describes in the "notes"

section of the lyric sheet (see Figure 3) when reviewing each verse. This review will reinforce

the chunked math explanation embedded in the verse. Chunking the material will help students

remember more, and give individuals the ability to readily access the stored information (Bodie,

Powers, & Fitch-Hauser, 2006) because it provides "the amount of information [students] can

deal with" (Miller, 1956, p. 95). Eventually, hearing a particular melody will cue the text and

math concept learned, since new information integrates with the students' prior knowledge

(Bodie, Powers, & Fitch-Hauser, 2006) via a familiar song melody, creating a powerful

connection. Wallace's (1994) four music experiments reinforce music mnemonics and the idea

that "hearing a melody to a well-known song can cue the text and, vice versa, hearing the text can

cue the melody" (p. 1).

Implementation

IC Math is a separate curriculum, incorporating lessons and songs into an individualized

unit. Figure 2 provides the unit plan and overview; however, IC Math is versatile, allowing

teachers the opportunity to extrapolate individual lessons or songs to supplement the particular

math concept currently taught.

Summary

In using PBL's instructional method where students learn through facilitated problem

solving, IC Math attempts to help students gain flexible knowledge, self-directed learning skills

(Hmelo-Silver, 2004) and higher order thinking skills. Student success in math relies on

conceptual knowledge; hence, the inclusion of explicit instruction (Hudson & Miller, 2006) in IC

Math songs. Computational practice within lessons, explicit reviews of IC Math songs, and

generation of student examples of song verses provide practice of lower skills. Acquiring a solid

foundation of lower skills enable students to manage more complex tasks and activate attributes

I 32

of higher order thinking skills, such as: discussions that present multiple perspectives to solving

real life problems (Sousa, 2006) in IC Math, or demonstrating the comprehension of ideas by

defending them against questions and conflicting notions of others (Hiebert, et al., 1997).

Teaching concepts that present information in a relevant, problem solving perspective

allows a student to connect their prior knowledge to the new concept (Smith & Geller, 2004),

which is not true of isolated facts (Mulcahy, 2006). This also bridges the gap between current

student math performance and the skills needed for future success (Miller & Mercer, 1997) like

collaboration, communication and problem solving - effective problem solving skills yield the

capacity to transfer reasoning strategies to various problems (Hmelo-Silver, 2004). The

curriculum also tries to make learning math fun, since enjoyment makes acquiring learning goals

attractive, while concurrently enhancing the learning method (Dweck, 1989).

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Figure 2: The Restaurateur unit overview, chronological order of lessons, and lesson synopsis. Unit title: Unit Overview: Standards at a glance: The This unit provides - Math: geometry; algebra and function; and number Restaurateur students the sense opportunity to - Language arts: reading comprehension; writing; work individually listening and speaking; and presentation speaking Unit length: and collaboratively Science: digestive system and scientific method 4-6 weeks toward enhancing number sense and - Social studies: geography basic geometry. Lesson Lesson overview Gardener's Students help Garland the gardener design square and rectangular planters, while Dilemma determining the greatest possible area for each designed planter. Gardener's Students continue to help Garland in designing unique planters and calculating the Dilemma II perimeter for the desired area given. Gardener's Students will use the skills obtained in the first scenarios of Garland's dilemma to Dilemma III design a 3D fountain and calculate the surface area, and volume of each planter so that it may be appropriately filled. Number sense and computationalscenario practice How many Students will write the procedure of making a peanut butter and jelly sandwich sandwiches? before calculating the number of servings you get out of each ingredient needed to make a sandwich (see recommended serving size). United Students will generate a hypothesized list of ingredients that join to make a States' regional dish, justifying their hypothesis through the examination of each region's recipes habitat and agricultural practices. Be a food Students read a critic's food review deciphering what is fact vs. opinion before critic writing their own critique about either a dish they've eaten at home, or a restaurant. Ingredient Students will use the grocery guide and grocery list to determine how much a breakdown single serving of various ingredients will cost. What's a Students discover which is more cost effective, eating out, or staying at home better deal? cooking. Students examine individual costs, as well as the cost to feed a family of 3 -5. Students will also write a persuasive argument as to which option is better. Feeding an Each group will decide upon a menu for their assigned catering event, and army calculate the total cost to feed everyone. Assume that everyone at the function will only eat one serving of each meal. Business Students will develop their own business plan regarding their proposed proposal restaurants. The business plan must include: * Restaurant name . A floor plan including calculations of the area, perimeter, & surface area. * Type of food to be served - a menu * Estimated costs of serving each dish for I person * Projections of possible profit * Advertising techniques 35

Figure 3: This song provides an example of where students can take notes regarding the song verses. In this example, the notes portion provided students the opportunity to write all the numbers 1-50 and highlight those numbers that have multiple factors, and can be divisible by one of their factors.

Prime number By: Charity Tanaleon

This is a call and response song to the tune of Where is thumbkin?

Are you a prime number? Are you a prime number?

Yes I am. Yes I am.

'Cause I have no other factors 'Cause I have no otherfactors

Section where students Than 1 and myself i can take notes of math Than I and myself terms or examples of the math song concept ...... 11 Notes/examples 1 2 3 1 5 7 | M I 1 y 13 *17 19 * db23 U diiibey db 29 31 0 37 * 41 * 43 * * * 47 *

* divisible by 2 divisible by 3 divisible by 5

E divisible by 7 -

Chapter 4

SUMMARY, LIMITATIONS, AND RECOMMENDATIONS

Summary

Current math curriculums and educator practices emphasize storage and recollection of

unconnected facts (Caine & Caine, 1994). The choice of math curriculum reinforces the practice

of unconnected fact memorization. Current California math curriculums utilize a spiral

approach to introduce math skills; however, constantly reintroducing the same skill turns math

into a dry routine. The brain habituates; having brain cells fire less frequently, when repeatedly

given the same problem (Zambo & Zambo, 2008). Even expecting teachers to cover more math

topics (Schmidt, McKnight, & Raizen, 1996) leave students with a limited understanding of

basic math concepts and "deficient in their ability to apply mathematical skills [toward] simple

problems" (National Research Council, 2000, p. 4). Perhaps assigning busy work is detrimental

to learning because computational problems have no relevance to students. Students lack power

over their learning when personal meaning or sense of ownership is nonexistent (Caine & Caine,

1994). This project looks to remedy the lack of daily life relevancy basal curriculums tend to

omit from their programs by developing a problem based learning (PBL) curriculum entitled,

Integrated Collaborative Mathematics (IC Math).

Research within the literature review suggests that a PBL approach generates a more

equitable classroom due to collaborative opportunities and exchanges of mathematical

perspectives amongst peers (Boaler, 2006b). A PBL based curriculum may increase

interpersonal skills and enhance a student's ability to become a self-directed learner. This

student becomes competent in problem solving, analyzing informationa and developing an

efficient plan (Schmidt, Vermeulen, & Van der Molen, 2006). The literature also discusses

constructivist theories like Dewey and his belief that "only true education comes through

k.I I 37 stimulation of the child's powers by the demands of the social situations in which he finds himself' (Dewey, 1897, p. 36). Brain research implies that learning comes from a sequence of experiences, relections, abstractions and active testing (Zull, 2002, p. 13). A constructivist classroom provides students opportunities for self-exploration, to utilize prior knowledge to make new neural connections (Smith & Geller, 2004; Peele & Foster, 2001) and reformulate ideas to reach unique conclusions in an environment free from rote memorization (Gordon,

1998).

This project developed the curriculua, IC Math as a tool to infuse PBL lessons with direct instruction math songs. The integration (I) of subjects allow students the ability to practice and reinforce math and language art skills while decreasing the amount of time needed to address each standards independently. The collaboration (C) element provides communicative skill practices needed for the workforce and the opportunity to enhance their understanding of the material after interacting with peers (Knowlton, 2003). The math songs and lessons complete IC Math.

The math songs provide mathematical terminology and procedures that hopes to increase mathematical competence by facilitating a quicker recall of information. The PBL based lessons attempts to bridge the gap between acaemics and reality, especially since real-life situations do not lend themselves cohesively into California's mandated curriculum, textbooks or standarized tests (Gordon, 1998).

Limitations

IC Math is merely a mathematical tool that grade five classrooms may use; however, various factors contribute to a student's learning. Teachers play a significant role delivering math concepts and establishing a learning environment where a PBL derived curriculum can flourish (Boaler, 2002; Boaler, 2006a; Boaler, 2006b). Student attitude and aptitude also 38 contribute to IC Math's effectiveness. Foundational knowledge is a precursor to problem- solving skills (Knowlton, 2003); unfortunately, some students have gaps in their knowledge or have yet to develop specific math skills allowing them to feel successful learning mathematics.

Students who lack mastery of basic math skills are likely to experience augmented failure and frustration later in subsequent math classes (Witzel & Riccomini, 2007).

Mathematics is one of the most challenging subjects especially for students with learning disabilities (LD) (Cawley & Miller, 1989). Students with LD may have memory deficits, difficulty attending to key dimensions, and take a passive approach to learning, making a PBL based curriculum challenging for students with LD to engage fully (Miller & Hudson,

2006). Even with IC Math's collaborative component, collaborative tasks may not work effectively. Bottge, Heinrichs, Mehta, and Hung (2002) imply, students with LD who work with students without LD let their partner do most of the thinking and work, defeating the purpose of collaboration and learning from each other. Twenty-five years of intervention research, finds that cognitive strategy instruction and direct instruction are the most effective instructional practices that address the needs of students with LD (Kroesbar & Van Luit, 2002; Montague,

2007).

Students without LD may find the transition from the traditional teacher-driven approach to a student-centered one difficult (MacKinnon, 1999). This transition may be challenging because habits and beliefs are physiologically entrenched, making students resistant or slow to change. Traditional classroom settings are what many students are familiar with, and master the conditions of the classroom (Caine & Caine, 1994; MacKinnon, 1999; Zull, 2002) and dislike the effort needed to answer open-ended questions.

Even though IC Math attempts to decrease time spent on addressing individual standards, time may still pose a problem. Students may not have enough time to work through 39 the scenarios at their own pace, or, allowing students too much time trying to discover an algorithm that does not generalize itself to other problems (National Research Council. 2005).

Time utilizing IC Math may still be limited due to the positive learning environment a teacher should establish before collaboration becomes effective. Teacher modeling of appropriate behaviors and providing enough time for students to become proficient enough to demonstrate these behaviors takes time to establish. Students' presentations that demonstrate their knowledge, via presentations, may take additional time.

Recommendations

Teachers who implement appropriate cooperative and academic behaviors at the beginning of the year may alleviate the time limitations discussed in the previous section. Since collaboration is a component of this project's curricula, a professional development training utilizing cooperative learning will benefit teachers interested in using IC Math. Collaborative training before the school year will allow teachers the opportunity to implement collaborative and cooperative practices such as active listening, compromising, and coming to a consensus.

These practices established at the beginning of the year will alleviate the time limitation previously mentioned. The beginning of the year is also a good time to emphasize student responsibility toward managing their learning and establishing high expectations of success that focuses on effort over innate ability. Over time, the emphasis of effort in the classroom gives students the ability to persevere when confronted with challenging math concepts or problems.

Developing persistence enhances a student's ability to regulate their learning better. Self- regulation is a good predictor of success (Cerezo, 2004) and high self-regulated learners value problem analysis and reflection (Artino, 2008); hence, the importance of encouraging effort and developing a supportive learning environment. 40

The focus of research cited and discussed in prior chapters examine the use of PBL approaches regarding mature learners - students in high school, college, graduate and professional schools. There may be a bias toward PBL because these students may already be intrinsically motivated and eager to learn, making them adept to a PBL approach. These students have the capacity to adapt well and flourish to any learning environment (Hmelo-Silver,

2004). This researcher suggests that more examination of PBL approaches in urban public schools and the influence of school climate effects this non-traditional approach to instruction.

Researchers should also examine possibilities in developing a truly balanced mathematical approach to curriculum and instruction.

Zull (2002) suggests that,

students taught solely by a traditional method, gained understanding slowly and

at a low level, while students using the discovery approach alone seemed to

learn quickly at first, but their comprehension did not grow over time... students

using the balanced approach increased their understanding steadily and reached

levels significantly higher in the end. (p. 41)

IC Math may be a good curriculum to examine further since it attempts to create a balanced approach to instruction and aligns itself with California's grade five standards. This researcher hopes further research will lead to a paradigm shift. Too often educators recognize the validity in Hillard's observations.

Untracking is right. Mainstreaming is right. Decentralization is right.

Cooperative learning is right. Technology for all is right. Multiculturalism is

right. But none of these approaches or strategies will mean anything if the

fundamental belief system does not fit the new structures that are being created"

(as cited in Jamar & Pitts, 2005, p. 129). 41

Perhaps one day everyone will treat educators as the professionals they are, and have the

opportunity to incorporate meaningful methods of instruction and curriculum that engages

student learning.

hi 42

APPENDIX A

California 51I grade: Subject standards

Page

California 5th grade: Math standards ...... 43

California 5b grade: Reading-language arts standards...... 45

California 5 hgrade: Science standards ...... 50 California 5h grade: Social Studies standards...... 52 43

California 5 th grade: Mathematics standards

Start End KEY Math Description of math standards of of Standards the the year year Number sense 1.2 Interpret percents as a part of a hundred. Find decimal and percent equivalents for common fractions Compute a given percent of a whole number Explain why decimal and percent are equivalents for common fractions 1.4 |Determine prime factors 1-50 Write numbers as their prime factors 1.5 Identify decimals, fractions, and mixed #'s on a number line. 4Identif positive and negative integers on a number line. Graehsent decimals, fractions and mixed #'s on a number line. Reresent positive and negative integers on a number line. 2.1 |Add, subtract, multiply, and divide with decimals Add with positive and negative integers Subtra ct with positive and negative integers 2.2 Division proficiency with long division with multi-digit divisors. Division proficiency with decimals and fractions 2.3 Solve addition problems involving fractions and mixed #'s Solve subtraction problems involving fractions and mixed #'s 2.4 & 2.5 |Solve multiplication problems involving fractions and mixed

Algebra& Functfio 1.2 aBe Ter to rpent_ anukonnme Write ~~~a simple algebraic expression with one variable

=Graph ordered pairs in the four quadrants of the coordinate 1.5 Solve problems involving linear functions with integer values Write equations of linear functions with integer values Graph the linear function using ordered pairs. 44

Start End KEY Math Description of math standards of of Standards the the ear ear Measurement & Geomet 1.1 Understands and uses the formula for the area of a triangle

Unestands and uses the formula for the area of a quadri~~~~lateral _ 1L2 Construct a cube and rectangular box from 2-D3 pattern _ C~~~~~~~~ompute surface area of a cube and rectangular box

_L3 Understand the concept of volume Compute the volume of rectangular solids 2.1 l Measure, identify, and draw angles Measure, identify, and draw perpendicular and parallel lines 2.2 Know that the sum of triangle angles is 180 degrees Know that the sum of uadrilateral an les is 360 degees Statisti sii Data Anals, and ; ~~~~~Probabilit L 1.4 Identify ordered pairs of data from a graph Interpret the meaning of ordered pair data in terms of the situation depicted by the graph 1.5 Know how to write ordered pairs correctly; for example, (xy) 45

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California 5th grade: Science standards

Standard Description of standards Physical Elements and their combinations account for all the varied types of matter in science the world. As a basis for understanding this concept: l.a Students know that during chemical reactions the atom in the reactants

____ rearrange to form products with different properties L.b Students know all matter is made of atoms, which may combine to form molecules. L.c Students know metals have properties in common, such as high electrical and thermal conductivity. Some metals, such as aluminum (Al), iron (Fe), nickel (Ni), copper (Cu), silver (Ag), and gold (Au), are pure elements; others, such as steel and brass, are composed of a combination of elemental metals. L.d Students know that each element is made of one kind of atom and that the elements are organized in the periodic table by their chemical properties. L.e Students know scientists have developed instruments that can create discrete images of atoms and molecules that show that the atoms and molecules often occur in well-ordered arrays. L.f Students know differences in chemical and physical properties of substances are used to separate mixtures and identify compounds. L.g Students know properties of solid, liquid, and gaseous substances, such as sugar (C 6H 206), water (H2 0), helium (He), oxygen (02), nitrogen (N2), and carbon dioxide (CO.). L.h Students know living organisms and most materials are composed of just a few elements L.i Students know the common properties of salts, such as sodium chloride '-I Life Sciences Plants and animals have structures for respiration, digestion, waste disposal, and transport of materials. As a basis for understanding this concept: 2.a Students know many multicellular organisms have specialized structures to

____ support the transport of materials. 2.b Students know how blood circulates through the heart chambers, lungs, and body and how carbon dioxide (CO2 ) and oxygen (02) are exchanged in the lungs and tissues. 2.c Students know the sequential steps of digestion and the roles of teeth and the mouth, esophagus, stomach, small intestine, large intestine, and colon in the function of the digestive system. 2.d Students know the role of the kidney in removing cellular waste from blood and converting it into urine, which is stored in the bladder. 2.e Students know how sugar, water, and minerals are transported in a vascular plant. 2.f Students know plants use carbon dioxide (CO2) and energy from sunlight to build molecules of sugar and release oxygen 2.g Students know plant and animal cells break down sugar to obtain energy, a process resulting in carbon dioxide (CO,) and water (respiration).

.a I Stuaents Know most or Earth s water is present as salt water in the oceans, which cover most of Earth's surface. 51

Students know when liquid water evaporates, it turns into water vapor in the air and can reappear as a liquid when cooled or as a solid if cooled below the freezing point of water. Students know water vapor in the air moves from one place to another and can form fog or clouds, which are tiny droplets of water or ice, and can fall to Earth as rain, hail, sleet, or snow. 3.d Students know that the amount of fresh water located in rivers, lakes, underground sources, and glaciers is limited and that its availability can be extended by recycling and decreasing the use of water. 3.e Students know the origin of the water used by their local communities. Energy from the Sun heats Earth unevenly, causing air movements that result in changing weather patterns. As a basis for understanding this concept: 4.a Students know uneven heating of Earth causes air movements (convection currents). 4.b Students know the influence that the ocean has on the weather and the role that the water cycle plays in weather patterns. 4.c Students know the causes and effects of different types of severe weather. 4.d Students know how to use weather maps and data to predict local weather and know that weather forecasts depend on many variables. 4.e Students know that Earth's atmosphere exerts a pressure that decreases with distance above Earth's surface and that at any point it exerts this pressure equally in all directions. Solar system The solar system consists of planets and other bodies that orbit the Sun in predictable paths. As a basis for understanding this concept: 5.a Students know the Sun, an average star, is the central and largest body in the solar system and is composed primarily of hydrogen and helium. 5.b Students know the solar system includes the planet Earth, the Moon, the Sun, eight other planets and their satellites, and smaller objects, such as asteroids and comets. 5.c Students know the path of a planet around the Sun is due to the gravitational attraction between the Sun and the planet. Investigating Scientific progress is made by asking meaningful questions and conducting & careful investigations. As a basis for understanding this concept and Experiments addressing the content in the other three strands, students should develop their own questions and perform investigations. Students will: 6.a Classify objects (e.g., rocks, plants, leaves) in accordance with appropriate criteria. 6.b Develop a testable question. 6.c Plan and conduct a simple investigation based on a student-developed question and write instructions others can follow to carry out the procedure. 6.d Identify the dependent and controlled variables in an investigation. 6.e Identify a single independent variable in a scientific investigation and explain how this variable can be used to collect information to answer a question about the results of the experiment. 6.f Select appropriate tools (e.g., thermometers, meter sticks, balances, and graduated cylinders) and make quantitative observations. 6.g Record data by using appropriate graphic representations (including charts, graphs, and labeled diagrams) and make inferences based on those data. 52

California 5 th grade: Social Studies standards

Standards I Description of standard 5.1 Students describe the major pre-Columbian settlements, including the cliff dwellers and pueblo people of the desert Southwest, the American Indians of the Pacific Northwest, the nomadic nations of the Great Plains, and the woodland peoples east of the Mississippi River. 1. Describe how geography and climate influenced the way various nations lived and adjusted to the natural environment, including locations of villages, the distinct structures that they built, and how they obtained food, clothing, tools, and utensils. 2. Describe their varied customs and folklore traditions. 3. Explain their varied economies and systems of government. 5.2 Students trace the routes of early explorers and describe the early explorations of the Americas. 1. Describe the entrepreneurial characteristics of early explorers and technological developments that made sea exploration by latitude and longitude possible. 2. Explain the aims, obstacles, and accomplishments of the explorers, sponsors, and leaders of key European expeditions and the reasons Europeans chose to explore and colonize the world. 3. Trace the routes of the major land explorers of the United States, the distances traveled by explorers, and the Atlantic trade routes that linked Africa, the West Indies, the British colonies, and Europe. 4. Locate on maps of North and South America land claimed by Spain, France, England, Portugal, the Netherlands, Sweden, and Russia. 5.3 Students describe the cooperation and conflict that existed among the American Indians and between the Indian nations and the new settlers. 1. Describe the competition among the English, French, Spanish, Dutch, and Indian nations for control of North America. 2. Describe the cooperation that existed between the colonists and Indians during the 1600s and 1700s. 3. Examine the conflicts before the Revolutionary War. 4. Discuss the role of broken treaties and massacres and the factors that led to the Indians' defeat, including the resistance of Indian nations to encroachments and assimilation. IEM_. . _1311. 53

APPENDIX B

The RestaurateurUnit

Page

A quick glance at the unit ...... 54

Gardener's Dilemma ...... 57

Gardener's Dilemma 11 ...... 61

Gardener's Dilemma m ...... 70

How much is that sandwich?...... 73

United States' recipes...... 76

Be a Food critic ...... 78

Ingredient breakdown...... 79

What's a better deal? ...... 86

Feeding an army ...... 92

The business proposal...... 98

Student jobs and descriptions ...... 102

Area and perimeter calculations ...... 103

Teacher's Grocery Guide with completed calculations...... 107 -

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Teacher's plan: Gardener's Dilemma I

Math pre-requisite: Students have a Math concept: Possible math song: basic conceptual & procedural * Calculating area Area andperimeter - understanding of arithmetic * Calculating perimeter easy tofind Key standards: * M.G 1.1: Understands and uses the formula for the area of a triangle and quadrilateral * M.G 2.1: Measure, identify, and draw angles, perpendicular lines and parallel lines * N.S 2.1: Add, subtract, multiply and divide with whole and decimal numbers Vocabulary: Materials: * Area: the measure of a surface expressed in square units * Closedfigure: a geometrical figure where the beginning point of Graph/ dot paper a line and ending point of a line meet. PDAG (Perimeter, * Height: the vertical measurement of an object dimensions, area * Length: the horizontal measurement of an object grid) * Perimeter: The distance around a simple closed figure Procedure:

1. Introduce/ review the math song, Area andperimeter - easy to find 2. Have students read Garland's scenario and begin creating planters. Inform students that there are multiple ways to configure the planks to create a closed shape, and uses all the planks. Possible plank amounts you can assign student groups: A B C D E 12 18 14 8 10 24 20 22 16 24 26 24 24 24 30 40 32 36 48 34

3. Have students collaborate and come up with a persuasive argument as to which design, corresponding to the plank amount, Garland should present to her clients. *Note: encourage students to present a mathematical argument when discussing their

design, instead of providing the answer, ". . .because it would just be good." Have students justify their answer using mathematical terms. 58

Student scenario from teacher's lesson: Gardener's Diemina You've been hired to assist Garland, a gardener and landscape architect, with her dilemma. Garland's business has been extremely busy, and she doesn't have the time to design garden planters anymore or determine which planter design provides the greatest area for the plants to grow. Your job is to help Garland design as many planters as possible using the set amount of Iby I foot planks provided for each job. Garland's clients have been rather particular about their planters. Her numerous clients insist that their planters consist of: * Four 90° angles * Uses all the planks provided * Creates the maximum space (area) Use the dotted paper below to design possible planters for Garland to utilize, calculate the area for each design, and provide a brief recommendation for Garland as to which design would be the best for her to present to her clients. iI i f . -.... lo~---.--1-.....----- i -- ---.------...... i i I i i . I...... I ------F i i r i i i

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Gardener'sDilemma You've been hired to assist Garland, a gardener, with her dilemma. Garland's business has been extremely busy, and she does not have the time to design garden planters anymore or determine which planter design provides the greatest area for the plants to grow. Your job is to help Garland design as many planters as possible using the set amount of I by 1-foot planks provided for each job. Garland's clients have been rather particular about their planters. Her numerous clients insist that their planters consist of: * Four 900 angles * Uses all the planks provided * Creates the maximum space (area) Use the dotted paper below to design possible planters for Garland to utilize, calculate the area for each design, and provide a brief recommendation for Garland as to which design would be the best for her to present to her clients.

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I 61

Teacher's plan: Gardener's Dilemma II

Math pre-requisite: Students have a Math concept: Possible math song: basic conceptual & procedural * Calculating area Area andperimeter - understanding of arithmetic * Calculating perimeter easy tofind Key standards: * A.F 1.2: Use a letter to represent an unknown number; write, and evaluate a simple algebraic expression with one variable substitution. * M.G 1.1: Understands and uses the formula for the area of a triangle and quadrilateral * M.G 2.1: Measure, identify, and draw angles, perpendicular lines and parallel lines * N.S 2.1: Add, subtract, multiply and divide with whole and decimal numbers Vocabulary: Materials: * Graph/ dot paper PDAG Procedure:

1. Introduce/ review the math song, Area andperimeter - easy to find 2. Have students read Garland's scenario and begin creating planters. Inform students there are multiple ways to configure the planks to create a closed shape that encompasses the specified areas.

Possible areameasurements you can assign student groups: 12 24 36 40 48 60 64 80 96 100

3. Have students collaborate and come up with a persuasive argument as to which two designs, corresponding to the plank design utilizing the assigned area, Garland should present to her clients. *Note: encourage students to present a mathematical argument when discussing their

design, instead of providing the answer, ". . .because it would just be good." Have students justify their answer using mathematical terms. 62

Student scenario from teacher's lesson: Gardener's Dilemma II Garland is having difficulty keeping up with all the business after the terrific work she did on the planters (with your help, of course). The planters were such a huge success that Garland expanded her business to include garden design. New clients approached Garland creating a garden in their backyard. You'll be assisting Garland with developing fabulous garden designs for her clients. Garland informed you that the clients' expect: * Utilize all the area allocated for the garden * Corners should only consist of 900 angles * Determine the perimeter (number of 1' by 1' planks needed) Using the space below, develop different garden designs, calculate the perimeter; and provide Garland with two design recommendations to be presented to the new clients. i ...... ___.._. I ____ _.1 _ .. .i i

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Name:

Gardener'sDilemma H Garland is having difficulty keeping up with all the business after the terrific work she did on the planters (with your help, of course). The planters were such a huge success that Garland expanded her business to include garden design. New clients approached Garland creating a garden in their backyard. You will be assisting Garland with developing fabulous garden designs for her clients. Garland informed you that the clients' expect: * Utilize all the area allocated for the garden * Corners should only consist of 900 angles * Determine the perimeter (number of 1' by 1' planks needed) Using the provided space below, develop different garden designs, calculate the perimeter, and provide Garland with two design recommendations that Garland can be present to the new clients.

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PADS grid Name:

Perimeter Dimensions Area of faces | Surface (# of planks) j (Length & width) | area 65

Teacher's plan: Gardener's Dilemma m

Math pre-requisite: Students have a Math concept: Possible math song: basic conceptual & procedural * area * Area and understanding of arithmetic * perimeter perimeter- * surface area easy tofind * volume * Surface area * Coordinate plane Key standards: * A.F 1.4: Identify and graph ordered pairs in the four quadrants of the coordinate plane * M.G 1.1: Understands and uses the formula for the area of a triangle and quadrilateral * M.G 1.2: Compute surface area of a cube and rectangular box * M.G 1.3: Understand the concept of volume; compute the volume or rectangular solids * M.G 2.1: Measure, identify, and draw angles, perpendicular lines and parallel lines • N 4 I 1 - Add siihtrart miltinlv and dividp with whn1o and decimal nuimber Vocabulary: Materials: * Surface area: the total area of a space figure Ruler * Volume: the measure of the space bounded by a closed space Graph paper figure. The unit of measure is called a cubic unit, for example, cubic foot or cubic meter. * Landscape architect: someone who arranges features of the landscape or garden attractively Procedure:

1. Introduce/ review the math song, Surface Area 2. Introduce vocabulary words 3. Have students read Garland's next dilemma. You may need to provide examples of calculating surface area and volume. 4. Have students calculate the area, perimeter, surface area, and volume (for the planters only) on each of their design choices before tabulating the total area used by specific tree, and vegetable choices. 5. Have students reflect in a journal, on how they began the process and why they made their aesthetic choices. 66

Student scenario from teacher's lesson: Gardener'sDilemma III The City commissioned Garland to build and design planter boxes to hold herbs within the community garden. The City would also like an artistic fountain, which Garland was to design, positioned at the center of the garden. However, Garland is too busy and hired you to tackle this project. The City wants to plant a variety of trees, and vegetation in the garden. Since they are particular, they included a list of how much space each plant requires in order to grow properly, so you are able to design the garden accordingly (see list A); while Garland's crew needs to know certain calculations seen in list B to build and paint the planters. LIST A: The City Requirements: * 4 - 6 planters for mint (each planter with a base area of 12 sq units * 64 sq. units used for gathering/ dining area. (area may not be split) * A total of 40 sq units used for 2 benches; area may be split. * A 3 by x path/paths that lead to the fountain At least 300 sq units is designated for growing crops. Design where you would plant the different vegetative life, YOU have full artistic control. Be sure to label each area with its proper symbol. Here's what you need to know: Vegetation Legend Area plants Vegetation Legend Area plants need to grow symbols need to grow symbols Lemon tree L 9 sq units Corn C 4 sq units Maple tree MP 9 sq units Eggplant E I sq unit Orange tree OR 9 sq units Garlic G 2 g's per sq unit Willow tree W 9 sq units Onions ON 2 onions per sq unit Peppers P 2 sq units Tomato T 4 sq units LIST B: Garland's crew: * Determine the area of each box side * Calculate the total surface area of each planter so they know how much waterproof stain to use * Calculate volume so the crew knows how much soil needed to cover the entire planter. * Design a geometric fountain, based on what you've learned about shapes, area and perimeter. a. Your fountain must have: i. The material the fountain should be built from ii. Dimensions needed to build the fountain iii. Surface area calculated so enough paint can be ordered iv. Volume so The City knows the maximum water capacity the fountain will use *Bonus: build a 3D scale model of what you envision your fountain to look like. 67

Possible extension - Designingyour space: Using the skills they learnedin this lesson, have students landscapetheir dream backyard of 600 square units. Have students consider the items they would place in their backyard, and what the area andperimeter of each item takes. Have students provide a brief synopsis of why they would include certain elements in their backyard, and what they plan to do when the yard is completed

Backyard items to consider: Swimming pool garden tree house dog house BBQ grill

basketball court trampoline

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Example of landscape choices: 69

Item Section x-coordinate y_ coordinate Fountain origin -3 to 2 -1 to I Dining area East 3 to 10 -4 to 3 Orange tree Southwest -5 to -3 -15 to -13 ORI Orange tree Southwest -9 to -7 -13 to -11 0R2

I I I I 70

Gardener'sDilemma III The City commissioned Garland to build and design planter boxes to hold herbs within the community garden. The City would also like an artistic fountain, which Garland was to design, positioned at the center of the garden. However, Garland is too busy and hired you to tackle this project. The City wants to plant a variety of trees, and vegetation in the garden. Since they are particular, they included a list of how much space each plant requires in order to grow properly, so you are able to design the garden accordingly (see list A); while Garland's crew needs to know certain calculations seen in list B to build and paint the planters. LIST A: The City Requirements: * 4 - 6 planters for mint (each planter with a base area of 12 sq units * 64 sq. units used for gathering/ dining area. (area may not be split) * A total of 40 sq units used for 2 benches; area may be split. * A 3 by x path/paths that lead to the fountain At least 300 sq units is designated for growing crops. Design where you would plant the different vegetative life, YOU have full artistic control. Be sure to label each area with its proper symbol. Here's what you need to know: Vegetation Legend Area plants Vegetation Legend Area plants need to grow symbols need to grow symbols Lemon tree L 9 sq units Corn C 4 sq units Maple tree MP 9 sq units Eggplant E 1 sq unit Orange OR 9 sq units Garlic G 2 g's per sq unit

tree _ _ _ _ _ Willow W 9 sq units Onions ON 2 onions per sq unit

tree__ _ _ _ Peppers P 2 sq units Tomato T 4 sq units LIST B: Garland's crew: * Determine the area of each box side * Calculate the total surface area of each planter so they know how much waterproof stain to use * Calculate volume so the crew knows how much soil needed to cover the entire planter. * Design a geometric fountain, based on what you've learned about shapes, area and perimeter. a. Your fountain must have: i. The material the fountain should be built from ii. Dimensions needed to build the fountain iii. Surface area calculated so enough paint can be ordered iv. Volume so The City knows the maximum water capacity the fountain will use *Bonus: build a 3D scale model of what you envision your fountain to look like. 71

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Teacher's Plan: How many sandwiches

Math pre-requisite: Students have a Math concept: Possible math song: basic conceptual & procedural * Adding and subtracting Decimal rhyme understanding of arithmetic with decimals * Basic multiplication and I__ _ _ _ division_ practice I Key standards: * N.S 2.1: Add, subtract, multiply and divide with whole and decimal numbers * W 1.2 Create an expository composition as events in a sequence or chronological order. Vocabulary: Materials: * Ingredient: a compound or mixture used to make something How many * Servings: a quantity of food or drink sandwiches chart

Grocery list Procedure:

1. Have students list ingredients needed to make a peanut butter and jelly sandwich 2. (LA) Have students write a descriptive, detailed procedure to making a peanut and butter jelly. 3. (Optional: have students analyze and evaluate their own procedures, after following each step explicitly.) 4. Have students calculate the number of sandwiches that can be made using each ingredient.

Example: Item Food item Food item Total number Amt. needed Total quantity cost (ingredients) of servings for one number of $ serving sandwiches 1 3.99 Rainbow Bread 32 slices 2 slices 32 + 2 = 16 1 3.79 Skippy peanut 16 ounces (oz)2oz 16 2 = 8 butter I 1 5.59 Smucker's 32 Oz 2 oz 32 -. 2 = 16 strawberry jelly

5. Have students continue practicing their basic division skills by having students see what type and how many of their chosen sandwiches can make.

Possible types of sandwiches students can Possible condiments students can include in evaluate: their sandwich: * Ham and cheese * Mayonnaise * Bacon, lettuce and tomato * Mustard * Roast beef * Pickles *Any ingredient that a student includes in their sandwich must be accounted for. 74

How many sandwiches? Name:

Date:

Type of sandwich:

Food item Total number of Amt. needed for one I Total number of (ingredients) servings serving sandwiches

Type of sandwich:

Food item Total number of Amt. needed for one Total number of (ingredients) servings serving sandwiches 75

Type of sandwich:

Food item Total number of Amt. needed for one Total number of (ingredients) servings serving sandwiches 76

Teacher's Plan: U.S. Recipes

Pre-requisite: * Students are capable of reading and writing independently * Students know how to develop a hypothesis that's ground in some theoretical knowledge * Students understand what creates a habitat Key standards: . W 1.1: Create multiple-paragraph narrative compositions . W 1.2: Create an expository composition as events in a sequence or chronological order. . WA 2.1: Write narratives . S 3.a: Students know most of Earth's water is present as salt water in the oceans, which cover most of Earth's surface. * S 3.b: Students know when liquid water evaporates, it turns into water vapor in the air and can reappear as a liquid when cooled or as a solid if cooled below the freezing point of water . S 3.c: Students know water vapor in the air moves from one place to another and can form fog or clouds, which are tiny droplets of water or ice, and can fall to Earth as rain, hail, sleet, or snow. . S 3.d: Students know that the amount of fresh water located in rivers, lakes, underground sources, and glaciers is limited and that its availability can be extended by recycling and decreasing the use of water. . S 4.d: Students know how to use weather maps and data to predict local weather and know that weather forecasts depend on many variables Vocabulary: Materials: * Recipe: a set of instructions as to combine or put together How many ingredients. sandwiches chart * Habitat: area or environment where an organism lives * Hypothesis: an educated guess Grocery list Procedure:

1. Ask students to consider which types of food are traditional regional cuisines. For example, the Midwest is known for having "buffalos roam," thus it would reasonable to speculate that traditional Midwest cuisines have buffalo as a main ingredient. 2. You may either assign students to examine different regions of the U.S, or give students names of dishes and have them decipher which areas renowned for that particular dish.

Example: Chicago = deep dish pizza

3. Have students use their knowledge of climate and weather, and locations to hypothesize these regional dishes, making sure students justify their choices. 77

Possible regions for students to examine: Possible dishes for students to hypothesize the recipes for: * Pacific Northwest * BBQ ribs/ chicken/ pork * West * Collard greens * Southwest * Pizza * South * California roll * Midwest * Hot tamales * East * Clam chowders * Fried chicken * Philly cheese steak NOTE: recipes can befound in various cookbooks, or atfoodnetworkcom

Possible extension activity: All around the worldfood exists. Each country is known for theirparticular delicacies. Japan is the home of sushi; Spainfor its tapas and Thailandfor its hot pad Thai. Have students choose a country and discover the foods most commonly associatedwith each place.

Other related activities: * Have students write a students' write a narrative o Possible story topics: * Life as a famous chef in • Kitchen culinary disaster * My first cooking experience * Favorite foods * Have students research and present information regarding an assigned region or state within a particular regional area. 78

Teacher's Plan: Food critic

Pre-requisite: * Students are capable of reading and writing independently * Students know the steps to evaluate an article Key standards: * W 1.1: Create multiple-paragraph narrative compositions * W 1.2: Create a multiple-paragraph expository composition * W 1.4: Create simple documents by using electronic media and employing organizational features * W 1.5: Use a thesaurus to identify alternative work choices and meanings * W 1.6: Edit and revise manuscripts to improve the meaning and focus of writing * WA 2.4: Write persuasive compositions Vocabulary: Materials: * Ambiance: an atmosphere created by the environment Food review * Critique: an evaluation or critical review of someone's work * Fact: something that can be proven with concrete evidence * Opinion: a belief or conclusion that one holds which cannot be proven with evidence.

Procedure:

1. Introduce vocabulary 2. Introduce the difference between fact and opinion 3. Explain what critics look for when they evaluate anything, like restaurants or food 4. Have students write a critique on a dish they have eaten before or a restaurant they have tried. * Explain to students that when they evaluate a restaurant, they should examine: o Ambiance o Customer service o Quality of food * Explain to students that when they evaluate food, they should examine: o Taste - utilize all five senses for a better description o Portion size o Correct meal 79

Teacher's plan: Ingredient breakdown

Overview: When it comes to cooking, only a portion of an ingredient is needed to complete a particular recipe. Students will use the grocery guide and grocery list to determine how much a single serving of various ingredients will cost. Procedure: I. Tell students: When it comes to cooking, sometimes only a portion of an ingredient is needed to complete a dish. You would not use an entire bottle of ketchup to dress a hot dog. As a perspective businessperson and restaurateur, knowing how many servings an ingredient provides will help you save money in the end. 2. Have students break down serving sizes of bulk ingredients *The cost of ingredients and portions derive from the products sold at Costco* 3. Optional:you may provide the conversion sizes of each ingredient, or have students collaborateto complete the entire chart. 4. Give students the grocery guide, some items have already been completed for them. 80

Single Serving size recommendations

Dish Ingredients Recommended Notes Serving Size Beverages Coffee 8- 10 oz Juice Soda Tea Water Burger Bread 2 slices Ground beef patty 6 - 8 oz Lettuce, tomato, pickle 2 -3 oz Sauce 0.5 oz- I oz French fries Potatoes 12 - 16 oz Hot dog Hot dog bun I bun Hot link 4 oz Ketchup 0.25 - 0.75 oz Mustard 0.25 - 0.75 oz relish 0.25 - 0.75 oz Macaroni Elbow noodles 8 - 10 oz and cheese Cheddar cheese 6 - 8 oz Cream or whole milk I - 2 oz Meats Beef 8-12 oz Chicken Pork Fish Omelet Eggs 2 eggs Hotlinks 2 oz Spinach 2 oz Onions I oz Cheese I - 1.5 oz Rice Rice 8 - 10 oz Salad Lettuce or spinach 6 oz Other vegetables 2-3 oz Salad dressing 4 - 6 oz Sandwiches Bread 2 slices Sliced deli meat 2-3 oz Sliced cheese 2-3 oz Vegetables 2-3 oz Mustard, mayo 0.5 - I oz Scrambled Eggs 2 eggs eggs Spaghetti Spaghetti 8- 10 oz Tomato sauce (Ragu) 10 - 12 oz Ground beef 6 - 8 oz Oil for frying Corn oil 32 - 40 oz Assume thatfrying oil can be used repeatedly, up to 10 times if used tofry the same dish only. Ex: FrenchfriesONLY; fried chicken in a different pot. NOTE: Students can also find a single serving size listed on the label of any ingredient 81

Grocery Guide

Item Total Total Total Cost Serving Cost weight Weight units cost per size per lbs/gallon oz/fl. oz _ $ lb/oz/unit serving PROTEIN Beef: Beef loin 15.52 lbs 58.82 Ground Beef 5.89 lbs 2.89/lb Rib-eye steak 4.00 lbs 27.96 6.99/lb Tri-tip 4.72 lbs 2.99/lb Chicken: Drumsticks 7.93 lbs 9.44 1.19/lb Whole 9.55 lbs 9.45 0.99/lb Eggs 15 doz 16.99 1.13/doz Eggs 2 doz 2.59 1.30/doz Pork: Bacon 4.00 lbs 8.99 Pork ribs 5.24 lbs 17.24 Pork tenderloin 7.38 lbs 13.95 1.89/lb Fish: Canned tuna 7oz/ can 1c2 10.99

______c a n s _ _ _ _ _ Catfish 2.72 lbs 14.93 5.49/lb Salmon 1.71 lbs 13.66 7.99/lb Other: Sliced ham 1.87 lbs 6.15 Sliced turkey 2.00 lbs 11.99 5.99 Slicked roast 1.36 lbs 6.79 4.99 beef ______Thin sliced 30 oz 7.99 turkey 82

Item Total Total Total Cost Serving Cost weight Weight units cost per size per lbs/gallon oz/fl. oz $ lb/oz/unit serving PRODUCE

Asparagus 2.25 lbs 4.99 2.21/lb

Broccoli 3.00 lbs 3.99 1.33/lb Garlic 3.00 lbs 4.79 0.96/lb Lettuce 6 2.79 Oranges 13.00 lbs 6.99 0.54/lb Potatoes 20.00 lbs 7.99 Red peppers 6 4.99 Spinach 2.50 lbs 3.79 Yellow onions 10.00 lbs 3.29 0.33/lb

Frozenfruits Blueberries 3.00 lbs 13.63 Festival fruit: 6.00 lbs 7.79 pineapple, mango, papaya, Strawberries 6.00 lbs 8.79 Tri-berries: 4.00 lbs 10.99 blueberries, raspberries marionberries Fruitjuices Apple juice 1 gallon 4.39 Apple, peach, 96 oz/ 2 5.99 passion fruit bottle Orange juice 2 gallons 9.99 Strawberry 96 oz 7.79 kiwi 83

Item Total Total # Total Cost Serving Cost weight Weight units cost per size per lbs/gallon oz/fl. oz $ lb/oz/unit serving DAIRY 2 % milk 2 gallons 3.79 Soy milk 1.5 gallon 7.49 Cheddar 2.5 lbs 6.59 chevse Butter 4.00 lbs 6.00 Carnation milk 12 oz/can 12 9.85

He ~~~~~~~~~cansESi

Elbow noodles 1.10 lbs 6 7.99

Rice 50 lbs 25 25.99 Spaghetti 1.10 lbs 8 8.29 packs

Demi baguettes 6 4.49 Hamburger 24 1.99 Hot dog buns 24 1.99 Sliced bread 2.00 lbs 48 3.75 slices FATS/OILS Corn oil 2.5 gallons 16.29 Olive oil 1.05 20.00 gallons MISCELLANEOUS

Flour 50 lbs 12.99 0.41/lb Salt 25 lbs 3.99 0.26/lb Sugar 50 lbs 20.49 0.16/lb Peanut butter 32 oz 5.59 Strawberry 16 oz 3.99 jelly 84

Item Total Total Total Cost Serving Cost weight Weight units cost per size per lbs/gallon oz/fl. oz $ lb/oz/unit serving OTHER Balsamic .26 10.99 vinegar gallons Chocolate 72 oz 7.79 0.1 /oz chips Ketchup 44 fl. oz 4.99 Mustard 44 fl. oz 5.99 Ranch dressing 40 fl. oz/ 2 8.99 bottle Relish 30 fl. oz 2.39 Spaghetti sauce 44 fl. 3 7.69 oz/bottle Teriyaki sauce 63 fl. oz 6.89 BEVERAGES Coffee beans 3.00 Ibs 10.99 Soda 36 9.19 0.25/can cans 85

Name

Date _

Grocery List >

Item Total Total # Total Cost Serving Cost per weight Weight units cost per size serving lbs/gallon oz/fl. oz $ ib/oz/unit

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Teacher's plan: What's a better deal

Math pre-requisite: Students have a Math concept: Possible math song: basic conceptual & procedural * Add, subtract, multiply Decimal rhyme understanding of arithmetic and divide with whole Fractionsong and decimal numbers Key standards: * NS 2.1: Add, subtract, multiply and divide with whole and decimal numbers * AF 1.2: Use a letter to represent an unknown number; write and evaluate a simple algebraic expression with one variable substitution. Vocabulary: Materials: * Bargain: something purchased at a cheap price Competitor comparison Completed Grocery guide Grocery list Serving size recommendation Overview: Students discover which is more cost effective, eating out, or staying at home cooking. Students examine individual costs, as well as the cost to feed a family of 3 -5. Students will also write a persuasive argument as to which option is better. Procedure: 1. Tell students: When it comes to cooking, sometimes only a portion of an ingredient is needed to complete a dish. You would not use an entire bottle of ketchup to dress a hot dog. As a perspective businessperson and restaurateur, knowing how many servings an ingredient provides will help you save money in the end. 2. Have students break down serving sizes of bulk ingredients - You may have students round to the nearest dollar; tenth of a cent; or cent. The same goes for the weight of each product. *The cost of ingredients and portions derive from the products sold at Costco* 3. Optional: you may provide the conversion sizes of each ingredient, or have students collaborateto complete the entire chart. 4. Give students the grocery guide; some items are already completed for them. 5. Here's an example of the comparing: *Note: Let students that they can only make as many food items as the least ingredient you have. For example, a student cannotfly more than 20 servings of Frenchfries because there aren 't enough potatoes, even though there is stillplenty of oil left. Also, let students know that oil can be reused at least 4-6 times before discarding, as long as the same recipe gets fried - i.e. fries in one pot only; chicken in anotherfryer. 87

6. Have students determine which is most cost effective for an individual, and a 4-5 member family. 7. Have students examine: * $5 foot long sandwich at Subway * Various types of smoothies from Jamba * Burgers and combination meals from any burger place 8. Have students reflect on their findings in a journal, and write a persuasive article as to which is most cost effective.

Things to consider: What meals are the most cost effective? Is there a better time to eat out? When? How many people would it be more cost effective for?

Competitor chart

Homemade VS In - N - Out

Baffle: French fries

Ingredient Total Initial cost # of Cost per Total cost Total cost Competitor Cost needed amount of for servings serving for initial of dish for cost ($) for difference ingredient ingredient ($) ingredients single single For single ($) serving serving serving

Potatoes 20 lbs 8 20 0.40

Cornoil 32oz 10 20 0.25 18 0.65 ' 1.19 0.54

A single serving of homemade fries = 0.65; while it would cost $18 for 20 servings of French fries The difference for 20 servings of fries is: (1.19 X 20)- 18 = 5.80 88

Single Serving size recommendations

Dish Ingredients Recommended Notes Serving Size Beverages Coffee 8 - 10 oz Juice Soda Tea Water Burger Bread 2 slices Ground beef patty 6 - 8 oz Lettuce, tomato, pickle 2 -3 oz Sauce 0.5 oz- I oz French fries Potatoes 12 - 16 oz Hot dog Hot dog bun I bun Hot link 4 oz Ketchup 0.25 - 0.75 oz Mustard 0.25 - 0.75 oz relish 0.25 - 0.75 oz Macaroni Elbow noodles 8 - 10 oz and cheese Cheddar cheese 6 - 8 oz Cream or whole milk 1 - 2 oz Meats Beef 8- 12 oz Chicken Pork Fish Omelet Eggs 2 eggs Hotlinks 2 oz Spinach 2 oz Onions I oz Cheese I -1.5 oz Rice Rice 8-10 oz Salad Lettuce or spinach 6 oz Other vegetables 2-3 oz Salad dressing 4 - 6 oz Sandwiches Bread 2 slices Sliced deli meat 2-3 oz Sliced cheese 2 - 3 oz Vegetables 2-3 oz __Mustard, mayo 0.5 - I oz Scrambled Eggs 2 eggs eggs Spaghetti Spaghetti 8 - 10 oz Tomato sauce (Ragu) 10- 12 oz Ground beef 6 - 8 oz Oil for frying Corn oil 32 - 40 oz Assume thatfrying oil can be used repeatedly, up to IO times if used tofry the same dish only. Ex: Frenchfries ONLY; fried chicken in a different pot. NOTE: Students can also find a single serving size listed on the label of any ingredient 89

Name

Date _

Grocery List

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Teacher's plan: Feeding an army

Math pre-requisite: Students have a Math concept: Possible math basic conceptual & procedural * Add, subtract, multiply song: understanding of arithmetic and divide with whole Decimal rhyme and decimal numbers Average sound off Key standards: * NS 2.1: Add, subtract, multiply and divide with whole and decimal numbers * AF 1.2: Use a letter to represent an unknown number; write and evaluate a simple algebraic expression with one variable substitution. Vocabulary: Materials: . budget: a sum of money allocated for a particular purpose. Completed Grocery * quantity: how much there is or how many there are of guide something that you can quantify Grocery list * variable: a symbol representing an unknown quantity recommendationServing size Dish deconstruction table Overview: Students' knowledge of the amount of servings each ingredient allows them to determine the quantity of each ingredient they would need to cater different events with x amount of people. Each group will decide upon a menu for their assigned catering event, and calculate the total cost to feed everyone. Assume that everyone at the function will only eat one serving of each meal. Procedure: 1. Tell students that their catering company has been hired to cater a large event in a month; and must begin planning accordingly. Tell students they will need to calculate the cost of making each menu item and the quantity of each necessary ingredient. 2. You may have students randomly choose the event they will plan to cater, or you may assign them. 3. You may have students randomly choose a number of guests that will be in attendance, or allocate a specific number(s) to them, perhaps having students practice with a small gathering of 2 and 4 before assigning larger numbers of 325 or more. Frenchfries because there aren 't enough potatoes, even though there is still plenty of oil left. Also, let students know that oil can be reused at least 4-6 times before discarding, as long as the same recipe gets fried - i.e. fries in one pot only; chicken in anotherfryer 93

Possible events students can cater: Event Summary On the menu Breakfast A breakfast event Pancakes/ toast; bacon/sausage; scrambled bonanza eggs; orange juice/coffee Egg-celent A breakfast event consisting of Scrambled eggs/ omelets extravaganza cooked egg variations - Spinach, onion mix - Spinach, onion, cheese mix - Hot links, onion - Hot links, spinach, onion - plain Sensational Lunch consisting of various Sandwich combinations (all include lettuce, sandwiches sandwich combinations tomato, mayo and/or mustard): - ham & cheese - turkey - bacon, lettuce, tomato (BLT) - roast beef - veggie Pizza party Lunch consisting of Pizza combinations (all pizzas have a personalized pizza (6" tomato sauce base and a layer of cheese): diameter) - cheese only - pepperoni - mushroom, pepper, olive - pepperoni & sausage - pineapple & bacon Simple BBQ Easy to grill foods Hot dogs, hamburgers, salad, chips, and soda BBQ A BBQ including more time Beef tri-tip; pork ribs; chicken drumsticks; throw-down consuming foods to grill for a green salad; mashed potatoes; vegetable picnic feast at a park. platter; and soda Spaghetti Spaghetti dinner for a Spaghetti, garlic bread, spinach salad Spectacle community _-EM ei nt 94

Challenge: Instead of allowing students to an open-ended amount, you can restrict their budget to $300 - $5000 depending on the attendancefor the assignedevent to see how creative students will be in their menu development. Have students try to generalize aformula to determine the totalfood cost with 'xV'people. Example: If 1 soda = 0.25. then, 3 sodas = (0.25)(3) Iff hotdog costs = 0.98 then, 3 hotdogs = (0.98) (3) Total cost for a hotdog and sodafor 1 person = 0.25 + 0.98 = $1.23 Total cost for a hotdog and sodafor 3 people: Method 1: 1.23 x 3 = 3.69 Method 2: (0.25 x 3) + (0.98 x 3) = 3.69 Method 3: 3 (0.25 + 0.98) = 3.69 Generalizationfor a hotdog with ketchup, mustard, and soda: Using method 3: x(0.25 + 0.98) = M let M = total cost of a hotdog meal. 95

Single Serving size recommendations

Dish Ingredients Recommended Notes Serving Size Beverages Coffee 8 - 10 oz Juice Soda Tea Water Burger Bread 2 slices Ground beef patty 6 - 8 oz Lettuce, tomato, pickle 2-3 oz Sauce 0.5 oz- I oz French fries Potatoes 12- 16 oz Hot dog Hot dog bun I bun Hot link 4 oz Ketchup 0.25 - 0.75 oz Mustard 0.25 - 0.75 oz relish 0.25 - 0.75 oz Macaroni and Elbow noodles 8 - 10 oz cheese Cheddar cheese 6 - 8 oz Cream or whole milk 1 - 2 oz Meats Beef 8 - 12 oz Chicken Pork Fish Omelet Eggs 2 eggs Hotlinks 2 oz Spinach 2 oz Onions I oz Cheese I - 1.5 oz Rice Rice 8 - 10 oz Salad Lettuce or spinach 6 oz Other vegetables 2-3 oz Salad dressing 4 - 6 oz Sandwiches Bread 2 slices Sliced deli meat 2-3 oz Sliced cheese 2 - 3 oz Vegetables 2 - 3 oz Mustard, mayo 0.5 - Ioz Scrambled Eggs 2 eggs eggs Spaghetti Spaghetti 8 - 10 oz Tomato sauce (Ragu) 10 - 12 oz Ground beef 6- 8 oz Oil for frying Corn oil 32 - 40 oz Assume thatfrying oil can be used repeatedly, up to IO times if used tofry the same dish only. Ex: Frenchfries ONLY; fried chicken in a different pot. NOTE: Students can also find a single serving size listed on the label of any ingredient 96

Name

Date _

Grocery List

Item Total Total Total Cost Serving Cost weight Weight units cost per size per lbs/gallon Ozlfl. Oz S lb/ozlunit serving 97

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km 98

Teacher's plan: Business proposal

Math pre-requisite: Students have a basic conceptual & procedural Possible math understanding of arithmetic songs: Math concept: Area andperimeter * Calculation of area, perimeter, surface area, and volume. Surface area * Add, subtract, multiply, divide with whole numbers and Decimal rhyme decimals. Averages Sound Off * Present information to gain investor backing * Write a expository compositions regarding the goals of the restaurant * Understand propaganda and how it is used as an advertising technique

Key standards: * AF: 1.2; MG: 1.1; 1.2, 1.3;N.S: 2.1; SDAP: 1.4; RC: 2.2; 2.5; RRA: 3.7W: 1.1; 1.2; 1.4; 1.5;1.6 Vocabulary: Materials: * Conjunction: bring together Business plan template * Colleagues: peers who you collaborate with; members who work in the same field. * Investor: a person who provides financial backing of an idea or company Overview: Students will develop their own business plan regarding their proposed restaurants. Students business proposal must include: * Restaurant name * Floor plan with calculations of area, perimeter, surface area, and volume (if applicable) * Menu * Estimated cost of a single serving of each dish * Budget * Perspective amount that your company intends to charge customers * Advertising strategies and techniques Procedure: 1. Tell students that this is their culminating experience, which they utilize all the lessons and skills they learned throughout the entire unit. 2. Provide students with time to work on the project. Suggest that students work on their projects at home as well as during class. 99

3. Have students present their proposals. Those students in the audience take on the role of investors and decide if the presenting group deserves their financially backing or not. o Perspective investors should consider: * Was the group's proposal clear and concise? * Were menu prices realistic and reasonable? * Are the calculations correct? * How persuasive were the presenters? * Would you invest in this company? Why? Or Why not? 4. Have the presenting students evaluate themselves and their group. Have students write about: * What role they played in the group? * How well they feel their group worked together? * What, if any issues arose and how they were resolved? * Did everyone contribute? * What worked well? What would they do differently? * What did they learn from this experience? * What grade do they feel is fair for themselves, and what grade they feel their group deserves? 100

Student scenario Utilizing all the mathematical concepts, you have learned thus far to create a business proposal. You and a few colleges collaborate and combine your talents together to opening up a business. Unfortunately, none of you has enough money to open up your business without assistance. One of your colleagues was able to schedule an appointment to meet with potential investors who are financially willing to support your idea; however, you and your colleagues must develop a persuasive business proposal, and present a brief PowerPoint to these investors. You and your colleagues have one chance to impress the board, so you want to make sure your plan contains the necessary information.

Business proposal should include: > Cover page o Plan name o Company name o Persons included in the company > Company summary: o This is where you discuss who the partners are and how they will contribute to the plan and the business. > Business information: o The name of the business? o Floor plan: including the needed area, and the dimensions (perimeter). Include any appliances necessary (ex: chairs, table, freezer, stove, etc) > Menu information: o Name of dishes with a listing of ingredients o Cost of a single serving of each dish > Who is your target of customers? (demographics) o How do you plan getting them to frequent your business? o Advertising techniques or strategies o Possible extension: actually creating an advertisementforyour buisness > Financial information: o Projected monthly budget o How much you plan on charging customers with justification o Projected profit

6 101

The Business Proposal Utilizing all the mathematical concepts, you have learned thus far to create a business proposal. You and a few colleges collaborate and combine your talents together to opening up a business. Unfortunately, none of you has enough money to open up your business without assistance. One of your colleagues was able to schedule an appointment to meet with potential investors who are financially willing to support your idea; however, you and your colleagues must develop a persuasive business proposal, and present a brief PowerPoint to these investors. You and your colleagues have one chance to impress the board, so you want to make sure your plan contains the necessary information.

Student job roles and their descriptions

Student job Shortcut Description ofjob/role symbol Architect A This student is responsible for illustrating the final draft of any floor plans, or other necessary drawings for any projects. The final designs must be drawn to scale utilizing straight clean lines. California public CPA This student assists other members who have difficulties accountant with calculations. This student assists their peers by demonstrating the procedure, or re-explaining the concept. This student IS NOT RESPONSIBLE for doing all computational work for the group. Computer CE Student uses the computer to generate group presentations engineer after the completion of the final draft. This student is proficient in the use of PowerPoint and Word. Editor Ed This student double-checks the final written draft to ensure that the document is grammatically sound - punctuations after every sentence; capitalizing appropriately. This student puts together a cover for any proposal. This student helps facilitate the discussion of topics. Event coordinator EC Student who takes notes of key ideas and relevant suggestions that made during collaboration. This student also ensures that time is being utilized wisely by keeping track of time allotted to work on collaborative assignment. Production analyst PA This student acquires all the necessary materials from the teacher for each group member. This student is also responsible for returning any supplies borrowed. This student is also responsible for turning any assignments or needed drafts to the teacher. Public relations PR Student who asks teacher for guidance after collaboration has been thoroughly exhausted. Student also makes sure that group members know when Technology Tech Student assists with the creation of any document using support technology. This student is proficient with ALL Microsoft applications: Word, PowerPoint, and Excel. This student assists engineer in typing out any -

103

Area and perimeter calculations

x _ per area x _ per | area x | y |_ per area 1 2 2 4 6 2 2 4 2 4 8 4 3 6 3 6 12 9 1 2 3 6 8 3 2 4 3 6 10 6 3 6 4 8 14 12 1 2 4 8 10 4-2-4 4 8-12 8 3-6 5 10 16 15 1 2 5 10 12 5 2 4 5 10 14 10 3 6 6 12 18 18 1 2 6 12 14 6 2 4 6 12 16 12 3 6 7 14 20 21 1 2 7 14 16 7 2-4 7 14 18 14 3 6 8 16 22 24 2 8 16 18 8 2 4 8 16 20 16 3 6 9 18 24 27 1 2 9 18 20 9 2 4 9 18 22 18 3 6 10 20 26 30 _ 2 10 20 22 10 2 4 10 20 24 20 3 6 11 22 28 33 1 2 11 22 24 11 2 4 11 22 26 22 3 6 12 24 30 36 1 2 12 24 26 12 2 4 12 24 28 24 3 6 13 26 32 39 1_ 2 13 26-28 13-2 4 13 26-30-26-3-6 14 28-34 42 1 2 14 28 30 14 2 4 14 28 32 28 3 6 15 30 36 45 1 2 15 30 32 15 2 4 15 30 34 30-3-6 16 32 38 48 1 2 16 32 34 16 24 16 32 36 32 3 6 17 34 40 51 1 2 17 34-36 17 2 4 17 34 38 34-3-6 18 36-42 54 12 18 36 38 18 2 4 18 36 40 36 3 6 19 38 44 57 1 2 19 38 40 19 2 4 19 38 42 38 3 6 20 40 46 60 1 2 20 4042 20 2420 40 44 40 3 6 214248 63 1 2 21 42 44 21 2 4 21 42 46 42 3 6 22 44 50 66 1 2 22 44 46 22 2 4 22 44 48 44 3 6 23 46 52 69 1 2 23 46 48 23 2 4 23 46 50 46 3 6 24 48 54 72 1 2 24 48 50 24 2 4 24 48 52 48 3 6 25 5056 75 1 2 25 50 52 25 2 4 25 50 54 50 3 6 26 52 58 78 1_ 2 26 52-54 26 24 26 52-56 52 3-6 27 54-60-8 1 1 2 27 54 56 27 2 4 27 54 58 54 3 6 28 56 62 84 1 2 28 56 58 28 2 4 28 56 60 56 3 6 29 58 64 87 1 2 29 58 60 29-2 4 29 58-62 58-3-6 30 60-66 90 1 2 30 6062 30 2430 60 64 60 3 6 3162 68 93 1 2 3162 64 3 12431 62 66 62 3 6 32 64 70 96 1 2 32 64 66 32 2 4 32 64 68 64 3 6 33 66 72 99 2 33 66 68 33 2 4 33 66 70 66-3-6 34 68 74 102 1 2 34 68 70 34-2 4 34 68 72 68 3-6 35 70 76 105 1 2 35 70 72 35 2 4 35 70 74 70 3 6 36 72 78 108 1 2 36 72 74 36 2 4 36 72 76 72 3 6 37 74 80 I11 1 2 37 74 76 37 2 4 37 74 78 74 3 6 38 76 82 114 1 2 38 76 78 38 2 4 38 76 80 76 3 6 39 78 84 117 1 2 39 78 80 39 2 4 39 78 82 78 3 6 40 80 86 120 1 2 40 80 82 40 2 4 40 80 84 80 3 6 41 82 88 123 2 4 1 82 84 41 2 4 41 82 86 82 3 6 42 84 90 126 2 42 84 86 42 2 4 42 84 88 84 364386 92 129 1 2 43 86 88 43 2 4 43 86 90 86 3 6 44 88 94 132 l1 2 I 44 I 88 90 44 I 2 I 4 I 44 I 88 92 88 I 3 I 6 45 90 96 135 I_ i452i 90 92 445 1214[45 190 941 90 _ 104

Area and perimeter calculations

x - __ pr area x - _ Per area x __ per area 4 84 8 16 16 510 5 10 20 25 612 6 12 24 36 4 8 510 18 20 510 6 12 22 30 612 714 26 42 4 8 6 12 20 24 5 10 7 14 24 35 6 12 8 16 28 48 4 8 7 14 22 28 5 10 8 16 26 40 6 -12 9 18 30 54 4 8 8 16 24 32 5 10 9 18 28 45 6 12 10 20 32 60 4 8 9 18 26 36 5 10 10 20 30 50 6 12 11 22 34 66 4 8 10 20 28 40 5 10 11 22 32 55 6 12 12 24 36 72 4 811 22 30 44 510 12 24 34 60 61213 26 38 78 4 812 24 32 48 510 13 26 36 65 61214 28 40 84 48i13 26 34 52 510 1428 38 70 612 1530 42 90 48 14 28 36 56 510o15 30 40 75 61216 32 44 96 4 815 30 38 60 510 1632 42 80 61217 34 46 102 4 8 16 32 40 64 5 10 17 34 44 85 6 12 18 36 48 108 4 8 17 34 42 68 5 10 18 36 46 90 6 12 19 38 50 114 4 8 18 36 44 72 5 10 19 38 48 95 6 12 20 40 52 120 4 8 19 38 46 76 5 10 20 40 50 100 6 12 21 42 54 126 4 82040 48 80 510 21 42 52 105 61222 44 56 132 4 8 21 42 50 84 5 10 22 44 54 110 6 12 23 46 58 138 4 8 22 44 52 88 5 10 23 46 56 115 6 12 24 48 60 144 4 82346 54 92 510 24 48 58 120 61225 5062 150 4 8 24 48 56 96 5 10 25 50 60 125 6 12 26 52 64 156 4 8 25 50 58 100 5 10 26 52 62 130 6 12 27 54 66 162 4 826 52 60 104 510 27 54 64 135 61228 56 68 168 4 8 27 54 62 108 5 10 28 56 66 140 6 12 29 58 70 174 4 8 28 56 64 112 5 10 29 58 68 145 6 12 30 60 72 180 4 8 29 58 66 116 5 10 30 60 70 150 6 12 31 62 74 186 4 83060 68 120 510 31 62 72 155 612 32 64 76 192 4 8 31 62 70 124 5 10 32 64 74 160 6 12 33 66 78 198 4 8 32 64 72 128 5 10 33 66 76 165 6 12 34 68 80 204 4 8 33 66 74 132 5 10 34 68 78 170 6 12 35 70 82 210 4 8 34 68 76 136 5 10 35 70 80 175 6 12 36 72 84 216 4 83570 78 140 510 36 72 82 180 61237 74 86 222 4 83672 80 144 510 37 74 84 185 612 38 76 88 228 4 8 37 74 82 148 5 10 38 76 86 190 6 12 39 78 90 234 4 8 38 76 84 152 5 10 39 78 88 195 6 12 40 80 92 240 4 8 39 78 86 156 5 10 40 80 90 200 6 12 41 82 94 246 4 8 40 80 88 160 5 10 41 82 92 205 6 12 42 84 96 252 4 8 41 82 90 164 5 10 42 84 94 210 6 12 43 86 98 258 4 I 8 I 42 84 92 168 1 5 1 10 1 43 86 96 215 1 6 1 12 1 44 1 88 I nn 264 4 I 8 I 43 GA 172 1 5 1 10 I 44 88 98 220 6 270 4- 176 5 225 4 180 105

Area and perimeter calculations

x _ y _ per area x Y per area x - per area 7 14 7 14 28 49 8 16 8 16 32 64 9 18 9 18 36 81 7 14 8 16 30 56 8 16 9 18 34 72 9 18 10 20 38 90 7 14 9 18 32 63 8 16 10 20 36 80 9 18 11 22 40 99 7 14 10 20 34 70 8 16 11 22 38 88 9 18 12 24 42 108 7 14 11 22 36 77 8 16 12 24 40 96 9 18 13 26 44 117 7 14 12 24 38 84 8 16 13 26 42 104 9 18 14 28-46 126 7 14 13 26 40 91 8 16 14 28 44 112 9 18 15 30 48 135 7 14 14 28 42 98 8 16 15 30 46 120 9 18 16 32 50 144 7 14 15 30 44 105 8 16 16 32 48 128 9 18 17 34 52 153 7 14 16 32 46 112 8 16 17 34 50 136 9 18 18 36 54 162 7 14 17 34 48 119 8 16 18 36 52 144 9 18 19 38 56 171 7 14 18 36 50 126 8 16 19 38 54 152 9 18 20 40 58 180 7 14 19 38 52 133 8 16 20 40 56 160 9 18 21 42 60 189 7 14 20 40 54 140 8 16 21 42 58 168 9 18 22 44 62 198 7 14 21 42-56 147-8 16 22 44 60 176 9 18 23 46 64 207 7 14 22 44 58 154 8 16 23 46 62 184 9 18 24 48 66 216 7 14 23 46 60 161 8 16 24 48 64 192 9 18 25 50 68 225 7 14 24 48 62 168 8 16 25 50 66 200 9 18 26 52 70 234 7 14 25 50 64 175 8 16 26 52 68 208 9 18 27 54 72 243 7 14 26 52 66 182 8 16 27 54 70 216 9 18 28 56 74 252 7 14 27 54 68 189 8 16 28 56 72 224 9 18 29 58 76 261 7 14 28 56 70 196 8 16 29 58 74 232 9 18 30 60-78 270 7 14 29 58 72 203 8 16 30 60 76 240 9 18 31 62 80 279 7 14 30 60 74 210 8 16 31 62 78 248 9 18 32 64 82 288 7 14 31 62 76 217 8 16 32 64 80 256 9 18 33 66 84 297 7 14 32 64 78 224 8 16 33 66 82 264 9 18 34 68 86306 7 14 33 66 80 231 8 16 34 68 84 272 9 18 35 70 88315 7 14 34 68 82 238 8 16 35 70 86 280 9 18 36 72 90324 7 14 35 70 84 245 8 16 36 72 88 288 9 18 37 74 92 333 7 14 36 72-86 252-8 16 37 74 90 296 9 18 38 76 94 342 7 14 37 74 88 259 8 16 38 76 92 304 9 18 39 78 96351 14 38 _R 76 90 266 l R I 6q 79 OA 'AI 0 12 n AO QfA) 0 AnU v v , v ovv u sv J i IU Jo J , I 0. I I 70. JVV 7 14 39 78 92 273 8 16 40 80 _ 320 9 i 18 1 411 82 100 369 7 14 40 80 94 280 8 16 41 82 328 _9 I 18 I 42 I 84 378 7 14 41 82 96 287 8 16 42 84 3361 9 118 43 387 7 14 42 84 98 294 8 16 43 86 ] 344 I 9 396 7 14 43 86 100 301 8 16 44 88 352 9 405 7 308 360 7 I* 315 106

Area and perimeter calculations x = = I per are x y per area x y Perp are 10 20 10 20 40 100 11 22 11 22 44 121 12 12. 24 48 144 10 20 11 22 42 110 11 22 12 24 46 132 12 13 26 50 156 10 20 12 24 44 120 11 22 13 26 48 143 12 14 28 52 168 10 20 13 26 46 130 11 22 14 28 50 154 12 15 30 54 180 10 20 14 28 48 140 11 22 15 30 52 165 12 16 32 56 192 10 20 15 30 50 150 11 22 16 32 54 176 12 17 34 58 204 10 20 16 32 52 160 11 22 17 34 56 187 12 18 36 60 216 10 20 17 34 54 170 11 22 18 36 58 198 12 19 38 62 228 10 20 18 36 56 180 11 22 19 38 60 209 12 20 40 64 240 10 20 19 38 58 190 11 22 20 40 62 220 12 21 42 66 252 10 20 20 40 60 200 11 22 21 42 64 231 12 22 44 68 264 10 20 21 42 62 210 11 22 22 44 66 242 12 23 46 70 276 10 20 22 44 64 220 11 22 23 46 68 253 12 24 48 72 288 10 20 23 46 66 230 11 22 24 48 70 264 12 25 50 74 300 10 20 24 48 68 240 11 22 25 50 72 275 12 26 52 76 312 10 20 25 50 70 250 11 22 26 52 74 286 12 27 54 78 324 10 20 26 52 72 260 11 22 27 54 76 297 12 28 56 80 336 10 20 27 54 74 270 11 22 28 56 78 308 12 29 58 82 348 10 20 28 56 76 280 11 22 29 58 80 319 12 30 60 84 360 10 20 29 58 78 290 11 22 30 60 82 330 12 31 62 86 372 10 20 30 60 80 300 11 22 31 62 84 341 12 32 64 88 384 10 20 31 62 82 310 11 22 32 64 86 352 12 33 66 90 396 10 20 32 64 84 320 11 22 33 66 88 363 12 34 68 92 408 10 20 33 66 86 330 11 22 34 68 90 374 12 35 70 94 420 10 20 34 68 88 340 11 22 35 70 92 385 12 36 72 96 432 10 20 35 70 90 350 11 22 36 72 94 396 12 37 74 98 444 10 20 36 72 92 360 11 22 37 74 96 407 12 38 76 100 456 10 20 37 74 94 370 11 22 38 76 98 418 12 39 78 102 468 10 20 38 76 96 380 11 22 39 78 100 429 12 40 80 104 480 10 20 39 78 98 390 11 22 40 80 102 440 12 41 82 106 492 10 20 40 80 100 400 11 22 41 82 104 451 12 42 84 108 504 10 20 41 82 102 410 11 22 42 84 106 462 12 43 86 110 516 10 1 20 1 42 84 104 1 420 11 22 43 I86 108 473 I12 528 01201 43I 861 106 1430 1 _i 44 i 88i1 1101 484 112 _ 540 10 120 144 £881 108 1440 l l . 495 450 107

Teacher's Grocery Guide

Item Total Total # Total Cost Serving Cost weight Weight units cost per size per lbs/gallon oz/fl. oz $ lb/oz/unit serving

PROTEIN Beef: 10 oz Beef loin 15.52 lbs 248.32 oz 58.82 3.79Ab 24.832 0.422 Ground Beef 5.89 lbs 94.24 oz 17.02 2.89/lb 9.424 1.806 Rib-eye steak 4.00 lbs 64 oz 27.96 6.99Ab 6.4 4.368 Tri-tip 4.72 lbs 75.52 oz 14.11 2.99/lb 7.552 1.868 Chicken: 10 oz Drumsticks 7.93 lbs 126.88 oz 9.44 1.19/lb 12.688 0.744 Whole 9.55 lbs 152.8 oz 9.45 0.99/lb 15.28 0.618 Eggs 15 16.99 1.13/doz 90 0.188 doz Eggs 2 doz 2.59 1.30/doz 12 0.215 Pork: 10 oz Bacon 4.00 lbs 64 oz 8.99 2.25/ lb 6.4 1.404 Pork ribs 5.24 lbs 83.84 oz 17.24 3.29/ lb 8.384 2.056 Pork tenderloin 7.38 lbs 118.08 oz 13.95 1.89/ lb 11.808 1.181

Fish: I0 oz Catfish 2.72 lbs 43.52 oz 14.93 5.49Ab 4.352 3.430

Salmon 1.71 lbs 27.36 oz 13.66 7.99/lb 2.736 4.992

Canned tuna 5.25 lbs 7 oz/ can 12 10.99 2.11/lb 28 0.392 84 oz cans Other: 3 oz Sliced ham 1.87 lbs 29.92 oz 6.15 3.29/ lb 9.97 0.616 Sliced turkey 2.00 lbs 32 oz 11.99 5.99 /lb 10.66 1.124 Slicked roast 1.36 lbs 21.76 oz 6.79 4.99/ lb 7.25 0.936 beef Thin sliced 1.875 lbs 30 oz 7.99 4.26/ lb 10 0.799 turkey 108

Item Total Total # Total Cost Serving Cost weight Weight units cost per size per lbs/gallon oz/fl. oz $ lb/oz/unit serving PRODUCE Asparagus 2.25 lbs 36 oz 4.99 2.21/lb

Broccoli 3.00 lbs 48 oz 3.99 1.33/lb Garlic 3.00 lbs 48 oz 4.79 0.96/lb Lettuce 6 2.79 0.46 each Oranges 13.00 lbs 208 oz 6.99 0.54/lb Potatoes 20.00 lbs 320 oz 7.99 0.40/ lb Red peppers 6 4.99 0.83 each Spinach 2.50 lbs 40 oz 3.79 1.51/ lb Yellow onions 10.00 lbs 160 oz 3.29 0.33/lb Frozen fruits 12 oz Blueberries 3.00 lbs 48 oz 13.63 4.54/ lb 4 3.40 Festival fruit: 6.00 lbs 96 oz 7.79 1.29/ lb 8 0.973 pineapple, mango, papaya, Strawberries 6.00 lbs 96 oz 8.79 1.47/ lb 8 1.098 Tri-berries: 4.00 lbs 64 oz 10.99 2.75/ lb 5.33 2.061 blueberries, raspberries marionberries Fruitjuices 10 oz

Apple juice 1 gallon 128 oz 4.39 0.034/ oz 12.8 0.435 Apple, peach, 1.5 96 oz/ 2 5.99 0.031/oz 19.2 0.595 passion fruit gallons bottle 192 oz Orange juice 2 gallons 256 oz 9.99 0.039.oz 25.6 0.998 Strawberry 0.75 96 oz 7.79 0.081/oz 9.6 0.779 kiwi gallons 109

Item Total Total # Total Cost Serving Cost weight Weight units cost per size per lbs/gallon oz/fl. oz $ lb/oz/unit serving DAIRY 10 oz 2 % milk 2 gallons 256 oz 3.79 0.014 25.6 0.148

Soy milk 1.5 gallon 192 oz 7.49 0.039 19.2 0.39 Cheddar 2.5 Ibs 40 oz 6.59 0.165/oz 20 0.329

cheevj _ __ _ _ Butter 4.00 Ibs 64 oz 6.00 0.094/oz 32 0.187 Carnation milk 9.00 Ibs 12 12 9.85 0.068/ oz 72 0.137 oz/can cans GRAINS/ BREADS 8 oz Elbow noodles 1.1 IObs 17.6 oz 6 7.99 1.03/lb 2.2 3.63 packs Rice 50 lbs 800 oz 25.99 0.519/lb 100 0.25 Spaghetti 1.10 lbs 17.6 oz 8 8.29 7.53/lb 2.2 3.76 packs

Demi baguettes 6 4.49 0.748

Hamburger 24 1.99 0.83/unit 24 0.83 buns Hot dog buns 24 1.99 0.83/unit 24 0.83 Sliced bread 2.00 lbs 32 oz 48 3.75 0.78/unit 48 0.78 slices FATS/OILS Corn oil 2.5 gallons 320 oz 16.29 0.051/oz

Olive oil 1 gallons 128 oz 20.00 0.156/oz

MISCELLANEOUS

Flour 50 lbs 800 oz 12.99 0.41/lb Salt 25 lbs 400 oz 3.99 0.26/lb Sugar 50 lbs 800 oz 20.49 0.16/lb Peanut butter 4 cups 32 oz 5.59 0.175/oz 16 0.35 Strawberry 2 cups 16 oz 3.99 0.25/oz 8 0.50 jelly 110

Item Total Total # Total Cost Serving Cost weight Weight units cost per size per lbs/gallon oz/fl. oz $ lb/oz/unit serving OTHER Balsamic 0.26 33.28 oz 10.99 vinegar gallons Chocolate 4.5 lbs 72 oz 7.79 0.1 1/oz chips Ketchup 0.34375 44 fl. Oz 4.99 0.1 I/oz gallons Mustard 0.34375 44 fl. Oz 5.99 0.13/oz gallons Ranch dressing 0.625 40 fi. oz/ 2 8.99 0.1 l/oz gallons bottle 80 fl oz Relish 0.234375 30 fl. Oz 2.39 0.079/oz gallons Spaghetti sauce 1.03125 44 fi. 3 7.69 0.1 74/oz gallons oz/bottle

132 oz Teriyaki sauce .4921875 63 fl. Oz 6.89 0.1 1/oz gallons BEVERAGES Coffee beans 3.00 lbs 48 oz 10.99 0.22/oz Sodat 3.375 12oz/can 36 9.19 0.25/can

______jgallons 432 oz cans 111

APPENDIX C

IC Math songs

Page

Angle Sound Off ...... 112

Area and Perimeter - easy to find ...... 113

Are you a prime number? ...... 114

Averages Sound Off ...... 115

Decimal Rhyme ...... 116

Fraction Operations ...... 118

The Surface Area ...... 119

Three Triangles Ode ...... 120 112

Angle Sound Off By: Charity M Tanaleon

Angles are measured by degrees Notes! examples: Use a protractor to look and see. The angle type that you found Then look to see how the angle's bound.

Sound off: angles Sound off: measurement Sound off: 1, 2, 3, 4 - degrees!

Ray, line segments, and two lines Form four types of angle signs. Straight, right, acute, obtuse Each angle has its own use

Sound off: angles Sound off: straight angle Sound off: 1, 2, 3, 4 -180 degrees!

To find the angle you look to see The question asked of specific vertices The middle letter, will dictate The vertex needed to evaluate

Sound off: angles Sound off: right angle Sound off: 1, 2, 3, 4-90 degrees!

Acute angles are determined by The cuteness of the angle's smaller size Degrees less than 90, but more than 1 Make acute angles, so much fun

Sound off: angles Sound off: acute angles Sound off: 1,2,3, 4 - less than 90!

"Ah my tooth!" sounds like obtuse A mnemonic device for your use Remember obtuse angles are very wide Greater than 90 degrees in size. Sound off: angles Sound off: obtuse angles Sound off: 1, 2, 3, 4- greater than 90 113

Area and perimeter - easy to find By: Charity M Tanaleon

Sung to the tune of Twinkle, twinkle little star

Area and perimeter is easy to find in geometrical shapes that are closed and defined.

To calculate area, multiply length and width, it measures the surface expressed in square units.

Area and perimeter is easy to find in geometrical shapes that are closed and defined.

Perimeter is the distance around a closed shape, Add up all the sides, find the sum ... it's great.

Area and perimeter is easy to find in geometrical shapes that are closed and defined.

Notes/examples: 114

Are you a prime number? By: Charity M Tanaleon This is a call and response song to the tune of Where is thumbkin?

Are you a prime number? Are you a prime number?

Yes I am. Yes I am.

'Cause I have no other factors 'Cause I have no otherfactors

Than I and myself Than I and myself

Notes/examples 115

Averages Sound Off By: Charity M Tanaleon

Three mathematical ways are what you'll find to evaluate averages, making them one of a kind. Averages come both large and small calculate them, and have a ball.

Sound off: mean Sound off: median and mode Sound off: 1, 2, 3, 4 - averages!

The median lies in the middle Arrange the numbers from large to little. From both extreme ends, just count back in and soon you'll discover the median.

If two numbers compete for the middle add 'em up divide by two, it's simple. Now you know what the median is, It'll help you on an average math quiz.

Sound off: averages Sound off: in the middle Sound off: 1, 2, 3, 4- median

The mean is the one that's more challenging two procedures is what you'll have to bring. Add up all the numbers in the data set, next divide the sum with the total terms present.

Sound off: two procedures Sound off: add then divide Sound off: 1, 2, 3, 4-found the mean

The last average is quick and easy use your eyes, and you'll soon see me. I'm the number which appears the most 'cause an average like me just likes to boast.

Sound off: likes to boast Sound off: appears the most Sound off: 1, 2, 3, 4 - it's the mode! 116

Decimal Rhyme By: Charity M Tanaleon

When you look at a problem, and you find Decimals in numbers, just use this rhyme. Notes/examples

When you add or subtract Line the decimals up, before you act.

After everything is lined up ever so neat, Bring the decimal down, and you're almost complete 'cause all that's left is a bit - of arithmetic.

1,2,3,4

Use the decimal rhyme To help you find The proper place For the decimalface

1,2,3,4

Now, when you multiply You have to find The number of digits that are behind The funky face, of the decimal place

1,2,3,4

And with the total number of digits that you find It must be equivalent to the final product In line

1, 2,3, 4

Use the decimal rhyme To help you find The proper place For the decimalface 1,2,3,4

So in reverse of multiply Count spaces in the divisor Before you divide 117

1,2,3,4

Move the decimals from the divisor to the very end Then move the same number of decimals Notes/examples In the dividend

1,2,3,4

You're almost ready for the very last step Just bring the decimal up And then you're set

Plain as day, and out of sight The last thing you do Is strictly divide.

1,2,3,4

Use the decimal rhyme To help you find The properplace For the decimalface 118

Fraction Operations By: Charity M Tanaleon

Add, subtract, multiply, divide e Practicingfraction operations Notes! examples: Make them come alive

Common denominators are a fact In fraction addition Or when you subtract

Add, subtract, multiply, divide Practicingfractionoperations Make them come alive

Multiplication itself takes 2 quick steps Multiply, reduce And then you're set

Add, subtract, multiply, divide Practicingfractionoperations Make them come alive

Division requires a change in sign Reverse the division to multiply And the fraction by its side

Multiply, reduce, then you're done With fraction division Its endless fun

Add, subtract, multiply, divide Practicingfraction operations Make them come alive 119

Surface Area By: Charity M Tanaleon Surface Area song to the tune Three Blind Mice

Sur... face area, sur... face area Of a cube and.. rectangular prism Are the areas of all.. .face sides Front, back, top bottom, side, side Add them up and obtain the prize Of total surface size

Notes/examples: 120

Three Triangles Ode By: Charity M Tanaleon

Three Triangles Ode to the tune of Ode to Joy

Three triangles, three triangles, equalateral, scalene, isosceles equal sides are what is on an equa ... lateral tri.. .angle

Three triangles, three triangles, equilateral, scalene, isosceles with my two eyes, I see two equal sides on triangle isosceles

Three triangles, three triangles, equilateral, scalene, isosceles scales lean too much, all sides, unequal, on a scalene triangle

Three triangles, three triangles, equilateral, scalene, isosceles.

Notes! examples 121

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